Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters

to the editor should be sent to the

editor-in-chief, Chandler Davis.

Letter to the Editor

Patti Wilger Hunter's article on Abra­ham Wald in the Winter 2004 issue

nicely illustrates how a mathematician

can be stimulated by, and respond to, challenges from beyond mathematics per se. Your readers may not know that Wald's sequential probability ratio test

(SPRT), which was independently dis­covered by George Barnard in the U.K. [1] and used by Turing's group in their code-breaking work at Bletchley Park, also illustrates unexpected applica­tions of existing mathematics.

In the 1960s psychologists, led by

Stone and Laming [2], proposed that people responding to stimuli in highly

constrained choice tasks with only two alternatives, do so by accumulating ev­idence and responding when a thresh­old is crossed, just as in SPRT. Subse­quently, Ratcliff [3] used a constant drift-diffusion process, the continuum limit of SPRT and perhaps the simplest stochastic differential equation,

dx = A dt + c dW, (DD)

to fit human behavioral data-specifi­cally, reaction-time distributions and error rates. (Here A denotes the drift term and c the variance of the Wiener process W.) Moreover, recent neural recordings from oculomotor brain areas of monkeys performing choice tasks has shown that firing rates of groups of neurons selective for the "chosen" of the two alternatives rise toward a threshold that signals the onset of mo­tor response in a manner that seems to match sample paths of (DD) [4].

As pointed out in [5], this suggests an

intriguing possibility. SPRT is the opti­mal decision-maker, in the sense that,

for a predetermined error rate, it mini­mizes the expected time required to make a decision among all possible tests. (Human reaction times also in­clude durations required for sensory and motor processing, and these must be al­lowed for in interpreting behavioral data.) Thus, if one wishes to optimize one's overall performance in completing a series of trials, one would do well to

4 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+Busrness Medra, Inc.

employ SPRT or (DD) with thresholds chosen to maximize the reward rate:

RR=

(Expected

fraction of correct

responses)

(Average time between

responses).

(RR)

Since the numerator (1 - Error Rate) and denominator of (RR) are simple expressions of the drift rate A, noise variance c, and threshold for (DD), it is an exercise in calculus to compute op­timal thresholds and derive an "optimal performance curve" relating reaction time to error rates. This appears to be

the first theoretical prediction of how best to solve the well-known speed-ac­curacy tradeoff: it is not optimal to try to be always right, since that makes re­action times too long; nor is it good simply to go fast, since then error rates are too high.

We are currently assessing the abil­ity of human subjects to achieve this theoretical optimum performance. While some of our subjects (Princeton undergraduates) appear more con­cerned to be correct than to be fast, the overall highest-scoring group indeed lies close to the optimal perforn1ance curve, although slightly on the conser­vative (high-threshold) side. Tests are planned with monkeys in which direct neural recordings will also be made.

Did the subconscious, with the help of evolution, discover SPRT long be­fore Wald and Barnard? Stay tuned.

REFERENCES [ 1 ] Barnard, G. A Sequential tests in industrial

statistics . J. Roy. Statist. Soc. Suppl. 8:

1 -26, 1 946. DeGroot, M. H. A conversa­

tion with George A Barnard. Statist. Sci. 3: 1 96-2 1 2, 1 988.

[2] Stone, M. Models for choice-reaction tirne.

Psychometrika 25: 251 -260, 1 960. Larning,

D. R. J. Information Theory of Choice-Reac­

tion Times. Acadernic Press, New York. 1968.

[3] Ratcliff, R. A theory of rnernory retrieval.

Psych. Rev. 85: 59-1 08, 1 978. Ratcliff, R . ,

Van Zandt, T., and McKoon, G. Connec-

tionist and diffusion models of reaction time.

Psych. Rev. 1 06 (2): 261 -300, 1 999.

[4] Roitman, J. D. and Shadlen, M. N. Re­

sponse of neurons in the lateral interparietal

area during a combined visual discrimina­

tion reaction time task. J. Neurosci. 22 ( 1 ) :

9475-9489, 2002. Ratcliff, R , Cherian, A, and Segraves, M. A comparison of

macaque behavior and superior colliculus

neuronal activity to predictions from mod­

els of two choice decisions. J. Neurophys­

iol. 90: 1 392-1 407, 2003.

[5] Gold, J. 1., and Shadlen, M. N. Banburis­

mus and the brain: Decoding the relation­

ship between sensory stimuli, decisions,

and reward. Neuron 36: 299-308, 2002.

Philip Holmes

Program in Applied and Computational

Mathematics and Center for the Study of

Brain , Mind and Behavior

Princeton University

e-mail: pholmes@math.princeton.edu

Rafal Bogacz

Department of Computer Science

University of Bristol

Bristol, UK

e-mail: r.bogacz@bristol .ac .uk

Jonathan Cohen

Department of Psychology and Center for the

Study of Brain, Mind and Behavior

Princeton University

e-mail: jdc@princeton.edu

Joshua Gold

Department of Neuroscience

University of Pennsylvania

e-mail: jigold@mail . med . upenn . edu

Is Escher's Art Art?

In his review of M. C. Escher's Legacy: A Centennial Celebration, Helmer

Aslaksen writes, "It is also important to realize that arts specialists do not share our fascination with Escher. Many of them simply don't consider him to be an artist!"

This is sad but true, and I am afraid there is a very simple explanation for this. Although Escher was not a math­ematician, his art has deep mathemat­ical ideas, as some of the articles about H. S. M. Coxeter in the same issue of the Intelligencer, which mention Cox­eter's and Escher's relationship, make clear. On the other hand, many people

in the arts have no mathematical train­ing (much less mathematical interest).

When confronted with something, even something beautiful, that one doesn't understand, there are two com­mon human reactions. One is admira­tion and wonderment, and a desire to learn more about it. The second is to belittle and denigrate the work so as not to have to admit one's ignorance. There is only a fine line between this latter attitude and outright hostility, and the line is easily crossed.

I am afraid that the second reaction is by far the most common one in the art world. Perhaps the most egregious example of this is the January 21, 1998, review by New York Times art critic Roberta Smith of a wonderful Escher exhibition at the National Gallery of Art, where the reviewer's overt hostility cul­minated in her statement, " . . . one won­ders if Fascism, which Escher detested, hadn't also contaminated his art."

Steven H. Weintraub

Department of Mathematics

Lehigh University

Bethlehem, PA 1 801 5-31 74

USA

e-mail: steve.weintraub@lehigh.edu

Where are the Women?

I am a junior at St. Cloud State Uni­versity in Minnesota. While studying

to become a mathematics educator, I came across The Mathematical Intel­ligencer, vol. 25, no. 4 (Fall 2003). I think The Intelligencer will be a good resource for me as a future educator.

However, I was sorry to see that at most one of the nine articles was writ­ten by a female. Traditionally, math is thought of as consisting mostly of men. I think it is important that students see that females are as prominent in the field as males. As Ian Law said in "Adopting the Principle of Pro-Feminism" in the book Readings for Diversity and Social Justice (see p. 254), many men think they need to be "dominating the airspace mak­ing sure it is [their] voice and views that get heard." The ideas of males as domi­nant and females as subordinate need to be challenged. Another article in the san1e book, "Feminism: A Movement to End Sexist Oppression" by bell hooks,

emphasizes that overcoming the thought of men dominating women "must be solidly based on a recognition of the need to eradicate the underlying cultural bi­ases and causes of sexism and other group oppression" (p. 240).

This image of male dominance is given to readers when they see an issue in which no woman has a voice. Also, having more female authors will help provide female role models, which will help inspire fe­male students in their love of math and encourage them to pursue it.

Christina Green

1 303 Roosevelt Road

St. Cloud, MN 56301

USA

e-mail: grch01 04@stcloudstate.edu

The Editor Replies:

The exact number of women authors in the issue you chanced to read first is zero. This is low, for us: many issues be­fore and since it have numerous women authors (though I note that vol. 26, no. 2 again has none-sorry). It is Intelli­gencer policy to encourage participation by mathematicians of whatever sex, whatever nation, whatever background. The policy has been stated in print be­fore, and your letter is a welcome occa­sion to state it again.

As you say, we try to give women a voice. We also try to spread awareness of their achievements; and we provide a forum for discussion of ways to re­move the barriers to their full partici­pation in the profession.

I must say, though, that I hope it was inadvertent that you said women are now equally prominent in mathematics. So far, no. We observe that more than half of the best mathematics is done by men, and we ask, are women being dis­couraged from studying it? are they be­ing eliminated by unfair grading? are they being refused jobs at the level they have earned? We fmd that all of these deterrents sometimes operate, and we struggle to eliminate them. In order to do it effectively, we need to acknowl­edge the nature of the imbalance.

I hope that as an educator you will help more girls become enthusiastic about mathematics. (Don't feel bad if you engage some boys too.)

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1, 2005 5

ERIC GRUNWALD

Eponymphomania "But if the arrow is straight And the point is slick, It can pierce through dust no matter how thick. "

-Bob Dylan [ 1]

he Mathematical Intelligencer is full of delightful surprises. Eric C. R. Hehner, in his

paper "From Boolean Algebra to Unified Algebra" [2], claims that terminology that

honors mathematicians is sometimes wrongly attributed, is used deliberately to lend

respectability to an idea, and even when the intention is genuinely to honor the

eponymous person, the effect is to make the mathematics forbidding and inaccessible.

As I perused Hehner's paragraph (I use this term de­scriptively, not honorifically), I found myself in general agreement with him, with perhaps one or two caveats. Per­sonally, I would preserve Abelian groups: decent mathe­matical jokes are rare, and "What's purple and commutes? A commutative grape" seems to lose something in the trans­

lation. I would also vote in favour of topological spaces whose points are hausdorff from one another (and salads

whose ingredients are waldorf). And I certainly advocate that we continue to remember Norbert Wiener for his sem­inal invention of the schnitzel. But the biggest exception to Hehner's generally sensible rule should surely be made for an eponymous term of astonishing beauty to be found to­wards the end of his paragraph. It appears that there ex­ists something called the Peirce arrow.

Mr. Peirce's arrow is surely worth keeping. As Bob Dy­lan pointed out, it penetrates dust no matter how thick. It is inspiring: just as Sir Karl Popper regarded Darwin's evo­

lutionary theory as a "metaphysical research program," so the Peirce arrow was a metaphysical research program for me. I determined to find out more about Mr. Peirce and his

arrow. Googling intrepidly through hundreds of thousands of references, using both Dylan's and Hehner's spellings, I uncovered these three pearls:

x = y - Sheffer stroke, NAND; x # y - Pierce arrow, NOR. [3]

. . . The questions also refer to "Sheffer's stroke" and "Pierce arrow" (not an antique car!) operators . . . [4]

6 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Scrence+Busrness Medra, Inc

Queries with disjunction are first converted to disjunc­

tive nor-mal form (disjunction of conjunctions) . . . [5]

These gnomic utterances raised the following important research questions:

(a) Why ARE certain WORDS written in upper case for no APPARENT reason? This is surely much more off­putting than any mere use of an honorific name. I feel I'm being yelled at by NAND and NOR, and I dislike them already.

(b) Why is something written "#" called the "Peirce arrow"

rather than the "Peirce sharp sign" or the "Peirce waffle­iron"? Further internet research revealed that this sym­bol appears variously as "#", "D", and "!". Whether this is a bitter controversy amongst logicians, or whether it's a consequence of the inadequacy of my computer (run­ning on low-octane Windows 66) I don't know, but I sus­pect that unless Mr. Peirce was an extremely poor archer

he probably meant "!", so I'll go with that one. (c) If we are going to be told that the Peirce arrow is not an

antique car!, why aren't we told that Sheffer's stroke is not a medical condition!? Or that it's not a sexual tech­nique!? In fact the author told us (the Sheffer stroke is not a medical condition!)! (the Sheffer stroke is not a sexual technique!). Or should that read "the Sheffer stroke is (a medical condition)! (a sexual technique)!"? Or should I actually have written "(the author told us the Sheffer stroke is not a medical condition!)! (the author told us the Sheffer stroke is not a sexual technique!)"?

(d) It's good to know that the Peirce arrow can be used to generate a disjunction of conjunctions. But if you want a term truly guaranteed to put off any aspiring student,

"disjunctive nor-mal form" must be it: unlike the erudite readers of this journal, the student might become dan­

gerously disoriented when trying to distinguish between

a disjunction of conjunctions and a conjunction of dis­

junctions. After all, when Polonius so wisely advised "(a

borrower)! (a lender) be" [6], was he speaking con­junctively or disjunctively? As with so much of the the­oretical output of that particular author (for example his paper To Be or Not To Be? The Law of the Excluded Middle [7]), I can't fully get to grips with it, so giving my pen a long, lingering, disjunctive gnaw I must pass on.

Let's give two cheers for the Peirce arrow. It may be elit­ist and off-putting. It may use an author's prestige to lend respectability to an unremarkable idea. It may, for all I know, be attached to the wrong bloke altogether. But it's beautiful. It's poetic. It inspires research. And unlike other rival names, it doesn't give my eyes disjunctivitis. So please don't shoot the arrow away. You may chuck away at a stroke all reference to Sheffer. I would shed no tears at the demise of Banach spaces or Sylow's theorem. But Peirce's arrow deserves to thrive, along with all the other beautiful terms that enrich mathematics: wonderful expressions like Weyl integrals, Killing fields, the Gordan knot, the Roch group, Jordan delta functions, Plateau's plane, Taylor cuts, the Schur Certainty Principle, Abel-Baker-Chasles-Lie sym­bols, and, since I'm feeling rather eponymous just now, the

Grunwaldian or recursive citation. [8]*

REFERENCES [ 1 ] Bob Dylan, Restless Farewell, 1 964.

[2] E. C . R. Hehner, "From Boolean Algebra to Unified Algebra," The

Mathematical lntelligencer, vol. 26, no. 2 , 2004.

[3] http:/ /rutcor.rutgers.edu/pub/rrr/reports2000/32 .ps. Rut cor Research

Report

[4] web.fccj.org/�1 dap991 1 /COT1 OOOUpdate. html

AU T H O R

ERIC GRUNWALD Perihelion Ltd.

1 87 Sheen Lane London SW14 8LE

UK

e-mail: EricGrunwald@compuserve.com

Eric Grunwald received his doctorate in mathematics from Ox­

ford. Since then he has been employed in the chemical, en­

ergy, and health-care industries, and has become expert in

advising organizations on their future planning. He has, sadly,

not found anyone else in the field of future thinking who knows

much about mathematics; if there are others, he would like to

meet them.

[5] iptps03 .cs. berkeley .edu/final-papers/result_ caching. pdf

[6] W. Shakespeare, Hamlet, act 1 , scene 3, 1 601 .

[7] W. Shakespeare, private communication.

[8] E. J. Grunwald, "Eponymphomania," The Mathematical lntel ligencer,

vol. 27, no. 1 , 2005.

[9] K. D0sen, "One More Reference on Self-Reference," The Mathe­

matical lntelligencer, vol. 1 4, no. 4, 4-5, 1 992.

'As a true Hehnerian, or eponymous honorific, the term "Grunwaldian" not only attempts to add weight to a pointless concept, it is also elegantly misattributed. The

ngorous process of peer review through which this paper was extruded has revealed that the recursive citation has appeared previously in the literature [9].

Quantitative Programmer Leading UK hedge fund is looking for a quantitative an­alyst-programmer

Supporting major strategic research initiatives, you will

be involved in development of complex simulation soft­

ware and creation of advanced analytical tools for re­

gression analysis. You are a graduate with background

in Math/Statistics/Engineering/Computer Science. You are

a highly quantitative C++ programmer with at least 2 years of commercial experience. You are interested in

application of data analysis to modelling of financial

markets. If you have always wanted to combine statis­

tical modelling with programming, this is a fantastic

opportunity.

Location: London

Competitive salary + bonus

Apply to: research.position@btconnect.com

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 27, Number 1 , 2005 7

W. M. PRIESTLEY

Plato and Anaysis

he Statesman, a late work of Plato's, begins with a playful allusion to mathematics.

The setting is an ongoing inquiry ostensibly intended to complete the delineation of

the true natures of the Sophist, the Statesman, and the Philosopher, but more basic

philosophical issues are raised as well. As the scene opens we find Socrates thanking

Theodorus, an elderly mathematician, for having brought

to Athens with him his young student Theaetetus and an

unnamed philosopher visiting from Elea, the Greek town

in southern Italy that is home to Zeno and his paradoxes.

The "Eleatic Stranger"-the appellation given this name­

less visitor in older translations of Plato-may suggest to

us the archetypal masked man who descends upon the ac­

tion from nowhere to round up the outlaws and establish

order.

Sure enough, the Stranger has already gone after the

Sophist earlier in the day, using a dichotomizing technique

that closely resembles the modern analyst's bisection

method of successive approximations. In the words of a

modern commentator [P2, p. 235], he "first offers six dis­

tinct routes for understanding the [S]ophist, by systemati­

cally demarcating specific classes within successively

smaller, nested ... classes of practitioners; these subclasses

are then identified as the [S]ophists." Then, following a

lengthy discussion to introduce a "change of coordinates,"

the Stranger resumes his search and finally obtains neces­

sary and sufficient conditions to characterize the slippery

Sophist. Socrates expresses delight.

Here, in Benjamin Jowett's nineteenth-century transla­

tion, are the opening lines that follow in the Statesman.

8 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Scrence+Busrness Medra, Inc.

[SocRATES] I owe you many thanks, indeed, Theodorus, for

the acquaintance both of Theaetetus and of the Stranger.

[THEODORUS] And in a little while, Socrates, you will owe

me three times as many, when they have completed for

you the delineation of the Statesman and of the Philoso­

pher, as well as of the Sophist.

[SocRATES] Sophist, statesman, philosopher! 0 my dear

Theodorus, do my ears truly witness that this is the es­

timate formed of them by the great calculator and geo­

metrician?

[THEODORUS] What do you mean, Socrates?

[SocRATES] I mean that you rate them all at the same

value, whereas they are really separated by an interval,

which no geometrical ratio can express.

[THEODORUS] By Ammon, the god of Cyrene, Socrates,

that is a very fair hit; and shows that you have not for­

gotten your geometry. I will retaliate on you at some

other time . . . . (Statesman 257a-b)

What is the "hit" by Socrates that provokes Theodorus's

oath? Some commentators on Plato say that Socrates is

alluding to the existence of incommensurables in geome­

try, something that Plato was fond of mentioning in other

contexts. Thus, Socrates would seem to be implying that

IIOAITIKOI.

TA TOT �IAAOfOT IlPOl:OIJA

�OKPATH�, 8EO�OPO�, ;a'ENO�, �OKPATH� 0 NEOTEPO�.

"'II 7rOAA�V xaptv o¢etAw UOt Tij� 8eatT�TOU yvwpl-.,. 8 Is;;. d \ � � C I uew�, w eouwpe, ap.a Kat TTJ� Tou c;Evou.

8EO. Taxa M ye, cJ �wKpaTEf, o¢etA�UEt� Tav-"i I J S;o \ I "i \ J I TTJ� Tpt7rAautav, E7rEtuav TOV TE 7rOAtrtKov a7repya-

uwvral UOt Kat TOll if>tAOUoif>ov, �0. Elev· ovTw Tovro, cJ ¢lA.e 8eo8wpe, ¢�uop.ev

CtKTJKOOTE� e'lvat TOV 7rEpt Aoytup.ov� Kat Ta yewp.E-' I TptKa Kparturou;

8EO. llro�, cJ �wKpaTe�; �n T � . s;;. � ,., *fJ ,

* � , , t� 4.:..1.. wv avopwv EKauTov EIITO� TTJ� LUTJ� ac;ta�, ot TV np.fi 7rAEOll [ aAA�I\wv] acf>EuTaUlV � KaTa T�V avaA.oy{av T�v T�� vwr/pa� rEXIITJf.

8EO. Eo ye v� Tov �p.ITepov fJeov, cJ �wKpaTe�, \ ,A ' � I ' I \ '9 Tov .n.p.p.wva, Kat utKatw�, Kat 1ravu p.ev ouv !J.VTJ!J.OVt-

� ' , l c' ' ' ' "i ' • ' Kw� E7rE7rATJc;a� p.ot TO 7rEpt TOU� Aoytup.ou� ap.ap-, , , , , I ' ""'() I \ TT}p.a. Kat ue p.ev avrt TOUTWV Etuau t� p.eretp.t' uu

8 �p.'iv, cJ Elve, p.TJ8ap.ror Ct7r0Kap.y� xapt(op.evo�, aA.A.' '�:� Jl \ "i \ >I s;;. I Jl \ Ec;TJ�, EtTE Tov 7T'OI\trtKOV avopa 7rpOTEpov ELTE TOV ¢tA.ouo¢ov 7rpoatpe'i, 1rpoeA.op.evo� 8tlteA.Oe.

-E T � . .,. 8 Is;;. , ' I " t � • auT , W EOuwpe, 7rOLTJTEOV' E7rEL7rfEp a1rac; .J I J J I \ <Ito ' ,.. re EYKEXELPTJKap.Ev, OVK a7rOUTaTEOV 7rptv av avTWV

Only a third part of our task is done: nay, not a third, for the States­man rises above the Sophist in value and the Philo­sopher above the Statesman in more than a geo­metrical ratio.

The beginning of Plato's Politikos-Politicus in Latin, Statesman in English.

these three types of individuals have incommensurable

natures.

Another reading suggests itself.

Existence Questions

Socrates might be referring to the classical geometric ver­

sion of what is now familiar to us as the Archimedean property. To see why this is plausible, though, requires a

digression. In the modem setting of ordered fields, the

property states that if a and b are positive elements, then

there exists some natural number n such that na exceeds

b. In modem terms, a is not infinitely small (i.e., not an "in­

finitesimal") relative to b. "Every little bit counts," as Paul

Halmos once quipped. In a complete ordered field, this

property is a simple consequence of (primarily) the com­

pleteness axiom, for if the elements of the set { na) were all

bounded above by b, then a contradiction follows in an el­

ementary way as soon as this axiom is invoked. The

Archimedean property is the main feature of standard, as

opposed to non-standard, treatments of modem analysis.

Infinitesimals and their reciprocals (infinitely large num­

bers) do not exist among the positive elements of the "stan­

dard" real number system.

How could Plato (427-347 BCE) have known of the prop­

erty bearing the name of Archimedes, who was born later?

The answer is that this property, whose eponymous name

is relatively recent-0. Stolz's 1883 paper [St] may mark its

first appearance-was familiar to mathematicians about a

hundred years before Archimedes (287-212 BCE) drew at­

tention to its fundamental role. In fact, the property (see

[Bo2, p. 129]) seems to have been introduced by Eudoxus,

who studied, perhaps briefly, in Athens as a penniless youth

and later, having become a noted mathematician and as­

tronomer, returned with his own students around 365 BCE

to meet with Plato on more nearly equal terms. The dates

of Eudoxus's life are uncertain, although it is now thought

© 2005 Spnnger Science+ Bus1ness Media, Inc .. Volume 27, Number 1. 2005 9

[Gu, p. 447] that he was born about 395 BCE and outlived

Plato by a few years.

Eudoxus, according to tradition, formulated the subtle

definition that vastly extended the applicability of the the­

ory of proportions by making it possible to decide when

two ratios are the same, even when each of them is a ratio

of incommensurable magnitudes. Some commentators

read the language used in Parmenides 140c as indicating

Plato's awareness of the theory of Eudoxus.

It was after Eudoxus's return to Athens that Plato com­

pleted his Theaetetus-Sophist-Statesman trilogy, which is

intimately concerned with ontology, that is, with general

philosophical questions regarding existence.

I hold that the definition of being is simply power. - The Eleatic Stranger (Sophist 247e)

As we shall see, the Stranger's association of being with

power has something to do with the existence of incom­

mensurables.

Plato's Academy, founded in Athens around 385 BCE, had

exceptional scholars to reflect upon philosophical and

mathematical questions. Aristotle (384-322 BCE) joined

Plato's Academy at an early age-about the time of Eu­

doxus's return-and would eventually rival, if not surpass,

his teacher. And until his death around 369 BCE, there was

Theaetetus, who may have been the main proponent of an

alternative treatment of ratios that I shall mention later. In

Plato's dialogue dedicated to him it is implied that Theaete­

tus, whose investigations advance as smoothly as "a stream

of oil that flows without a sound," was first to prove that

all positive integers for which there exist rational square

roots must be perfect squares.

Plato acted in the important role of catalyst for the math­

ematical investigations of others, but he, like Aristotle

(whose interest in logic far exceeded his interest in math­

ematics), seems to have proved no new theorems of his

own. Of Plato's contemporaries, Theaetetus and Eudoxus

contributed most heavily to the material collected around

300 BCE in Euclid's Elements.

Arithmetization: Dedekind and Eudoxus

(and Plato?)

While Benjamin Jowett, at Oxford, was busily translating

and analyzing Plato's dialogues (the first edition of Jowett's

massive project appeared in 1871), a remarkable new

movement in mathematics was developing on the Conti­

nent. In 1858 Richard Dedekind (1831-1916) realized that,

in a sense, the key to the modem "arithmetic" foundations

of real analysis had been in Eudoxus's hands some 2200

years earlier.

What needed to be done to obtain a purely numerical

theory, Dedekind saw, is to retain Eudoxus's insight, but

to remove all reference to the geometric magnitudes whose

existence the Greeks took for granted. In fact, as Dedekind

remarked later in a letter to Rudolf Lipschitz, the Euclid­

ean theory of ratios cannot encompass the complete sys­

tem of real numbers required by modem analysis because

1 Q THE MATHEMATICAL INTELLIGENCER

only algebraic numbers can result from Euclidean con­

structions. (See [Fe, p. 132] and [De, pp. 37-38].)

As is now well known, Dedekind [De, p. 15] declared

that a real number is defined-or "created"-by a cut (Schnitt), by which he meant, essentially, a partition of the

rational numbers into a pair of nontrivial segments. He first

showed [De, pp. 13-14] that there exist infinitely many cuts

not produced by rational numbers by giving a clever proof,

using the well-ordering principle, of Theaetetus's result that

square roots of non-square positive integers are irrational.

He then observed that the expected algebraic properties

(and ordering) of the real number system can be made to

follow from properties of the integers by defining arith­

metic operations (and an order relation) on cuts in a nat­

ural way. More importantly, he showed how the complete­

ness property of the real number system flows smoothly

from these considerations. Dedekind used the word conti­nuity (Stetigkeit) to describe the crucial property that is

more commonly called completeness or connectedness by

mathematicians today.

Dedekind [De, p. 22] insisted that such a "theorem" as

V2 V3 = V6 can be given a "real proof' only after we at­

tach to these symbols their appropriate numerical mean­

ings in terms of cuts (see [Fo1]). By going far beyond the

Greeks in appealing to infinite processes, Dedekind melded

the discrete and the continuous, freeing analysis to be de­

veloped independent of its geometric origins. In fact, much

of geometry could now be made dependent upon analysis

by identifying geometric points inn-dimensional space with

n-tuples of real numbers. The foundations of mathematics

thus began to shift decisively from geometry toward arith­

metic and set theory, to which Dedekind and his great

friend Georg Cantor (1845-1918) began to devote much at­

tention. While it may be too much to call Dedekind "the

West's first Modernist" [Ev, p. 30], he certainly helped to

foster a movement that is about as close to a paradigm shift

as the history of mathematics can provide. Dedekind withheld publication of his radically modem

ideas until he realized in 1872 that other mathematicians

(he names Heine, Cantor, Tannery, and Bertrand) were also

ready to face squarely the ontological question of the ex­

istence of "real" numbers [De, p. 3]. In 1887 Dedekind ac­

knowledged his ancient source by writing that Euclid's

Book V sets forth "in the clearest possible way" his own

conception that an irrational number-if it is presented as

a ratio of magnitudes-can be defined by the specification

of all rational numbers that are greater and all those that

are smaller [De, pp. 39-40].

Intriguingly, "the Great and the Small" happens to be a

phrase used by Aristotle (Metaphysics 987-988) to refer

to an idea, apparently puzzling to Aristotle, whose impor­

tance Plato emphasized in lectures given late in life. This

has led some scholars to speculate, once Dedekind's ideas

had become well understood, that Plato in his later years

might have been thinking along similar lines. Interest in

such speculation has been heightened by the juxtaposition

of two curious facts: ( 1) Plato wrote on more than one oc­

casion that some things cannot be expressed in writing and

might be more accurately conveyed only through the (still

imperfect) give-and-take of (oral) dialectic; and (2) "the

great and the small" is a phrase used by Plato (in States­man 283e, for example) but is seemingly never applied

anywhere in his writings in the manner described by Aris­

totle [Sa, p. 96]. According to Aristotle, Plato asserted,

among other things, that numbers come "from participa­

tion of the Great and the Small in Unity." I shall suggest

below what Plato might have meant by this cryptic pro­

nouncement.

No one would suggest, of course, that an ancient Greek

could have foreseen all of our present basis for real analy­

sis. A sea change (see [Cr]) had to occur before modem

mathematicians even began to look for a numerical, as op­

posed to geometrical, underpinning to their discipline. The

"arithmetization of analysis" did not take place until the late

nineteenth century with the amalgamation of results of

Cauchy, Balzano, Weierstrass, Dedekind, Heine, Borel,

Cantor et al. [Bo2, p. 560ff.]

Before considering what Plato's ideas might have to do

with those of modem mathematical analysis, however, let

us return to Socrates and Theodorus.

Flatland in 350 acE In Euclid V a ratio (logos) is described as "a sort of relation

in respect of size between two magnitudes of the same

kind." Two geometric magnitudes are then said to have a

ratio if and only if some (positive, integral) multiple of each

exceeds the other. Thus, for example, one cannot speak in

Euclidean geometry of the ratio of a line segment to a square

because no (finite) number of copies of a given line seg­

ment can make up an area exceeding that of a given square.

Nor can one speak of the geometrical ratio of a square to a

cube, for these are likewise not "of the same kind."

By now it must be clear what this has to do with

Socrates's "hit" in the opening lines of the Statesman. The

Sophist, Statesman, and Philosopher represent three types

whose relative statures differ greatly. In the opening ex­

change of the Sophist the first two are spoken of as "ap­

pearances" or, in Comford's translation, "shapes" that the

philosopher can assume in the eyes of others. As the dia­

logue reveals, Plato sees the devious Sophist, with his pen­

chant for demagoguery, as essentially nothing in compari­

son to the noble Statesman, who will himself cut a small

figure when placed alongside the truly wise Philosopher. If

two magnitudes were said to be separated by an interval

that no geometrical ratio can express, a geometer like

Theodorus would immediately infer that the larger magni­

tude exceeds the smaller by every (finite) multiple. In

other words, given the context, Socrates's remark implies

that the worth of one statesman exceeds that of any (arbi­

trarily large) number of sophists; similarly, one philosopher

is worth more than any number of statesmen. A sophist is

thus infinitesimal in comparison to a statesman, who is him­

self infinitesimal in comparison to a philosopher.

Was Plato thinking of the sophist, statesman, and

philosopher as analogous to one-, two-, and three-dimen­

sional magnitudes, respectively, perhaps along the lines of

the beings brought to life in Edwin Abbott's Victorian ro­

mance, Flatland [Ab]? It might have been the other way

around. Was Abbott (1838-1926), whose field was classics

and who introduces his own "Stranger" to lead a playful di­

alogue about "Spaceland" [Ab, p. 65], borrowing from Plato

the idea of one-dimensional and two-dimensional beings?

A classical analogy between persons and magnitudes does

suggest itself, for in the Greek of Plato's day the word for

magnitude referred not only to line segments, rectangles,

cubes, etc.; it also, as Salomon Bochner points out [Bol, pp.

278-79], carried an older connotation (circa 775 BCE) from

Homer:

. . . [T]he Greeks did not have real numbers but, in its

place, a notion of "magnitude" [megethos] . In Homer this

noun still means: personal greatness or stature (of a hero,

say); and it is remarkable that for instance in the French

noun grandeur and the German noun Grosse the two

meanings of personal greatness and of mathematical

magnitude likewise reside simultaneously.

Perhaps there is evidence in Plato's other writings to

suggest that he might have been in the habit, as some of us

are today, of thinking of narrow-minded people as being in

some sense "one-dimensional." In the Republic (587d) we

find the remark, accompanied by an obscure explanation,

that the philosopher is 729 times happier than the tyrant.

But 729 is the cube of 9; this seems to hint at the "three-di­

mensionality" of the "solid" philosopher and the relative

shallowness of the tyrant.

What is Analysis?

Whatever one intends by the meaning of a proposition, it

surely involves the collection of statements implied by that

proposition in some universe of discourse reflecting a con­

text either explicitly given or implicitly understood. The

completeness axiom, for example, states that, given a bounded, nonempty set, there exists a least upper bound. If we were asked what this "really" means, we might reply

that in the context of an ordered field it means a host of con­

sequences-that the system is unique up to isomorphism,

that the real numbers naturally form a connected topologi­

cal space, that every non-empty convex subset is an inter­

val, that there exists a point common to a collection of

nested, bounded, non-empty closed intervals, etc. [01].

Some 700 years after Plato's death the mathematician

Pappus of Alexandria described a "method of analysis" dat­

ing from Plato's time (see [Kat, pp. 184-5]) that seems to

flow from this observation about meaning. To test the truth

of a proposition, Pappus says, deduce implications from it.

Should one deduce an implication that is self-evidently true,

then a synthetic proof-as in Euclidean geometry-is said

to be obtained if the steps in this deduction can be reversed

so as to obtain the given proposition as a logical conse­

quence of self-evident truths. Pappus's usage of the term

analysis is criticized by Wilbur Knorr [Kn2, pp. 354-360].

Stephen Menn [Me, p. 194] remarks that the neo-Pla­

tonists are conscious that they are speaking metaphorically

© 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc . . Volume 27, Number I, 2005 1 1

• 1'0/NTL.fND

0 SI'AC£LAND

"Fie, fie, how franticly I square my talk!" Title page of Flatland, embellished by art work of "A. Square"-the pseudonym of Edwin Abbott Abbott.

in extending the term analysis from geometry to philoso­

phy. The word, coming from the Greek analyein, meaning

"to break up," is never used by Plato in his writings [Me, p.

196], but Aristotle uses it, and-more importantly-Aristo­

tle describes Plato as ever watchful to see whether an ar­

gument is proceeding to or from first principles [Me, p. 193].

The word analysis has been used in different ways over

the centuries, but today mathematicians use it, of course,

to refer to the modem branch of mathematics whose first

principles involve the notions of number and limit.

Plato's early dialogues typically recount how Socrates

disproves the unexamined philosophical assertions of oth­

ers by deducing from them absurd implications (reductio ad absurdum). Socrates repeatedly claims to know only

how to examine carefully the assertions of others and not

how to advance a thesis of his own. This "ignorance of

knowledge" pervades the early dialogues, where philo­

sophical questions are raised and conventional answers

found wanting. Perhaps the object is to point us in the right

direction by examining in which way(s) our approxima­

tions miss the mark. One is reminded of the remark by John

von Neumann [vN] near the end of The Mathematician that

truth is much too complicated to allow anything but ap­

proximations.

In his so-called middle period Plato begins to apply to

philosophy something like the method of analysis described

12 THE MATHEMATICAL INTELLIGENCER

by Pappus. In Books II-X of the Republic he has Socrates

no longer criticize his interlocutors' ideas, but instead, as a

modem commentator puts it [P2, p. 972], to proceed

in a spirit of exploration and discovery, proposing bold

hypotheses and seeking their confirmation in the first in­

stance through examining their consequences. He often

emphasizes the tentativeness of his results, and the need

for a more extensive treatment.

In Plato's later writings the role of Socrates is dimin­

ished. The Eleatic Stranger and Timaeus, a Pythagorean

(apparently fictional characters, both), are introduced to

discuss ontology and cosmology-philosophical subjects

not associated with the historical Socrates. The self-criti­

cal analysis in the Parmenides seems to hint at Plato's need

for a new voice. Here we are cautioned to pay close at­

tention in discussions to the implications of the negation of the proposition in question. If one of these should be

false, then as logicians from Aristotle onward would em­

phasize, the original proposition is true by reductio ad ab­surdum, provided that we accept the law of the excluded

middle. Plato notes what is less often emphasized, that in

this way we uncover sufficient conditions for a proposition

to hold. (If not-p implies q, then not-q is a sufficient con­

dition for p.) Thus, with foresight, we can use the analysis

of implications to determine sufficient as well as necessary

conditions for a proposition to hold. This observation must

have been quite surprising when first noted.

About mathematicians we hear that they move "down­

ward" in deducing theorems from accepted hypotheses,

while philosophers should, as well, learn to move "upward"

from hypotheses to an ontological level at which the hy­

potheses themselves are seen to be justified. What do we

think about this today? When the axioms for a complete

ordered field were "justified" on the basis of the existence

of Dedekind cuts satisfying these axioms, was it mathe­

matics or philosophy that was done?

The power of analysis had been strikingly felt (around 430

BCE, as dated by Knorr [Kn1, p. 40]) when the Pythagorean

presumption that every ratio can be expressed as a ratio of

(whole) numbers was tested and proved false by reductio ad absurdum. Aristotle indicates (Prior Analytics i23),

perhaps too laconically, that this follows from the simple

fact that an odd number cannot be even. Most beginning

students of mathematics today know how to use this fact

to deduce that V2 is irrational.

Here is a less familiar proof of this ancient result: If the

ratio of the diagonal of a square to its side were express­

ible in lowest terms as (a + b )Ia, a ratio of positive inte­

gers with b < a, then (a + b )2 = 2a2 by the Pythagorean

theorem. But this implies that (a - b)2 = 2b2, so the origi­

nal ratio is expressible in strictly lower terms as (a - b )/b, which is a contradiction. The algebra here may at first ap­

pear contrived, but the geometry behind it is natural-as

van der Waerden [vdW, p. 127] explains-and would prob­

ably have been familiar to Theaetetus. It was well known

in Plato's time, and soon thereafter codified (Euclid X,

Proposition 2), that if the Euclidean algorithm never comes

to an end when applied to two line segments, then the seg­

ments are incommensurable.

Number and Measure in Ancient and

Modern Mathematics

Can the square root of two be expressed in terms of ratios

of integers? Theodorus, along with latter-day Pythagoreans,

would have said no because there is no single ratio that can

measure it. Our answer today, of course, is yes, it can be

measured precisely in terms of a cut in the set of all ratio­

nal numbers. Would Plato's colleagues, including Eudoxus

and Theaetetus, agree with us? It may be useful to consider

the barriers to the ancients' taking our point of view.

In classical times the word number was restricted to

"positive, whole number": a number is "a multitude com­

posed of units" according to Euclid VII. The unit itself was

not considered to be a number because it is not a multi­

tude, and the unit chosen in practice might be different in

different contexts, depending upon whether one were mea­

suring length or measuring area, for example. One speaks

of ratios of numbers, however, just as one speaks of ratios

of geometric magnitudes. As Plato and his colleagues were

acutely aware, there are more of the latter than of the for­

mer-a fact from which some might infer that geometry is

a "higher science" than arithmetic.

The ancient Greeks would have spoken of the ratio of

the diagonal to the side of a square rather than the "square

root of two," which only later denotes a numerically mea­

sured quantity. Their problem was to come to grips with

such ratios in the first place-and once this was done, to

check, for example, that the ratio of diagonal to side in one

square is the same as the ratio of diagonal to side in an­

other. But how can our numerical understanding of ratio

possibly be extended to incommensurable magnitudes?

Here is Eudoxus's definition of proportionality ("sameness

of ratio") from Euclid V, given two pairs of magnitudes,

each of which is assumed to have a ratio:

Magnitudes are said to be in the same ratio, the first to

the second and the third to the fourth, when, if any equi­

multiples whatever are taken of the first and third, and

any equimultiples whatever of the second and fourth, the

former equimultiples alike exceed, are alike equal to, or

alike fall short of, the latter equimultiples respectively

taken in corresponding order.

The convoluted phrasing may remind us of our first en­

counter with Cauchy's epsilon-delta definition of a limit.

Augustin-Louis Cauchy (1789-1857) is sometimes called the

nineteenth-century Eudoxus, for giving precise numerical

significance to a subtle concept essential to future

progress-although Dedekind is thought by some to be

even more deserving of this title.

To see what is going on here, let us consider a famous

case that Euclid left for Archimedes to study. Suppose that

the first and second magnitudes are the area A and the

square of the radius r2 of a circle, while the third and fourth

are the circumference C and the diameter D. Eudoxus's def­

inition given above says that the ratio A : r2 is the same as

C : D if and only if, for arbitrary natural numbers m and n,

(1) nA >mr � nC>mD (2) nA = mr2 � nC = mD (3) nA < mr2 � nC < mD.

In proofs involving proportionality (such as Euclid V,

Proposition 8) Euclid assumes what Archimedes later (see

[Di, p. 146 and p. 43lff.]) states explicitly as an axiom: that

if two magnitudes are unequal, then some integral multiple

of their difference (the magnitude by which one exceeds the

other) exceeds either. Perhaps Euclid's readers are expected

to infer that equality of a pair of like magnitudes should be

understood to mean that their difference has no ratio to ei­

ther member of the original pair-thus offering justification,

if needed, for the familiar fact that ratios of like magnitudes

are generally unchanged by the inclusion or exclusion of por­

tions of their boundaries. Euclid's silence on this issue makes

it difficult to determine whether he would consider condi­

tions (1), (2), and (3) to be independent.

Theaetetus must have been among those who first

thought deeply about how to treat equality in this setting, for

Plato pictures him (Theaetetus 155c) as being remarkably

concerned, even as a youth, with the problematic nature of

this seemingly transparent notion. Historians tend to credit

© 2005 Spnnger SC1ence+Bus1ness Med1a, Inc., Volume 27, Number 1, 2005 13

Eudoxus with the approach finally adopted, which (as clar­

ified by Archimedes) is beautiful in its simplicity. Properties

following from inequality are postulated explicitly, so that

"being equal" (in the case of continuous magnitudes) really

means "not being unequal." Here we have an inspired use of

litotes, the rhetorician's term for the expression of an affrr­

mative by the negative of its opposite.

By taking this indirect approach to the notion of "equal­

ity" of continuous magnitudes, the Greeks were able to han­dle many limits to their apparent satisfaction without be­

coming embroiled in (modern) concerns about existence

and uniqueness (and precise definitions) of limits. The

Greek method of exhaustion typically proves equality of

areas or of solids by simply showing that the assumption

of inequality leads to contradiction. If exhaustion is taken

to mean elimination (of possible answers that are too large

or too small), then the similarity of this method to the

rhetorician's artful use of litotes becomes quite clear.

Eudoxus's brilliant use of number to clarify the notion

of exactness in proportions is sometimes said [Bo2, p. 88)

to have led to "Platonic reform" in mathematics, but its ef­

fect upon Plato himself has drawn surprisingly little atten­

tion. The Greek word for proportion ( analogia) has a

broader meaning that encompasses analogy as well, and

Plato may have tried to extend Eudoxus's method to this

wider realm. As we shall see in the next section, Plato re­

gards the rhetorical treatment of "being" and "non-being"

as worthy of serious attention having to do, among other

things, with the demonstration of "exactness itself."

Let us first recall, however, the oft-noted connection be­

tween Eudoxus's condition specifying sameness of ratios

of geometric magnitudes and our modern condition for

equality of real numbers. This becomes apparent as soon

as we identify the ratios A : r2 and C : D with the real num­

bers that we conventionally symbolize by Ali!- and CID. Re­

quirements (1), (2), and (3) can then be interpreted to say

that the numbers Ali!- and C/D are equal if and only if the

condition that Ali!- is greater than (respectively, less than, or equal to) an arbitrary rational number min implies that

C/D is greater than (respectively, less than, or equal to) min as well. Eudoxus's condition is thus closely related to

the modern criterion that real numbers are equal if and only

if there is no rational number lying in between. It is easy

for a modern analyst to see how Dedekind's contemplation

of the condition of Eudoxus in Euclid V might have led him

to consider the reification of cuts.

When Archimedes demonstrated (by the method of ex­

haustion) in his Measurement of the Circle [Di, Chap. VI]

that A is equal to the area of a right triangle with legs C

and r, it became easy to prove that A : r2 is indeed the same

as C : D. Archimedes went on to show how one can effi­

ciently compute ratios, both greater and smaller, that ap­

proximate C : D to any accuracy desired, and in fact, he

proved that

223 : 71 < C : D < 22 : 7.

In an analysis class today it might be anachronistic, but cer­

tainly not hyperbolic, to say that Archimedes gave us an ef-

1 4 THE MATHEMATICAL INTELLIGENCER

fective algorithm to construct the Dedekind cut corre­

sponding to 7T. Nevertheless, the "ratio of the circle to its

diameter" seems not to have achieved firm status as a "num­

ber" until the advent of Indo-Arabic decimal fractions, il­

lustrating the vast difference between our modern numeri­

cal outlook and the geometrical framework it replaced. This

famous ratio was not called 7T before 1706, by which time

it was already known to many decimal places [Bo2, p. 442].

Expressing the Inexpressible: Is Non-Being a Form

of Being?

Did Plato inquire in what sense we can ascribe real nu­

merical significance to a ratio of incommensurables? The

answer implicit in Eudoxus's-or Dedekind's-approach is

surprising today, and would have been startling in Plato's

Academy in 350 BCE. We cannot say, for example, what the

numerical ratio of the diagonal to the side of a square is,

except to say what it is not. And this, the pair of segments

consisting of rational numbers greater and smaller, re­

spectively, as Dedekind has taught us, is all we need to deal

effectively with it as a number. While superficially similar,

perhaps, to the reification of "negative space" in art,

Dedekind's insight is considerably deeper. Non-being in the universe of rational numbers, when specified by such a dyad of segments, constitutes being in the real num­bers. The phrase "shadowy forms," which Dedekind used

to describe integers defined by his new approach to num­

ber theory [De, p. 33], seems even more apt to describe ir­

rational numbers defined by cuts.

We would know a lot about how much Plato anticipated

Dedekind's approach if the lengthy discussion in the

Sophist of the tricky relation between "being" and "non­

being" had ever been specialized to ratios of whole num­

bers. The Eleatic Stranger, however, pores over this puzzle

only in the most general terms before finally concluding

(Sophist 258) that Non-Being is-as Otherness-a Form.

Later on in the sequel, however, the Stranger remarks that

. . . just as with the sophist we compelled what is not

into being, . . . so now we must compel the more and the

less, in their turn, to become measurable . . . in relation

to the coming into being of what is due measure. (States­man 284b)

In context the practical point of this remark seems to

be that, for example, we should not simply say of a politi­

cian that he is being too heavy-handed or too wishy-washy,

but, more importantly, we should recognize and affirm the

existence of the precise "mean" attitude ("the Good") that

he should try to attain in the case at hand. The ostensible

theme of the Statesman, after all, is the delineation of the

character of the true political leader.

But the Stranger remarks immediately (284d) that he

may someday require this notion of a mean for the demon­

stration of exactness itself. Perhaps this is a hint as to the

content of a projected dialogue entitled the Philosopher, a

sequel to the Statesman that Plato never wrote. Could this

remark be based upon an underlying assumption by Plato

that the coming into being of a number ("due measure") is

coextensive with the specification of all ratios greater and

smaller? And is Plato here metaphorically associating the

existence of the Good with the existence of a point of unity

or harmony that brings opposing tendencies into proper

balance?

This speculation encounters a problem. Plato's commit­

ment to such a metaphor should force him as well to as­

sociate Evil with the dyad of the Great and Small that comes

into being simultaneously with the due measure, or "Unity,"

of the Good. Aristotle (Metaphysics 988a14-15) maintains,

however, that Plato did indeed assign "the causation of

Good and Evil" to these very elements. Aristotle's words

here do not carry all the moral connotations of their coun­

terparts in English. "Good" here means (the character of

being) "in good condition." "Evil" has the opposite mean­

ing, and is thus naturally associated with all ratios greater

or smaller than what is due. For Plato, ignorance-lack of

knowledge of what is right-is indeed the cause of evil.

"I hold that the definition of Being is simply

dynamis." (Sophist 247e)

The Greek word dynamis is usually translated power. We

get our words dynamic and dynamite from it, and dyne comes from it too, although this is a unit of force, rather

than power, in physics. The association of "being" with

"power" is one of the Stranger's most striking observations.

We might be tempted today to say that a real number ex­

ists simply by virtue of its power to cut the rationals in two.

The way the Stranger "cuts out" the Sophist by conjoining

appropriate properties (or by taking intersections of sets,

as we might describe it now) may be intended to suggest

that Platonic Forms exist by virtue of their power to com­

bine.

Near the beginning of the Theaetetus-Sophist-States­man trilogy, Plato has Theaetetus introduce the word dy­namis with some fanfare, giving it a special mathematical

meaning by associating it with the simplest irrationalities

like the one we call V2. Such a usage would be consistent

with modem terminology when we speak of both squares

and square roots as "powers," and many commentators

have understood the word dyna.mis to refer here to such

a "quadratic surd." This view, however, has been challenged

in more recent years. In [Kn1, pp. 65-69] Wilbur Knorr sum­

marizes the arguments on both sides and then defends his

claim that dynam·is can only mean "square," concluding

that its use as "square root" is not required by the text and

does not contribute to our understanding of the Theaete­tus.

It is still possible, however, that the mathematical usage

of dynamis as "square root" might contribute to our un­

derstanding of the famous line in the Sophist. It seems in­

disputable that the Stranger's "Being is dynamis" is in­

tended to mean "Being is power"-that is, that "being" is

coextensive with possessing the capacity to affect another

or to be affected. Plato might have foreseen, however, that

some of his readers would still associate dynamis with the

mathematical meaning imprinted upon it earlier in the tril­

ogy. The deftness of Plato's writing, with his occasional

penchant for puns and wordplay, sometimes enables him

to appeal in the same words to readerships of very differ­

ent sophistication. Plato left no doubt that he expected his

most serious readers to be serious students of contempo­

rary mathematics (Republic, Book VII). What would "Be­

ing is (something like) a quadratic surd" suggest to such

readers?

The historical figure Theaetetus himself is thought to

have been familiar with the information stored within our

modem continued fraction representation of quadratic

surds. One way we prove their irrationality today is to ob­

serve that their continued fractions are periodic and thus

unending, and Theaetetus probably knew something equiv­

alent to this method (see [Kat, p. 80]). Fowler [Fo2] argues

that the mathematicians of Plato's Academy were deeply

concerned with such analysis (anthyphairesis).

Irrational Exuberance?

A possible connection between square roots and "the dyad

of the Great and Small" (Aristotle's phrase to describe

Plato's conception) was put forward in 1926 by the philoso­

pher A. E. Taylor, who called attention to the familiar con­

tinued fraction for V2: 1

1 + ------1

2 + ----,..1-2 + --

2 + . . .

Taylor observed that the convergents 1, 1 + 1h 1 + 1/(2 + 1/2), . . . are (as Theaetetus almost certainly knew) alter­

nately less and greater than their limit of V2, and Taylor

identifies this "endlessness" with the dyad of the Great and

Small. As evidence Taylor cites Aristotle, who heard Plato

lecture for twenty years, and who implies (Physics A 192a)

that Plato identifies "the Great and Small" with the non-be­

ing mentioned by the Stranger in the Sophist [Ta, pp.

510-11].

Taylor is convinced that Plato is close to Dedekind's sub­

tle idea that conveys real numerical meaning to measure­

ments of ratios in the case of incommensurables. The con­

nection goes something like this. Dedekind's pair of

segments of rationals is identical, says Taylor, with the dyad

of the Great and the Small, which Aristotle says is the "non­

Being" in Plato's Sophist, which-as suggested above-is

really Being in disguise, defining Plato's "due measure" of

the quantity in question. Taylor thus refers exuberantly to

Plato as "the first thinker who had formed the concept of

a 'real' number" [Ta, p. 513].

Few mathematicians would grant so much. For one

thing, Plato does not discuss the accompanying algebraic

structure that modem analysts expect numbers to enjoy.

The Greeks did not associate ratios with our conception

of common fractions, so while it is natural to "compound"

(multiply) ratios, it is not so natural to add them. Could

Plato possibly have taken addition of ratios for granted? In

the Greater Hippias (303c) we find the remark that when

© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005 1 5

W a s fi n b unb w as fo l l tn b i t

ll a� l tn?

!Bon

�id)arb l) e b eiinb , Srojrj[or o tt � r r l r dJ u i ilf�r n � o lfl fdJ u l r a u !! u u tt flil m t iB·

� r a u n f cJ1 w e i g , �rud unb !Btrlag non {Yritbridj !Biewtg unb @'io�n.

1 8 8 8.

Title page o f Dedekind's famous monograph o n the natural numbers. The Greek motto "Man eternally arithmetizes" invites comparison with

the phrase "God eternally geometrizes" commonly (though perhaps mistakenly [Fo2, p. 293]) ascribed to Plato. Dedekind viewed mathemat­

ics as the science of number and, unlike Plato, viewed number as a free creation of the human mind.

the sum of two numbers is even, the numbers themselves

may be both even or both odd; but "when each of [the un­

specified things] is inexpressible, then both together may

be expressible, or possibly inexpressible." Some commen­

tators (see [Knl, p. 278]) interpret this to mean that Plato

was aware of the possibility that the sum of irrationals may

be rational. Moreover, Karl Popper [Po, pp. 250-253] gives

reasons for speculating that Plato must have conjectured,

in effect, that 7T = V2 + V3. Archimedes, however, whose

calculations with a 48-sided polygon circumscribing a cir­

cle effectively disproves this, makes no mention of such a

cof\iecture. Whatever Plato thought about the possibility

of adding ratios, it would be a very long time indeed be­

fore negative numbers and zero would be accepted as

"real."

16 THE MATHEMATICAL INTELLIGENCER

Creation versus Discovery

What should we make of the startling thesis that Plato sub­

stantially anticipated Dedekind's definition of a real num­

ber? In arriving at this thesis, A. E. Taylor seems to gloss

over apparent disparities between Plato's writings and Aris­

totle's account of Plato's teachings. Such disparities have

led some scholars to suggest that Plato left significant oral

remarks unwritten. Others, such as Kenneth Sayre, resist

speculating about Plato's "unwritten teachings" and un­

dertake instead a more careful reading of his extant work

Sayre discusses the interplay of ideas of Plato, Eudoxus,

and Dedekind in [Sa, p. lOOff. ] .

Whatever similarities w e may see, however, in their

views regarding real numbers, we can hardly fail to notice

the marked difference between Dedekind and Plato re-

garding ontology. According to Ferreir6s [Fe, p. 134],

Dedekind was always convinced that in mathematics we

create notions and objects. Here Dedekind writes of our

power to create the reals from the rationals:

We are of divine stock and there is no doubt that we have

creative power not only in material things (railways,

telegraphs) but in particular in spiritual things . . . . I pre­

fer to say that I create something new (different from the

cut) . . . . We have every right to adjudge ourselves such

creative powers.

-Letter to H. Weber in 1888, translated by Artmann

[Ar, p. 129]

Plato takes almost the opposite point of view regarding

creation and discovery. In the Philebus, beginning at 16c,

Plato has Socrates speak in praise of a "divine method" of

dialectic by which we ordinary human beings might dis­

cover the existence of (permanent, unchanging) things-a

method perhaps intended [Sa, p. 133] to be the reverse

counterpart of the (divine) method of creation by which

the system of Platonic Forms is originally composed.

Dedekind cuts, of course, are still considered somewhat

esoteric outside the world of mathematics, while Plato's

late works are not well known outside philosophical cir­

cles. This leaves a small group of readers who might try to

formulate and defend a carefully considered version of Tay­

lor's thesis.

Whatever the final verdict, at least we surely find

phrases from Plato that suggest Dedekind cuts or other no­

tions of a limit. Why do such phrases recur, particularly in

Plato's late dialogues? How did Plato's evolving ontology

draw him toward mathematical ideas that were difficult and

"modern" two millennia later?

Cuts, or Transition Points, as

Expressions of Limits

Limits are crucial to modern analysis, but Dedekind's point

of view sometimes enables us to get the job done without

mentioning them. A Riemann integral, for example, is noth­

ing but the Dedekind cut defined by all the lower sums and

all the upper sums off on [a, b ] . Thus, the number that Rie­

mann denoted by fif(x)d:x: is as simple (or as complicated)

as the number denoted by V2 would be to Theaetetus or

the number denoted by 1T would be to Archimedes. In each

case it is the transition point between the (rational) num­

bers that are too great and those that are too small to re­

flect its due measure. (Surprisingly, one can also avoid

mentioning limits in defining the derivative, by using ap­

propriate transition points instead. See [Ma] or [Prl] for de­

tails.)

In the Parmenides (156d) Plato discusses briefly the

idea of a transition point in time such as the instant be­

tween the states of rest and motion, but the Greek word

for limit (peras ), as we would expect, is never suitably de­

fined in a modern sense. From its opposite (apeiron), which seems to refer to indeterminacy in some sense, we

can gather that peras has to do with specifying something

precisely or exactly. When Plato uses peras in the Par­menides he seems to mean the defining edge, in a spatial

sense, of a thing or construction; but in the Philebus he

seems to mean the sorts of ratios that fix, in relations to

one another, apportionments (today we might say convex combinations) of opposites. The conflation, under the

rubric of limits, of such geometrical and numerical con­

ceptions as these is not unlike what we do in calculus

classes today.

When Plato speaks-alas, too vaguely-of how "due

measure" relates to the More and the Less, he may be close

to the key idea behind our modern understanding of lim­

its, but only so long as we are working in one dimension.

It is awkward, as Marsden and Weinstein [Ma, p. 180] note,

to move to a discussion of limits in higher dimensions with­

out giving up transition points in favor of something like

Cauchy's conception.

One wishes that Eudoxus could have returned to Athens

sooner to give Plato an earlier start on these new ideas.

Plato was in his seventies when he wrote the Philebus. The

extent to which he might have seen a connection between

one-dimensional cuts and higher-dimensional limits seems

beyond conjecture.

Forms and Sensibles

Nowadays, mathematicians generally regard the "dual na­

ture" of a set as requiring little or no explanation. Mathe­

maticians have no problem-and they expect their students

to have no problem-in thinking of a set S contained in a

universal set U both as a plurality of points and as a unit

unto itself, that is, as a "point" in the power set of U. In

Plato's time, however, the unity-in-plurality or one/many problem engendered a degree of interest that may remind

us of the modern fascination with the wave-particle dual­

ity of quantum mechanics. A Form was to be conceived of

simultaneously as a unit, an indivisible member of the

"world of being," and yet also as a plurality by virtue of its

capacity at any particular moment to manifest itself as a

multitude of sensibles in the "world of becoming."

A natural correspondence between Forms and sets

seems to suggest itself. In the Republic [P3, p. 213] Plato

speaks of the distinction

between the multiplicity of things that we call good or

beautiful or whatever it may be and, on the other hand,

Goodness itself or Beauty itself and so on. Correspond­

ing to each of these sets of many things, we postulate a

single Form or real essence, as we call it. . . . Further,

the many things, we say, can be seen but are not objects

of rational thought; whereas the Forms are objects of

thought, but invisible. (Republic 507b)

The "multiplicity of things" contains, however, only things

in our sensory world, and all such things are in flux. Some­

thing that is beautiful today may fade tomorrow, giving a

time-dependence to the "set" of beautiful things. The Form

of Beauty, on the other hand, is conceived to be a fixed ob­

ject of real knowledge and therefore unchanging in time.

© 2005 Spnnger Sc1ence+BuS1ness Media, Inc., Volume 27, Number 1, 2005 1 7

The Stranger [Pl, p. 408] lauds the ability of "the Philoso­

pher, pure and true" to

. . . see clearly one form peiVading a scattered multitude,

and many different forms contained under one higher form;

and again, one form knit together into a single whole and

peiVading many such wholes; and many forms existing

only in separation and isolation. This is the knowledge of

classes which determines where they can have commu­

nion with one another and where not. (Sophist 253d)

No modern analyst could read this passage without men­

tally picturing Venn diagrams of sets illustrating the notions

of union, intersection, set inclusion (or the notion of one

propositional function implying another), and disjoint or

overlapping sets (or the notions of inconsistency or con­

sistency of propositional functions).

In his late writings Plato begins to picture philosophical

knowledge as a great web of connections accessible to the

intellect, yet mediated by the senses. Plato's late interest

in sensibles is a departure from his earlier view of the

senses as an impediment to knowledge, their fallibility be­

ing a main source of false opinion. His early writings in­

troduce us to a realm of changeless Forms in which So­

cratic ideals (Virtue, Justice, etc.) float in splendid isolation

above the changing world of the senses. This is why many

readers still associate Platonism with a vague or mystical

conception of otherworldly existence that is cut off from,

and perhaps disdainful of, the transitory experiences of

everyday life. Plato's less-familiar late ontology, however,

is concerned with the problem of the interaction of these

discrete, fixed Forms with themselves and with our flow­

ing, continuous world. " [T]he thesis of radical separation

[of Forms and sensibles] is expressly rejected in the Par­men ides, is absent in the Statesman, and is replaced in the

Philebus by a contrary theory" [Sa, p. 255] .

Accounting for a causal interaction of Forms with sen­

sibles would seem to require something like the notion of

a limit. A modern analyst might envisage a simplex whose

extreme points represent the Forms and whose interior

points represent states of possible sense experience. Since

each interior point is expressible as a unique convex com­

bination of extreme points, we might think of measuring

how much a state "participates" (to use Plato's term) in a

Form, or in each of a collection of Forms, by the relative

sizes of its barycentric coordinates with respect to them.

In the Philebus, as already remarked, Plato speaks of some­

thing like a convex combination of opposites, and says in

effect (at 24d, for example) that varying combinations re­

flect the varying degrees of extremes that we momentarily

sense. As time goes on, these "coordinates" continually

change because the objects of sense perception are in con­

tinual flux, yet the Forms remain our fixed points of refer­

ence on the boundary. The world of becoming is thus rep­

resented by the interior points, while on the boundary lies

the world of being that holds sway above the flux. Perhaps

we might push the analogy further to obseiVe that, just as

1 8 THE MATHEMATICAL INTELLIGENCER

the boundary points are limits of interior points, we can

know the Forms as limits of our possible sense experiences.

Needless to say, Plato, whose mathematics counte­

nanced no simplex more complicated than a tetrahedron,

could barely begin such an analogy. In fact, an unembel­

lished simplex model fails because not all the Forms are

independent-Three-ness implies Oddness, for example.

Another shortcoming is that some Forms have opposites

(Hot/Cold, for example) and there is no naturally distin­

guished vertex in a simplex that is opposite a given vertex.

If we wish to indulge ourselves in modeling Plato's late on­

tology by using ideas from modern mathematics, we should

begin with something else-a Hilbert cube, perhaps.

In fact, Plato never wrote down a systematic and co­

herent account of his ontology. Here is Benjamin Jowett's

judgment: "At the time of his death he left his system still

incomplete; or he may be more truly said to have had no

system, but to have lived in the successive stages or mo­

ments of metaphysical thought which presented them­

selves from time to time" [Pl, p. 558] .

Nevertheless, Plato's final thoughts on the greatest Form

of all may best reveal why we see traces of limits and cuts

in his late writings.

Mathematics and "the Good"

In the Philebus Plato allows Socrates to take center stage

for the last time to discuss the Good, although "his manner

is more like that of the Eleatic visitor than of the ironic

Socrates we know" [Gu, p. 197]. This very late dialogue is

sometimes seen, because of a remark made by Aristotle

(Nicomachean Ethics 1 172), as written in part to counter

the philosophical views of Eudoxus, who claimed that the

Good consists in pleasure. Mter highlighting the notion of

a mixed state-a conception that, ironically, may owe much

to the mathematical views of Eudoxus-Socrates finally

concludes (Philebus 65-67) that the Good cannot be cap­

tured in one Form, but is a mixture of three-Beauty, Pro­

portion, and Truth-and is located "ten thousand times

nearer" to Wisdom than to Pleasure [Pl, p. 629].

On the Good is, in fact, the title of Plato's enigmatic fi­

nal lecture (or series of lectures), which he is thought to

have given about the time he reached the age of eighty. Ac­

cording to the testimony of some of those in attendance,

Plato surprised and confused his listeners by speaking

mainly about mathematics. How could Plato have failed to

make himself understood? Presumably, he used mathe­

matical imagery intended to evoke the euphoric idea of the

Good, which he had in his writings described as lying be­

yond knowledge, beauty, and being, and yet the cause of

all these. Could Plato have been trying to explain how the

Forms might be approached through limits? His thesis on

this occasion was "that Limit is the Good, a Unity" ac­

cording to one translation of words in a contemporary re­

port [Gu, p. 424] . Others, however, read the same words

quite differently [Ga, p. 5] .

While we may never know many details of Plato's last lec­

ture (see the beginning of [Ga] for a summary of what we

now know), we can speculate more confidently about why

his last writings contain suggestions of Dedekind cuts. Forms

can have opposites whose mixtures we sense, but sensible

instances of the Good are given by certain precise appor­

tionments that justly balance these opposites-those result­

ing, for example, in the divisions of the Pythagorean musical

scale (Philebus 26a). The essence of the Good thus involves

its power to cut, in exactly the right places, each of these con­

tinua joining opposites. "The mean or measure is now made

the first principle of good," as Jowett puts it [P1, p. 558].

Although the Form of the Good itself remains nebulous

in the writings of Plato, he seems to suggest that Wf' can

approximate its sensible manifestations arbitrarily closely

in much the same way that Theaetetus approximates \12. It was Plato's need to make sense of "good" measurements

on a continuum of possibilities that took him down the path

that Dedekind would explore so much more fully some

2200 years later.

But far more excellent, in my opinion, is the serious

treatment of these things, the treatment given when

one practices the art of diaJectj.c. Discerning a kin­

dred soul, the dialecti.cian plants and sows speeches

infused with insight, speedles that are capable of de­

fending themselves and the one who plants them, and

that are not barren but have a seed from which there

grow up different speeches in different characters.

Thus the seed is made immortal and he who has it

is granted well-being in the fullest measure possible for mankind. (PfuJedrus 276e-277a, translated by

Mitchell H. Miller, Jr., from Plato's 'Parmenides, ' Princeton University Press, 1986, p. vii.)

Continuing the Conversation

Seventeen-year-old Mark Kac experienced something like

an epiphany at the beginning of his calculus class at the

University of Lwow in 1931. Kac was expected to be fa­

miliar with Dedekind cuts, which he had never heard of,

and a young junior assistant named Marceli Stark recom­

mended something for him to read.

So I went home and read, and as I read, thf' beauty of

the concept hit me with a force that sent me into a state

of euphoria. When, a few days later, I rhapsodized to

Marceli about Dedekind cuts-in fact, I acted as if I had

discovered them-his only comment was that perhaps I

had the makings of a mathematician after all [Kac, p. 32] .

Like Kac, I remember the thrill that I felt as a college

student upon first understanding Hermann Weyl's succinct

explanation of the relation between Dedekind cuts and the

condition of Eudoxus [We, p. 39] . The present article, how­

ever, derives more from a reconsideration of the following

remark of mine:

What was a limit, before it was given a name? Before

Cauchy gave a precise significance to the notion, the an­

swer might have been an oracular utterance alluding to

the Greek method of exhaustion, like "that which re­

mains when everything to which it is not equal is elimi­

nated." If anything was ever airy nothing, this is it [Pr2,

p. 18].

Here I was close to existence problems that challenged

Plato, but I knew nothing then of what he had said about

such things in his late writings. The refutation of a single

wrong opinion, as Socrates discovers in Plato's early writ­

ings, can leave us still in the dark, but the elimination of

all wrong answers, as Socrates finally learns from the

Eleatic visitor, can carry us to the threshold of enlighten­

ment. Thus we have the power, so to speak, to tum wrong

answers inside out and make them tell the truth. When first

brought to light in Plato's Academy, this subtle observation

must have generated great excitement.

In modem mathematical analysis this kind of argu­

ment is familiar and its value can scarcely be overstated.

If all possibilities for the answer to a problem should lie

in a one-dimensional continuum, the key is often simply

the existencf' and uniqueness of the "right answer" after

the elimination of all numbers that are too great or too

small. Devoting a little time in an analysis class to study­

ing questions deliberately framed in this fashion can lead

both to an understanding of real numbers through

Dedekind cuts and to a quick grasp of "one-dimensional"

limits. The Eleatic visitor, thanks to Dedekind's kindred

insight, should be a stranger to mathematicians no

longer.

Epilogue

A discussion of possible connections between the ideas of

Dedekind, Eudoxus, and Plato might help to re-stimulate

interest among analysis instructors and their students in

the topic of Dedekind cuts and to arouse more interest in

the history and philosophy of mathematics. See [Kat] and

[Sh] , for example. Such a discussion could involve classi­

cists, historians, philosophers, and mathematicians in a

richly collaborative endeavor that would be valuable in it­

self. Among these groups are a multitude of fine scholars,

and some of them-including Chandler Davis, Hardy Grant,

Mitchell H. Miller, Jr. , and Jan Zwicky-have generously

given me much help and encouragement.

This paper is dedicated to the memory of Hugh Harris Cald­

well, Jr. , whose philosophy classes introduced me both to

Plato and to Dedekind.

REFERENCES [Ab] E. Abbott, Flatland, A Romance of Many Dimensions, sixth ed ,

Dover, New York, 1 952.

[Ar] B. Artmann, Euclid- The Creation of Mathematics , Springer-Ver­

lag, New York, 1 999.

© 2005 Spnnger Sc1ence+ Business Med1a, Inc , Volume 27. Number 1 . 2005 19

[Bo1 ] S. Bochner, The Role of Mathematics in the Rise of Science,

Princeton Univ. Press, Princeton, 1 966.

[Bo2] C. B. Boyer, A History of Mathematics, rev. by U. Merzbach, Wi­

ley, New York, 1 991 .

[Cr] A. W. Crosby, The Measure of Reality: Quantification and Western

Society, 1250-1 600, Cambridge Univ. Press, Cambridge UK, 1 997.

[De] R. Dedekind, Essays on the Theory of Numbers, trans. W. W. Be­

man, Open Court, Chicago, 1 901 .

[Di] E. J. Dijksterhuis, Archimedes. Princeton University Press, Prince­

ton, 1 987.

[Eu] Euclid, Euclid's Elements, trans. T. L. Heath, Green Lion Press,

Sante Fe, NM, 2002.

[Ev] W. R. Everdell, The First Moderns, University of Chicago Press,

Chicago, 1 997.

[Fe] J . Ferreir6s, Labyrinth of Thought: A History of Set Theory and its

Role in Modern Mathematics, Birkhauser, Boston , 1 999.

[Fo 1 ] D. Fowler, Dedekind's Theorem: v2 x V3 = v6, Amer. Math.

Monthly 99 (1 992), 725-733,

[Fo2] D. Fowler, The Mathematics of Plato's Academy, second ed. ,

Clarendon Press, Oxford, 1 999.

[Ga] K. Gaiser, Plato's Enigmatic Lecture 'On the Good, ' Phronesis 25

(1 980), 5-37.

[Gu] W.K.C. Guthrie, A History of Greek Philosophy, Vol. V, The Later

Plato and the Academy, Cambridge University Press, Cambridge UK,

1 978.

(Kac] M . Kac, Enigmas of Chance, Harper & Row, New York, 1 985.

[Kat] V.J. Katz, A History of Mathematics, second ed. , Addison-Wes­

ley, Reading MA, 1 998.

[Kn1 ] W. R. Knorr, The Evolution of the Euclidean Elements, Reidel,

Dordrecht, The Netherlands, 1 975.

[Kn2] W. R . Knorr, The Ancient Tradition of Geometric Problems,

Boston, Birkhauser, 1 986.

[Ma] J. Marsden and A. Weinstein, Calculus Unlimited, Benjamin Cum­

mings, Menlo Park, CA, 1 981 .

[Me] S. Menn, Plato and the Method of Analysis, Phronesis 57 (2002),

1 93-223.

[01] J. M. H. Olmsted, The Real Number System, Appleton-Century­

Crofts, New York, 1 962.

[P1 ] Plato, The Dialogues of Plato, Vol. I l l , trans. B. Jowett, Clarendon

Press, Oxford, 1 953.

[P2] Plato, Complete Works, ed. J. M. Cooper, Hackett Publishing, In­

dianapolis, 1 997.

[P3] Plato, The Republic of Plato, trans. F. M. Cornford, Clarendon

Press, Oxford, 1 948.

[Po] K. R . Popper, The Open Society and Its Enemies, fourth ed , Prince­

ton University Press, Princeton, 1 963.

[Pr1] W. M. Priestley, Review of Calculus Unlimited, Math. lntelligencer

4 (1 982), 96-97.

20 THE MATHEMA11CAL INTELLIGENCER

AU T H O R

W. M. PRIESTLEY Department of Mathematics and Computer Science

University of the South

Sewanee, TN 37383

USA

e-mail: wpriestl@sewanee.edu

William McGowen Priestley, known as "Mac," graduated from

the University of the South and returned there to teach in 1 967.

He received his Ph.D. at Princeton with a thesis directed by

Edward Nelson. He and his wife Mary, a botanist and now cu­

rator of the Sewanee Herbarium, have raised three children in

Sewanee. His persistent efforts to put together a one-semes­

ter calculus course for humanities majors have led to a dis­

tinctive textbook, Galculus: A Liberal M (Springer, 1 998).

[Pr2] W. M . Priestley, Mathematics and Poetry: How Wide the Gap?

Math. lntelligencer 12 (1 990), 1 4-19 .

[Sa] K . M . Sayre, Plato 's Late Ontology, Princeton University Press,

Princeton , 1 983.

[Sh] S. Shapiro, Thinking about Mathematics, Oxford University Press,

Oxford, 2000.

[St] 0. Stolz, Zur Geometrie der Allen, insbesondere uber ein Axiom

des Archimedes, Mathematischen Annalen 22 (1 883), 504-51 9.

[Ta] A. E. Taylor. Plato: The Man and His Work, Methuen and Co. , Lon­

don, 1 926.

[vdW] B. L. van der Waerden, Science Awakening, trans. A. Dresden,

Wiley, New York, 1 963.

[vN] J. von Neumann, The Mathematician, pp. 227-234 of Mathemat­

ics: People, Problems, Results, Vol. I , ed. D. M. Campbell, J . C. Hig­

gins, Wadsworth, Belmont CA. 1 984.

[We] H. Weyl, Philosophy of Mathematics and Natural Science, Prince­

ton University Press, Princeton, 1 949.

l*@ii•i§i•@hi%11fJ.i,ir:iii,hitfj Marjorie Senechal , Ed itor I

Herman Muntz: A Mathematician's Odyssey Eduardo L. Ortiz and Allan Pinkus

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater

than one. What we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 01 063 USA

e-mail: senechal@minkowski .smith.edu

In 1885 Weierstrass [ 1 ] proved that

every continuous function on a com­

pact interval can be uniformly approxi­

mated by algebraic polynomials. In other words, algebraic polynomials are

dense in C[a,b] (for any -x < a < b < + x ). This is a theorem of major im­

portance in mathematical analysis and

a foundation for approximation theory.

One of the first outstanding gener­

alizations of the Weierstrass Theorem

is due to Ch. H. Muntz, who answered

a col\iecture posed by S. N. Bernstein

in a paper [2] in the proceedings of the

1912 International Congress of Mathe­

maticians held at Cambridge, and in his

1912 prize-winning essay [3]. Bernstein

asked for exact conditions on an in­

creasing sequence of positive expo­

nents an, so that the system {X"" ]� =O is

complete in the space C[0, 1 ] . Bernstein

himself had obtained some partial re­

sults. On p. 264 of [2] Bernstein wrote

the following: "It will be interesting to

know if the condition that the series

:k llan diverges is necessary and suffi­

cient for the sequence of powers

{X"" )� =O to be complete; it is not cer­

tain, however, that a condition of this

nature should necessarily exist."

It was just two years later that

Muntz [M7] was able to confirm Bern­

stein's qualified guess. What Muntz

proved is the following:

Theorem. The syste:m {x"o, x"', X"', . . . ] , where 0 ::::: ao < a1 < a2 < . . . , is complete in C[0, 1 ] if and only if ao =

0 and

1 I - = X. n = l a,

Today there are numerous proofs and

generalizations of this theorem, widely

known as the "Muntz Theorem." In fact

a quick glance at Mathematical Re­views, that is, at papers from 1940,

shows nearly 150 papers with the name

Muntz in the title. All these articles

mention Muntz's name in reference to

the above theorem, except one refer­

ring to his thesis. Muntz's name with

his theorem appears in numerous

books and papers. In addition there are

Muntz polynomials, Muntz spaces,

Muntz systems, Muntz-type problems,

Muntz series, Muntz-Jackson Theo­

rems, and Muntz-Laguerre filters. The

Muntz Theorem is at the heart of the

Tau Method and the Chebyshev-like

techniques introduced by Cornelius

Lanczos [4] . In other words, Muntz has

come the closest a mathematician can

get to attaining a little piece of immor­

tality.

Notwithstanding, a quick search of

the mathematical literature will also

show that essentially nothing is known

about Muntz, the person and the math­

ematician. The purpose of this paper is

to try to redress this oversight. Muntz's

life, mathematically and otherwise,

was an illuminating and dramatic jour­

ney through the first half of the twen­

tieth century. It is unfortunate that it

was not a more pleasant journey.

Early Years (1884-1 9 1 4)

Herman Muntz1 (officially named

Chaim) was born in .t6dz on August

28, 1884. Muntz's family was bourgeois

and Jewish, though not religious. At

that time .t6dz was a part of "Congress

Poland" under Russian rule. It was an

important industrial city at the western

boundary of this area. In the last

decades of the nineteenth century,

when Muntz was born, it had a vibrant

Jewish community, mainly engaged in

textiles and other related trades, as

well as in business in general. In offi­

cial documents, Muntz's father is de­

scribed as "in trade," with the sugges­

tion that he was an estate agent. The

Eduardo L. Ort1z thanks the Royal Society, London, for its financial support while researching this paper.

1 The file on Muntz preserved at the Society for the Protection of Science and Learning, now at the Bodleian

Library, Oxford, provided a valuable start in the search for other sources included. On the Muntz files there

see Ortiz [5].

22 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+ Bus1ness Med1a. Inc.

Herman MOntz

family name was spelt in the German

manner rather than the more common

Mine. Herman was the eldest of five

children, all of whom (except for the

youngest) were sent to study at Ger­

man and Swiss universities. The tur­

bulent economic times were such that

the family was generally, though not al­

ways, comfortably well off. A notice­

able decline was associated with the

depression of the late 1920s. Muntz

started his studies at the Hohere Gewerbeschule in L6dz, the top tech­

nical high school in that city, with a

bias toward textiles, textile machinery,

and chemistry. He was fluent in Polish

and had a reasonably good command

of German and Russian.

In 1902 Muntz went to Berlin to

study at the Friedrich-Wilhelms-Uni­

versitat, generally referred to as the

University of Berlin (called Humboldt

Universitat Berlin since 1948), where

he studied mathematics, the natural

sciences, and philosophy. In 1906 he

earned his matriculation degree. He

named Frobenius, Knoblauch, Landau,

Schottky, and Schwarz as his teachers,

singling out Frobenius and Schwarz as

his main influences.

From 1906 to 1910 Muntz was in

Berlin, where he worked, wrote, and

studied. In 1912 he married Magdalena

(Magda) Wohlman who was from the

area of Zlotk6w near Poznan, an area

of Poland under German control. Magda

had come to Berlin to study biology.

While the marriage would remain child­

less, it was, by all accounts, an unusu-

ally harmonious union. During this early

period Miintz was involved in the pri­

vate teaching of mathematics. Money

was always a pressing problem. For

much of his life Muntz remained en­

gaged in pedagogy in one form or an­

other, "teaching elementary and higher

mathematics, partly in private schools

and partly as a private tutor."2

Mi.intz was an intellectual who was

intensely interested in philosophy, po­

etry, art, and music. He was especially

taken with Goethe, but more particu­

larly with Nietzsche's philosophy,

which was to have a profound influ­

ence on him. He attended university

lectures by the philosopher Alois Riehl,

and he seems to have written a thesis

on Nietzsche.

In these years he also became in­

terested in a reassessment of Jewish

culture and the position of Jews in so­

ciety. In 1907 he published a 124-page

book called Wir Juden (We Jews) [6],

dedicated to Friedrich Nietzsche and

showing the influence of his Also sprach Zamthustm. The book con­

cerns the need for a basic reform of the

Jewish people in the post-orthodox pe­

riod, and a reconsideration of the po­

sition of Jews in society.

Mi.intz discussed in detail what he

called the "new Jew," and the contribu­

tion Jewish people had made and could

make to humanity. He characterized

Jews not as a pure race but as a diver­

sity of many peoples, emphasizing the

past and present connections between

Jews and a variety of other people. The

book aspired to help the young Jewish

generation of the time to achieve reli­

gious and political self-definition. It em­

braced a view of Zionism not uncommon

at the time, in which socialist viewpoints

were discernible. There were remarks in

Miintz's text that are very much race­

based, which may make it discomforting

to read today. But they should be un­

derstood in the context of the era. The

book was advertised in the Berlin Jew­

ish/Zionist weekly Jiidische Rundschau in its list of "Zionistische Literatur."

These advertisements continually mis­

spelt the author's name as "Miintzer,"

which might be considered as a measure

of the perceived importance of the book

Aside from a mathematical text men­

tioned later, this was the only book by

Miintz that was ever published. How­

ever, we have found various items of

correspondence indicating that he also

wrote at least three other (non-mathe­

matical) texts. All written from about

1911 to 1924, they were: Ober Ehe und Treue (On marriage and fidelity); a book

about the Psalms; and Der Judische Staat (The Jewish state). The three

manuscripts were sent to different pub­

lishers, but for a variety of reasons, in­

cluding the war and lack of paper, none

seems to have been published. How­

ever, parts of the last-named book ap­

peared as articles in a journal.

Despite these varied activities,

Muntz's main focus in the period 1906-

1910 was his mathematical studies, un­

der the supervision of Hermann Aman­

dus Schwarz. His first results were of

a geometric character, having to do

with rational tetrahedra. However, he

soon began to produce results on the

main topic of his doctoral dissertation,

namely, minimal surfaces defined by

closed curves in space, that mathe­

matically involved the approximate so­

lution of non-linear partial differential

equations. On October 1, 1910, Muntz

was awarded a doctorate, Dr. Phil.,

magna cum laude. His official review­

ers were Schwarz and Schottky. His

dissertation, under the title "On the

boundary-value problem of partial dif­

ferential equations of minimal sur­

faces," was published in Grelle's jour­

nal [M1] . This work is still occasionally

referenced.

In this thesis Mi.intz studied the

Plateau problem in some detail. He

used potential theory and the method

of successive approximation, two tools

he would return to in subsequent pa­

pers. When Muntz was near the end of

this dissertation work, Schwarz ad­

vised him that Arthur Korn, who was

working in the same area, had submit­

ted for publication a paper on the sub­

ject of his thesis, which was later pub­

lished [7) . In his Grelle paper Muntz

acknowledged Korn's work Although

their results had a common ground, the

techniques used and the final results

were sufficiently different to merit in-

·-- - --- ---

2Muntz to Geheeb, March 1 , 1 91 4. Archtve of the Ecole d 'Humantte

© 2005 Spnnger Sc1ence+Bus•ness Media. Inc , Volume 27. Number 1 . 2005 23

dependent publication. Muntz seems to

have been the last of Schwarz's doc­

toral students. Other doctoral students

of Schwarz included Leopold Fejer,

Ernst Zermelo, Paul Koebe, and Leon

Lichtenstein. The latter became a close

friend of Muntz.

In late 191 1 Muntz went to Munich

to give a lecture at the seminar of Fer­

dinand von Lindemann. He was also ac­

cepted into Aurel Voss's circle. These

were two of the three mathematics pro­

fessors at the Karl Ludwig-Maximilians

Universitat in Munich; the third was Al­

fred Pringsheim. The Muntzes decided

to move to Munich primarily on the ba­

sis of this visit, which seemed to open

some opportunities. But they were also

undoubtedly influenced by the fact that

two of Muntz's brothers were also then

residing in Munich.

Miintz's aim, and that of any young

aspiring mathematician in Germany at

this stage of his career, was to secure a

position as a "Privatdozent." The next

stage was to gain a Habilitation and

eventually an academic position at a uni­

versity. At that time (and the same is es­

sentially true today) the Habilitation was

necessary for a professorship, and a pro­

fessorship is what Miintz wanted then

and throughout his life. According to his

correspondence, Miintz, who was not

the only candidate, obtained the support

of the three mathematics professors. It

seems, however, that there were also

what he termed some "strange regula­

tions," and serious formal problems. The

matter dragged on. In the end, Miintz

was unsuccessful in gaining the dozent

position.

While in Munich Muntz was again

earning his living privately as a teacher

at various levels. His wife also worked

part-time and there was some financial

help from the family. Muntz attended

lectures and seminars given by von Lin­

demann and Voss and was actively en­

gaged in mathematics research. From

1912 to 1914 he published four papers

in the field of modem projective geom­

etry and the axiomatics of geometry,

two of which appeared in Mathematis­che Annalen. His 1912 paper on the con­

struction of geometry on the basis of

only projective axioms was read by

Voss at a meeting of the Bavarian Acad­

emy. In 1913 he published two notes in

Comptes Rendus in connection with the

use of iterative techniques for the solu­

tions of algebraic equations. It is very

possible that Miintz was the first to de­

velop an iterative procedure for the de­

termination of the smallest eigenvalue

of a positive definite matrix. It certainly

predates the more generally quoted re­

sult of R. von Mises of 1929 [8]. In 1914

he published an additional two papers

on approximation theory. The first is a

note on properties of Bernoulli polyno­

mials published in Comptes Rendus. The other is the paper in which the

Muntz Theorem appeared. This last

work was written as a contribution to

the Festschrift in honour of his teacher

Hermann Schwarz's 70th birthday.

In this period reference is already

made in Muntz's correspondence to se­

rious problems in one of his eyes. Eye

problems would plague him through­

out his life.

Boarding Schools and Martin

Buber ( 1 9 1 4-1 9 1 9)

In early 1914, probably through social­

ist and feminist common friends,

Muntz started a correspondence with

the pedagogue Paul Geheeb, who ran a

boarding school called the Odenwald­

schule near Heppenheim in southern

Hessen. Muntz moved to Geheeb's

school in 1914 as a mathematics

teacher, with the understanding that he

would be able to devote a considerable

amount of his time to his mathemati­

cal research. It was agreed that he

would have at most three hours of

teaching a day. This was to be the first

time he taught very young children.

In a letter to Geheeb written by

Mario Jona, who interviewed him for

the position, there is the following pas­

sage:3 "He [Muntz] is perfectly aware

of what he is worth and shows it, which

face to face is not so unpleasant as in

writing. As it was I imagined him from

his letter to be much more terrible. He

is short, pleasant and with a very seri­

ous appearance and sometimes a little

clumsy in politeness, . . . For him the

most important thing is his scientific

work. He is in a period of important sci-

3Jona to Geheeb, March 1 0, 1 91 4. Archive of the Ecole d'Human1te.

24 THE MATHEMATICAL INTELLIGENCER

entific activity, but would like also to

work in a school like ours if he also has

time to work for himself."

Geheeb was a liberal humanist, pro­

feminist, and much opposed to anti­

Semitism. He and his schools hold a

special place in the history of progres­

sive education in Germany. At one of

his earlier schools, in Wickersdorf, he

had established the first co-educa­

tional boarding school in Germany. His

wife, Edith Cassirer, was a progressive

young teacher, the daughter of the

wealthy Berlin Jewish industrialist

Max Cassirer. With his father-in-law's

financial backing, Geheeb founded the

Odenwaldschule in 1910. It was a large

boarding school with modem or spe­

cially modernized buildings. Co-educa­

tion, an emphasis on physical educa­

tion, and flexibility in the curriculum

were among its innovations. The new

school was run with a fair amount of

self-government. The teachers, and es­

pecially Geheeb, supposedly guided

rather than led. The students were

called Kameraden, "comrades," and

the teachers Mitarbeiter, "co-work­

ers." In 1914, there were 68 full-time

students, many of whom were children

of the liberal, affluent German intelli­

gentsia. The children of Thomas Mann

and of other noted writers and artists

were among the pupils and were not

necessarily easy to handle. Much has

been written about this school and

Geheeb. The school survived both wars

and exists today, but the Geheebs left

in 1934 when the influence of Nazi ac­

tivists reached the school, and they

moved to Switzerland. There, he and

his wife established a school of a re­

lated character: the Ecole d'Humanite. According to some, Muntz included,

life in Odenwaldschule seemed anar­

chic on occasion. Muntz and Geheeb

parted ways in the summer of 1915.

Nonetheless Muntz kept in touch with

some of the school's faculty and re­

mained on speaking terms with

Geheeb. Muntz then found a similar po­

sition at another school, Durerschule,

which does not exist today, in Hoch­

waldhausen also in Hessen. Muntz

seems to have enjoyed his teaching,

and developed very definite opinions

on the teaching of mathematics and

science to younger children. Another

teacher who joined him at the Dur­

erschule was his friend and brother-in­

law Herman Schmalenbach, married to

his sister Sala, who later became a Pro­

fessor of Philosophy at the University

of Basel.

The war was having its impact.

Muntz was an "alien," with Hessian res­

idency but no German citizenship, and

he was generally restricted in his trav­

els. This prevented a move to Heidel­

berg planned in 1915. In a letter dated

August 1917, Muntz wrote that he had

to stay in Hessen to avoid difficulties

with the authorities. However, as an

"alien" he did not take part in the war.

Although happy at the school, he was

forced to leave after an open meeting

in 1917 where the headmaster, G. H.

Neuendorff, called him a "little Polish

Jew." See Butschli [9].

Many pupils, especially the Jews, also

did not return to Dtirerschule after the

holiday. Muntz felt he had a responsi­

bility for some of these children and de­

cided to return as a private scholar to

Heppenheim where he had friends, but

not to Odenwaldschule. With his wife,

he managed a small boarding house for

students: a Schiilerpensionat. Despite his many obligations and

worries, Muntz still managed to carry

on with his mathematics research. Dur­

ing this period he published five more

papers, concerning problems in pro­

jective geometry, and the solution of al­

gebraic equations and algebraic eigen­

value problems.

While still at the Odenwaldschule,

Muntz had begun to correspond with

Martin Buber, the enlightened and

broad-minded philosopher, Zionist

thinker, and writer, who was then in

Berlin. Buber was the spiritual leader

of an entire generation of German­

speaking Jewish intellectuals. He ad­

hered to a form of tolerant utopian so­

cialism he called "Hebrew humanism."

In 1915 Muntz helped Buber find a

house4 in the town of Heppenheim,

where Buber and his family lived from

1916 until 1938. Buber then left for

Palestine to take a chair in Social Phi­

losophy at the Hebrew University, and

subsequently had a distinguished ca­

reer there. During the First World War

the two families kept in close contact

and exchanged fairly intense and in­

teresting correspondence.

In 1915 Buber founded and co­

edited a journal called Der Jude [ 10]

that for eight years was the most im­

portant organ of German-reading Jew­

ish intellectuals. In a letter dated in No­

vember of that year, Buber invited

Muntz to become one of his collabora­

tors on this journal. He wrote, "You are,

of course, amongst the first whom I am

asking to participate."" Muntz wrote 18

articles and notes for this journal,

some quite lengthy, under the pseudo­

nym of Herman Glenn. It is an indica­

tion of the way in which Muntz's con­

tributions were valued that in the very

first issue of Der Jude, the first article

was signed by Buber, while the second

was signed by Glenn (Muntz).

GoHingen and Berlin (1 91 9-1 929)

Around 1919 or 1920 Muntz seems to

have had a nervous breakdown and

was placed at a sanatorium in Gander­

sheim (now called Bad Gandersheim)

near Gi:ittingen. We do not know ex­

actly how long Muntz was in the sana­

torium. The few letters available from

this period are rather bleak. In a letter

to Buber in September 1923 Muntz re­

called that he suffered a personal col­

lapse in 1919-1920 and said he learnt

from the experience to look at things

from a distance, and in "this way they

are no longer dangerous to me."6

Toward the end of 1920 Mtintz and

his wife moved to his wife's family farm

in Poland to recuperate for some eight

to ten months. Letters show that during

this period the Muntzes, together with

his wife's brothers, considered emi­

grating to Palestine. But the economic

situation there was far from encourag­

ing and the idea was dropped. As he re­

cuperated, Muntz took up mathematics

again and from the farm traveled to

Warsaw to attend seminars and to lec­

ture on his research. This activity is re-

fleeted in a number of publications in

the journal of the then recently founded

Polish Mathematical Society.

In October 1921 the Muntzes re­

turned to Germany, moving into a

boarding house in Gi:ittingen. At that

time the Schmalenbachs also lived in

that city. It is not clear how the

Muntzes supported themselves in Gi:it­

tingen-probably again through pri­

vate teaching and with the help of their

family. While in Gi:ittingen Muntz did

considerable mathematical research.

During this time he wrote eleven pa­

pers, published between 1922 and

1927. They cover a number of topics,

including integral equations, the n­body problem, summability, Plateau's

problem, and quite a few papers on

number theory, possibly under the in­

fluence of Edmund Landau. One of his

results from this period is quoted in

Titchmarsh [ 1 1 ] .

By this time Muntz seems to have

made a name for himself within both

mathematical and Jewish/Zionist cir­

cles. He was a member of the editorial

board of the mathematics and physics

section of the short-lived journal

Scripta Universitatis founded by Im­

manuel Velikovsky in Jerusalem. The

one and only issue of the mathematics

and physics section was edited by Ein­

stein and published in 1923. He also co­

operated with Hertz, Kneser, and Os­

trowski in a German translation of

some lectures of Levi-Civita [M21 ] .

In April 1924 the Muntzes moved to

Berlin, while maintaining scientific

contacts in Gi:ittingen. Muntz also re­

turned to writing on Jewish matters.

He sent contributions to Der Jude, some of which were excerpts from the

third of his unpublished books on Jew­

ish matters.

Muntz, who never did much collab­

orative research, did have at least one

student in this period. Divsha Amira

(nee Itine) from Palestine was a

geometer who officially obtained her

doctorate from the University of

Geneva in 1924. She worked with

Muntz whilst residing in Gi:ittingen. In

1925 she published a memoir [ 12] on a

�� - - - -� - ------�--- �---------�-----------

4Today called The Martin Buber House, it is home to the Internal Council of Christians and Jews.

5Buber to Muntz, November 1 1 , 1 91 5, Buber Archives, JNUL, Jerusalem.

6Muntz to Buber, September 18 , 1 923, Buber Archives, JNUL, Jerusalem

© 2005 Spnnger SC1ence+Bus1ness Media, Inc .. Volume 27, Number 1 , 2005 25

projective synthesis of Euclidean

geometiy. Mtintz had attempted a con­

struction of algebraic Euclidean geome­

tiy using what he called Basisfiguren. In her memoir Divsha extended Mtintz's

ideas to the general Euclidean plane,

considering, instead, sets of straight

lines. She discussed elementary con­

structions, congruence axioms, and the

axiomatic construction of geometiy. Di­

vsha was generous in her remarks to

Mtintz and to his research. Although not

her formal thesis supervisor, Mtintz

clearly was her mentor. Divsha's hus­

band Bel\iamin, who also obtained his

doctorate from the University of Geneva,

was a student of Edmund Landau.

The first meeting of the board of

governors of the new Hebrew Univer­

sity in Jerusalem took place in April of

1925. At that meeting it was decided to

establish an institute devoted to re­

search in pure mathematics, staffed by

one professor and two assistants. The

board of governors also authorized the

President and Chancellor to offer Ed­

mund Landau, then still in Gottingen,

the professorship in pure mathematics.

The involvement of Landau in the He­

brew University started well before the

First World War, and lasted into the

1930s. He had the major say on who

would be appointed in mathematics. At

a meeting of the board of governors in

September 1925, Landau was asked to

draw up plans for the establishment of

a mathematical institute to be opened

as soon as funding became available.

At the suggestion of Landau, it was de­

cided to appoint Benjamin Amira as the

first assistant.

In October of 1925 Mtintz was in

Berlin and was busy trying to find a po­

sition, either in or outside Germany. The

creation of the Hebrew University un­

doubtedly interested him as a mathe­

matician, as someone without proper

employment, and as a Zionist. Mtintz

saw himself as the professor and thus

the director of this new mathematical

institute. In a rather manipulative way,

he used his connections, particularly

Schmalenbach's everlasting good dispo­

sition towards him. He asked his well­

positioned brother-in-law to contact Bu­

ber, Landau, and Courant on his behalf.

Schmalenbach reported that Courant

was emphatic in stating that he had no

doubts as to Mtintz's qualifications,

which he exhibited in his papers and in

his lectures at the meetings of the local

mathematical society. However, he in­

dicated that not having a Habilitation

was a serious drawback The Jerusalem

matter was resolved negatively in early

November. In retrospect, it seems Mtintz

had misread the entire situation. At this

early stage in 1925 Landau probably was

saving the professorship for himself. In any case, the academic leadership of the

Hebrew University was looking for an

established star to take up the profes­

sorship-or at least for someone with a

Habilitation and also a chair somewhere

else. They were looking indeed for a per­

son like Landau, who did move to

Jerusalem with his family for the initial

academic year of 1927-28. However, for

various reasons things did not work out

and he returned to Gottingen the fol­

lowing year.

Throughout this period Muntz was

constantly seeking an academic ap­

pointment, while at the same time at­

tempting to obtain his Habilitation. In

1925 Voss and von Lindemann recom­

mended him for a Habilitation in

Giessen, but nothing came of it. That

same year it appears he was recom­

mended, by A A Fraenkel, for a posi­

tion at the University of Cairo. Again

he was unsuccessful.

As we said, according to Courant,

the fact that Mtintz had not been given

the Habilitation in Gottingen was not

as a consequence of his lack of qualifi­

cations. There were other reasons. On

the one hand there was the question of

his origin, which he did not try to hide.

On the other hand there was Gottin­

gen's hierarchy. As in the cases of

Bernays, Hertz, and E. Noether, if he

were not to be called by a university,

Gottingen would feel morally obliged

to provide for his maintenance. Muntz

had heard essentially the same from

Hilbert years earlier. To this Muntz jus­

tifiably complained that he was in an

impossible situation. If he had a guar­

anteed position then he would have no

problem being given Habilitation, but

without the Habilitation it was almost

7MOntz to Buber, October 30, 1 927, Buber Archives, JNUL, Jerusalem.

26 THE MATHEMATICAL INTELLIGENCER

impossible to obtain a position. His

case was not unique.

One source of income for Mtintz we

have identified is the Jahrbuch iiber die Fortschritte der Mathematik (FdM).

This annual review, published from 1869

until the end of the Second World War,

was in the format adopted by the Zen­

tralblatt fur Mathematik, and later

shared by Mathematical Reviews, ex­

cept that it appeared each year as a sin­

gle volume. As a consequence of the

First World War, the work on the annu­

als was severely backlogged and re­

mained so for many years thereafter. A

count of the reviews shows Mtintz wrote

nearly 800 reviews for the FdM, mainly

during the mid-1920s. He was still regis­

tered among the journal's regular re­

viewers up to 1929, when he had already

left Germany. Reviewers were paid 1

Reichsmark per review. The average

salary at the time seems to have been

about 120 Reichsmark per month.

The reviews by Mtintz cover an ex­

tensive mathematical, as well as linguis­

tic, area. Besides the languages he was

brought up in, namely Polish, German,

and Russian, Mtintz reviewed papers in

English, French, Italian, Dutch, and

Swedish. The topics, besides function

theory and differential equations, were

probability theory, fluid mechanics, the

theory of electricity and magnetism, in­

cluding its geophysical applications, nu­

merical methods of calculation, and the

history of mathematics.

At the end of 1927 Muntz wrote to

Buber that for several months he had

been the professional scientific collab­

orator of Einstein, and added, "This, of

course, compensates me a great deal

for what has been in Germany an al­

most impossible situation, as the offi­

cial professionals are more 'official'

than 'professional.' " 7 He was probably

alluding to the fact that he had been

unable to obtain a Habilitation. It is not

clear when Muntz started to work with

Einstein and for how long this collab­

oration continued. From his wife's cor­

respondence we learn he met Einstein

socially in January, 1927. For much of

the time that Mtintz was working

with/for Einstein, another subsequently

well-known mathematician, Cornelius

Lanczos, did so too. Both seem to have

been supported by grants from a fund,

the Notgemeinschaft Deutscher Wis­senschaftler, supporting prom1smg

"young" scientists. Muntz published no

joint papers with Einstein, but Ein­

stein's archive has extensive corre­

spondence between Muntz and Ein­

stein on a range of mathematical ideas.

Moreover Mtintz and Lanczos are men­

tioned and thanked in two of Einstein's

papers on distant parallelism.

In describing to his sister his work

under Einstein in September of 1927,

Muntz indicated that it was "running

'normally' and for reasons of conve­

nience I have 'submitted' myself; for in

these fields it is he who is the extraor­

dinary master while I am only the 'tech­

nical' assistant. Nevertheless I am very

happy to be working with him." On

more than one occasion, however, Ein­

stein politely expressed reservations

regarding Muntz's work. Einstein

sometimes indicated that he did not be­

lieve there were sufficient reasons for

Muntz's assumptions, or he did not re­

gard Muntz's reasoning as being justi­

fied, or he did not think that Muntz's

arguments made "any obvious experi­

mental-physical sense."

Toward the end of 1928 Muntz was

again considering the possibility of tak­

ing a chair outside Germany. However,

he purposely kept these discussions

from many of his close friends and col­

leagues, including Lichtenstein and

Einstein,8 which suggests that he still

expected that they might be able to

help him find a job within Germany.

Leningrad (1 929-1 937)

In May of 1929 Muntz finally obtained

an academic appointment, something

for which he had yearned for many

years. He was invited to fill the posi­

tion of Professor of Mathematics and

Head of the Chair of Differential Equa­

tions at the Leningrad State University.

In Leningrad Muntz was also put in a

group of "exceptional scientists," and

given a "personal salary." From 1933 he

is listed as Head of the Chair of Dif­

ferential and Integral Equations.

In a letter written a few years later,

Muntz stated that in 1927 he had been

offered the Lobachevsky Chair in

Kazan, and during the technical period

of waiting was offered the Chair for

Higher Analysis in Leningrad. Accord­

ing to Muntz, he "exchanged" the chair

in Kazan with that in Leningrad (ini­

tially offered to Bernstein), while his

friend N. G. Chebotarev took the Kazan

Chair. We have found no direct docu­

mentation to support this claim, but

the fact is that Chebotarev became pro­

fessor at Kazan University in 1928 af­

ter having been offered posts at both

Kazan and Leningrad.

G. G. Lorentz, who was an under­

graduate at the time, recalled [ 13] that

in 1930, shortly after his arrival in

Leningrad, Muntz was called upon to

present a lecture sponsored by the

Leningrad Physical and Mathematical

Society on the so-called crisis of the ex­

act sciences. The subject was the foun­

dational debate in mathematics, and

Hilbert's attack on the intuitionism of

Brouwer and W eyl. Muntz was an ideal

candidate to deliver the lecture be­

cause of his research background on

the foundations of mathematics; and

having recently arrived from Germany,

he was perceived as the carrier of the

latest advances on this controversy.

The lecture was well attended. Of

course Muntz stated that the crisis was

only in the foundations and did not in

any way affect the work of most math­

ematicians. However, because of an

underlying power struggle between N.

M. Gunter, V. I. Smirnov and Ya. V. Us­

penskyi, on the Society's traditional

side, and L. A Lefert and E. S. Rabi­

novich, of an alternative young Com­

munist league, the meeting turned

rowdy and undisciplined. The Leningrad

Physical and Mathematical Society sub­

sequently ceased to exist in its previous

form, being amalgamated into a new

organization under Rabinovich.

The row does not seem to have af­

fected Mtintz's subsequent career. From

1931 Mtintz was also in charge of math­

ematical analysis at the Scientific and

Research Institute in Mathematics and

Mechanics (Nauchny'i Jssledovatelski'i Institul Mathematiki i Mehanik� or NI­

IMM) at Leningrad State University. Fur­

thernlOre, in 1932, Mtintz's position in

Russia must have been quite firm, for

he was given the singular distinction of

being sent to the International Con­

gress of Mathematicians in Zurich as

one of the Soviet Union's four official

delegates. The other three were Cheb­

otarev, representing Kazan State Uni­

versity, who gave a plenary lecture on

Galois Theory (on the occasion of the

centenary of the death of Galois), the

famous topologist P. S. Aleksandrov

from Moscow State University, who

talked about Dimension Theory, and E. Ya. Kol'man, a mathematician and a

member of the Communist Academy in

Moscow. The Academy was an institu­

tion created in 1918 which had been

given the task of developing Marxist

views in the fields of philosophy and

science. Kol'man, the ideologist in this

delegation, gave two talks, the first

about quaternions, and the second about

the foundations of differential calculus

in the works of Karl Marx. Mtintz, rep­

resenting NIIMM at Leningrad State Uni­

versity, read a paper on Boundary Value

Problems in Mathematical Physics

[M29].

While Muntz had been unable to ob­

tain his Habilitation in Germany, he

was far more successful in Russia. In

1935, at the recommendation of

Leningrad State University, VAK, the

committee that gave these higher (or

second) doctorates in Russia, awarded

Muntz a higher degree without requir­

ing the submission of a written thesis.

Muntz would later write that he had

been awarded an honorary doctorate,

and that could be one possible inter­

pretation of this degree. In sum, with­

out doubt Muntz held a senior position

at Leningrad State University and had

the respect of his colleagues. He had

fulfilled his ambition.

Muntz was active administratively,

pedagogically, and mathematically. In

a later letter to Einstein he wrote about

working on a uniform theory of the so­

lutions of non-stationary boundary­

value problems in homogeneous and

non-homogeneous spaces. However he

also talked about the heavy teaching

and administrative load, and the un­

fortunate state of his eyes that hin­

dered him greatly. In 1934 he published

8"E1nstein (as well as Lichtenstein) as well as the rest of the 1ns1der world shall not learn anything of th1s." Muntz to Schrnalenbach, Decernber 5, 1 928.

© 2005 Spnnger SC1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1 , 2005 27

a textbook on Integral Equations [M32], which is still sometimes refer­

enced, and in 1935 he edited a Russian

edition of Lyapounov's important

monograph on General Problems of Stability of Motion [M35]. The fact that

Muntz was given this task, of historical

as well as scientific importance, is an­

other indication of the high regard in

which he was held. He also wrote some

half-dozen research papers, mainly on

boundary-value problems, integral

equations, and Mathematical Physics.

Furthermore, he was asked to write a

review of his own work for the Second

All-Union Mathematical Congress held

in Leningrad in 1934. The latter was a

definite honour, awarded at a time

when he began to recover from further

eye problems.

While in the Soviet Union Muntz kept

a low, neutral profile vis-a-vis internal

politics. Although he kept his German

citizenship, obtained in 1919, at some

stage he was given a "former foreignern

status within the Soviet Union. He also

traveled abroad widely in the company

of his wife, visiting Finland, Germany,

Switzerland, and Poland. Generally vis­

its were vacations or had to do with

mathematics research or meetings, but

sometimes they were motivated by his

and his wife's health.

While visiting Berlin on vacations,

by March 1930 Muntz9 was again work­

ing under Einstein. His work related to

a question "of compatibility of partial differential equations" which Einstein

indicated had been solved by Cartan,

in a wonderful fashion, but had not yet

been published. He said to Muntz, "You

will take pleasure from it."

Magda had suffered a cerebral thrombosis in 1934. Miintz himself, as

has been mentioned, suffered from se­

vere problems in one eye, and around

1934, at the beginning of term, he suf­

fered damage in the retina of the other

eye. This kept him from his academic

duties for several months. As he began

to recover, his department provided a

secretary to help him with his research,

and when he was able to teach, his stu­

dents helped him by writing on the

blackboard before each lecture the for­

mulae he needed.

9Letter to his sister Sala. March 28. 1 930.

28 THE MATHEMATICAL INTELLIGENCER

F. I. Ivanov, who was a graduate stu­

dent in the 1930s, recalls that Muntz

conducted a seminar in Mathematical

Physics. Ivanov remembers Muntz as

being very actively occupied with sci­

ence, but both accessible and sociable.

He writes that Muntz was stout, with

light-grey hair and very strong glasses

because of his poor eyesight. For this

reason Muntz's wife would bring him

to the seminar.

According to Muntz, he also helped,

sometimes directly and sometimes in­

directly, mathematicians from Central

Europe obtain positions in the Soviet

Union. He said that the appointments

of S. Cohn-Vossen, a former collabora­

tor of Hilbert who died of pneumonia

shortly after arriving in Moscow, and

of the number theorist A. W alfisz, who

went to Tbilisi, were the result of his

suggestions. With others, he helped

both A I. Plessner and Stefan Bergman

obtain positions.

In October 1937 Miintz was expelled

from the Soviet Union without, ac­

cording to him, any apparent reason.

This was part of a wide movement

sweeping the country, which turned

the tide against foreigners, including

teachers and engineers. It even af­

fected those, like Muntz, who had

helped develop areas of academic ac­

tivity in the Soviet Union. The Muntzes

were given a few weeks to leave.

Sweden (1 937-1 956)

Leaving the Soviet Union was a shock

to Muntz, who was then 53 years old.

He had lost the position he had worked

most of his life to obtain. The Muntzes

were permitted to take their personal

possessions with them, but no financial

recompense was offered for his eight

years as a professor at Leningrad State

University.

Travelling long distances was risky

because of his wife's thrombosis. The

Muntzes left Russia and first went to

Tallinn in Estonia. According to Muntz,

the Mathematics-Mechanics Faculty of

the Technical University in Tallinn of­

fered him a visiting professorship for

the spring semester. But neither Ger­

man nor Russian was an acceptable

teaching language- ministry regula-

A portrait of Herman Muntz during his years

in Sweden.

tions required lectures to be given in

Estonian-so this was not a possibil­

ity. In February of 1938, the Muntzes

moved to Sweden, where they later re­

quested political asylum.

Immediately after being expelled

from Russia, Muntz asked for help in ob­

taining an academic position from a

number of colleagues and former teach­

ers. He also asked family and personal

friends for financial help. Miintz's file at

the Society for the Protection of Science

and Learning and files on him in other

scientific refugee organizations indicate

that he contacted, among others, Har­

ald Bohr, Einstein, Landau, Levi-Civita,

Volterra, Weyl, and Courant.

Finding an academic position for a

scientist of Muntz's age was not easy.

Technically he was not a refugee from

Hitler's Germany, having left in 1929.

This put him outside the purview of re­

lief organizations such as the Notge­meinschaft Deutscher Wissenschaftler im Ausland. Furthermore, he was al­

ready in Sweden, a relatively safe

place. At this late date, competition for

academic positions in Europe and in

the United States was fierce and in­

volved a large number of parameters to

overcome the resistance and anti-for­

eign feeling, often as intense as the gen­

erosity of those many prepared to help.

Seniority in the field, age of the candi­date, area of research, even personal­ity, without leaving aside the strength of his personal network of scientific contacts, were among these parame­ters.

While Mi.intz was referred to as a "mathematician of the highest rank" in the dossiers of agencies dealing with dis­placed scientists, he was not considered a star. There were other drawbacks in Muntz's prospects for employment. Cru­cially, Einstein had concerns regarding Muntz's personality, dating from the late 1920s. In a letter to a colleague, Einstein explained his reservations in terms of what he regarded as Mi.intz's inability to submit his ideas to a proper level of crit­ical analysis and his previous mental ill­health. Einstein thought that, in a period of general distress, he should reserve his influence for more clear-cut cases. An imbalance in his personality, probably associated with his nervous breakdown of 20 years earlier, was now a serious drawback in Muntz's prospects for em­ployment.

Mi.intz resented Einstein's attitude and especially Einstein's suggestion (in 1938!) that Muntz should look for work in his native Poland. Muntz indicated that he was now in exile from Poland, Germany, and the Soviet Union, and in Germany he had not been forgiven for his former cooperation with Einstein. The tremendous competitiveness for jobs in America at the time may not have been entirely clear to Muntz, as the job description of his aspirations sug­gests. In a lost letter, Einstein may have pointed to the shortage of openings. Only Muntz's reply to this letter is avail­able. Muntz responded that it was painful and unjust after so many years "to conclude that my present fate is to be judged only by the statistics of sup­ply and demand." Fair enough, but the job market was clearly not in Einstein's hands.

In Sweden, as previously in Ger­many, Muntz had less success in pene­trating official academic circles than in the Soviet Union. However, from the first days of his arrival there he had the support of Professor Gi:iran Liljestrand, chairman of the funding committee for

exiled intellectuals, who helped him fi­nancially in 1939 and 1940. A number of distinguished Swedish academics also offered their help and friendship, among them Professors Folke K.-G. Odqvist, a mechanics specialist, Hugo Valentin, a physicist, Marcus Ehrenpreis, a pedia­trician surgeon, and David Katz, a psy­chologist.

Initially, in 1940-42, he received re­search grants from the Karolinska In­st'itutet for work on mathematical prob­lems related to haemodynamics, the study of the dynamics of blood flow, which involved the solution of complex non-linear partial differential equations. Research on this subject was carried out at the Maria Hospital, in Stockholm, in collaboration with a young medical doctor, Dr. A. Aperia. Muntz published a note on haemodynamics in Cornptes Rendus, submitted by Hadamard, that appeared in the February 1939 issue. In 1942 Aperia died and this research came to an abrupt end.

Although he was on good terms with some leading Swedish scientists and in­tellectuals, Muntz was not able to forge a working contact with the small but ac­tive Swedish mathematical community ofthe time. Nevertheless, for some years he remained interested in various math­ematical problems. In correspondence with Einstein and others he indicated he was interested in problems of integral equations, turbulence, knot theory, ac­tuarial mathematics, and of course haemodynamics. However, possibly due to the severity of his circumstances, no scientific papers of Mi.intz have come to light from this last period.

While in Stockholm Muntz returned to private teaching, and the couple moved to an apartment in Solna, a pleasant district of Stockholm. They had a telephone in their name, which suggests that their financial circum­stances had improved. It seems that they did have some outside financial resources. Muntz later received a small pension from the Warburgfonden, a foundation controlled by the "Mosaic" community in Sweden.

His wife Magdalena died of hemi­plegia on January 19, 1949. Muntz be­came a Swedish citizen in 1953 and

1 00dqvist is wrong in this po1nt, Muntz was granted Swedish Citizenship 1n 1 953. R1ksark1vet, Stockholm.

died on April 17, 1956, at the age of 71 . He was blind for the last few years of his life. But for an obituary in Svenska Dagbladet, the leading Swedish news­paper, written by Odqvist, his death passed almost unnoticed to the math­ematical community of Sweden and the rest of the world. In this obituary Professor Odqvist summarized the last years of Muntz's life in a short but poignant paragraph: "Herman Muntz is dead. In spite of the fact that he lived in Sweden for 18 years, the last five years10 as a Swedish citizen, there are probably not many Swedes outside his nearest circle of acquaintances, that knew that we had among us a mathe­matician of international fame who was thrown up on our calm shore by the storms of the times, his life saved but with his scientific activities bro­ken." He ended the obituary with these words: "Herman Muntz lived in an ex­ceptionally harmonious marriage and his wife Magda meant much to him, not in the least in order to keep his float­ing spirit down to earth. After her death in 1949 he only seldom saw his friends and he went every day to her grave in the Jewish cemetery with fresh flow­ers as long as he could. Now he is gone. Let this be a modest flower of memory from his Swedish friends. May his memory be blessed."

Acknowledgements

A paper such as this could not have been written without the help of many, many people and of various institu­tions. While it is impossible to name them all here we hope to acknowledge them by name at a later opportunity.

HERMAN MUNTZ: LIST OF MATHEMATICAL

PUBLICATIONS

[M 1 ] Zum Randwertproblem der partiellen Dif­

ferentialgleichung der Minimalflachen, J.

Reine Angew. Math. , 139 ( 19 1 1 ), 52-79.

[M2] Aufbau der gesamten Geometrie auf

Grund der projektiven Axiome allein, MUnch­

ener Sitz . , (1 91 2) , 223-260.

[M3] Das Euklidische Parallelenproblem, Math.

Ann . , 73 (1 9 1 3), 241 -244.

[M4] Das Archimedische Prinzip und der Pas­

calsche Satz, Math. Ann. , 74 (1 9 1 3) ,

301 -308.

© 2005 Spnnger Sc1ence+Bus1ness Med1a. Inc . . Volume 27. Number 1. 2005 29

[M5] Solution directe de !'equation seculaire et

de quelques problemas analogues tran­

scendants, C. R. Acad. Sci. Paris , 1 56

(1 9 1 3) , 43-46.

(M6] Sur Ia solution des equations seculaires et

des equations integrales, C. R. Acad. Sci.

Paris , 1 56 ( 1 9 1 3) , 860-862.

[M7] Ober den Approximationssatz von Weier­

strass, in H. A. Schwarz-Festschrift, Berlin,

1 91 4 , 303-3 1 2 .

[M8] Sur une propriete des polyn6mes de

Bernoulli, C. R. Acad. Sci. Paris , 158 (1 9 1 4) ,

1 864-1866.

[M9] Ein nichtreduzierbares Axiomensystem

der Geometrie, Jber. Deutsch. Math. Verein,

23 (1 9 14), 54-80.

[M 1 0] Approximation willkurlicher Funktionen

durch Wurzeln , Archiv Math. Physik, 24

(1 9 1 6) , 31 0-316.

[M1 1 ] Zur expliziten Bestimmung der Haupt­

achsen quadratischer Formen und der

Eigenfunktionen symmetrischer Kerne, Gott.

Nachr. (1 9 1 7) , 1 36-140.

(M1 2] On projective analytical geometry (in Pol­

ish and German), Prac. Mat. -Fiz. , 28 (1 9 1 7) ,

87-1 00.

[M 1 3] The problem of principal axes for quad­

ratic forms and symmetric integral equations

(in Polish and German), Prac. Mat. -Fiz. , 29

(1 9 18), 1 09-1 77.

A U T H O R S

�-..

• '> w:·�>> . � ..,. , . � .. .,

"

(M 1 4] A general theory for the direct solution

of equations (in Polish), Prac. Mat.-Fiz. , 30

( 191 9), 95-1 1 9.

[M1 5] Die Ahnlichkeitsbewegungen beim allge­

meinen n-K6rperproblem, Math. Z. , 1 5

(1 922), 1 69-1 87 .

[M1 6] Allgemeine independents Aufl6sung der

lntegralgleichungen erster Art, Math. Ann . ,

87 (1 922), 1 39-149.

[M1 7] Beziehungen der Riemannschen ?-Funk­

lion zu willkurlichen reellen Funktionen, Mat.

Tidsskrift 8, (1 922). 39-47.

[M 1 8] Absolute Approximation und Dirichletsches

Prinzip, G6tt. Nachr. , 2 (1 922), 12 1-1 24.

(M 1 9] Allgemeine Begrundung der Theorie der

h6heren ?-Funktionen, Abhdl. des Sem.

Hamburg, 3 (1 923), 1 -1 1 .

[M20] Der Summensatz von Cauchy in beliebi­

gen algebraischen Zahlkorpern und die

Diskriminante derselben, Math. Ann . , 90

(1 923), 279-291 .

[M21 ] Fragen der klassischen und relativistis­

chen Mechanik. Vier Vortrage gehalten in

Spanien in January 192 1 , by T. Levi-Civita;

authorized translation by P. Hertz, H. Kneser,

Ch. H. Muntz, and A. Ostrowski , pp. vi +

1 1 0, J. Springer, Berlin, 1 924.

[M22] Umkehrung bestimmter Integrals und

absolute Approximation, Math. Z. , 21 (1 924),

96-1 1 0.

[M23] Ober den Gebrauch willkurlicher Funk­

tionen in der analytischen Zahlentheorie,

Sitzungsberichte der Berliner Math.

Gesellschaft, 24 (1 925), 8 1-93.

[M24] Die L6sung des Plateauschen Problems

uber konvexen Bereichen, Math. Ann . , 94

(1 925), 53-96.

[M25] Zur Gittertheorie n-dimensionaler Ellip­

soids, Math. z. , 25 (1 926), 1 50-1 65.

[M26] Zum Plateauschen Problem. Erwiderung

auf die vorstehende Note des Herrn Rad6,

Math. Ann. , 96 (1 927), 597-600.

[M27] Ober die Potenzsummation einer En­

twicklung nach Hermiteschen Polynomen,

Math. Z. , 31 (1 929), 350-355.

[M28] Sur Ia resolution du problems dynamique

de l 'elasticite, C. R. Acad. Sci. Paris , 194

(1 932), 1 456-1 459.

[M29] Ober die L6sung einiger Randwertauf­

gaben der mathematischen Physik, Ver­

handlungen des lnternationalen Mathe­

matiker-Kongress ZOrich 1932, Dr. Walter

Saxer, ed. , Zurich, 1 932, 1 09-1 10 .

[M30] lntegralgleichungen der Elastodynamik,

Rec. Math. Moscou, 39, 4 (1 932), 1 1 3-132.

[M31 ] Zum dynamischen Warmeleitungsprob­

lem , Math. Z. , 38, 3 (1 934), 323-337.

[M32] Integral Equations, Vol. I, Volterra 's Lin­

ear Equations, (in Russian), 330 pages,

Leningrad, 1 934.

EDUARDO L. ORTIZ

Department of Mathematics

Imperial College London, South Kensington Campus

London, SW7 2f.Z.

ALLAN PINKUS

Department of Mathematics

Technion

United Kingdom

e-mail: e.ortiz@ic.ac.uk

Eduardo L Ortiz did his doctoral work under the supervision of

Mischa Collar in Buenos Aires, and subsequently went to Dublin

for research under Cornelius Lanczos. Since 1 963 he has been at

Imperial College London, where he is now Professor. He has writ­

ten prolifically on functional analysis and its applications and on

history of mathematics. He has held visiting positions at Harvard

and the universities of Orleans and Rouen.

30 THE MATHEMATICAL INTELLIGENCER

Haifa, 32000 Israel

e-mail: pinkus@tx.technion.ac.il

Allan Pinkus, a native of Montreal, did his undergraduate work at

McGill University and his doctoral work at the Weizmann Institute

under Samuel Karlin's supervision. Since 1 977 he has been at the

Technion. His research interests center on approximation theory.

He was for ten years an Editor-in-Chief of the Journal of Approx­imation Theory.

[M33] Sur les problemes mixtes dans l 'espace

heterogene, Equation de Ia chaleur a n di­

mensions, C. R. Acad. Sci. Paris , 199 (1 934),

821 -824.

[M34] Functional Methods for Boundary Value

Problems (in Russian), Works of the 2nd All­

Union Mathematical Congress, Leningrad,

Leningrad-Moscow, 1 (1 935), 31 8-337 .

[M35] General problems of stability of motion,

by A. Lyapounov, (in Russian) , Ch. H. Muntz,

ed. , Leningrad-Moscow, 1 935.

[M36] Zur Theorie der Randwertaufgaben bei

hyperbolischen Gleichungen, Prace Mat. ­

Fiz. , (Gedenkschrift fur L. Lichtenstein), 43

(1 936) , 289-305.

[M37] Les lois fondamentales de l 'hemody­

namique, C. R. Acad. Sci. Paris , 280 (1 939),

600-602 .

REFERENCES

[ 1 ] K. Weierstrass, Uber die analytische

Darstellbarkeit sogenannter willkurlicher

Funktionen einer reellen Veranderlichen,

Sitzungsberichte der Akademie zu Berlin,

1 885, 633-639 and 789-805.

[2] S. Bernstein, Sur les recherches recentes

relatives a Ia meilleure approximation des

functions continues par des polyn6mes,

Proceedings of the Fifth International Con­

gress of Mathematicians, (Cambridge,

22-28 August 1 91 2) , E. W. Hobson and

A. E. H . Love, eds . , Cambridge, 1 91 3 , Vol.

I, 256-266.

[3] S. N. Bernstein, Sur l 'ordre de Ia meilleure

approximation des functions continues par

les polyn6mes de degre donne, Mem. Cl.

Sci. Acad. Roy. Be/g . , 4 ( 1 9 1 2) , 1 -1 03.

[4] E . L. Ortiz, "Canonical polynomials in the

Lanczos' Tau Method, " B. P. K. Scaife,

ed. , Studies in Numerical Analysis , New

York, 1 97 4, 73-93, on 75.

[5] E. L. Ort1z, "The Society for the Protection

of Science and Learning and the Migration

of Scientists in the late 1 930s," Panel 's

Chairman's lecture, Proceedings of the

1 13th annual meeting of the American His­

torical Association, Washington, 93 (1 999),

1 -28.

[6] Ch. Muntz, Wir Juden, Oesterheld and Co. ,

Berl in , 1 907.

[7] A. Korn, Uber Minimalflachen, deren Rand­

kurven wenig von ebenen Kurven abwe­

ichen Abhdl. Kg/. Akad. Wiss., Phys-math,

Berl in , (1 909), 1 -37.

[8] R . von Mises and H . Pollaczek-Geiringer,

Praktische Verfahren der Gleichungsaufl6-

sung, Zeitschrift fur Angewandte Mathe­

matik und Mechanik, 9 (1 929), 58-77 and

1 52-164.

[9] L. Butschli, HochwaldhauserDiary, 39, 39a.;

quoted in Karl-August Helfenbein, Die

Sozialerziehung der Durerschule Hochwald­

hausen, Hochhausmuseum and Hohha­

subibliotek, Lauterbach, 1 986, p. 1 5.

[1 0] Der Jude, Judischer Verlag, Berl in ,

1 9 1 6-1 928.

[1 1 ] E. C. Titchmarsh, The Theory of the Rie­

mann Zeta-Function, Oxford, 1 951 , p. 28.

[1 2] D. Amira, La Synthese Projective de Ia

Geometrie Euclidienne, ltine and Shoshani,

Tel-Aviv, 1 925.

[1 3] G. G. Lorentz, Mathematics and Politics in

the Soviet Union from 1 928 to 1 953, Jour­

nal of Approximation Theory, 1 1 6 (2002),

1 69-223.

r

clg�Scient if ic WorkPlace· Mathematical Word Processing • J!.T,EX Typesetting • Computer Algebra

Version 5 Sharing Your Work Is Easier

• Typeset PDF in the only oftware that allow you to tran form Iff.TE)( file to PDF, fully hyp rlinked and with embedded graphic in over 50 format

• Export document a RTF with editable mathematic (M icro oft Word and MathType compatible)

• Share documents on the web a HTML with mathematics a MathML or graphic

The Gold Standard for Mathematical Publ ishing

Scientific WorkPlace make writing, haring, and doing mathematic ea ier. A click of a button al lows you to typeset your documents in lt-TEX . And, you can compute and plot solutions with the integrated computer algebra engine, MuPAD 2.5.

Tol�free: 877-724-9673 • Phone: 360-394-6033 • Email: info@mackichan.com

Vr.;� our website for free trial versions of all our software.

© 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc., Volume 27, Number 1 , 2005 31

KELLIE 0. GUTMAN

Quando Che 'l Cubo

e n the history of mathematics, the story of the solution to the cubic equation is as

convoluted as it is significant. When I first read an account of it in William Dun­

ham's Journey Through Genius1 in 2000, I was captivated by the personalities, the

intrigues, and the controversies that were part of mathematics in sixteenth-cen-

tury Italy. For those unfamiliar with it, the story runs as follows:

In the early 1500s, the mathematician Scipione del Ferro of the University of Bologna discovered how to solve a de­pressed cubic-one without its second-degree term-but in the style of the day he kept his discovery to himself. On his deathbed in 1526 he divulged the solution to his student Antonio Fior. 2

Eight years later Niccolo Fontana, known as "Tartaglia" ("Stutterer"), hinted that he knew how to solve cubics that were missing their linear term. Fior publicly challenged Tartaglia to a contest in February of 1535, sending him a set of thirty depressed cubics to solve. At first Tartaglia was stumped, but with the deadline approaching, he fig­ured out how to solve depressed cubics, thus winning the challenge.

In Milan, the mathematician/physician Gerolamo Car­dano heard about Tartaglia's grand accomplishment. For several years, he pleaded with Tartaglia to tell him his se­cret. Finally in 1539, Tartaglia traveled to Milan from Venice and told Cardano the solution, but made him swear never to publish it.

With continued research, Cardano figured out how to re­duce a general cubic to a depressed one, thus completely solving the classical problem of the cubic. Then his assis­tant Lodovico Ferrari extended this string of discoveries by solving fourth-degree problems, but both men refrained

from publishing their results because they were based on Tartaglia's solution.

On a hunch, Cardano and Ferrari traveled to Bologna in 1543 to look at the papers of Fior's master, Scipione del Ferro, who they must have reasoned also knew the solu­tion to depressed cubics. They found Scipione's original al­gorithm and it was identical to Tartaglia's.

Finally, Cardano felt released from his oath to Tartaglia Giving full credit to both Scipione and Tartaglia, he published the solution to the depressed cubic, his own solution to the general cubic, and Ferrari's solution to the quartic, in 1545, in a huge tome, Ars Magna. This widely dispersed work is con­sidered by many to be the first book ever written entirely about algebra In it, Cardano devoted little space to the solution of the quartic, because a fourth power was considered a mean­ingless concept, not corresponding to any physical object.

Tartaglia was enraged. The following year, in his own book Quesiti et inventioni diverse, Tartaglia presented his version of a long conversation between himself and Car­dano from their encounters six years earlier, in which he made it clear that his "invention" was not to be disclosed. He then presented his solution in a poem, saying this was the easiest way for him to remember it.

* * *

' Dunham, William. Journey Through Genius . New York: John Wiley & Sons, Inc., 1 990.

2Many of the historical facts came from the Mac Tutor History of Mathematics archive of the School of Mathematics and Statistics, University of St Andrews, Scotland

Created by John J. O'Connor and Edmund F. Robertson

http://www-history.mcs.st-andrews.ac.uk/history/index.html

32 THE MATHEMATICAL INTELLIGENCER © 2005 Springer Scrence+ Busrness Medra, Inc

Quando che'l cubo3

Quando che'l cubo con le cose appresso Se agguaglia a qualche numero discreto

Trovar dui altri differenti in esso.

Dapoi terrai questo per consueto Che'l lor produtto sempre sia eguale

AI terzo cubo delle cose neto,

El residuo poi suo generate Delli lor lati cubi ben sottrati Varra la tua cosa principale.

In el secondo de cotesti atti Quando che'l cubo restasse lui solo Tu osservarai quest'altri contratti,

Del numer farai due tal part'a volo Che l'una in l'altra si produca schietto

El terzo cubo delle cose in siolo

Delle qual poi, per commun precetto Torrai li lati cubi insieme gionti

Et cotal somma sara il tuo concetto.

El terzo poi de questi nostri conti Se solve col secondo se ben guardi Che per natura son quasi congionti.

Questi trovai, e non con passi tardi Nel mille cinquecente, quatro e trenta Con fondamenti ben sald'e gagliardi

Nella citta dal mar' intorno centa.

Any Italian who encountered this poem would have im­mediately recognized it as being written in the celebrated form known as terza rima, invented by Dante Alighieri and used in his masterwork, La Divina Commed·ia. Like Dante, Tartaglia wrote in Italian, which was the language of liter­ature, not Latin, which was the main language of science: this was because Tartaglia did not know Latin. Terza rima is made up of eleven-syllable, or hendecasyllabic, lines. Each line is iambic with five stressed and six unstressed syllables. It is an especially fitting form for a poem about cubic equations because there are two sets of threes con­tained in it: the poem is written in tercets, or three-line stan­zas, and all the rhymes, except at the start and finish of the poem, come in triplicate, with the center line of each ter­cet rhyming with the outer lines of the tercet following it,

3Tartaglia. Niccol6, Ouesiti et inventioni diverse de N1ccol6 Tartalea Bris01ano.

[Stampata in Venetia per Venture Rotflnelli, 1 546.]

thus propelling the poem forward. This form is extraordi­narily well-known by Italians.

* * *

In the early sixteenth century, algebra was rhetorical­that is, variables, the equal sign, negative numbers, and the concept of setting something equal to zero did not exist. Everything was described solely through words. Instead of writing "X3 + mx = n" one would write cuba con cosa ag­guaglia ad un numero or "cube and thing are equal to a number." It was a cumbersome system, and calculations and proofs were difficult to follow.

When I saw Tartaglia's poem for the first time in early 2004, I was so taken with it that I had to translate it, but I soon found myself faced with a dilemma. Either I could translate it literally as he wrote it, and have it be as obscure as his was (and it is obscure), or I could do a modern trans­lation and essentially say, "This is what he meant, though it is not what he said." The second way would make it very clear for today's reader. Neither of these felt quite right to me. Instead, I decided to bridge the two worlds of Renais­sance mathematics and modern mathematics, attempting to retain the poem's ancient flavor along with its terza rima, but using variables where Tartaglia used only words.

Because the vast majority of Italian words end in an un­stressed syllable, it is natural to have iambic lines of po­etry with eleven syllables. It is slightly more difficult in Eng­lish. In my translation I have used an alternating pattern of masculine rhymes, with the stress and rhyme on the final syllable, and feminine rhymes, which rhyme on the stressed penultimate syllable.

* * *

When X Cubed

When x cubed's summed with m times x and then Set equal to some number, a relation Is found where r less s will equal n.

Now multiply these terms. This combination rs will equal m thirds to the third; This gives us a quadratic situation,

Where r and s involve the same square surd. Their cube roots must be taken; then subtracting Them gives you x; your answer's been inferred.

The second case we'll set about enacting Has x cubed on the left side all alone. The same relationships, the same extracting:

----- - -----

Quesito XXXII I I. Fatto personalmente dalla eccellentia del medesimo messer H1eronimo Cardano 1n Millano in casa sua adi. 25. Marzo.1 539

"Quando chel cubo con le cose apresso . . . " - begins leaf 1 23 recto

. . . Nella citta dal mar' intorno centa " - ends leaf 1 23 verso

(Also reproduced on the following Web site:

http· I I digi lander . libero. itlbasecinqueltartaglia/ eq uacu bica. htm)

© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005 33

Seek numbers r and s, where the unknown rs will equal m-on-3 cubed nicely, And summing r and s gives n, as shown.

Once more the cube roots must be found concisely Of our two newfound terms, both r and s, And when we add these roots, there's x precisely.

The final case is easy to assess: Look closely at the second case I mention­It's so alike that I shall not digress.

These things I've quickly found, they're my invention, In this year fifteen hundred thirty-four, While working hard and paying close attention,

Surrounded by canals that lap the shore.

So what exactly is Tartaglia saying? He's saying that when ,i3 + mx = n, two other numbers, r and s, can be found such that r - s = n and rs = (m/3)3. Mathematicians of his day knew that when they were told the values of a product and a difference (or sum) of two unknown numbers, they had what I have called a "quadratic situation" (there was no such thing as a quadratic equation). They had an algorithm, which was tricky but manageable, to fmd the solutions to such sit­uations. In fact, because they didn't recognize negative num­bers, they had a set of variants of what we would think of as one single thing, namely the quadratic formula. Using the applicable variant, one could solve for rand s. Next, Tartaglia is telling his readers to take the cube roots of the numbers r and s, and to subtract the cube root of s from that of r. This will be x, the solution to the given cubic.

He then moves on, in the fourth stanza, to what was con­sidered a different situation, when .i3 = mx + n, and he gives the solution again. The third case, when .i3 + n = mx, he says, in the seventh stanza, is almost exactly like the second, and so he leaves that for the reader to figure out. He con­cludes with a flourish by claiming credit for the discovery, and telling his readers he found the solution in Venice.

* * *

Tartaglia discovered his solution by thinking about an actual physical cube. To him, and most likely to Scipione as well, the solution to a problem involving a cubic was em­bodied in a real cube. Seven hundred years earlier, in Bagh­dad, Al-Khwarizmi (from whose name comes the word "al­gorithm") thought about a square when working on problems involving quadratics. He came up with a formula for "completing the square" to solve such problems.

An equation of the type x2 + mx = n can be pictured by first drawing a square of side x (see Figure 1). Next make two congruent rectangles of length x and width m/2, and attach them to two adjacent sides of the square. The di­mensions m/2 and x are picked for very good reasons­two rectangles of this size together make up an area of mx, to add to the original square of the area x2, and these three together have a joint area of n, giving x2 + mx = n.

34 THE MATHEMATICAL INTELLIGENCER

Completing the Square m I

X

Figure 1 . A version of AI-Khwarizmi's completion of the square. Mov­

ing left to right, the equation can be read directly off the diagram.

The picture looks like a square cardboard box from above, with two adjacent flaps open. It calls out for one other square, of side length m/2, to be drawn in, in order to complete the larger square. Let's call the side of this new big square t, and the side of the new little square u. When we combine the area n with the area u2, which is (m/2)2, we get the area of the larger square, t2• The square root of this square area-that is, the square root of n + ( m/2)2-gives us the side length t. But t is equal to x + m/2, so x equals V n + ( m/2)2 - m/2. Thus by completing the square, Al-Khwarizmi solved the quadratic.

In a similar fashion to Al-Khwarizmi, Tartaglia envisioned "completing the cube" to solve the depressed cubic. He took Al-Khwarizmi's drawing into a third dimension (Fig. 2).

With an equation of the form .i3 + mx = n, he started by imagining a cube of side x (this corresponded to the square of side x in two dimensions). He then looked for analogous volumes to play the role of the two rectangles flanking the square of side x, but since he was in three di­mensions he instead imagined three slabs. Each had one side of length x, and two other sides of unknown lengths, which we will call t and u. These three slabs fit neatly

Completing the Cube

Figure 2. Tartaglia's completion of the cube. Once again the equa­

tion can be read directly off the diagram.

3tux

Figure 3. Like a Necker cube, this picture flips between two inter­

pretations. In the intended interpretation, one sees three slabs, each

of volume tux, swirling counter-clockwise around a (missing) cube

of side u. In the other interpretation (and this came as a complete

and lovely surprise to me) one sees a cube of side u sitting nestled

in one corner of a cutaway cube of side t, and thanks to the colors

painted on the large cube's walls, one cannot help "seeing" (though

they are missing) the three slabs of volume tux, once again swirling

counter-clockwise about the little cube of side u.

around the cube of side x, thus giving him a larger cube of side t, but (as before) with one crucial piece missing. In or­der to complete the larger cube, Tartaglia added one last cube of side u (corresponding to the little square of side u that completed Al-Khwarizmi's square; Fig. 3).

Each of the three slabs has sides of length t, u, and x, and so the total volume of the slabs is 3tux. Now the vol­umes of the two interior cubes are x3 and u3, so the total volume of the big cube is .x3 + 3tux + u3, but of course it is also t:1. In symbols,

.i3 + 3tux + u:l = (l. We can imagine Tartaglia striving to imagine the di­

mensions of a physical cube that would represent the so­lution to an actual depressed-cubic problem posed by his challenger Fior. In Al-Khwarizmi's quadratic, the value of u is known instantly without calculation. But in the case of the cubic, things are not so simple, because one doesn't know the value of either t or u. In the realm of all possible cubes, Tartaglia needed to find the one cube with the ex­act dimensions that satisfy his problem. He had to imagine the lengths u and t both changing (the overall cube grow­ing and shrinking, and also the cube of side x changing size because it is determined by t and u, its side being t - u) . It seemed as if the search for the proper cube could only be carried out by trial and error, without any formula, and thus it was not really a mathematical solution.

At this point, though, rather than giving up, Tartaglia has a brilliant insight. Looking at his equation (above), he re­alizes that if he merely moves u3 to the right side, it will give him a new equation that precisely embodies Fior's de­pressed cubic _Tl + m:J.: = n, with 3tu playing the role of m and t3 - u3 playing the role of n.

x3 + 3tux

I I

x3 + mx n

This is a breakthrough moment for Tartaglia, because it tightly connects the unknowns, t and u, with the knowns, m and n:

3tu = m,

This is very promising, but he is not there yet, because he doesn't know how to solve these equations for t and u in terms of m and n. As he considers these equations, however, Tartaglia sees that he has a situation that comes very close to being a quadratic in t and u, but just misses-namely, he has a product and a difference involving t and u, but one of them involves their cubes. Thus provoked, Tartaglia has another in­sight. He gives names to the two cubic volumes, calling t3 "r" and ua "s," knowing that in this way he will obtain a genuine quadratic situation (involving a difference and a product) with his new variables r and s. Now his equations are

1· - s = n rs = (m/3)3.

The last equation is an immediate consequence of the def­inition of r and s. From 3tu = m it follows that tu = m/3, and thus, cubing both sides, t3u3 = (m/3)3.

Now he is operating in familiar territory. He can easily find his quadratic by eliminating r as follows: r = n + s and therefore rs = s(n + s), giving

s2 + ns = (m/3)3.

Tartaglia has at last come full circle. Mter starting out with Al-Khwarizmi's model of completing the square in or­der to come up with his own model of the cubic, he now applies Al-Khwarizmi's square-completing method to solve this quadratic for r and s; having gotten those, he can then take their cube roots to obtain the values of t and u. Then he merely subtracts u from t, and x has been found.

* * *

When Cardano published Ars Magna, rather than giving a general proof, he illustrated the solution to this particu­lar cubic: ;il + 6x = 20. Following the poem's directions, here is how it is solved.

.i3 + 6x = 20

r - s = 20

rs = (6/3)3 = 23 = 8 r = 20 + s and therefore s(20 + s) = 8 s2 + 20s = 8 s2 + 20s - 8 = 0.

Using the quadratic formula to solve for s, we get

s = c-20 ± V4oo + 32)/2

= - 10 ± v'i08 = v1o8 - 10

r = s + 20 = v'i08 + 10.

© 2005 Springer Sc•ence+Bus1ness Med1a. Inc . . Volume 27, Number 1 . 2005 35

Numerically,

r = 20.3923 and s = .3923.

Then, taking these numbers' cube roots,

x = Vr - Vs X = 2. 73205 - . 73205 X = 2.

If we plug this back into the original equation x3 + 6x = 20, we find that it is correct: 8 + 12 = 20. The method works, although it must be admitted that it makes it look fortuitous that the answer is a simple integer.

* * *

Finding a solution by radicals to the cubic was a monu­mental accomplishment. However, it led to a thorny ob­stacle: in the case of a cubic equation that had only one real root (back then, mathematicians would have said the equation had only one root at all, for no one suspected that all cubics have three roots), the algorithm always yielded that root. By contrast, in the case of a cubic that had three real roots, the algorithm seemed to yield nonsense. Even if the three real roots were already known, it led to expres­sions featuring negative numbers under the square-root sign, a situation that Cardano dubbed the casus irre­ducibilis, reflecting the fact that Renaissance mathemati­cians were not comfortable with negative numbers, let alone their square roots.

The Bologna mathematician Rafael Bombelli took Car­dana's casus irreducibilis very seriously and tried to make sense of the square roots of negative numbers. He figured out how to do the four standard arithmetical operations not only with negative numbers but also with their "imaginary" square roots, and shortly before his death in 1572, he pub­lished a book on this topic titled Algebra, in which he pre­sented an early symbolic notation system. Although he never found out how to take cube roots of complex num­bers in general, he was able to determine the complex cube root called for by Cardano's algorithm in one specific case, and he showed that the two imaginary contributions to the fmal answer canceled each other out, leading to a purely real root. More details of Bombelli's work will be found in a recent scholarly article in this journal by Federica LaNave and Barry Mazur; see vol. 24, no. 1 (2002), 12-21 .

Despite this accomplishment, Cardano's formula pro­vided Bombelli with only one of the equation's three roots, and it took another 40 years until Fran<;ois Viete figured out how to find the other two real roots, and then a further 300 years until mathematicians penetrated the mystery of the casus irreducibilis and finally understood why com­plex numbers were needed to express the real roots to cu­bic equations through radicals.

When Ferrari based his solution of the quartic equation on that of the cubic, just as Tartaglia had based his solu­tion of the cubic on that of the quadratic, it seemed as if this clever method could go on indefinitely: lower the de-

36 THE MATHEMATICAL INTELLIGENCER

AU T H O R

KELLIE 0. GUTMAN

75 Gardner Street

West Roxbury, MA 021 32-4925

USA

e-mail: RKLGutman@corncast.net

Kellie Gutman has studied mathematics and audiology-and,

since 1 999, poetry. One piece of mathematical research she

wrenched into poetic form, quite impressively; see The Math­

ematical lntelligencer 23 (2001 ), no. 3, 50. Wrth her husband,

Richard Gutman, she is co-owner for 25 years of a company

specializing in audio-visual presentations for museum installa­

tions; co-au1hor of two books; and parent of Lucy.

gree of an equation by one, and use this new equation's formula to help solve the original. But when mathemati­cians tried to solve the quintic equation in this way, they hit a brick wall. It wouldn't yield.

For the next 250 years, mathematicians struggled to solve quintics by radicals. Finally in 1 799, Paolo Ruffini, an­other mathematician/physician, wrote a book Teoria Gen­erate delle Equazioni, offering a proof that fifth-degree equations-indeed, all equations of degree greater than four-were in general unsolvable by radicals; but almost no one accepted his claims. Twenty-two years later the dis­tinguished French mathematician Cauchy wrote to Ruffini, praising his proof, but few people agreed with Cauchy. In a few years, however, Niels Henrik Abel in 1825 and Evariste Galois in 1830 published works on the unsolv­ability of the quintic equation and equations of higher or­der, and their discoveries, which were centered on the sym­metry groups of the roots, were widely accepted.

For the thousand or so years between the destruction of the Library of Alexandria and the Renaissance, European mathematics, with a few notable exceptions, had made slow progress. But the Italian mathematicians who worked on solving the cubic initiated a series of events that led to the use of negative numbers, complex numbers, powers and dimensions higher than the third, and symbolic alge­bra, with its highly efficient system of symbol manipula­tion. This work, spanning roughly one hundred years, rein­vigorated mathematics and led directly to many of the discoveries of the modem era.

M a them a tic a l l y Bent

The proof is in the pudding.

Opening a copy of The Mathematical

Intelligencer you may ask yourself

uneasily, "What is this anyway-a

mathematical journal, or what?" Or

you may ask, "Where am I?" Or even

"Who am I?" This sense of disorienta­

tion is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It's

mathematical, it 's a humor column,

and it may even be harmless.

Column editor's address: Colin Adams,

Department of Mathematics, Bronfman

Science Center, Williams College,

Williamstown, MA 0 1 267 USA

e-mail: Colin.C.Adams@williams.edu

C o l i n Adam s , Ed itor

Tria l and Error Colin Adams

Please be seated, Mr. Phipps. Actually, it's Dr. Phipps. Really? Are you a medical doctor? No. But I have a Ph.D. That means

I have a doctorate. So I should be ad­dressed appropriately.

And tell the court, Mr. Phipps, do you often insist on being called doctor?

I prefer to be called that. Perhaps because of some insecurity

on your part? A need to assert your au­thority through a title?

I want my students to know I am in control.

And do you take some pleasure in that control, Mr. Phipps?

Objection, this is an irrelevant line of questioning.

Sustained. I retract the question. Now tell me

Mr. Phipps, were you the instructor for Math 105 Multivariable Calculus at Freedmont College this last fall?

I was the professor for that course, yes.

And was there a student named Jeffrey Foible in that class?

Yes, there was. And do you see him here today in

the courtroom? Yes, he is sitting over there next to

his mother. And can you tell us how Jeffrey did

in your class? He received a C + . Was h e close to a B - ? Yes. But h e was clearly in the C +

range. I see. And how many students were

in the course, Mr. Phipps? About 1 50. 150? With that many students, is it

difficult to keep track of individual stu­dents, and how they are doing in the course?

I have teaching assistants.

I see. And so they keep track of the individual students, relieving you of the necessity to do so?

No, I pay attention, too. I review the grades on the homework and I do all the grading on the exams.

Really? How much time does that take?

On the two midterms, I spent about twelve hours grading each of them. Then the final took longer. About twenty hours.

Twenty hours? That's a tremendous amount of time to be sitting, staring at student work. It must be exhausting. Is it hard to keep up your concentration that long?

I take breaks. You mean like getting a drink of

juice, using the bathroom, maybe watching a little TV?

Yes, that's right. And of course, you aren't going to

complete twenty hours of grading in a day. I suppose it is stretched over a pe­riod of several days. How many days did it take you on the exam for this course?

As I remember it, about three days. I see. That's quite a bit of time. So

you might finish for one day, and then have a 12- or 15-hour break before resuming.

Yes. Now, correct me if I'm wrong. After

all, you're the math teacher. But 20 hours means 1200 minutes. Divide by 150 and that means an average of 8 minutes per exam.

Yes, that's right. And how many pages were there on

the final? Eight pages. So, a minute per page. Yes. And how many problems are on a

page? About four, if each part of a multiple-

part question is counted separately. So 15 seconds a problem. About that, yes. And do you give partial credit?

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1 , 2005 37

Oh, yes.

Do you feel that in the 15 seconds

you apportion to a given problem, you

can fairly determine the appropriate

amount of partial credit?

Some problems take less than 15

seconds and some take more. When I

need to think about how much to give,

I take longer.

Oh, and so some of the time when

you are grading, you aren't thinking at

all. Kind of on automatic pilot.

There are some problems where

there isn't much partial credit to give.

Either they have the answer right or

they do not.

And do you consider yourself a con­

sistent grader?

I try.

Do you know a Ms. Elaine Plepp­

meyer?

I believe she was a student in the

course.

Good for you, Mr. Phipps. It must

not be easy getting to know all the

students.

I do my best.

And can you tell me, is she seated

in the courtroom right now?

Well, urn, I'm not sure.

But I thought you knew her.

Is that her over there, seated by

Foible?

No that is Jeffrey Foible's sister.

How about the blonde woman by

the door?

No, that is not her.

Then I am not sure.

In fact, Mr. Phipps, Elaine Plepp­

meyer is not here today. However I do

have her final exam right here. Can you

verify that this is her final exam from

the course?

Well, yes, it does appear to be her

exam.

I would like to enter this as Exhibit

A. Can you tell the court, please Mr.

Phipps, what grade appears at the top

of the exam?

It is a B - .

And the numerical score?

An 81.

I see, and can you verify for the

court that here I have a copy of Mr.

Foible's fmal exam?

Yes, that appears to be the exam.

I would like to enter this as Exhibit

38 THE MATHEMATICAL INTELLIGENCER

B. Can you tell the court what grade

and score is on this exam?

It is a C + , with a score of 79. Interesting. Not so different, is it?

Now tell me Mr. Phipps, and please feel

free to consult your roll book, but other

than this difference in their final ex­

ams, how close were the grades for

these two students coming into the

final?

Well, Foible did slightly better on

the homeworks, and Pleppmeyer did

slightly better on the two midterms.

They were both borderline between a

B and a C.

So you would characterize their per­

formance up to the final as being es­

sentially equivalent as far as their ulti­

mate grade is concerned?

Yes, that is true.

Oh, dear, so it seems that this minor

two-point difference between their

scores on the final made the difference

between Jeffrey Foible receiving a B

and a C.

Actually, a B - and a C + .

Whatever. I t is a big difference, Mr.

Phipps. Perhaps the difference be­

tween getting into law school and not

getting into law school?

I don't know about that. I just give

students the grades they deserve.

Do you have any idea of the poten­

tial earning power that a lawyer has,

Mr. Phipps? Do you realize that this

tiny two-point difference may have

cost Mr. Foible five million dollars over

his lifetime?

That isn't my concern.

Well, perhaps it should be, Mr.

Phipps, because if you made a mistake

in grading the exam, that could be a

very costly mistake.

What do you mean?

Please open the exam to page 4. Let's take a look at problem 1 1. Could

you please read the problem to the

court?

It says, "Find the volume of the

tetrahedron in the first octant of space

bounded by the three coordinate

planes and the plane x+2y +3z = 6." Thank you, Mr. Phipps. Perhaps you

could explain the problem to us. After

all, we are not all experts in calculus,

as you are.

Well, the three coordinate planes

are the xy, the yz and the xz plane,

which are three orthogonal planes in­

tersecting pairwise along the coordi­

nate axes.

I'm sorry. I must be slower than

your students. Orthogonal? Intersect­

ing pairwise?

Umm. It's like the corner of a box.

Three planes meeting perpendicularly.

Then a fourth plane slices through and

cuts off the corner of the box.

Thank you. That makes a bit more

sense. Now you say the plane x+2y

+3z = 6. So x+2y +3z = 6 is a plane.

Yes, that is the equation of a plane.

Oh, so it's not itself a plane. It is the

equation of a plane. That seems like an

important distinction.

If you are going to be picky, then

yes, it would be slightly more correct

to say that x+2y +3z = 6 is the equa­

tion of a plane, rather than a plane

itself.

Well, perhaps when five million dol­

lars is at stake, it would pay to be picky.

Now, as I understand it, the tilted plane

intersects the x, y, and z axes at 6, 3,

and 2, respectively. Is that correct?

Yes.

Then the base is a right triangle with

the two edges at the right angle of

lengths 6 and 3. So it has area 9. That is correct.

And the height of the tetrahedron is

2, so it appears you want the students

to apply the famous formula for the

volume of a tetrahedron, which is one­

third of the area of the base times the

height. In this case, we obtain 6. Did I

do that right?

That is the correct answer, but no,

I did not want them to use the formula

for the volume of a tetrahedron. The

whole point of the class is to learn cal­

culus. They were supposed to create a

double integral that gives the volume

of the solid.

You mean as in this solution given

by Ms. Elaine Pleppmeyer, which we

see on this overhead slide.

Yes, that is what I intended. Only the

upper limit of integration on that outer

integral should be 6, not 3.

I see, and that's why you took off

two points out of the fifteen possible.

Yes, that's right.

Seems fair enough.

It is a standard amount I took off for

a mistake like that.

Very good. Now, shall we look at

Jeffrey Foible's solution? I have it on

this next overhead slide. Well, look at

that. He appears to have the correct so­

lution 6. And he appears to have done

it using the exact method I described.

Can you tell us how many of the fifteen

points you took off, and what it is you

wrote on the exam?

Umm, it looks as though I took off

5 points. And I wrote in the margin,

"Use calculus to solve these problems,

not memorized formulas."

So, correct me if I misunderstand.

Ms. Pleppmeyer got the answer wrong,

and you deducted two points, and Mr.

Foible got the answer right, and you de­

ducted five points.

Well, yes. But he didn't use calculus.

Oh, and can you point me to where

on the exam it says that all problems

must be solved using calculus?

That was implied.

Oh, I see. Implied?

I did put on the front of the exam,

"In all cases, grade is determined by the

instructor."

I see. And tell me, Mr. Phipps, if you

wrote on the exam, "In all cases, grade

to be determined by size of student's

butt," would that then make it accept­

able to determine the grade in that

manner?

Objection, your honor.

Sustained. The jury is instructed to

ignore the word "butt. " Strike it from

the record.

Let me rephrase the question. Are

instructors not bound by some code of

ethics? Should it not be the case that

if students get the right answer, they

should get the points they deserve?

This was a calculus course. Stu­

dents were supposed to learn calculus.

We had done problems like that on

the homework They knew they were

supposed to do the problem using

calculus.

And because Jeffrey Foible had this

additional knowledge, because he had

taken the time in his previous mathe­

matical career to learn this formula, to

remember this formula, you deducted

5 points?

I don't believe he knew the formula.

What do you mean?

I think he coped it from Karen

Lapala's paper.

What makes you think that?

Karen Lapala is an A student. She is

the only other student in the entire

class who used the formula to solve

that problem. I think Foible didn't

know how to do that problem, looked

at her paper, and wrote down the an­

swer. I think he cheated.

Really? And do you know if Jeffrey

Foible was sitting anywhere near Ms.

Lapala during the exam?

I can't be sure, but I have a vague

recollection he was sitting near her.

Really? If you are right, you must

have one of the most prodigious mem­

ories known to humankind. Let's see.

With 150 people sitting in the audito­

rium, each person sitting next to say,

two other people, how many pairs does

that make? That's approximately 150

pairs that you would need to remem­

ber. You looked out at that mass of

people and immediately memorized

150 pairs. Is that the case?

No, I don't remember who everyone

was sitting next to, but I think Foible

was sitting next to Lapala.

Forgive me if I seem skeptical that

you could possibly remember that. But

let me ask you this. Might your suspi­

cion that Jeffrey Foible cheated off

Ms. Lapala have influenced how many

points you took off on his exam? Or

perhaps the better question is, how

many points did you take off on this

problem on Ms. Lapala's exam?

Urn, I think I took off 3.

Three? But that isn't fair. After all,

she and Mr. Foible had the same an­

swer. Why should she lose only 3

points when he lost 5? For all you

know, she could have cheated off

Mr. Foible.

She had a picture of the tetrahe­

dron.

But nowhere in the problem did it

say to draw the tetrahedron. Why are

you giving points for it?

It demonstrates that she understood

what was going on. I gave partial credit

for that.

Come now, Mr. Phipps, do you re­

ally expect the jury to believe that by

drawing random pictures on an exam,

pictures that were not asked for, a stu­

dent can receive points?

Well, yes, because . . .

If I draw a picture of a person for

my Abnormal Psychology essay ques­

tion on schizophrenia, should I get par­

tial credit?

Of course not, but . . .

The truth, Mr. Phipps, is that you

didn't take off more points for Mr.

Foible because he failed to draw a

tetrahedron. You took off the extra

points because you had it in for him.

Had it in for him? I barely knew him.

My client recalls waking up one day

in lecture, and you fixing him with a

particularly malevolent stare. It was at

this moment that he realized that you

had it in for him.

I didn't have it in for him. And even

if I did, I couldn't have graded his exam

more harshly. I grade blind.

Do you mean to tell us that you

grade your students' exams with your

eyes closed?

No, of course not. I mean that I flip

over the cover sheet with the student's

name on it before I ever start grading.

Then when I grade a page, I have no

idea which student is which. I cannot

be influenced by my impressions of

their abilities.

Well, isn't that a clever idea. So

you never know who it is that you are

grading?

That's right.

But I suppose that once in a while,

there is a student who has very dis­

tinctive handwriting, and sometimes,

you are aware of who it is.

Maybe once in a while.

Perhaps, that was the case with Jef­

frey Foible?

I don't believe so.

I would like everyone to direct your

attention to the screen at right. On the

overhead I have a sample of Jeffrey

Foible's handwriting. Note how the c's

have an unusually sharp curvature.

And notice the angle between the two

lines making up the x. That angle is

27.3245 degrees, approximately. How­

ever the national average is 29.2234.

Am I right, Mr. Phipps, that x's come

up a lot on your exams?

Yes, they do. But you can't believe

that I would be able to recognize

© 2005 Spnnger Sctence+ Bus1ness Medta, Inc., Volume 27, Number 1 , 2005 39

Foible's handwriting from these minor

variations?

Perhaps not consciously, Mr.

Phipps, but the human mind is an in­

credibly intricate and subtle device. It

is capable of much more than we give

it credit for. Do you know that they say

we use less than 10% of our possible

brain capacity? What do you think the

rest of our brain is doing?

I have no idea.

No, it does not appear that you do.

Thank you, Mr. Phipps. You may step

down.

Members of the jury, this case now

falls into your capable hands. And from

my point of view, it is a relief to see it

there. Because I believe you under­

stand the critical importance of this

trial.

Take a look at Jeffrey Foible, sitting

there. Look at him. There sits a boy

who was one of the best and the bright­

est, on the verge of manhood, ready to

embark on his future. But what future

is that? What future is left to him now?

Who was it that destroyed his con­

fidence, that crushed his dreams, that

thwarted him from following his right­

ful path? I think you know the answer

to that. It was the defendant, Mr.

Phipps.

Fifteen seconds! That's how long

Mr. Phipps put into scoring each prob-

40 THE MATHEMATICAL INTELLIGENCER

lem. Fifteen seconds. That's the amount

of time he bequeathed, in his magna­

nimity, to determining the difference

between the B grade that would get

Jeffrey into law school and the C grade

that shuts him out forever. Fifteen sec­

onds. Does that sound fair to you? No,

I'm betting it doesn't.

Now, of course, this case isn't just

about compensating Jeffrey Foible for

the suffering he has incurred. It is not

about one student and the disastrous

results of his professor's incompe­

tence.

No, it is about how systematically,

across this country, faculty destroy

their student's hopes and dreams. It is

about how the whim or mood of a pro­

fessor can change the course of a stu­

dent's life. How insults exchanged in a

heated department meeting can gener­

ate emotions that alter the distribution

of points on an exam. How a poorly di­

gested bean burrito can cause a stu­

dent to be barred from the career to

which he has always aspired.

What happens when you put ab­

solute power into the hands of a

despot? When checks and balances

don't exist? When one person has the

unbridled authority to capriciously de­

termine the fates of others?

I am not just asking you to chastise

Mr. Phipps. I am asking you to send a

message to all the teachers out there.

To tell them that the age of tolerance

for their misdeeds is over, once and

for all.

Each and every one of you members

of the jury knows what I am talking

about. Because each and every one of

you knows that when you were a stu­

dent, you were unfairly graded. You did

not receive the points you deserved,

perhaps because your teacher had a

head cold, or even worse, because your

teacher didn't like the way you looked.

Do the right thing. Do it for the millions

of students who have gone before. Do

it for the millions of students yet to

come. And do it for Jeffrey Foible.

Thank you for your attention.

Doug Phipps jerked awake, his heart

pounding. Throwing off the covers, he

leaped out of bed and flicked on the

desk lamp. He rifled through the pile of

papers strewn about the desk until he

found Foible's exam.

Flipping it open to the fourth page,

he crossed out his written remarks and

the circled - 5, replacing them with a

check mark Then he flipped the exam

closed, turned out the light, and settled

back into bed. But sleep eluded him, and

four hours later, as light began to stream

through his windows, he looked forward

to the day's grading with dread.

ULRICH DAEPP, PAUL GAUTHIER, PAMELA GORKIN, AND GERALD SCHMIEDER

A ice i n Switzerland · The Life and Mathemat ics of A ice Roth

lice Roth, now well known for her work in rational approximation theory,

was the first woman to win the Silver Medal at the Eidgenossische Tech-

nische Hochschule (ETH) in Zurich, Switzerland and the second woman to

obtain a Ph.D. in mathematics there. (The next female recipient of the Ph.D.

in mathematics at the ETH would appear over 20 years

later.) Though Roth published her thesis in 1938, it would

be many years before her work was rediscovered and ap­

preciated. And it would be 35 years before she was able to

concentrate on her research again, but then-at the age of

66-her influential work in the field of rational approxi­

mation and function theory would become the focus of her

life and remain so until the day she died.

Three of her discoveries were outstanding for complex

approximation theory: her celebrated "Swiss cheese," be­

fore which the very raison d'etre of qualitative approxi­

mation was in doubt; her extension of approximation the­

ory from bounded to unbounded sets; and her discovery of

fusion, making it possible to simultaneously approximate

two different functions on two different sets by a single

function, even though these sets may overlap.

Dubbed "Alice in Switzerland" by her friend and co­

author, Paul Gauthier, she was agreed to be a remarkable

woman. Why? We tum to the life and work of this woman

who, in all areas of her life, was always too early to bene­

fit from her talent, determination, and strong will.

The Early Years

Alice Roth was born on February 6, 1905, in Bern, Switzer­

land. Her father, Conrad Roth, was then the director of the

Gaswerk und Wasserversorgung der Stadt Bern (Gas and

Water Supply for the City of Bern), a prestigious and in­

fluential post that he was elected to when only 27 years of

age.

Conrad Roth came from a family of boatmen, but his fa­

ther died when Conrad was just ten years old. He appren­

ticed four years as a mechanic, took evening courses on the

© 2005 Spnnger Science+ Bus1ness Media, Inc., Volume 27, Number 1, 2005 41

Alice's baptism at the Gaswerk Bern.

side, and attended the Technikum Wi nterth:u.r where he

earned a diploma as a machine technician. Alice's mother,

Marie Landolt, was a daughter of the mayor of Zi.ilich-Enge

(before it was incorporated into the city of Zi.irich). Marie

Roth-Landolt was described as a warm and loving woman

who kept house perfectly and was an excellent hostess,

skills practiced in tum by Alice. Alice's parents married in 1902, and one year later they had a son, Conrad. Alice was

born two years later. Four years later tht:> family was com­

pleted with the arrival of a second son, Waltt>r.

In 1911 Alice's father took a new position in the man­

agement (at the national level) of gasworks and coal sup­

ply. This required the family to move to Zi.irich, where Marie

Roth-Landolt had grown up. Alice, who started her school­

ing in Bern, continued her education in Zurich, but changed

schools once more when her family moved into their newly

built house in Zollikon. The house, situated in a suburban

community of Zurich and overlooking the lake, must have

been a peaceful place for a young girl to grow up. Alice's

older brother Conrad studied forestry at the ETH in Zi.irich,

and received an assistantship there while writing his doc­

toral dissertation. He eventually took a position as forPstry

superintendent in the canton Aargau. Alict:>'s younger

brother, Walter, did an apprenticeship as a bookselkr in

Zurich and later emigrated to Rio de Janeiro in Brazil where

he opened a bookstore.

After finishing the mandatory school years in Zollikon,

Alice commuted daily to Zi.irich where she attended the

Gymnasialabteilung der Hoheren Tochterschule der Stadt Zi.iri.ch. Schooling in Switzerland is the responsibility of the

canton, but preparatory schools for the university (Gym­nasien) were only open to boys. Some of the larger Swiss

cities had excellent schools for girls and, fortunately, the

Hohere Tochterschule der Stadt Zurich was one of them.

As Alice Roth mentions, she had excellent teachers at this

school. Her mathematics teacher, Prof. Dr. William Brun­

ner, instilled in her the desire to continue with her study

of mathematics. He resigned (to become professor of as­

tronomy at the ETH and director of the Federal Observa­

tory in Zurich) just a few years later. At his retirement from

the Hohere Tochterschule he was thanked:

. . . he worked at our school for many years (since 1908). In hin1 we had a truly outstanding mathematics teacher. In a

way fuat only a few teachers are able, he made his subject­

one that is, in general, difficult to grasp-understandable

and easily comprehensible to girls. [ 17, 1925/'26 p. 13]

The curriculum at the school was typical for a Real­gymnasium at this time in Switzerland. Latin took up the

most weekly hours of any single subject. Mathematics,

physics, chemistry, and other sciences were balanced witl1

German and two modern foreign languages (French and ei­

ther English or Italian). In the spring of 1924, Alice passed

tht:> Matura , the examination that entitled her to admission

to a university.

Graduate Years- ETH and Silver Medal

Alice Roth's goal was clear: she wanted to study mathe­

matics. Her mother, a practical woman, had nothing against

this study but wanted her daughter first to learn the basics

of household management. (This request was most likely

influenced by a national trend that led, in some places, to

the introduction of mandatory domestic training for girls

[ 12, pp. 361-:365] .) Thus it was that Alice went to Schloss Ralligen, on the shore of Lake Thun. This medieval castle

was home of a Haushaltungsschule for well-to-do daugh­

ters. During that year, Alice also took courses in needle­

work and dressmaking, and helped in the family household.

She then spent the summer semester of 1925 at the Urt'i­

versiUit Zilrid�, where she prepared for her entrance to

The Roth family house in Zollikon. Alice with her mother.

42 THE MATHEMATICAL INTEUIGENCER

Schloss Ralligen, 2004.

the department for Fachlehrer in Ma,thematik und Physik

of the ETH in Zurich. In the fall, she entered the ETH. From

1925 to 1929 her major field of study was mathematics,

physics her first minor, and astronomy her second.

The ETH is Switzerland's premier university for the sci­

ences and technology. It was very much dominated by men;

there were very few female students and there were no

women on the faculty. (In 1910, a woman wrote her Habil­

itation in mineralogy, but she died in 1916. For the forty

years following, no other woman was on the faculty at the

ETH [31 , p. 163] .) Nevertheless, Alice Roth did very well.

Her grades for the diploma exam and on her Diplmnnrbeit

were exceptionally high. Though Alice was surrounded by

men, most of her lasting friendships wpre with women very

much like herself. One of these was her colleague at the

ETH, Hanna Bretscher-Greminger. In HJ30 Alice completed

her Diplmrwrbeit (Master's thesis), entitlPd Ausdeh rrung des Wwiw·stra.ss 'schen Approxima/.ionssatze,-; a.uf das kmnple.r:e Geb·iet und a,uf ein u nendl iches Interval! (Ex­

tension of Weierstrass's Approximation Theorem to the

complex plane and to an infinite intt>rval), under the di­

rection of Professor George P6lya.

Following the completion of her diploma as Pachlehrer

in mathematics and physics, Roth spent 10 years as sub­

stitute teacher and Hiifslehrer at various middle schools in

Zurich and St. Gallen. She taught mathematics, physics,

Alice Roth during her graduate school years.

arithmetic, geometry, bookkeeping, business mathematics,

zoology, and anthropology. All but one of the schools (the

Freies Gymnasium in Zurich, where she was Hiifslehrer for

mathematics, April-December 1939) were girls' schools.

This was hardly a coincidence as we learn from the fol­

lowing quote.

True coeducation, in the sense that at all levels a mixed

student body is taught by a mixed teaching staff, hardly

exists anywhere in Switzerland. As long as female teach­

ers are almost completely excluded from the coeduca­

tional higher level Primar-schule, the co-educational

Sekundarschule, and the Gymnasien, and as long as they

are relegated almost exclusively to being in charge of

girls' schools, in the interest of working women, we will

hardly be prepared to tum existing girls' schools into co­

educational schools. [ 12, p. 399]

Her primary place of employment was at the school she

had attended, which was now renamed Tdchterschule der Sta.d/. Zurich. In April 1930, Alice Roth was elected Hilfs­

lehnrrin fiir Buchhaltung, Rechnen, Mathema,tik und Geometri.e (temporary teacher of accounting, arithmetic,

mathematics, and geometry) in the Abteilung I. This part

of the school had more than 600 students-all girls-of

whom slightly less than half were in the Gymna,sium, and

a faculty of 36 male and 28 female teachers. This was the public school for academically inclined girls in Zurich. It

had high standards and an exceptional faculty. Many of the

faculty members had connections to the two universities

in the city, the ETH and the Universitat Ziirich.

With respect to the teaching staff, the following tradition

can be noted with satisfaction and approval: Time and

time again, the school gave young academics the oppor­

tunity to fulfill their teaching obligations while preparing

themselves for work at the universities or to teach Uni­

versity Extension classes. [30, p. 5 1 ]

Alice Roth seems to have been well integrated into the

faculty. She participated in social and recreational func­

tions of the school. In the winter of 1932-33 she led 36 girls

to a ski camp in Arosa. One of her companions on this trip

(and the main leader) was Dr. Alfred Aeppli, a long-time

mathematics teacher at the school and a former Ph.D. stu­

dent of Professor P6lya at the ETH. The following year, Al­

ice Roth again helped lead a ski vacation week, this time

with Dr. Boller, another mathematics teacher and former

Ph.D. student of Professor P6lya. Alice Roth's colleague

and friend, Hanni Bretscher, was also a substitute teacher

at this school. After completing this stage in her education

and until the age of 35, Alice Roth still lived under her par­

ents' roof.

In looking at the education of her colleagues, it must

havP seemed to Alice that she would be able to obtain a

satisfying permanent position teaching mathematics only if

she earned a doctoral degree. In fact, in the academic year

1932-33, of the 42 teachers at the Tochterschule, 32 had a

© 200!1 Spnngf!r Sr.•ence+ Bus1ness Mecha, Inc . Volume 27, Number 1. 2005 43

Ph.D. (1 1 of these were women). The remaining ten taught

non-academic subjects, except for one Latin teacher.

Roughly half of the Hiifslehrer had Ph.D.'s and, in light of

the above quote, we suspect that many of them were in the

process of earning one. Since Alice was an excellent stu­

dent at the ETH, loved mathematics, and had a good rela­

tionship with her diploma advisor, it is not surprising that

she decided to continue her studies.

Thus, while still teaching, she worked on her disserta­

tion in function theory with P6lya once again as advisor.

Roth completed her thesis in 1938, becoming the second

woman to earn a Ph.D. in mathematics at the ETH. 1

Alice Roth's thesis, Approximationseigenschajten und Strahlengrenzwerte meromorpher und ganzer Funktio­nen (Properties of approximations and radial limits of

meromorphic and entire functions), was recognized as ex­

cellent. In it, she answered a question suggested by P6lya

and Szego [20, volume 2, p. 33] , and much more. As Pro­

fessor Heinz Hopf, in his report as co-referee, wrote,

In my opinion, both the main theorem, which is pre­

sented first and which in such a nice and simple manner

characterizes radial limits, as well as the approximation

theorems I just described, which indicate the role of

fairly general point sets in the theory of analytic func­

tions, are new, interesting, and important; and I consider

Fraulein Roth's achievement of having discovered and

proved these theorems truly laudable. The presentation

is also clear and lucid. I therefore recommend the work

for acceptance as a dissertation. [ 15]

Roth's thesis was singled out as worthy of special recog­

nition by her advisor and Hopf. The university had a prize

that would allow them to recognize Roth's work, but it was

a prize that had never been awarded to a woman: the ETH

Silver Medal.

The medal has a curious past. In the beginning of the

school's history, prize questions were posed every year and

a sum was paid to students who answered them or made

significant progress toward a solution. At some point, it was

believed that a medal would be a more suitable reward.

On August 10, 1866, the Swiss school board decided to

replace the hitherto existing monetary prizes by medals.

The medal will be cast in gold and silver. The gold medal

is intended only for solutions of prize questions that are

in every respect worthy of the attribute "outstanding,"

while the silver medal together with a correspondingly

higher or lower additional payment will be provided for

the main prize or secondary prize. This modus operandi

was first used during the year under review. [3, p. 6]

The prize questions were no longer posed, and the medal

and the money of the Kern Stijtung were used to reward

outstanding Diplomarbeiten or dissertations. Over the

The Silver Medal of the ETH with the inscription: "FRAULEIN ALICE

ROTH VON KESSWIL THURGAU."

years, less than 1% of these were so honored. Incidentally,

in our studies of the Berichte des eidgenossischen Poly­

technikums (later ETH), we were unable to locate an in­

stance between 1870 and 1940 in which a student was

awarded a gold medal.

On July 14, 1938, the Conference of the Department of

Mathematics and Physics requested that a Kern prize of 400

francs plus the silver medal be awarded to Alice Roth for

an excellent doctoral thesis. Records of the deliberation

and decision appeared in Protokoll des Schweizerischen Schulratesfiir das Jahr 1938 [21 , pp. 310-3 1 1 ] , from which

we now quote:

In the spring of 1930, Fri. Roth was granted a diploma for

Subject Teacher in Mathematics with a grade point aver­

age of 5.43; she received a grade of 5. 75 for her thesis. Her

doctoral thesis was examined by Professors Polya and

Hopf and was judged as outstanding. By motion of the pres­

ident it was decided [that] . . . For the outstanding doctoral

thesis a sum of Fr. 400.-from the Kern foundation and the

silver medal of the ETH will be given to Fri. Dip!. Fachl.

Mathern. A. Roth of Kesswil (Thurgau). [21, S. 310-311 ]

For a young mathematician the atmosphere at the ETH

was exciting. During her studies, Roth took a course from

Wolfgang Pauli. Rolf Nevanlinna and Lars Ahlfors were

both at the ETH when she was completing her Diplomar­beit. George D. Birkhoff also appeared for a brief visit, and

Roth told the following story about his visit: "P6lya said

that if I sat next to Birkhoff and was a pleasant dinner com­

panion, Birkhoff might help me get a job in the United

States. So I did, and P6lya got a job in the United States."

All the same, P6lya was a hero for Alice Roth. Unfortu­

nately for her, he left the ETH for the United States the

same year she left Zurich. P6lya eventually took a position

at Stanford, where Roth and he were to meet again in the

early 1970s. They maintained contact over the years, and

met whenever P6lya and his wife visited Zurich, which was

quite frequently. Several of their letters to each other have

survived, and P6lya's final letter to Roth reached her shortly

before her death. (See [ 1 ] for more about P6lya.)

' Elsa Frenkel, who worked in geodesics, was the first woman to earn a Ph.D. in mathematics at the ETH. The Referent and Koreferent of her thesis were A. Wolfer

and A. Einstein, respectively [7, p. 31 ] .

44 THE MATHEMATICAL INTELLIGENCER

The Next 30 Years- Humboldtianum

Curiously, after her success at the ETH, in 1940 Alice took

a position at a private gymnasium in Bern, the Institu/. Humboldtianum, where she became Hauptlehrerin fur Darstellende Geometrie, Mathematik und Physik. This po­

sition required her to work more hours than at a state

school, and she was less well-paid than her colleagues at

state schools. According to one student,

. . . in fact, the wages of a teacher at the Humboldtianum

are lower than the wages of a teacher at a public Gym­nasium. The fact that the number of weekly hours of

teaching is at least 28, while teachers at a public gym­

nasium teach at most 25 hours per week should also not

be overlooked. Therefore, it can be said that teachers in

private schools are rather badly off financially.

In this school it is difficult to cultivate a lively rela­

tionship between students and teachers at the Gymna­sium level (and also at other levels; for example, in the

trade school). Lack of time as well as stressful situations

(caused by the shorter preparation time of three years

for the Matura) prevent the realization of a relationship

beyond the traditional one of teacher as teacher and stu­

dent as the subordinate. [ 16, Urs GrabPr, p. 102)

It is important to note that while students in the public

gymnasium in Bern are more or less unifornlly well-preparE>d

for study, students at the Humboldtianurn had extremely

varied background and ability. As one student wrote,

I was fascinated by the composition of the student body:

"Sons and daughters by profession," for whose parents

the private gymnasium was the last hope that their off­

spring would reach an academic career; children of diplo­

mats and Swiss abroad with an aura of distance around

them; young people with some sort of handicap who

were better accommodated in a private school than in a

public school; latecomers like myself who had at least

gone through an apprenticeship already and who knew

exactly what they wanted. [ 16, Rita Liitzelschwab

1951/52, pp. 106-107]

Fraulein Dr. Alice Roth with her godchild Verena Gloor at the Sech­

seUiuten 1 939.

lnstitut Humboldtianum, Schlosslistrasse, Bern.

While Alice Roth appeared to be content in her work at

the Humboldtianum, in approximately 1959, she applied for

a position at the Stiidtisches Gymnasium in Bern, but her

application was unsuccessful. Roth (as well as many of her

colleagues) believed that had she not been a woman, she

would have had a chance at a better position. On the other

hand, one must note that the years in which Roth was look­

ing for her teaching positions were not good years for

Switzerland.

The years 1933-1940 were labeled crisis years. In

Switzerland, there was also general unemployment and

a shortage of food. [30, p. 70]

As a teacher, Roth was very influential. She remained

close friends with some of her students long after they left

the Humboldtianum and, in some cases, until her death.

She was frequently lovingly referred to as ''Mammeli Roth" by her students, sometimes warmly referred to as

''Rotchiipph" (Little Red Riding Hood), and sometimes less

lovingly as ''Rotkappe" (Big Red Riding Hood). The students

repeatedly mention her ability and desire to explain things

many different ways. Though many students clearly ap­

preciated her efforts in this direction, others report that she

would explain things several different times, in several dif­

ferent ways, and often the fourth or fifth explanation was

so complicated that even those who understood at the be­

ginning were confused. As one student described it,

Then there was the review in the subject of mathemat­

ics. Here the "fairy-tale" atmosphere was literally not

missing, for we had ''Rotkappe, " namely Fri. Dr. Roth who

managed, not infrequently, to confuse us in the most lov­

ing manner. [ 16, Erhard Erb und Gattin Vreni (Isen­

schmid) 1952-54, p. 140]

In the words of a second student,

I do not know of a single Humber colleague who did not

adore Fraulein Dr. Alice Roth, our "Mammi," who un­

fortunately died too early. In her lectures we started to

understand that mathematical problems could not only

be solved like this or that, but also in one way or another.

CO 2005 Spnnger Sc1ence 1 Business Med1a, Inc , Volume 27, Number 1, 2005 45

Whoever did not understand this and asked, received a number of other possibilities to choose from. Often, we feared that our dear "Mammi" would tangle up her arms while explaining-her gestures were so graphic. . . . "Mammi" always succeeded in hiding the intellectual woman behind a refreshing, natural, and cheerful per­sonality. I always looked forward to her lectures or to a private visit with her. [ 16, Jiirg Scharer, 1965-67, prepa­ration for agricultural studies at the ETH, pp. 157-160]

While by all accounts Roth was a much-loved, success­ful, and apparently happy teacher at the Humboldtianum, she also seems to have become more disillusioned with teaching as time went on. While clearly influenced by P6lya and his teaching methods, she also became frustrated by her lack of opportunity to implement them. In a letter to P6lya on the occasion of his 80th birthday (dated Novem­ber 29, 1967) Roth wrote:

On the other hand, it certainly depresses me to think that I so inadequately follow your teaching methods; meth­ods with which I have been familiar for such a long time. In particular, in more recent years I use so much time in my courses explaining important concepts and material for the exams to my students, most of whom were at most 1 1/2 years away from their examinations, that all too little time remains to correct the tendency of the less talented students to misconceive mathematics as simply an application of formulas. Unfortunately, unlike former times at the Humboldtianum, I am no longer given the opportunity to teach courses at the lower or middle level while retaining my "reduced" (but linked with so much work to correct) weekly teaching load of 24 hours. And right now, when it would be particularly important for me that the final years of my teaching be enjoyable, I am especially depressed by the disparity between effort and effect, whereby I am aware that this is not only caused by the exterior circumstances of our school, but is in some degree my own failure; on the other hand, there are, of course, several things about my school work that do not look so negative. Please excuse the personal out­burst that is so completely out of place in a congratula­tory letter, and that, at best, shows that aside from very successful students, you also have an aging student whose effectiveness is rather problematic. 2

It appears that during her employment at the Humbold­tianum with such a heavy work load Roth had little chance to keep up with mathematics. However, as Roth put it, she always dabbled in mathematics. Friends suggest that it was the only way that she could deal with her disillusionment with her teaching at the Humboldtianum. Roth filled her life in other ways. She was an accomplished pianist, she enjoyed hiking and skiing, she took frequent trips, and as a result of her early training she was an excellent cook She was also a determined, complicated, and strong woman. She em-

Flowers for Dr. Alice Roth to celebrate 30 years at lnstitut Hum­

boldtianum in 1970. On the left is colleague Hans Roder and on the

right Rektor Dr. Donald Keller.

phatically insisted on being called "Fraulein Dr. Roth" as op­posed to "Frau Dr. Roth," which to a German speaker indi­cates that the degree is her own rather than that of her hus­band. Roth was a long-time friend of Marie Boehlen, a Bemese lawyer, family rights activist, suffragette, and soon­to-be Grossrahn. And Roth herself was a strong supporter of women's right to vote. She often vented her frustration with a system that forced her to pay taxes, but allowed her no say in governance. Swiss women received the right to vote in 1971-the year of her retirement.

Retirement-A New Beginning

Alice Roth remained at the Humboldtianum until her re­tirement in 1971. It appears that shortly before retirement she had begun her transition back to work in mathematics. After announcing her plans to return to research to friends and relatives, she was told by one of them that in his field of medicine it would be impossible to return after so long an absence. Surely, most mathematicians would agree that it is impossible in the field of mathematics as well. After all, as G. H. Hardy said,

No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's gan1e . . . . Galois died at twenty-one, Abel at twenty­seven, Ran1anujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fun­damental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself. [ 13, p. 70]

And so Alice Roth would seem an unlikely candidate for success. Yet much had changed in the thirty years that she

·----· · ·-- · ----· -- --- -- ---- -- -

2ETH-Bibliothek Zurich, Hs 89:580/39.

46 THE MATHEMATICAL INTELUGENCER

had been teaching. In particular, Roth's area of research­

begun over thirty years earlier-had become fashionable.

Alice Roth re-entered mathematical research with great

enthusiasm and pleasure. She could count on the help of

her former student at the Humboldtianum, Peter Wilker,

who had become a professor of mathematics at the Uni­versitdt Bern. Though Wilker was in a different field of

mathematics, he helped her to obtain access to some of the

papers that she needed. A second mathematician who was

influential in Roth's new work was another student of

P6lya, Professor Albert Pfluger, who was now teaching at

the ETH.

For the first time, Roth was in the right place at the right

time. Her first work that appeared after her retirement

found its way to Paul Gauthier, then a young mathemati­

cian at the Universite de Montreal. Impressed, Gauthier be­

gan a correspondence with Roth. The two were highly com­

patible and a close friendship developed. They began joint

work, which resulted in an invitation to lecture in Montreal.

At the age of 70, a very excited Alice Roth left for her first

mathematical trip outside Switzerland.

Roth also spent some time with her friend Hanni

Bretscher in her secluded mountain retreat in the southern

Swiss alps translating the Anhang from P6lya's book Math­

ematics and Plausible Reasoning. She clearly enjoyed this,

as she describes to Albert Pfluger in a (much more upbeat)

letter dated October 31 , 1973:

We are having fun doing it, because we not only trans­

late, we also discuss the mathematical content.:>

At last Alice Roth had time on her side and was able to

put her mathematical creativity to work. She was now "am chnobble" (pondering a problem) full-time, gave talks to

other mathematicians at universities, and made good

progress-at the cutting edge of contemporary mathemat­

ics. But this happy last period of her life was cut short. In

1976 she began to suffer from an illness eventually diag­

nosed as cancer. What did she manage to do in the very

short time she devoted to mathematics?

Approximation of All Continuous Functions

on a Closed Set

We must begin with a brief overview of complex approxi­

mation theory. What follows is a "fusion" of the authors'

perspectives, but we believe it to be a good approximation

of Alice Roth's own mathematische Weltanschauung. We shall be dealing with complex-valued functions de­

fined on subsets of the complex plane C All topological

notions (such as closure, etc.) are with respect to the com­

plex plane except for a few instances, where we shall ex­

plicitly state that we are dealing with the Riemann sphere

C We are interested in approximating a continuous func­

tion on a compact set of the plane uniformly by polynomi­

als in the complex variable z.

There are many sets on which polynomial approxima-

3ETH-Bibliothek Zunch, Hs 1 446: 1 75

Between Osco and Calpiogna in 1 968. Alice Roth (on the right) with

unidentified friend.

tion fails. For example, consider the functionj(z) = liz on

the unit circle. Now, suppose that for each natural number

n there is a polynomial Pn such that :j - Pn1 < lin on the

circle. Multiplying by z we have that I I - ZPn(z) l < lin for

each z on the unit circle. Applying the maximum principle

we see that the same inequality holds in the open unit disc

[D. In particular, for z = 0 we have 1 < lin, a contradiction.

Thus, polynomial approximation fails on the unit circle. We

tum now to cases in which it is possible to approximate

continuous functions by polynomials.

The most famous positive result in polynomial ap­

proximation is the celebrated theorem of Karl Weierstrass

(1885), which states that on a closed and bounded inter­

val of the real line, each continuous function can be ap­

proximated uniformly by polynomials. In 1926, Joseph L.

Walsh [33] proved that in the Weierstrass theorem, we

may replace the closed interval by a compact Jordan arc

(homeomorphic image of a compact interval). In 1927,

Torsten Carleman [5] extended the Walsh, and hence the

Weierstrass, theorem: Let us define an unbounded Jordan

arc to be a homeomorphic image of the real line, such that

both "ends" tend to infinity. Carleman asserted that on an

unbounded Jordan arc, each continuous function can be

approximated uniformly by entire functions. To see how

Walsh's theorem follows from Carleman's theorem, take

a continuous function on a compact Jordan arc and ex­

tend it to a continuous function on an unbounded arc.

Apply Carleman's theorem to obtain an approximating

function that is also entire. The polynomial required by

Walsh's theorem can now be obtained by taking partial

sums of the Taylor series that represents the entire func­

tion. A particular case is that each continuous function

on the real line can be uniformly approximated by entire

functions.

In fact, this is the only case that Carleman actually

© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc., Volume 27, Number I, 2005 47

proved. He left the general proof to the interested reader. In this case, it seems the reader was Alice Roth. The fre­quency with which Roth mentioned the work of Carleman in conversations, and the fact that she took over where Car­ternan left off, suggest that Carleman and his work may have been the initial source of inspiration that led to one of Alice Roth's major contributions, the extension of Runge's theorem (see below) to unbounded sets. Indeed, in the very first sentence of her dissertation, she proclaims that the theorem of Carle man is the Ausgangspunkt of her work.

It is not surprising that Roth would be attracted by Car­ternan's work, as his theorem has exciting consequences. For example, Carleman's theorem makes it relatively easy to construct an entire function such that the image of the real line under that function is dense in the plane: Let M denote a countable and dense subset of the complex plane, and consider a bijective mapping u : 1L � M. The function IP : � � C defined by

ip(x) = u(n + 1)(x - n) + u(n)(1 - x + n) (x E [n,n + 1), n E 7L)

connects all points of M by line segments and is obviously continuous. For f(x) = exp(x2)1P(x) there exists, by Carle­man's result, an entire function F with IF(x) - f(x)l < 1 , and thus I F(x)exp( -x2) - ip(x)l < exp( -x2) for all x E �­The function g(z) = F(z) exp( -z2) is entire, and we see from the last inequality that g(�) is a dense subset of the plane.

We now consider approximation by rational or mero­morphic functions. In the previous paragraphs we men­tioned that on a Jordan arc each continuous function can be approximated uniformly by polynomials. We also proved that polynomial approximation fails on a Jordan curve (homeomorphic image of a circle). In the same paper in which Walsh proved that one could approximate continu­ous functions on Jordan arcs by polynomials, he also showed that one can approximate continuous functions on Jordan curves by rational functions. A closer look at the history of rational approximation will lead us directly to Roth's work.

In 1931 , Friedrich Hartogs and Arthur Rosenthal proved the very nice result that one can approximate continuous functions on compact sets of Lebesgue measure zero by ra­tional functions [ 14 ] .

In view of the Hartogs-Rosenthal theorem, one might be­lieve that continuous functions on more general compact sets can always be approximated by rational functions. The next step would be to investigate nowhere dense sets of positive measure. This is precisely what Roth did in 1938, and the result was a crucial example of a compact set on which not every continuous function can be approximated uniformly by rational functions-the so-called Swiss cheese. Among counterexamples in the field of function algebras, the Swiss cheese is hard to match.

The set Roth considered is just slightly more compli­cated than the example that is now �nerally attributed to her: take K to be the closed unit disc ID with infinitely many

48 THE MATHEMATICAL INTELUGENCER

Figure 1 . Alice Roth's Swiss cheese.

holes tJ.1, where the tJ.1 are open discs in ID with pairwise disjoint closures such that the sum of their radii is less than 1 and the remaining set has no interior points.

To see that the Swiss cheese does what needs to be done, we show why the construction implies that there ex­ists a function that is continuous on K but cannot be uni­formly approximated by rational functions with poles off K.

Choose K so that 0 a= K and recall that the open discs tJ.1 were chosen so that the sum of their radii is less than 1 . From Cauchy's theorem, any rational function R with poles off K satisfies

t!= l R(z)dz = � L.J R(z)dz,

where all circles are positively oriented. Thus, if we can ap­proximate a function g uniformly by rational functions with poles off K, the function must satisfy

J g(z)dz = L J g(z)dz. lz = l j <ll!.j

Now consider the function j(z) = lzl!z, which is contin­uous on K. For this function, flzl= l f(z)dz = 2m, while the integral 'i.1 fat>Jft:z)dz is less than 2 7T in modulus. Therefore the continuous function f cannot be uniformly approxi­mated by rational functions with poles off K. (See Gaier [8, chapter 3, section 3] for more details.)

On an unbounded closed set, there is no hope of ap­proximating every continuous function by rational func­tions, because rational functions are continuous at infin­ity (as functions from the Riemann sphere to the Riemann sphere). Thus, for example, it is impossible to approxi­mate, on the entire real axis, the function sin x by ratio­nal functions. On unbounded sets, the proper generaliza­tion of rational functions are meromorphic functions, just as entire functions are the proper generalization of poly­nomials. In 1938, Alice Roth extended the Hartogs-Rosen­thal theorem to unbounded sets by showing that, on closed sets of measure zero, one can approximate con­tinuous functions by meromorphic functions. This result, interesting in itself, was a key ingredient in her general­ization of the Runge theorem (which we will discuss be­low) to unbounded sets.

Approximation of All Holomorphic Functions

on a Closed Set

Runge's theorem on approximation by polynomials or ra­

tional functions can be regarded as the starting point of

complex approximation. Carl Runge published it in 1885 (see [29]), the same year that the Weierstrass approxima­

tion theorem appeared. When we speak of functions holo­

morphic on a closed set, we will always mean holomorphic

on a neighborhood of the closed set.

Theorem 1 (Runge). Let K be a compact subset of C Then each function holommphic on K can be uniformly approximated by rational functions.

Furthermore, unifo'rrn approx·imation by polynomials is possible if and only if C\K is connected.

The following theorem of Alice Roth extends Runge's

theorem to unbounded closed sets. It gives approximations

by entire or meromorphic functions, and it was proved in

her thesis [24, Satz III and the Zusatz p. 1 10] .

Theorem 2 (Roth-Runge theorem). Let E be a closed subset of the complex plane. Then each function holo­morphic on E can be uniformly approximated by .func­tions merommphic on C

Furthermore, U1!:_iform approximation by entire func­tions is possible ifC \E is connected and locally connected in C

It was Roth's discovery that the topological condition

just mentioned is sufficient for uniform approximation by

entire functions. This was also independently introduced

one year later by M. Keldysh and M. Lavrentieff. In fact, the

requirement that the complement be connected and locally

connected is also necessary for approximation of this type,

as in Arakelian's theorem (below). We now turn to an ap­

plication of the Roth-Runge theorem that is related to yet

another result of Roth.

In 1925, George P6lya and Gabor Szego published their

famous exercise book [20] ; a book that was to inspire much

research in analysis in the decades to come. In her thesis

[24, §4], Roth completely answered a question arising from

[20, vol. 2, Abschnitt IV, Nr. 187, p. 33 and p. 2 12] . Let F(z) = F(rei<P) be an entire or meromorphic function

with the property that the limit (which she called the

Strahlengrenz1nert)

f(ei<P) : = lim F(rei<P) 1"------).X

exists (where oo is allowed) for all r.p. Alice Roth found3 characterization of the corresponding functions.f : iJ[]) � C

The Roth-Runge theorem can be used to obtain an ex­

ample of an entire function F such that the corresponding

functionf has the property thatf(e; '�') = 1 for all r.p but F is

not constant. This is in contrast to Liouville's theorem,

which says that every bounded entire function must be con­

stant. We turn to the creation of this "unusual" function F. So let E be the closed set indicate_<"! in Figure 2, and let

A = E U [ 0, 1 } . Then A� closed, and IC\A is connected and

locally connected in C Consider a function g holomor-

E

Figure 2. The closed set E.

phic on (a neighborhood of) A such that g(z) = z on E U [O J and g(l) = 2. By the Roth-Runge theorem, we can ap­

proximate this function uniformly on A by entire func­

tions. In particular, there exists an entire function h such

that h(z) - g(z)l < 112 on A. Now define a function F by

F(z) = (h(z) - h(O))Iz. Then F is entire, and we will show

that it has Strahlengrenzwert 1 everywhere and is non­

constant.

By our construction, g(O) = 0 and on E U [ 0} we have

jh(z) - z l = h(z) - g(z)l < 112. So for z E E we have

jh(z) - z - h(O)j F(z) - 1 = lz,

jh(z) - g(z) + jh(O) - g(O)j :::; lz1

Now, for each r.p E [0,2 7T) there is some r0(r.p) such that

rei<P E E for r 2:: r0(r.p). Therefore, f(r.p) = lim,._,x F(rei'�') =

1 for all r.p. Now, if F were constant, then it would have to be the

constant function 1; in other words,

1 = F(z) = (h(z) - h(O))Iz.

In this case we have h(l) = 1 + h(O), and consequently

!h(l) - 21 = jh(O) - 1 1 2:: 1 - h(O)! > 1/2.

But we know that ih(l) - 2 = jh(l) - g( 1) < 112, and this

establishes the contradiction. Therefore, the function F is

nonconstant.

Intermezzo

While Alice Roth taught, research in her area continued.

The Swiss cheese was rediscovered by Mergelyan and was

"known affectionately as Mergelyan's Swiss Cheese." [34, p. 69] Mergelyan's name was frequently found attached to

the Swiss cheese up until about 1968 and occasionally even

later (e.g., [32] and [6]). As E. L. Stout writes in his Math­ematical Review of Vitushkin's 1975 [32] paper,

It should be noted that the author attributes to Mergelian

the first example of a nowhere dense compact set E C C for which C(E) =F R(E). This is an unfortunate though com­

mon error. Alice Roth gave an example of such a set in

1938 [24].

© 2005 Spnr1ger Sc1ence 1 Bus1ness Media, Inc , Volume 27. Number 1 , 2005 49

Apparently, as Roth began working her way back into mathematics, she found Zalcman's notes [34] and recog­nized Mergelyan's example as her own. By 1969 the error was corrected [9] , and most mathematicians felt the name "Swiss cheese" could not have been more appropriate.

Roth's past as well as future work was to have a strong and lasting influence on mathematicians working in this area. Her Swiss cheese has been modified (to an entire va­riety of cheeses); see e.g., Gaier [8, pp. 103-106] or Gar­diner [ 10] . We now tum to Roth's fusion lemma, which ap­peared in her 1976 paper [27] and influenced a new generation of mathematicians worldwide.

Roth's Fusion Lemma

Let K1 and K2 be disjoint compact subsets of C, and as­sume that the rational functions r1 and r2 are close on some compact set K, in the sense that 1r1 - d is small on K. Is there a rational function r that approximates r1 on K1 U K and simultaneously approximates 1·2 on K2 U K? Of course, such an r cannot approximate both r1 and r2 on K much better than r-1 and r2 approximate each other on this set.

If K1 U K2 U K has a decomposition into two disjoint compact sets, with one containing K1 and the second con­taining K2, then this problem can be solved by Runge's the­orem. But, if K is a "bridge" connecting K1 and Kz, as in Figure 3, then Runge's theorem will not answer this ques­tion for us.

Before turning to the solution, we consider the question from a real perspective. Consider the case in which we choose real intervals for the compact sets Ki> K2, and K and real functions r·1 and r-2, as in Figure 4.

The function r that we wish to find should approximate r1 on K1 U K and rz on K2 U K, and the error bounds, lh - �IK1 uK and l lr-z - �IKzuK, should both be close to lh - r2IIK· Even in this real case, it is not altogether obvi­ous that such a rational function exists. However, the ex­istence of such an approximation is guaranteed by Roth's fusion lemma [27].

Lemma 3 (Fusion of rational functions). Let Ki> K2, and K be compact subsets of the extended plane with K1 and K2 d·isjoint. If r-1 and r2 ar-e any two r-ational func­tions satisfying, for some E > 0,

hCz) - r-2(z)! < E, .for z E K,

then the1·e is a positive number a, depending only on K1 and K2, and a r-ational function r- such that for-j = 1, 2,

' I r(z) - rj(z) < aE, for z E Kj U K.

Figure 3. K is a bridge between K1 and K2•

50 THE MATHEMATICAL INTELLIGENCER

-·----·-· rl - - - - - - r2 -- r

Figure 4. Fusing r1 and r2 with r.

The fusion lemma is a powerful tool that can be used to extend approximation on compact sets to obtain results on closed sets. The proof is based on techniques used in study­ing smooth (not necessarily analytic) functions. The so­called Pompeiu formula for differentiable functions with compact support plays an important role. This formula is similar to Cauchy's integral formula, but there is an extra term reflecting the "extent" of non-analyticity in terms of the Cauchy-Riemann operator. The miracle that allows Al­ice Roth to obtain results concerning meromorphic func­tions from these differentiable techniques is an implicit use of a clever trick that has become more and more fashion­able, namely solving a non-homogeneous Cauchy-Riemann equation. In this way, she is able to get rid of the non-ana­lytic part that arises from the Pompeiu fommla.

Roth used her fusion lemma to prove Bishop's localiza­tion theorem on compact sets. To see how this goes, let f be a continuous function defined on a compact set E. The localization theorem of Errett Bishop [4] states that f can be approximated by rational functions if and only if for each point z E E there is a closed disc Kz centered at z such that the restriction off to E n Kz can be approximated by ra­tional functions. Of course, one direction of this theorem is obvious. In [27] , Alice Roth showed how to derive the (interesting) half of Bishop's theorem from her fusion lemma. The idea is quite simple:

Having chosen the sets Kz as above, return to each point z E E and choose a disc kz that is again centered at z, but has half the radius of the disc Kz. Since the set E is com­pact, we can find finitely many of the smaller discs, kzp . . . , kzp' such that E is contained in the union of these p sets. We show how to obtain Bishop's localization theorem from the fusion lemma by using each of the larger sets, KzJ n E, to find an approximating rational function Tj and then fus­ing the smaller sets kzj n E to obtain one rational function approximating f on all of E.

So suppose that we have a compact subset F1 of E on which we can approximate F uniformly and we also have a disc kzJ from above. (In the first step F1 = kz1 n E and kzJ = kz2, but in the steps thereafter, F1 is simply a compact set on which uniform approximation is possible, and kzJ is a disc we have not yet "fused" into FJ.) If F1 and kzJ hap­pen to be disjoint, Runge's theorem will imply that there is a rational function that approximates f on F1 U (kzJ n E), and the interested reader can find the details in [8, p. 1 14].

If F1 n kzj =F 0, then we must use the fusion lemma, which we do as follows:

Let SJ denote the radius of the larger disc, KzJ' and con­sider the three compact sets E1 = F1 n [w: lw - Zjl 2: 3sj l4 l , E2 = kzj n E = (w E E : w - zj s sj 12 ) and K = F1 n K,j = F1 n [w: w - zjl s sj). Then E1 and E2 are disjoint and, for future use, we note that F1 U (kzj n E) = E1 U

E2 U K. Given E > 0, by our assumption we can approximate our

function.fon F1 to within El4 by a rational function r1 • Now K,j was our bigger disc on which approximation was pos­sible. Therefore, we can find a rational function r2 ap­proximating/uniformly to within E/4 on K::j n E. Now each of these rational functions is within E/4 off on F1 n K::j = K, so r1 - r2 < El2 on K. By the fusion kmma, there is a constant a (depending on E1 and E2 alone) and a rational function r such that for m = 1,2 we haVf• r - 1"111 1 < aE on

E111 U K. Since f\ U (k::j n E) = E1 U E2 U K, we have r ­

.f ! s r - 1"11,' + rm -I I < aE + E on F1 U (kzj n E). Fur­themwre, E is arbitrary, so we are able to approximate.( on the set F1 U (kzj n E), and the fusion of F1 and kzj n E is complete. To finish the proof, add one disc to the set ob­tained from the previous step and repeat the argument above until all sets have been fused.

In 1976 Roth [27], via her fusion lemma, also extended the Bishop localization lemma to unbounded sets to obtain Roth's localization theorem on closed sets. Namely, she showed that if f is a function defined on a closed set E, then f can be approximated by meromorphic functions if and only if for each point p in E, there is a closed disc K1, centered at p such that the restriction off to E n Kp can be approximated by rational functions. Again, one direc­tion is obvious. But the other direction had profound con­sequences as Alice Roth noted. We present these in the fol­lowing section.

Complete Solution for the Class of Plausibly

Approximable Functions

In earlier sections we attempted to approximate all func­tions belonging to a natural class of functions on E, namely, the class of functions continuous on E or the class of func­tions holomorphic on E.

Becausl:' the uniform limit of continuous functions is continuous and the uniform limit of holomorphic functions on an open set is holomorphic, it follows that if a particu­lar function f can be approximated on a sl:'t E by entirl:' functions or by meromorphic functions having no poles on E, then nt>cessarily f is continuous on E and holomorphic on the interior of E. Hence, the most natural class of func­tions to try to approximate on a set E is tlw class A(E) of functions continuous on E and holomorphic on the interior of E. This class combines the attributes, continuity and holomorphy, of the two cla.•;;st>s we considered earlier, in just the right dosage. We will refer to ACE) as the cla.'>s of "plausibly" approximable functions. The most natural ques­tion on approximating a class of functions is then to dt>­termine those sets E for which the plausibly approximable functions are in fact approximable. For approximation by

entire functions, this question was answered in 1964 by Norair Arakelian [2].

Theorem 4 (Arakelian). Let E be a closed set in C. The following are equivalent:

(1) Each function continuous on E and holomorphic on the interior of E can be uniformly approximated by entire functions.

(2) The complement of E in C is connected and locally connected.

In 1972, Ashot Nersessian answered this question for meromorphic approximation [ 19].

Theorem 5 (Nersessian). Let E be a closed set in C. The following are equivalent:

(1) Each function continuous on E and holomorphic on the interior of E can be uniformly approximated by functions meromorphic on C .

(2) For each closed disc K, each function continuous on E n K and holomorphic on the interior of E n K can be unUonnly approximated b:IJ rational functions.

As fundamental as these two theorems are, giving a char­acterization of those closed sets E on which the class of all plausibly approximable functions can be approximated, a still more fundamental question is to decide when an indi­vidual function can be approximated. Roth's localization theorem does this, and is so powerful that it yields the suf­ficiency (2 implies 1) in both of the preceding theorems. In the cast> of Nersessian's theorem, this is obvious.

We now derive the sufficiency in Arakelian's theorem. Suppose, then, that the complement of E in C is connected and locally connected, and let f be continuous on E and holomorphic on the interior of E. Let K be any closed disc. The complement of the compact set E n K is connected and so, by a famous theorem of Mergelyan [ 18], each func­tion in A(E n K), in particular, the restriction off to E n K, can be uniformly approximated by polynomials. From the Roth localization theorem, for each E > 0, there is a function h meromorphic on C such that .f - hi < E/2 on E. By the Roth-Runge theorem (Theorem 2), there is an entire function g such that .h - gl < El2 on E. From the triangle inequality, If - g < E on E, which gives the sufficiency in the theorem of Arakelian.

The theorems of Arakelian and Nersessian are best pos­sible (in the sense indicated), but for most applications the analogous Runge-type theorems of Alice Roth suffice.

In her second period of creative mathematical work, af­ter retirement, Alice Roth extended to arbitrary plane do­mains her thesis results on holomorphic and meromorphic approximation on closed subsets of the plane. Over the past 30 years, attempts were made to further extend these re­sults and in particular to Riemann surfaces. These attempts were successful with respect to extending fusion on com­pact sets, but attempts to approximate on closed subsets of Riemann surfaces encountered major obstacles still not resolved. On the other hand, Roth's work inspired highly successful research in potential theory. At the present time,

ID 2005 Spnnger Sc1ence • Bus1ness Media, Inc , Volume 27. Number 1, 2005 51

advances are also being made on such questions for solu­tions of differential operators other than the Cauchy-Rie­mann and Laplace operators.

Looking Back at Roth's Work and Life

Alice Roth's 1938 thesis was largely overlooked, but by 1969 mathematicians were learning that the Armenian "Swiss cheese" was, in fact, discovered by a Swiss woman. As V. P. Ravin writes in his Mathematical Review of Roth's 1973 paper Meromorphe Approximationen:

In 1938, the author published an article that contains im­portant ideas of the later theory of approximation by ra­tional functions. We mention only the first example of a nowhere dense compact set E c C on which not every continuous function can be approximated uniformly by rational ones (this set was named the "Swiss cheese" much later) or the formulation of questions that antici­pated the later excellent work (by Arakelian, for exam­ple) on tangential approximation (in the sense of Carle­man). This earlier work seems to have fallen into almost complete oblivion, a common (but unfair) punishment for anyone who dares to enter a subject too early, with­out waiting until it is accepted an1ong experts and has even become fashionable.

All of Alice Roth's publications are cited in our refer­ences [ 1 1,22,23,24,25,26,27,28] . A good account of her work and influence in approximation theory appears in Gaier's book [8].

As her friend and former student at the Humboldtianum, Prof. Peter Wilker, wrote in an obituary in the Bernese newspaper Der Bund on July 29, 1977,

In Switzerland, as elsewhere, women mathematicians are few and far between . . . Alice Roth's dissertation was awarded a medal from the ETH, and appeared shortly af­ter its completion in a Swiss mathematical journal . . . . One year later war broke out, the world had other worries than mathematics, and Alice Roth's work was simply forgotten. So completely forgotten, that around 1950 a Russian math­ematician re-discovered sinlilar results without having the slightest idea that a young Swiss woman mathematician had published the same ideas more than a decade before he did. However, her priority was recognized.

Alice Roth was hospitalized in Bern in 1977. Although she was very ill, she continued to work on mathematics. She completed her work on her final paper Uniform Ap­proximation by Meromorphic Functions on Closed Sets with Continuous Extension into the Boundary in the In­selspital in Bern, and Wilker helped her with the English translation as well as the typing. As Wilker continued,

Alice Roth died a mathematician . . . As her last work lay finished before her, she was as pleased with it as she was

4Coined by Jean-Paul Berrut. Universitat Freiburg, Switzerland.

52 THE MATHEMATICAL I NTELLIGENCER

Fraulein Dr. Alice Roth, 1 975 (photo by A. Ballinger-Roth).

with the many flowers that decorated her room, and if someone had asked her whether or not an obituary for her were justified, she surely would have said, "Don't make a fuss about me-but my approximation [by] mero­m orphic functions on closed sets, about those you may write a story."

Her last paper arrived at the Canadian Journal of Math­ematics on July 1 1, 1977. Alice Roth, "die bekannteste un­bekannte Schweizermathematikerin,"4 died on July 22, 1977. She is buried in the Roth-Landolt family grave at Friedhof Bergli, in the medieval Swiss town Zofingen.

Acknowledgments

We thank Professors Jiirg Ratz and Hans-Martin Reimann for their assistance as well as encouragement. We are also particularly grateful to Judith Fahrlander for library ser­vices customized to our needs, and to the friends and fam­ily of Alice Roth, on whom we depended so much for in­formation, recollections, and photographs: Verena and Alfred Ballinger-Roth, Deli Roth, Verena de Boer, Johanna Meyer, Roland Weisskopf, and Irene Aegerter. We thank each of these people for their generosity and willingness to assist us. We are also grateful to the Universitat Bern for its support, to the Gosteli Archiv and Angela Gastl at Archiv der ETH Zurich for their assistance, and to Professor David Coyle for carefully reading t11e manuscript and for helpful suggestions. Aside from the photo of Schloss Ralligen, all photos are in the possession of the Roth family, except the

picture with her godchild, which is from Verena De Boer­Gloor. We thank both parties for allowing us to reprint

them.

REFERENCES

[ 1 ] G. L. Alexanderson. The random walks of George P61ya. MAA

Spectrum. Mathematical Association of America, Washington, DC,

2000.

[2] N. U. Arakelian. Uniform and asymptotic approximation by entire

functions on unbounded closed sets (Russian). Dokl. Akad. Nauk

SSSR, 1 57 :9-1 1 , 1 964. English translation: Soviet Math. Ookl. 5,

(1 964) , 849-851 .

[3] Bericht des eidgenbssischen Polytechnikums uber das Jahr 1 870.

Zurich, 1 870. Archiv der ETH Zurich, P 92899 P.

[4] E. Bishop. Boundary measures of analytic differentials. Duke Math.

J . , 27:331 -340, 1 960.

[5] T. Carleman. Sur un theoreme de Weierstrass. Ark. Mat. Astr. och

Fys. , 20 B(4) : 1 -5 , 1 927.

[6] R . B. Crittenden and L. G. Swanson. An elementary proof that the

unit disc is a Swiss cheese. Amer. Math. Monthly, 83(7):552-554,

1 976.

[7] Dlssertationenverzeichnis 1909-1971 . Number 15 in Schriftenreihe

der Bibliothek. Eidg. Techn. Hochschule, Zurich, 1 972.

[8] D. Gaier. Vorlesungen uber Approximation im Komplexen.

Birkhauser Verlag, Basel , 1 980. English translation: Lectures on

Complex Approximation, Birkhiiuser Verlag, Boston, 1 987.

[9] T. W. Gamelin. Uniform Algebras. Prentice-Hall Inc. , Englewood

Cliffs, N. J , 1 969.

[1 0] S. J. Gardiner. Harmonic approximation, volume 221 of London

Mathematical Society Lecture Note Series. Cambridge University

Press, Cambridge, 1 995.

[1 1 ] P. M. Gauthier, A. Roth, and J . L. Walsh. Possibility of uniform ra­

tional approximation in the spherical metric. Canad. J. Math. ,

28(1 ) : 1 1 2-1 1 5 , 1 976.

[1 2] M. Gosteli, editor. Vergessene Geschichte!Histoire oubliee. 11/u­

strierte Chronik der Frauenbewegung 1914-1963, volume 1 and

2. Stampfli Verlag , Bern, 2000. Articles in German, French, or Ital­

ian.

[1 3] G. H. Hardy. A Mathematician 's Apology. Canto. Cambridge Uni­

versity Press, Cambridge, 1 992, Reprinted 1 993.

[1 4] F. Hartogs and A. Rosenthal. Uber Folgen analytischer Funktio­

nen. Math. Ann . , 1 04:606-610 , 1 931 .

[1 5] H. Hopi. Korreferat uber die Dissertation von Fraulein A. Roth: Ap­

proximationseigenschaften und Strahlengrenzwerte meromorpher

und ganzer Funktionen. ETH-Bibliothek Zurich, Hs 620 : 1 07 , 9. VI I .

1 938.

[1 6] 75 Jahre Humboldtianum Bern: zum Geburtstag . Bern, 1 979.

[1 7] Jahresbericht der Hoheren Tochterschule der Stadt Zurich. Zurich,

1 925/26--1 927/28.

[ 1 8] S. N. Mergelyan. On the representation of functions by series of

polynomials on closed sets. Ooklady Akad. Nauk SSSR (N. S.) ,

78:405-408, 1 951 . English translation: Amer. Math. Soc. Transla­

tion 1 953 (1 953), no. 85, 8pp.

[1 9] A. H. Nersessian. Uniform and tangential approximation by mero­

morphic functions. lzv. Akad. Nauk Armyan. SSR Ser. Mat . ,

Vll :405-4 1 2 , 1 972. (Russian).

[20] G. P61ya and G. Szegb. Aufgaben und Lehrsatze aus der Analy­

sis . Springer-Verlag, 1 925. 2 Bande.

[2 1 ] Protokoll des Schweizerischen Schulrates fUr das Jahr 1 938.

Zurich, 1 938. Archiv der ETH Zurich, SR 2 .

[22] A. Roth. Ausdehnung des Weierstrass'schen Approximations­

satzes auf das komplexe Gebiet und auf ein unendliches Interval!.

Diplomarbeit, ETH , Abteilung fUr Fachlehrer in Mathematik u.

Physik, Zurich, 9 . November 1 929.

[23] A. Roth. Uber die Ausdehnung des Approximationssatzes von

Weierstrass auf das komplexe Gebiet. Verhandlungen der

Schweizer. Naturforschenden Gesel/schaft, page 304, 1 932.

[24] A. Roth. Approximationseigenschaften und Strahlengrenzwerte

meromorpher und ganzer Funktionen. PhD thesis, Eidgen6ssische

Technische Hochschule, Zurich, 1 938. Separatdruck aus Com­

ment. Math. Helv. 1 1 , 1 938, 77-1 25.

[25] A. Roth. Sur les limites radials des fonctions entieres (presentee

par M. Paul Montel) . Academie des Sciences, pages 479-481 , 1 4

Fevrier 1 938.

[26] A. Roth. Meromorphe Approximationen. Comment. Math. Helv. ,

48: 1 51 -1 76, 1 973.

[27] A. Roth. Uniform and tangential approximations by meromorphic

functions on closed sets. Canad. J. Math. , 28(1 ) : 1 04-1 1 1 , 1 976.

[28] A Roth. Uniform approximation by meromorphic functions on

closed sets with continuous extension into the boundary. Canad.

J. Math . , 30(6) : 1 243-1 255, 1 978.

[29] C. Runge. Zur Theorie der eindeutigen analytischen Funktionen.

Acta Math. , 6:229-244, 1 885.

[30] 1 00 Jahre Tochterschule der Stadt Zurich. Schulamt der Stadt

Zurich, Zurich, 1 975.

[31 ] Verein Feministische Wissenschaft Schweiz. Ebenso neu als kuhn,

120 Jahre Frauenstudium an der Universitat Zurich . eFeF-Verlag,

Zurich, 1 988.

[32] A G. Vitushkin. Uniform approximations by holomorphic functions.

J. Functional Analysis , 20(2): 1 49-1 57, 1 975.

[33] J . L. Walsh. Uber die Entwicklung einer Funktion einer komplexen

Veranderlichen nach Polynomen . Math. Ann . , 96:437-450, 1 926.

[34] L. Zalcman. Analytic Capacity and Rational Approximation. Lec­

ture Notes in Mathematics, No. 50. Springer-Verlag, Berlin, 1 968.

© 2005 Springer SC1ence+ Bus1ness Media, Inc , Volume 27, Number 1, 2005 53

A U T H O R S

Left: Paul Gauthier (with hat) and Gerald Schmieder. Right: Pamela Gerkin and Ul­

rich Daepp.

ULRICH DAEPP

Department of Mathematics

Bucknell University

Lewisburg, PA 1 7837

USA

e-mai l : udaepp@bucknell.edu

Ulrich Daepp was a student at the Sttidtisches Gymnasium Bem

and could have had Alice Roth as a teacher, had they hired her

when she applied. He crossed the Atlantic Ocean in his earty twen­

ties and completed his Ph.D. in algebraic geometry at Michigan

State University. He is now at Bucknell University in Pennsylvania,

where he has several joint projects with the third author, the most

important being their two children.

PAMELA GORKIN

Department of Mathematics

Bucknell University

Lewisburg, PA 1 7837

USA

e-mail: pgorkin@bucknell.edu

Pamela Gorkin received her Ph.D. from Michigan State Univer­

sity. She also studied at Indiana University for one year, and it was

during this year that she first heard about the Swiss cheese and

Alice Roth. Upon completing her doctorate, she began teaching

at Bucknell University. She has been there ever since with the ex­

ception of three sabbaticals, all of which have been spent at the

University of Bern, Switzerland. Her mathematical interests are pri­

marily function theory and operator theory. She enjoys watching

films, learning languages, cooking, and eating.

54 THE MATHEMATICAL INTELLIGENCER

PAUL M. GAUTHIER

Department of Mathemat ics and Stat ist ics Universite de Montreal

Montreal H3C 3J7

Canada

e-mail: gauthier@drns.umontreal.ca

Paul Gauthier was born and raised in the United States of French­

Canadian parents and, after completing his studies, chose to re­

turn to the "old country" (Quebec) for his career. He has six chil­

dren, and with his family has spent two years in the Soviet Union

and one year in China. His hobby is singing. Alice Roth was one

of Paul's most beloved friends and (to his surprise) he was men­

tioned in her will. One of his children was named after Alice and,

amazingly, only in writing this biography did he learn that she has

the same birthday as Alice Roth.

Publishing this, his first, paper in The lntelligencer fulfills an am­

bition of many years: to catch up with another of his daughters,

who published in The lntelligencer when she was still in grade

school (see volume 1 8, no. 1 , p. 7).

GERALD SCHMIEDER

Falkultat V. lnstitut fOr Mathematik

Universitat Oldenburg

261 1 1 Oldenburg

Germany

e·mail : GSchm ieder@t-online.de

Gerald Schmieder was born in Bad Pyrmont, Germany, and stud­

ied mathematics and physics in Hannover. He works on complex

approximation theory, Riemann surfaces, and geometric function

theory. His hobby is playing violin, mainly string quartets.

ll"@ii!i§j.fiii£11=tfii§#fii,j,i§.Jd M i chael Kleber and Ravi Vaki l , Editors

Gold bug Variations Michael Kleber

This column is a place for those bits of

contagious mathematics that travel

from person to person in the

community, because they are so

elegant, suprising, or appealing that

one has an urge to pass them on.

Contributions are most welcome.

Please send all submissions to the

Mathematical Entertainments Editor.

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380,

Stanford, CA 94305-21 25, USA

e-mail: vakil@math.stanford .edu

J im Propp bugs me sometimes. I'm usually glad when he does.

Today, Jim's bugs are trained to hop

back and forth on the positive integers: place a bug at 1, and with each hop, a bug at ·i moves to either i + 1 or i - 2. Of course, it might jump off the left edge; we put two bug-catching cups at 0 and - 1. Once a bug lands in a cup, we start a new bug at 1 .

What I haven't mentioned is how the bugs decide whether to jump left or right. We could declare it a random

walk, stepping in either direction with

probability 1/2, but the U. of Wisconsin

professor's bugs are more orderly than that. At each location i, there is a sign­post showing an arrow: it can point ei­ther Inbound, toward i - 2 and the bug

cups, or Outbound, toward infinity. The bugs are somewhat contrarian, so when a bug lands at i, it first .flips the arrow to point the opposite direction, and then hops that way (Fig. 1). Add

an initial condition that all arrows be­gin pointing Outbound, and we have a deterministic system. Let bugs hop till they drop (Fig. 2).

Well, what happens? Or, for those

who would like a more directed ques­tion: First, show that every bug lands in a cup (as opposed to going off to in­finity, or bouncing around in some bounded region forever). Second, find what fraction of the bugs end their journeys in the cup at - 1 , in the many­bugs limit. Go ahead, I'll wait. Do the first ten bugs by hand and look for the beautiful pattern. You can even skip ahead to the next section and read about another related bug, one with far more inscrutable behavior, and come back and read my solution another day.

I should mention, by way of a de­laying tactic, that the analysis of this

bug was done by Jim Propp and Ander Holroyd. A previous and closely related

bug of theirs, in which every third visit

to i leads to i - 1, appears in Peter

Winkler's new book Mathemal'ical Puzzles: A Connoisseur's Collection, yet another delightful mathematical of-

Figure 1 . The two bug bounces.

fering from publishers A K Peters [ 14]. The book itself is a gold-mine of the type of puzzles that I expect readers of this column would enjoy immensely.

Winkler's solutions are insightful, well­written, and often leave the reader with more to think about than before. The preceding is an unpaid endorsement.

Very well, enough filler; here's my answer. If you solved the problem without developing something like the

Figure 2. The bounces of (a) the first bug, (b)

the fourth bug.

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27. Number 1. 2005 55

theory below, please do let me know

how.

Before we get to the serious work, let's answer the question, can a bug hop around in a bounded area forever? It cannot: let i be the minimal place vis­

ited infinitely often by the bug-oops, half the time, that visit is followed by a trip to i - 2, a contradiction. So each bug either lands in a cup or, as far as we know now, runs off to infinity.

To motivate what follows, let's have a little inspiration: some carefully cho­sen experimental data. Perhaps you no­ticed that of the first five bugs, the cup at - 1 catches three, while the cup at 0 catches two. If that is not sufficiently suggestive, let me mention that after bug eight the score is 5 : 3, while after bug thirteen it is 8 : 5. On the specula­tion that this golden ratio trend con­tinues, let us refer to the bouncing in­sects as goldbugs.

Now the details. Suppose that r.p is­I can only imagine your surprise-a real number satisfying r.p2 = r.p + 1 ; when I care to be specific about which

root, I will write 'P± for (1 ± V5)/2. The goldbugs and signs are, in fact,

a number written in base r.p. Position i has "place value" r.pi, and the digits are conveniently mnemonic: Outbound is

0, Inbound is 1, and the bug itself is the not entirely standard digit r.p, which may appear in addition to the 0 or 1 in the "r.pi's place," where it contributes r.pi+ 1 to the total. Of course, numbers do not have a unique representation with this set-up, but that's quite delib­erate: the total value is an invariant, un­changed by bug bounces.

• Bounce left: Suppose the bug arrives in position i and there is an Out­bound arrow. The value of this part

of the configuration is ( r.pi X r.p) + ( r.pi X 0). After the bounce, position

i holds an Inbound arrow and the bug has bounced to position i - 2, for a total value of ( r.pi X 1) + (r.pi-2 X r.p). And these are the same,

since 'Pi- l + 'Pi = 'Pi+ 1 .

• Bounce right: Suppose the bug ar­rives in position i and there is an In­bound arrow. The value before the bounce is (r.pi X r.p) + (r.pi X 1) . After

the bounce, the arrow points Out­bound, value zero, and the bug is at

56 THE MATHEMATICAL INTELLIGENCER

i + 1 , for a value of r.pi+2, again un­

changed.

Now let's see what happens when we add a bug at 1, let it run its course, and then remove it after it lands in a bug cup. Placing a bug at 1 increases the system's value by r.p2. If it eventually lands in the cup at 0 and is then removed, the value

of the system drops by r.p, while if it lands in the cup at - 1 and is removed, the value of the system drops by r.p0

=

1. So the net effect of adding a bug at 1 , running the system, and then removing the bug is that the value of the system increases by <f!- - r.p = 1 for cup 0, and by r.p2 - 1 = r.p for cup - 1 .

Now we are well-equipped to an­swer the original questions, and then some.

• Can a bug run off to infinity? It can­

not: if we take r.p to mean 'P+ =

1 .61803, then each bug can increase the net value of the system by at most 'P+. and the positions far to the right are inaccessible to the gold­

bugs, because they would make the value of the system too high. So every bug lands in a cup.

• What is the ratio of bugs landing in the two cups? This time for r.p, think about 'P- = - .61803. Between bugs, when the system consists of just In­bound and Outbound flags, its mini­mum possible value is r.p1

_ + r.p3 _ +

r.p5 _ + · · · = - 1, while its maxi­

mum possible value is <f!-_ + r.p4 _ +

r.p6 _ + · · · = - 'P- = .61803, which I

will write instead as 1/ 'P+ to help us

remember its sign. So the value of the system is

trapped in the interval [ - 1, 11 'P+] , and as each successive bug passes through the system, the value either increases by 1 , or else increases by 'P-, i.e. decreases by li'P+ · For the

value to stay in any bounded region, it must, in the limit, step down 'P+ times as often as it steps up, and so

the bugs land in the - 1 cup 1/r.p+ of

the time. Moreover, this approxima-

tion is very good: if a bugs have

landed at - 1 and b have landed at 0, the system's value b - a/r.p+ must lie in our interval, so lb!a - 11'P+ i < 11a. Every single approximation bla is one of the two best possible given the denominator, and that denomi­nator grows as nlr.p+.

In fact, notice that the length of the interval [ - 1, 1lr.p+ ] is the sum of the two jump sizes, the smallest we could possibly hope for. If the value lies in the bottom 11r.p+ of the inter­val, it must increase by 1 , while if it lies in the top 1 of the interval, it must decrease by 11'P+ · This leaves a

single point, 1lr.p+ - 1 = - .38297, where the bug's destination cup isn't determined. But that value can be at­tained only with infinitely many In­bound signs: if we run a single bug

through a system with that starting value, its ending value would be ei­ther 11 'P+ or - l-and to attain those two bounds, we found above, you

must sum the infinite series of all positive or negative place values. So

for any initial configuration with only finitely many signs pointing In­bound, the configuration's numerical value alone determines the destina­

tion cup of every single bug; you don't need to keep track of the ar­rows at all.

• If you prefer integers to irrationals, consider instead the following in­

variant. Label the (bug cups and the) sites with the Fibonacci numbers (0, 1), 1 ,2,3,5,8, . . . , as in Figure 3. Give an Inbound arrow the value F; of its site, and give the bug there the value F; + 1 of the site to its right. This invariant, of course, is an appropri­ate linear combination of the 'P+ and 'P- ones above.

Adding a bug to the system now

increases the value by 2, but what's special is that removing the bug at ei­ther 0 or - 1 subtracts 1. So the bugs

implement an accumulator: after the

nth bug lands in its cup, we can look

u u 1)

0 I 0 I I 0 I I

(0 1 <--- ----->

2 3 <--- <--- <--- ----->

5 8 13 21 34 55 Figure 3. The signs after bug 1 1 7 passes through the system; 1 1 7 = 2 + 5 + 8 + 1 3 + 34 + 55.

Representations in base Fibonacci are not unique; ours is characterized by a lack of two con­

secutive zeros.

at the signs it has left behind, and read off the number n, written in base Fibonacci!

. . . Hold the presses! Matthew Cook has pointed out to me that this point of view can be taken further. We still get an invariant if we shift all those Fibonacci labels one site to the right. Now adding a bug increases the system's total value by 1, and re­moving it from the right bug cup de­creases the value by 1-but remov­ing it from the left bug cup decreases the score by 0. So after n bugs pass through, the value of the Inbound ar­rows they leave behind counts the number of bugs that ended their jour­neys in the left bug cup.

Furthermore we can shift the la­bels right a second time. Now when the bug lands in the left cup, its value must be the - 1st Fibonacci number, before 0--which is 1 again, if the Fi­bonacci recurrence relation is still to hold. So with these labels, the In­bound arrows will count the number of bugs which landed in the right cup. As an exercise, decode the ar­rows to learn that the 117 bugs lead­ing to Figure 3 split 72:45.

Once we know that the same set of arrows simultaneously counts the total, left-cup, and right-cup bugs, it's straightforward to see that the ratios of these three quantities are the same as the ratios of three consecutive Fi­bonacci numbers, in the limit.

These arrow-directed goldbugs are doing a great job of what Jim Propp calls "derandomization." It's straight­forward to analyze the corresponding random walk, in which bugs hop left or right, each with probability 1/z: Let Pi be the probability that a bug at place i eventually ends up in the cup at - 1, and solve the recurrence Pi = CPi-2 + Pi+l)/2 with P-1 = 1, Po = 0-oh, and make up for one too few initial condi­tions by remembering that all the Pi are probabilities, so no larger than 1. Here too we get p1 = 1/'P+·

So the deterministic goldbugs have the same limiting behavior as their ran­dom-walking cousins. But if we run the experiment with n random walkers and count the number of bugs in a cup, we'll generally see variation on the or-

der of Vn around the expected num­ber. Remember the sharp results of the 'P- invariant: n goldbugs, by contrast, simulate the expected behavior with only constant error!

There are more results on these and related one-dimensional not-very-ran­dom walks. But let's move on-these bugs long for some higher-dimensional elbow room.

The Rotor-Router

This time, we will set up a system with bugs moving around the integer points in the plane. It will differ from the above in that this will be an aggrega­tion model. Bugs still get added re­peatedly at one source, but instead of falling into sinks (bug cups), the bugs will walk around until they find an empty lattice point, then settle down and live there forever.

Generalizing the Inbound and Out­bound arrows of the goldbugs, we de­cree that each lattice site is equipped with an arrow, or rotor, which can be

.. * ...

* * t ... t ..

�

w .ffi__

.. _., · ·i ,

•

- -t� - - - - ......

... ..� - - - - - - - --, •

it t .. I

Figure 4. A bug's ramble through some ro­

tors: before and after.

rotated so that it points at any one of the four neighbors. (Propp uses the word "rotor" to refer to the two-state arrows in dimension 1 as well.) The first bug to arrive at a particular site occupies it forever, and we decree that it sets the arrow there pointing to the East. Any bug arriving at an occupied site rotates the arrow one quarter turn counterclockwise, and then moves to the neighbor at which the rotor now points-where it may find an empty site to inhabit, or it may find a new ar­row directing its next step. In short, the first bug to reach ( i, J) lives there, and bugs that arrive thereafter are routed by the rotor: first to the North, then West, South, and East, in that order, and then the cycle begins with North again (Fig. 4).

Once a bug finds an empty site to in­habit, we drop a new bug at the origin, and this one too meanders through the field of rotors, both directed by the ar­rows and changing them as a result of its visits. Every bug does indeed fmd a home eventually, and the proof is the same as for the goldbugs: the set of all sites which a particular bug visits infi­nitely often cannot have any boundary.

And so we ask, as we add more and more bugs, what does the set of occu­pied sites look like?

Let's take a look at the experimen­tal answer. The beautiful image you see in Figure 5 is a picture of the set of oc­cupied sites after three million bugs have found their permanent homes . The sites in black are vacant, still awaiting their first visitor. Other sites are colored according to the direction of their rotor, red/yellow for East/West and green/blue for North/South. On the cover is a close-up of a part of the boundary of the occupied region. At http:/ /www.math. wisc.edu/-propp/ rotor-router-1.0/ you can find a Java ap­plet by Hal Canary and Francis Wong, if you'd like to experiment yourself.

As you can see, the edge of the oc­cupied region is extraordinarily round: with three million bugs, the occupied site furthest from the origin is at dis­tance Y956609, while the unoccupied site nearest the origin is at distance Y953461, a difference of about 1.6106.

And the internal coloration puts on a spectacular display of both large-

© 2005 Springer Sc�ence+Business Media, Inc .. Vc>ume 27. Number 1 . 2005 57

Figure 5. The rotor-router blob after 3 million bugs.

scale structure and intricate local pat­terns. When Ed Pegg featured the ro­tor-router on his Mathpuzzle Web site, he dubbed the picture a Propp Circle, and to this day I am jealous that I didn't think of the name first: it con­notes precisely the right mixture of aesthetic appreciation and conviction that there must be something deep and not fully understood at work

Recall, for emphasis, that this was formed by bugs walking deterministi­cally on a square lattice, not a medium known for growing perfect circles.

58 THE MATHEMATICAL INTELLIGENCER

Moreover, the rule that governed the bugs' movement is inherently asym­metric: every site's rotor begins point­ing East, so there was no guarantee that the set of occupied sites would even appear symmetric under rotation by 90°, much less by unfriendly angles. On the other hand, while the overall shape seems to have essentially for­gotten the underlying lattice, the inter­nal structure revealed by the color­coded rotors clearly remembers it.

Lionel Levine, now a graduate stu­dent at U.C. Berkeley, wrote an under-

graduate thesis with Propp on the ro­tor-router and related models (11 ] . It contains the best result so far on the roundness of the rotor-router blob: af­ter n bugs, it contains a disk whose ra­dius grows as n114. Below I report on two remarkable theorems which do not quite prove that the rotor-router blob is round, but which at least make me feel that it ought to be. I have less to offer on the intricate internal struc­ture, but there is a connection to some­thing a bit better understood. Let's get to work

IDLA

Internal Diffusion Limited Aggregation

is the random walk-based model which

Propp de-randomized to get the rotor­

router. Most everything is as above­

the plane starts empty, add bugs to the origin one at a time, each bug occupies the first empty site it reaches. But in IDLA, a bug at an occupied site walks to a random neighbor, each with prob­ability 1/4.

The idea underlying IDLA comes

from a paper by Diaconis and Fulton

[6]. They define a ring structure on the

vector space whose basis is labeled by the finite subsets of a set X equipped with a random walk. To calculate the product of subsets A and B, begin with the set A U B, place bugs at each point

of A n B, and allow each bug to exe­cute a random walk until it reaches an outside point, which is then added to

the set. The product of A and B records the probability distribution of possible outcomes. This appears to depend on the order in which the bugs do their

random walks, but in fact it does not­we'll explore this theme soon.

Consider the special case where X is the d-dimensional integer lattice with the random walk choosing uni­formly from among the nearest neigh­bors. Then repeatedly multiplying the singleton {0} by itself is precisely the random-walk version of the rotor­router. A paper of Lawler, Bramson,

and Griffeath [9] dubbed this Internal Diffusion Limited Aggregation, to em­

phasize similarity with the widely-stud­ied Diffusion Limited Aggregation model of Witten and Sandler [ 13]. DLA simulates, for example, the growth of dust: successive particles wander in "from infinity" and stick when they reach a central growing blob. The re­sulting growths appear dendritic and fractal-like, but rigorous results are hard to come by.

In contrast, the growth behavior of

IDLA has been rigorously established. It is intuitive that the growing blob should be generally disk-shaped, since the next

particle is more likely to fill in an un­occupied site close to the origin than one further away. But the precise state­ment in [9] is still striking: the random walk manages to forget the anisotropy of the underlying lattice entirely!

Theorem (Lawler-Bransom-Grif­

feath). Let wd be the volume of the d­dimensional unit sphere. Given any E > 0, it is true with probability 1 that for all sufficiently large n, the d-di­mensional IDLA blob of wdnd particles will contain every point in a ball of radius (1 - E)n, and no point outside of a ball of radius (1 + E)n.

To be more specific, we could hope to define inner and outer error terms

such that, with probability 1 , the blob

lies between the balls of radius n -{lj(n) and n + 80(n). In a subsequent paper [10] , Lawler proved that these

could be taken on the order of n113• Most recently, Blachere [3] used an in­duction argument based on Lawler's proof to show that these error terms were even smaller, of logarithmic size. The precise form of the bound changes with dimension; when d = 2 he shows

that Mn) = O((ln n ln(ln n))112) and

80(n) = O((ln n)2) . Errors on that or­der were observed experimentally by Moore and Machta [ 12] .

So how does the random walk-based IDLA relate to the deter­ministic rotor-router? I start drawing the connection with one key fact.

It's Abelian!

Here's a possibly unexpected property of the rotor-router model: it's Abelian.

There are several senses in which this is true.

Most simply, take a state of the ro­tor-router system-a set of occupied sites and the directions all the rotors point-and add one bug at a point Po (not necessarily the origin now) and let it run around and find its home P1. Then add another at Q0 and let it run until it stops at Q1. The end state is the same as the result of adding the two bugs in the opposite order.

This relies on the fact that the bugs are indistinguishable. Consider the

(next-to-)simplest case, in which the

paths of the P and Q bugs cross at ex­actly one point, R. If bug Q goes first instead, it travels from Q0 to R, and then follows the path the P bug would

have, from R to P 1 . The P bug then goes

from P0 to R to Q1. At the place where their paths would first cross, the bugs effectively switch identities. For paths

whose intersections are more compli­

cated, we need to do a bit more work,

but the basic idea carries us through.

Taking this to an extreme, consider

the "rotor-router swarm" variant,

where traffic is still directed by rotors at each lattice site, but any number of bugs can pass through a site simulta­

neously. The system evolves by choos­ing any one bug at random and moving it one step, following the usual rotor rule. Here too the final state is inde­

pendent of the order in which bugs

move; read on for a proof. To create our rotor-router picture, we can place three million bugs at the origin simul­taneously, and let them move one step at a time, following the rotors, in what­

ever order they like.

In fact, even strictly following the rotors is unnecessary. The rotors con­trol the order in which the bugs depart for the various neighbors, but in the

end, we only care about how many bugs head in each direction.

Imagine the following set-up: we run the original rotor-router with three mil­lion bugs as first described, but each time a bug leaves a site, it drops a card there which reads "I went North," or whichever direction. Now forget about the bugs, and look only at the collec­

tion of cards left behind at each site. This certainly determines the final state of the system: a site ends up oc­

cupied if and only if one of its neigh­bors has a card pointing toward it.

Now we could re-run the system

with no rotors at all. When a bug needs to move on, it may pick up any card from the site it's on and move in the in­dicated direction, eating the card in the process. No bug can ever "get stuck" by arriving at an occupied site with no card to tell it a way to leave: the stack of cards at a given site is just the right size to take care of all the bugs that can possibly arrive there coming from all of the neighbors. (There is, however, no

guarantee that all the cards will get used;

left-{)vers must form loops.) A version of this "stacks of cards" idea appeared in

Diaconis and Fulton's original paper, in the proof that the random-walk version is likewise Abelian-i.e., that their prod­uct operation is well defined.

If the bugs are so polite as to take the cards in the cyclic N-W-S-E order

© 2005 Spnnger Science+ Business Media, Inc., Volume 27, Number 1 , 2005 59

in which they were dropped, then we simulate the rotor-router exactly. If we start all the bugs at the origin at once and let them move in whatever order they want-but insist that they always use the top card from the site's stack­we get the rotor-router swarm variant above; QED.

Rotor-Roundness

Now let me outline a heuristic argu­ment that the rotor-router blob ought to be round, letting the Lawler-Bram­son-Griffeath paper do all the heavy lifting.

I'd like to say that, for any c < 1, the n-bug rotor-router blob contains every lattice site in the disk of area en-as long as n is sufficiently large. My strat­egy is easy to describe. Just as we did four paragraphs ago, think of each lat­tice site as holding a giant stack of cards: one card for each time a bug de­parted that site while the n-bug rotor­router blob grew. Now we start run­ning IDLA: we add bugs at the origin, one at a time, and let them execute their random walks. But each time a bug randomly decides to step in a given direction, it must first look through the stack of cards at its site, find a card with that direction written on it, and destroy it.

As long as the randomly walking bugs always find the cards they look for, the IDLA blob that they generate must be a subset of the rotor-router blob whose growth is recorded in the stacks of cards. This key fact follows directly from the Abelian nature of the models.

So the central question is, how long will this IDLA get to run before a bug wants to step in a particular direction and fmds that there is no correspond­ing card available? Philosophically, we expect the IDLA to run through "almost all" the bugs without hitting such a snag: for any c < 1, we expect en bugs to ag­gregate, as long as n is sufficiently large. If we can show this, we are certainly done: the rotor-router blob contains an IDLA blob of nearly the same area, which in turn contains a disk of nearly the same area, with probability one.

To justify this intuition, we clearly need to examine the function d(iJ) which counts the number of depar-

60 THE MATHEMATICAL INTELLIGENCER

tures from each site. This is a nonneg­ative integer-valued function on the lattice which is almost harmonic, away from the origin: the number of depar­tures from a given site is about one quarter of the total number of visits to its four neighbors.

d(iJ) = ± (d(i + 1,J} + d(i - 1,J} + d(i,j + 1) + d(i,j - 1)) - b(iJ)

Here b(iJ) = 1 if (i,J} is occupied and 0 otherwise, to account for the site's first bug, which arrives but never departs. When ( i,J} is the origin, of course, the right-hand side should be increased by the number of bugs dropped into the system. Matthew Cook calls this the "tent equation": each site is forced to be a little lower than the average of its neighbors, like the heavy fabric of a cir­cus tent; it's all held up by the circus pole at the origin-or perhaps by a bundle of helium balloons which can each lift one unit of tent fabric, since we do not get to specify the height of the origin, but rather how much higher it is than its surroundings.

For the rotor-router, the approxima­tion sign above hides some rounding er­ror, the precise details of which encap­sulate the rotor-router rule. For IDLA, this is exact if we replace d by a, the ex­pected number of departures, and re­place b by b, the probability that a given site ends up occupied. (The results of [9] even give an approximation of d.)

Now, I'd like to say that at any par­ticular site, the mean number of de­partures for an IDLA of en bugs (for any c < 1 and large n) should be less than the actual number of departures for a rotor-router of n bugs. If so, we'd be nearly done, with a just a bit of easy calculation to show that the Vd-sized error term at each site in the random walk is thoroughly swamped by the (1 - c)n extra bugs in the rotor-router.

But this begs the question of show­ing that the rotor-router's function d and the IDLA's function d are really the same general shape. Their difference is an everywhere almost-harmonic func­tion with zero at the boundary-but to paraphrase Mark Twain, the difference between a harmonic function and an almost-harmonic function is the differ­ence between lightning and a lightning bug.

Simulation with Constant Error

After I wrote the preceding section, I learned of a brand-new result of Joshua Cooper and Joel Spencer. It doesn't tum my hand-waving into a genuine proof, but it gives me hope that doing so is within reach. Their paper [4] con­tains an amazing result on the rela­tionship between a random walk and a rotor-router walk in the d-dimensional integer lattice ll_d.

Generalizing the rotor-router bugs above, consider a lattice ll_d in which each point is equipped with a rotor­that is to say, an arrow which points towards one of the 2d neighboring points, and which can be incremented repeatedly, causing it to point to all 2d neighbors in some fixed cyclic order. The initial states of the rotors can be set arbitrarily.

Now distribute some finite number of bugs arbitrarily on the points. We can let this distribution evolve with the bugs following the rotors: one step of evolution consists of every bug incre­menting and then following the rotor at the point it is on. (Our previous bugs were content to stay put if they were at an uninhabited site, but in this ver­sion, every bug moves on.) Given any initial distribution of bugs and any ini­tial configuration of the rotors, we can now talk about the result of n steps of rotor-based evolution.

On the other hand, given the same initial distribution of bugs, we could just as well allow each bug to take an n-step random walk, with no rotors to influence its movement. If you believe my heuristic babbling above, then it is reasonable to hope that n steps of ro­tor evolution and n steps of random walk would lead to similar ending dis­tributions.

With one further assumption, this turns out to be true in the strongest of senses. Call a distribution of bugs "checkered" if all bugs are on vertices of the same parity-that is, the bugs would all be on matching squares if ll_d were colored like a checkerboard.

Theorem (Cooper-Spencer). There is an absolute constant bounding the divergence between the rotor and 'ran­dom-walk evolution of checkered dis­tributions in ll_d, depending only on

Figure 6. The greedy sand-pile with three million grains.

Figure 7. A non-greedy sand-pile. Here the dominant color is yellow, which again indi­

cates the maximal stable site, now with three grains. It is hard to see the interior pix­

els colored black, indicating sites which were once filled but are now empty, impossi­

ble in the greedy version.

© 2005 Spnnger Science+Bus1ness Media, Inc , Volume 27, Number i , 2005 61

the dimension d. That is, given any checkered initial distribution of a fi­nite number of bugs in 7l_d, the differ­ence between the actual number of bugs at a point p after n steps of ro­tor-based evolution, and the expected number of bugs at p after an n-step random walk, is bounded by a con­stant. This constant is independent of the number of steps n, the initial states of the rotors, and the initial dis­tribution of bugs.

I am enchanted by the reach of this result, and at the same time intrigued by the subtle "checkered" hypothesis on distributions. (Not only initial dis­tributions: since each bug changes par­ity at each time step, a configuration can never escape its checkered past.) The authors tell me that without this assumption, one can cleverly arrange squadrons of off-parity bugs to reorient the rotors and steer things away from random walk simulation.

Thus the rotor-router deterministi­cally simulates a random walk process with constant error-better than a sin­gle instance of the random process usually does in simulating the average behavior. Recall that we saw a similar outcome in one dimension, with the goldbugs.

There are other results which like­wise demonstrate that derandomizing systems can reduce the error. Lionel Levine's thesis [ 11 ] analyzed a type of one-dimensional derandomized aggre­gation model, and showed that it can compute quadratic irrationals with constant error, again improving on the Vn-sized error of random trials. Joel Spencer tells me that he can use an­other sort of derandomized one-di­mensional system to generate binomial distributions with errors of size In n instead of Vn. Surely the rotor-router should be able to cut IDLA's already logarithmic-sized variations down to constant ones. Right?

Coda: Sandpiles

All of the preceding discussion ad­dresses the overall shape of the rotor­router blob, but says nothing at all about the compelling internal structure that's visible when we four-color the points according to the directions of

62 THE MATHEMATICAL INTELLIGENCER

the rotors. When we introduced the function d( i,J), counting the number of departures from the (i,J) lattice site, we were concerned with its approxi­mate large-scale shape, which exhibits some sort of radial symmetry. The di­rection of the rotor tells you the value of d( i, J) mod 4, and the symmetry of these least significant bits of d is an en­tirely new surprise.

I can't even begin to explain the fine structure-if you can, please let me know! But I can point out a surprising connection to another discrete dynam­ical system, also with pretty pictures.

Consider once again the integer points in the plane. Each point now holds a pile of sand. There's not much room, so if any pile has five or more grains of sand, it collapses, with four grains sliding off of it and getting dumped on the point's four neighbors. This may, in tum, make some neigh­boring piles unstable and cause further topplings, and so on, until each pile has size at most four.

Our question: what happens if you put, say, a million grains of sand at the origin, and wait for the resulting avalanche to stop? I won't keep you hanging; a picture of the resulting rub­ble appears as Figure 6. Pixels are col­ored according to the number of grains of sand there in the final configuration. The dominant blue color correspond­ing to the largest stable pile, four grains. (This makes some sense, as the interior of such a region is stable, with each site both gaining and losing four grains, while evolution happens around the edges.)

This type of evolving system now goes under the names "chip-firing model" and "abelian sandpile model"; the adjective abelian is earned because the operations of collapsing the piles at two different sites commute. In full generality, this can take place on an ar­bitrary graph, with an excessively large sand-pile giving any number of grains of sand to each of its neighbors, and some grains possibly disappearing per­manently from the system. Variations have been investigated by combina­torists since about 1991 [2]; they adopted it from the mathematical physics community, which had been developing versions since around 1987

[ 1 ,5]. This too was a rediscovery, as it seems that the mechanism was first de­scribed, under the name "the proba­bilistic abacus," by Arthur Engel in 1975 in a math education journal [7,8].

I couldn't hope to survey the current state of this field here, or even give proper references. The bulk of the work appears to be on what I think of as steady-state questions, far from the effects of initial conditions: point-to­point correlation functions, the distri­bution of sizes of avalanches, or a mar­velous abelian group structure on a certain set of recurrent configurations.

Our question seems to have a dif­ferent flavor. For example, in most sandpile work, one can assume with­out loss of generality that a pile col­lapses as soon as it has enough grains of sand to give its neighbors what they are owed, leaving itself vacant. The ver­sion I described above is what I'll call a "greedy sandpile," in which each site hoards its first grain of sand, never let­ting it go. The shape of the rubble in Figure 6 does depend on this detail; Figure 7 is the analogue where a pile collapses as soon as it has four grains, leaving itself empty.

Most compelling to me is the fine structure of the sandpile picture. I'm amazed by the appearance of fractalish self-similarity at different scales de­spite the single-scale evolution rule; I think this is related to what the math­ematical physics people call "self-or­ganizing criticality," about which I know nothing at all. But personally, in both pictures I am drawn to the eight­petalled central rosette, the boundary of some sort of phase change in their internal structures.

Bugs in the Sand

So what is the connection between the greedy sandpile and the rotor-router? Recall the swarm variant of rotor­router evolution: we can place all the bugs at the origin simultaneously, and let them take steps following the rotor rule in any order, and still get the same final state.

Since we get to choose the order, what if we repeatedly pick a site with at least four bugs waiting to move on, and tell four of them to take one step each? Regardless of its state, the rotor

directs one to each neighbor, and we mimic the evolution rule of the greedy sandpile perfectly. If we keep doing this until no such sites remain, we re­alize the sandpile final state as one step along one path to the rotor-router blob.

Note, in particular, that the n-bug rotor-router blob must contain all sites in the n-grain greedy sandpile. Surely it should therefore be possible to show that both contain a disk whose radius grows as Vn.

More emphatically, the sandpile per­forms precisely that part of the evolution of the rotor-router that can take place without asking the rotors to break sym­metry. If we define an energy ftmction which is large when multiple bugs share a site, then the sandpile is the lowest-en­ergy state which the rotor-router can get to in a completely symmetric way.

When we invoke the rotors, we can get to a state with minimal energy but without the a priori symmetry that the sandpile evolution rule guarantees. And yet, empirically, the rotor-router final state looks much rounder than that of the sandpile, whose boundary has clear horizontal, vertical, and slope ::+:: 1 segments.

At best, this only hints at why the sandpile and rotor-router internal structures seem to have something in common. For now, these hints are the best I can do.

Acknowledgments

Thanks most of all to Jim Propp, who introduced me to this lovely material, showed me much of what appears here, and allowed and encouraged me to help spread the word. Fond thanks to Tetsuji Miwa, whose hospitality at Kyoto University gave me the time to think and write about it. Thanks also to Joshua Cooper, Joel Spencer, and Matthew Cook, for sharing helpful comments and insights.

REFERENCES

[ 1 ] Bak, Per; Tang, Chao; Wiesenfeld , Kurt.

Self-organized criticality. Phys. Rev. A (3)

38 (1 988), no. 1 , 364-374.

[2] Bjorner, Anders; Lovasz, Laszlo; Shor, Pe­

ter. Chip-firing games on graphs. European

J. Combin. 1 2 (1 991 ) , no. 4 , 283-291 .

[3] Blachere, Sebastien. Logarithmic fluctua­

tions for the Internal Diffusion Limited Ag­

gregation. Preprint arXiv:rnath.PR/01 1 1 253

(November 2001) .

[4] Cooper, Joshua; Spencer, Joel. Simulat­

ing a Random Walk with Constant Error.

Preprint arXiv:math C0/0402323 (Febru­

ary 2004); to appear in Combinatorics,

Probability and Computing.

[5] Dhar, Deepak. Self-organized critical state

of sandpile automation models. Phys. Rev.

Lett. 64 (1 990), no. 1 4 , 1 61 3-1 61 6.

[6] Diaconis, Persi; Fulton, William. A growth

model, a garne, an algebra, Lagrange in­

version, and characteristic classes. Com-

mutative algebra and algebraic geometry, II

(Turin, 1 990). Rend. Sem. Mat. Univ. Po­

litec. Torino 49 (1 991 ), no. 1 , 95-1 1 9 (1 993).

[7] Engel, Arthur. The probabilistic abacus.

Ed. Stud. Math. 6 (1 975), 1 -22.

[8] Engel, Arthur. Why does the probabilistic

abacus work? Ed. Stud. Math. 7 (1 976),

59-69.

[9] Lawler, Gregory; Bramson, Maury; Grif­

feath, David. Internal diffusion l imited ag­

gregation. Ann. Probab. 20 (1 992), no. 4 ,

21 1 7-21 40.

1 0 . Lawler, Gregory. Subdiffusive fluctuations

for internal diffusion limited aggregation.

Ann. Probab. 23 ( 1 995), no. 1 , 7 1 -86.

1 1 . Levine, Lionel. The Rotor-Router Model.

Harvard University senior thesis. Preprint

arXiv:math.C0/0409407 (September 2004).

1 2 . Moore, Christopher; Machta, Jonathan. In­

ternal diffusion-limited aggregation: paral­

lel algorithms and complexity. J. Statist.

Phys. 99 (2000) , no. 3-4, 661 -690.

1 3. Witten , T. A . ; Sander, L. M. Diffusion­

limited aggregation. Phys. Rev. B (3) 27

(1 983), no. 9, 5686-5697.

1 4 . Winkler, Peter. Mathematical Puzzles: A

Connoisseur's Collection. A K Peters Ltd,

Natick, MA, 2003.

The Broad Institute at MIT

320 Charles Street

Cambridge, MA 021 41

USA

e-mail: kleber@broad.mit.edu

© 2005 Springer Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1 , 2005 63

BRUNO DURAND, LEONID LEVIN, AND ALEXANDER SHEN

Local Ru les and G lobal Order , Or Aperiod ic Ti l 1 ngs

an local rules impose a global order? If yes, when and how? This is a philo­

sophical question that could be asked in many cases. How does local interaction

of atoms create crystals (or quasicrystals) ? How does one living cell manage to

develop into a pine cone whose seeds form spirals (and the number of spirals

usually is a Fibonacci number)? Is it possible to program locally connected computers in such a way that the net­work is still functional if a small fraction of the nodes is corrupted? Is it possible for a big team of people (or ants), each trying to reach private goals, to behave reasonably?

These questions range from theology to "political sci­ence" and are rather difficult. In mathematics the most prominent example of this kind is the so-called Berger the­orem on aperiodic tilings (exact statement below). It was proved by Berger in 1966 [1 ] . 1 In 1971 the proof was sim­plified by Robinson [7], who invented the well-known "Robinson tiles" that can tile the entire plane but only in an aperiodic way (Fig. 1 ).

Since then many similar constructions have been in­vented (see, e.g., [3, 6]); some other proofs were based on different ideas (e.g., [4]). However, we did not manage to fmd a publication which provides a short but complete proof of the theorem: Robinson tiles look simple, but when you

0 0 0 0 0 0

Fig. 1 . The Robinson tiles [reflections and rotations are allowed].

start to analyze them you have to deal with many technical details. ("This argument is a bit long and is not used in the remainder of the text, so it could be skipped on first read­ing," says C. Radin in [6] about the proof.)

It's a pity, however, to skip the proof of a nice theorem whose statement can be understood by a high school stu­dent (unlike the Fermat Theorem, you don't even need to know anything about exponentiation). We try to fill this gap

1 1n tact, the motivation at that time was related to the undecidability of a specific class of first-order formulas, see [2] .

64 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc.

and provide a simple construction of an aperiodic tiling with a complete proof, making the argument as simple as possible (at the cost of increasing the number of tiles).

Of course, simplicity is a matter of taste, so we can only hope you will find this argument simple and nice. If not, you can look at an alternative approach in [5].

Definitions

Let A be a finite (nonempty) alphabet. A configuration is an infinite cell paper where each cell is occupied by a let­ter from A; formally, the configuration is a mapping of type 7L2 ----> A. A local rule is an arbitrary subset L C A4 whose elements are considered as 2 X 2 squares: <a1, a2, a:l, a4) E L is a square

� [;I;] We say that these squares are allowed by rule L. A config­uration T satisfies local rule L if all 2 X 2 squares in it are allowed by L. Formally this means that

<T(i,j), T (i + 1, j), T(i,j + 1), T(i + 1, j + 1)) E L

for any i ,j E 7L. A non-zero integer vector t = (t1.t2) is ape­riod of T if the t-shift preserves T, i.e.,

for any x1,x2 E 7L.

The Aperiodic Tilings theorem

Theorem (Berger): There exist an alphabet A and a local rule L such that

(1) there are tilings that satisfy L; (2) any tiling satisfying L has no period.

To prove the theorem we need some auxiliary defini­tions.

Substitution Mappings

A substitution is a mapping s of type A ----> A4 whose val­ues are considered as 2 X 2 squares:

s: a � s1(a) s2(a)

s3(a) s4(a)

We say that a substitution s matches local rule L if two con­ditions are satisfied:

(a) all values of s belong to L; (b) taking any square from L and replacing each of the

four cells by its s-image, we get a 4 X 4 square that satis­fies L (this means that all nine 2 X 2 squares inside it be­long to L).

Remark. Consider a square X of any size N X N (filled with letters from A) satisfying L. Apply substitution s to each letter in X and obtain a square Y of size 2N X 2N. If the substitution s matches L, then Y satisfies L. Indeed, any

2 X 2 square in Y is covered by an image of some 2 X 2 square in X.

This is true also for (infinite) configurations: applying a substitution to each cell of a configuration that satisfies L, we get a new configuration that satisfies L (assuming that the substitution matches L).

Proposition 1. If a substitution s matches a local rule L, there exists a configuration T that satisfies L.

Proof Take any letter a E A and apply s to it. We get a 2 X 2 square s(a) that belongs to L. Then apply s to all let­ters in s(a) and get a 4 X 4 square s(s(a)) that satisfies L. Next is an 8 X 8 square s(s(s(a))) that satisfies L, etc. Us­ing a compactness argument, we conclude that there ex­ists an infinite configuration that satisfies L.

Here is a direct proof not referring to compactness. As­sume that substitution s is fixed. A letter a' is a descendant of a letter a, if a' appears in the interior part of some square s(s( . . . s(a) . . . )) obtained from a. Each letter has at least one descendant, and the descendent relation is transitive (if a' is a descendant of a and a" is a descendant of a' , then a" is a descendant of a). Therefore, some letter is a de­scendant of itself (start from any letter and consider de­scendants until you get a loop). If a appears in the interior part of sCn)(a), then sCn)(a) appears in the interior part of sC2n)(a), which appears (in its tum) in the interior part of sC3n)(a), and so on. Now we get a increasing sequence of squares that extend each other and together form a con­figuration. (Here we use that a appears in the interior part of the square obtained from a.)

Proposition 1 is proved. Now we formulate requirements for substitution s and

local rule L which guarantee that any configuration satis­fying L is aperiodic. They can be called "self-similarity" re­quirements, and guarantee that any configuration satisfy­ing L can be uniquely divided (by vertical and horizontal lines) into 2 X 2 squares that are images of some letters un­der s, and that these pre-image letters form a configuration that satisfies L. Here is the exact formulation of the re­quirements:

(a) s is injective (different letters are mapped into dif­ferent squares);

(b) the ranges of mappings s1.s2,s3,s4 : A ----> A (that cor­respond to the positions in a 2 X 2 square, see above) are disjoint;

(c) any configuration satisfying L can be split by hori­zontal and vertical lines into 2 X 2 squares that belong to the range of s, and pre-images of these squares form a con­figuration that satisfies L.

The requirement (b) guarantees that there is only one way to divide the configuration into 2 X 2 squares; the re­quirement (a) then guarantees that each square has a unique preimage.

Proposition 2. Assume that substitution s and local rule L satisfy requirements (a), (b) and (c). Then any configu­ration satisfying L is aperiodic.

Pmof Let T be a configuration satisfying L and let t = (t1 ,t2) be its period. Both t1 and t2 are even numbers. In­deed, (c) guarantees that T can be split into 2 X 2 squares,

© 2005 Spnnger SC1ence+ Bus1ness Media, Inc . Volume 27, Number 1 . 2005 65

and then (b) guarantees that the t-shift preseiVes these squares (since, say, an upper left corner of a square must go to another upper left corner).

Then (a) guarantees that pre-images of these 2 X 2 squares form a configuration that satisfies L and has pe­riod t/2. Therefore, for each periodic L-configuration with period t we have found another periodic L-configuration with period t/2. An induction argument shows that there are no periodic L-configurations.

Proposition 2 is proved. Using Propositions 1 and 2 we conclude that to prove

the Aperiodic Tilings theorem it is enough to construct a local rule L and substitution s matching L that satisfy (a), (b) and (c). This we now do.

Construction: An Alphabet

Letters of A are considered as square tiles with some draw­ings on them. We describe a local rule and substitution in terms of these drawings.

Each of the four sides of a tile (1) is dark or light (has one of two possible colors); (2) has one of two possible directions, indicated by

arrows; (3) has one of two possible orientations; this means that

one of two possible orthogonal vectors is fixed; we say that this orthogonal vector goes "from inside to outside". (Our drawings show the orientation by a gray shading inside.)

In this way we get three bits per side, i.e. , 12 bits for each tile. In addition to these 12 bits, a tile carries two more bits, so the size of our alphabet is 214 = 8192. These two additional bits are graphically represented as follows: we draw a cross (Fig. 2) in one of four versions (which differ by a rotation).

Fig. 2. One version of cross.

It is convenient to assign color, direction, and orienta­tion to the segments that forming a cross. Namely, two neighboring sides of a cross are dark, the other two are light. The direction arrows go from the center outward, and the orientation is shown by a gray stripe that shows the "in­side" part as indicated in the picture (gray stripes are in­side the dark angle).

This will be important when we define the substitution.

Substitution

To perform the substitutions, we cut a tile into four tiles. The middle lines of the tile become sides of the new (smaller) tiles, with the same color, direction and orientation. Before cutting we draw crosses on the small tiles in such a way that the dark angles form a square as shown (Fig. 3).

66 THE MATHEMATICAL INTELLIGENCER

. . . . . . . . . . . . . . . . . . . . . . .

Fig. 3. A tile split i n four parts.

It is immediately clear that conditions (a) and (b) of Proposition 2 are satisfied. Indeed, to reconstruct tile x from its four parts, it is enough to erase some lines, and the position of a tile in s(x) is uniquely determined by the orientation of its central cross. The condition (c) will be checked later after the local rule is defined.

Local Rule

The local rule (L) is formulated in terms of lines and their crossings. There are two types of crossings that appear when tiles meet each other. First, a crossing appears at the point where corners of four tiles meet; crossing lines are formed by the tile sides. Second, a crossing appears at the middle of tile sides, where middle lines of tiles meet the tile side. First of all, the following requirement is put:

if two tiles have a common side, this shared side has the same color, direction, and orientation in both tiles.

Therefore, we can speak about the color, direction and orientation of a boundary line between two tiles without specifying which of the two tiles is considered.

We also require that

all crossings (of both types) are either crosses or meeting points. A cross is formed by four outgoing arrows that have colors and orientation as shown in Fig. 4 (up to a rotation, so there are four types of crosses). In a meeting point, two arrows of the same color, the same orientation, but oppo­site directions, meet "face to face," and the orthogonal line goes through this meeting point without change in color, direction, or orientation. One more restriction is put: if two dark arrows meet, then the orthogonal line goes "outward" (its direction agrees with the orientation of the arrows).

Our local rule is formulated in terms of restrictions say­ing which crossings are allowed when lines meet. Formally speaking, the local rule is a set of all quadruples of tiles where these restrictions are not violated. Fig. 4 shows the first type of allowed crossing, a cross, in one of four pos­sible versions (which differ by a rotation). The second type

+ '

Fig. 4. A cross formed by outgoing arrows.

t ·········-!"·········

t Fig. 5. Arrows meet.

of allowed crossings (symbolically shown in Fig. 5) has

more variations: (a) the meeting arrows can be horizontal or vertical; (b) the vertical line can have two orientations; (c) the horizontal line can have two orientations; (d) the vertical line can have one of two colors; (e) the horizontal line can have one of two colors; and finally (f ) if two light arrows meet, the perpendicular line can go in either of two directions. So we get 2 · 2 · 2 · 2 · 3 = 48 variations in this way.

Remark. The Local rule ensures that the orientation of any horizontal or vertical line remains unchanged along the

whole line. (Indeed, the orientation does not change at

crosses or meeting points.)

Substitution and Local Rule

We have to check that the substitution matches the local

rule. Indeed, when tiles are split into groups of four, the old lines still form the same crossings as before, but new crossings appear. These new crossings appear (a) in the centers of new tiles (where new lines cross new ones) and (b) at the midpoints of sides of new tiles (where new lines

cross old ones). In case (a) we have legal crosses by defi­

nition. In case (b) it is easy to see that two arrows meet

creating a legal meeting point. See Fig. 6, which shows a tile split into four tiles, with all possible meeting places of new and old lines circled. The orientation matches because the orientation of the new crosses is fixed by s; all other requirements are fulfilled, too.

Fig. 6. New lines meet old lines.

Self-similarity Condition

It remains to check condition (c) of Proposition 2. Assume that we have a configuration that satisfies the local rule.

Step 1 . Tiles are grouped by fours.

Consider an arbitrary tile in this configuration and a dark

arrow that goes outward. It meets another arrow from a

neighboring tile, and this arrow must be dark by the local

rule. These two arrows must have the same orientation, therefore we get half of a dark square (Fig. 7), not a Z­shape. Repeating this argument, we conclude that tiles form

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 7. Two neighbor tiles.

groups of four tiles whose central lines form a dark square (Fig. 8).

:· · · · · · · · · · · · · · · · · · : · · · · · · · · · · · · · · · · · :

f EEl ! . . . . . . . . . . . . · . . . . . . . . . . . . . . . . . . · . . . . . . . . . . . . . . . . . . .

Fig. 8. Four adjacent tiles.

Step 2. These 2 X 2 squares are aligned.

If two groups (each forming a 2 X 2 square) were wrongly

aligned, as shown in Fig. 9, then the orientation of one of the lines (in our example, the horizontal line) would change along the line (recall that all crosses have fixed orientation of lines). Therefore, 2 X 2 squares are aligned.

Fig. 9. Bad placement.

Step 3. Each group has a cross in the middle.

What can be in the group center? The middle points of the sides of the dark square are meeting points for dark arrows. Therefore, according to the local rule, an outgoing arrow should be between them. So a meeting point cannot appear in the center of a 2 X 2 group, and the only possibility is a cross.

Step 4. Uniform colors on sides.

To finish the proof that each group belongs to the range of

the substitution, it remains to show that the color, direc­tion, and orientation do not change at the midpoint of a

side of a 2 X 2 group. This is because this midpoint is a meeting point for arrows perpendicular to the side.

Step 5. Pre-image tiles satisfy the local rule.

This is evident: the substitution adds new lines. So taking

the pre-image just means that some lines are deleted, and no violation of the local rule can happen.

The Aperiodic THings theorem is proved.

© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1, 2005 67

A U T H O R S

ALEXANDER SHEN

ul. Begovaya, 1 7- 1 4

Moscow, 1 25284

RusSia

e-mail: shen@mccme.ru

Alexander Shen has been since 1 982 at the

Institute for Problems of Information Trans­

mission in Moscow. He is known to lntelli­

gencer readers as aU1hor of several contri­

butions and as former Mathematical

Entertainments Editor. He has written text­

books based on courses at Moscow State

University and at the Independent Univer­

sity of Moscow: see ftp.mccme.ru/usersl

shen. Some have been translated into En­

gl ish: Algorithms and Programming: Prob­

lems and Solutions, Birkhauser, 1 997; and

two from the American Mathematical Society.

REFERENCES

BRUNO DURAND

Laboratoire d'lnformatique Fondamentale de

Marseille

CMI, 39 rue Joliot-Curie

1 3453 Marseille Cedex 1 3

France

e-mail: Bruno.Durand@lif.univ-mrs.fr

Educated in Computer Science at the

Ecole Normale Superieure de Lyon, Bruno

Durand is now both Professor at the Uni­

versite de Provence and Director of the

Laboratoire d'lnformatique Fondamentale

de Marseille. His research is on cellular au­

tomata, tilings, and complexity. He is an

editor of Theoretical Computer Science.

LENOID LEVIN

Department of Computer Science

Boston University

Boston, MA 0221 5-24 1 1

USA

Leonid Levin has worked rncstly in theory of

compU1ation. He was one of the originators

of the P-NP conjecture, whose monetary

price-tag has now grown to $1 CXXXXlO, bU1

whose scientific value is not compU1able.

Readers curious aboU1 his more recent

thoughts are invited to his Web site, http://

www.cs.bu.edu/fac/lnd.

[ 1 ] R. Berger, The undecidability of the domino problem, Memoirs

Amer. Math. Soc., 66 (1 966), 1 -72.

[4] J . Kari, A small aperiodic set of Wang tiles, Discrete Math. , 160

(1 996), 259-264.

[5] Leonid A Levin, Aperiodic Tilings: Breaking Translational Symme­

try, http:l/arxiv.org/abs/cs/0409024. [2] E. Borger, E. Gradel, Yu. Gurevich , The Classical Decision Problem,

Springer, 1 997. [Berger's theorem is considered in the Appendix

written by C. Allauzen and B. Durand.]

[3] B. Grunbaum, C. G. Shephard. Tilings and Patterns, Freeman, New

York, 1 986.

[6] C. Radin, Miles of Tiles, AMS, 1 999 (Student Mathematical Library,

vol. 1 ).

[7] R. Robinson, Undecidability and non periodicity of tilings in the plane,

Invent. Math . , 12 ( 1971 ) , 1 77-209.

MOVING? We need your new address so that you

do not miss any issues of THE MATHEMATICAL INTELLIGENCER

Please send your old address (or label) and new address to: Springer

Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485

U.S.A.

Please give us six weeks notice.

li,iM.�ffl!•i§u6'h£ili.Jiiijbl D i rk H uylebrouck, Ed itor I

The Home of Golden Numberism Roger Herz-Fischler

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the caje

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

If so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

e-mail: dirk.huylebrouck@ping.be

The expression "golden numberism"

refers to the set of claims con­

cerning the purported use of the golden number (division in extreme ratio, golden section, golden ratio, . . . ) in man-made objects (art, architecture, etc.) or its purported appearance in na­ture (human body, plants, astronomy, etc.). If we leave aside the vague state­ments by Kepler, the early nineteenth­century association of the golden num­ber with phyllotaxis, and a few other

extremely obscure comments, then we

can state that the beginning of golden

numberism is due to one person, the German intellectual Adolph Zeising (1810-1876).

What is of particular interest to the mathematical tourist is that the origin of golden numberism is associated with a particular time and place. Zeis­ing's father was a court musician in the tiny dukedom of Anhalt-Bernburg. This dukedom, as well as the other An­

halts-the various pieces of which will bring joy to any map colourist or

topologist interested in non-connected surfaces-were contained in what is now the German Land of Sachsen­Anhalt. Because of his father's occu­pation, Zeising was born in Ballenstedt, the location of the summer palace, but a visit there made it clear that he is now

completely unknown to local official­dom.

Where Zeising is known to some extent-though not in connection with the golden number-is in the city of Bernburg, which is some 75 kilometres northwest of Leipzig. Despite the rav­ages caused by events of the last sixty years, Bernburg, situated on the Saale river, remains a charming city. There are two edifices in Bernburg that should probably be jointly designated as the birthplace of golden numberism.

The first is the building formerly oc­

cupied by the Karls-Gymnasium. In

1842, after having completed a doctor­ate with a specialty in Hegelian philos­ophy, Zeising became a full-time pro­fessor at that institution, and it was

while he taught there that a combina­tion of readings in philosophy and

other fields inspired him to think of the golden number as an inherent rule of nature.

The other building of interest is the Bernburg castle. Zeising was a leader

of the liberal left during the German revolution of 1848-1849, and he was elected to the first Landtag, which met in the castle. Perhaps Zeising thought

about the golden number during some

long-winded speeches, but more im­

portantly, the castle represented polit­

ical power. By 1851 the power was firmly in the hands of a very reac­

tionary group, and Zeising was pen­

sioned off in December 1852. Using this money he went to Leipzig to do fur­ther research, and in 1854 he published his book, An Exposition of a New The­

ory of the Proportions of the Human

Body, Based on a Previously Unrec­

ognized Fundamental Morphological

Law which Permeates all of Nature

and Art, Together with a Complete

Comparative Overview of Previous

Systems.

In the period from 1855 to 1858 Zeis­ing continued to publish articles and a booklet dealing with the golden num­ber. Some of these were of a popular nature, in particular his articles "Hu­

mans and Leaves" and "Face Angles," which were published in the widely read science magazine Die Natur. His book and these articles ensured that his theory became widely known, and

by the time of his death in 1876 golden numberism was widespread in Ger­many. Philosophers debated the foun­dations of his theory, and authors sug­gested the use of the golden number in such fields as typography and fashion.

The polymath Gustave Fechner, in­spired by Zeising's claims concerning

the Sistine Madonna, started scientific investigations, which in tum laid the

foundations of the field of experimen­

tal aesthetics. By the end of 1855 Zeising had

moved to Munich, where he became

© 2005 Spnnger Science+Bus1ness Media, Inc., Volume 27, Number 1 , 2005 69

�but8fr .ft4rl��mufium ht b« 9unftt1Jellf 1 84 l -- 1882

Fig. 1 . The Karls-Gymnasium, Bernburg, Germany.

active in literacy circles and wrote nov­els and plays. He also wrote on philos­ophy-his 1855 Aesthetics, which pre­sented an overview of the systems of Hegel and his followers, was very highly regarded-and politics. After a hiatus of ten years Zeising again wrote

Fig. 2. Bernburg Castle.

70 THE MATHEMATICAL INTELLIGENCER

on the golden number, but, aside from an article on the Cologne Cathedral, these works were of a cultural or philo­sophical nature. A particular honour came in 1856: his admission to the Deutsche Akademie der Naturforscher.

There have been many interesting

twists and turns in the development of the myth of golden numberism. Thus the association of the golden number with virtually all the pyramids of Egypt except the Great Pyramid was made by Friedrich Rober in 1855, independent of Zeising. On the other hand the first example of golden numberism in En­glish dates from 1866 and deals with the golden number and the Great Pyra­mid-but in a manner not equivalent to that of Rober-and again this was in­dependent of Zeising's writings.

Mter having entered France and the English-speaking world from Germany in the early part of the twentieth century, golden numberism spread rapidly. By a careful examination of sources, it is pos­sible to trace the path travelled by the golden number myth. Aside from the topics of phyllotaxis and the Great Pyramid, virtually everything that has been written on the subject can ulti­mately be traced back to the influence of Zeising. The next time the reader hears another "historical" claim con­cerning the golden number he or she might care to glance at the accompa­nying photographs of the Gymnasium and the Bernburg castle, and remem­ber where the myth started.

I consider Zeising the most intellec­tual of authors on the subject of the golden number. Unlike others, he at­tempted to present a true foundation­in his case philosophical-for his argu­ments. Not that I fmd him convincing! I am fascinated by the "sociology of mathematical myths" and the history of ideas. Thus my most recent work, Adolph Zeising, started out as a few paragraphs and then an appendix to my forthcoming book The Golden Number. In Adolph Zeising I trace the spread of golden numberism from Nees von Es­senbeck in 1852 through 1876, the year of Zeising's death. The Golden Number will present a complete discussion of golden numberism, including the his­torical, sociological, and philosophical aspects. Parts of the story can be found in my other writings listed below.

Biographical Notes

Ebersbach's book discusses the period (1835-1852) when Zeising lived in Bernburg. In particular, there are sev-

eral references to Zeising's role in

the 1848-1849 revolution. The pho­

tographs (1864 for the castle and the

early part of the twentieth century for

the Gymnasium) were taken from

[Erfurth, p. 63) and [Schulgemeinschaft

Carolinum und Friederiken-Lyzeum,

p. 142] respectively. These photographs

are reproduced with the kind permission

of the Mittledeutsche Verlag (Halle).

Herz-Fischler, R. "How to Find the "Golden

Number' Without Really Trying . " Fibonacci

Quarterly 1 9 (1 981 ) , pp. 406-4 1 0.

mid. Waterloo, Wilfrid Laurier University

Press, 2000.

Herz-Fischler, R. Adolph Zeising (1 810-1876):

Herz-Fischler, R. A Mathematical History of

Division in Extreme and Mean Ratio . Wa­

terloo, Wilfrid Laurier University Press.

1 987. Re-printed as A Mathematical His­

tory of the Golden Number. New York,

Dover, 1 998.

The Life and Work of a German Intellectual.

Ottawa: Mzinhigan Publishing, 2004.

Herz-Fischler, R. The Golden Number. To ap­

pear. Ottawa: Mzinhigan Publishing, 2005.

Schulgemeinschaft Carolinum und Friederiken­

Lyzeum. Geschichte der h6heren Schulen zu

Bernburg: Friederikenschule von 181 0 bis

1 950, Kar/sgymnasium von 1 835- 1944,

Karls-Realgymnasium von 1 853-1945. Mu­

nich: Wedekind, 1 980.

Herz-Fischler, R . "A 'Very Pleasant' Theorem."

REFERENCES

College Mathematics Journal 24 (1 993), pp.

31 8-324.

Ebersbach, V. Geschichte der Stadt Bernburg,

vol. 1 . Dessau: Anhaltische Verlagsgesell­

schaft, 1 998.

Erfurth, H. Gustav V61ker/ing & die altesten Fo­

tografien Anhalts . Dessau: Anhaltische Ver­

lagsgesellschaft, 1 991 .

Herz-Fischler, R. "The Golden number, and Di­

vision in Extreme and Mean Ratio." in Com­

panion Encyclopedia of the History and

Philosophy of the Mathematical Sciences,

London, Routledge, 1 994, pp. 1 576-1 584.

340 Second Avenue

Ottawa, K1 S 2J2

Canada

Herz-Fischler, R. The Shape of the Great Pyra- e-mail: rhf@herz-fischler.ca

Mathematics and Culture Mathematics and Culture Michele Emmer, University of Rome 'La Sapienza', Italy (Ed.)

This book stresses the strong links between mathematics and culture, as mathematics links theater, literature, architecture, art, cinema, medicine but also dance, cartoon and music. The articles introduced here are meant to be interesting and amusing starting points to research the strong connection between scientific and literary culture. This collection gathers contributions from cinema and theatre directors,

musicians, architects, historians, physicians, experts in computer graphics and writers. In doing so, it highlights the cultural and formative character of mathematics, its educational value. But also its imaginative aspect: it is mathematics that is the creative force behind the screenplay of films such as A Beautiful Mmd, theater plays like Proof, musicals like Fermat's Last Tango, successful books such as Simon Singh's Fermat's Last Theorem or Magnus Enzensberger's The Number Devil.

2004/352 PP., 54 ILLU S./HARDCOVER/$59.95/ISBN 3-540-01 770-4

Springer www.springer-ny.com

Mathematics, Art, Technology and Cinema Michele Emmer, University of Rome 'La Sapienza', Italy; and Mirella Manaresi, University of Bologna, Italy (Eds.)

This book is about mathematics. But also about art, technology and images. And above all, about cinema, which in the past years, together with theater, has discovered mathematics and mathe­maticians. The authors argue that the discussion about the differences between the so-called rwo cultures of science and humanism is a thing of the past. They hold that both cultures are truly

L--"""'-'li..o..-.....J linked through ideas and creativity, not only through technology. In doing so, they succeed in reaching out to non-mathematicians, and those who are not particularly fond of mathematics. An insightful book for mathematicians, film lovers, those who feel passionate about images, and those with a questioning mind.

2003/242 PP./HARDCOVER/$99.00/ISBN 3-540-00601 -X

EASY WAYS TO ORDER: CALL Toll-Free 1 -800-SPRINGER • WEB www.springcr-ny.com E-MAIL order>(!l'spnnger-ny.com • WRITE to Spri nger-Verlag New York, Inc., Order Dept. 57805, PO Box 2485, Secaucus, NJ 07096-2485 V1SJT your local sCientific bookstore or urge your librarian to order for your department. Pnce� subject to change wtthour notice. Please mention 57805 when ordering to guarantee listed prices.

PromotiOn S7H05

© 2005 Spnnger Setence+Bus1ness Med1a. Inc . Volume 27. Number 1, 2005 71

D avid E . R owe , E d i t o r I

H i l bert's Early Career: Encounters with A l l ies and Rivals David E. Rowe

Send submissions to David E. Rowe,

Fachbereich 1 7 -Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

It seems to me that the mathemati­cians of today understand each other far too little and that they do not take an intense enough interest in one an­other. They also seem to know-so far as I can judge-too little of our classical authors (Klassiker ); many, moreover, spend much effort working on dead ends. -David Hilbert to Felix Klein, 24 July 1890

Probably no mathematician has been quoted more often than

Hilbert, whose opinions and witty re­marks long ago entered mathematical lore along with his legendary feats. Fame gave him a captive audience, but as the opening quotation illustrates, even before he attained that fame Hilbert had no difficulty expressing his views. When he wrote those words, in fact, he had just completed [Hilbert 1890], the first in an impressive string of achievements that would vault him to the top of his profession.

Initially, he made his name as an ex­pert on invariant theory, but Hilbert's reputation as a universal mathemati­cian grew as he left his mark on one field after another. Yet these achieve­ments alone cannot account for his sin­gular place in the history of mathe­matics, as was recognized long ago by his intimates ([Weyl 1932], [Blumen­thal 1935]). Those who belonged to Hilbert's inner circle during his first two decades in Gottingen pointed to the impact of his personality, which clearly transcended the ideas found between the covers of his collected works (see [Weyl 1944], [Reid 1970]).

Hilbert's name became attached to thoughts of fame in the minds of many young mathematicians who felt in­spired to tackle one of the twenty-three "Hilbert problems." Some of these he had merely dusted off and presented

anew at the Paris ICM in 1900, but they then acquired a special fascination. As

Ben Yandell puts it in his delightful sur­vey, The Honors Class, "solving one of Hilbert's problems has been the ro­mantic dream of many a mathemati­cian" [Yandell 2002, 3] .1

Hilbert's ability to inspire was clearly central to Gottingen's success, even if only a part. His leadership style fostered what I have characterized as a new type of oral culture, a highly competitive mathematical community in which the spoken word often carried more weight than the information con­veyed in written texts (see [Rowe 2003b] , [Rowe 2004 ]). Hilbert was an unusually social creature: outspoken, ambitious, eccentric, and above all full of passion for his calling. Moreover, he was a man of action with no patience for hollow words.

Thus, when in July 1890 he con­veyed the rather harsh views cited in the opening quotation to Klein, he was not merely bemoaning the lack of com­munal camaraderie among Germany's mathematicians; he was expressing his hope that these circumstances would soon change. At that time plans were underway to found a national organi­zation of German mathematicians, the Deutsche Mathematiker-Vereinigung, and Hilbert was delighted to learn that Klein would be present for the inau­gural meeting, which would take place a few months later in Bremen. Both knew that much was at stake; as Hilbert expressed it, "I believe that closer personal contact between math­ematicians would, in fact, be very de­sirable for our science" (Hilbert to Klein, 24 July 1890 [Frei 1985, 68]). Soon after his arrival in Gottingen in 1895, Hilbert put this philosophy into practice. At the same time, his opti­mism and self-confidence spilled over and inspired nearly all the young peo­ple who entered his circle.

1 1t was Hilbert's star pupil, Hermann Weyl, who called those who actually succeeded the "honors class"; he

also wrote that "no mathematician of equal stature has risen from our generation" [Weyl 1 944, 1 30].

72 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Science+ Busrness Medra, Inc.

Hilbert's impact on modem mathe­matics has been so pervasive that it takes a true leap of historical imagina­tion to picture him as a young man struggling to find his way. Still, many of the seeds of later success were planted in his youth, just as several episodes from his early career have since be­come familiar aspects of the Hilbert leg­end. In [Rowe 2003a] , I describe the quiet early years he spent in Konigs­berg, where he befriended two young mathematicians who influenced him more than any others, Adolf Hurwitz and Hermann Minkowski. By 1885 Hilbert emerged as one of Felix Klein's most promising proteges. In this role, he traveled to Paris to meet the younger generation of French mathematicians, especially Henri Poincare, reporting back all the while to Klein, who avidly awaited news about the Parisian math­ematical scene. Here I pick up the story at the point when Hilbert returned from this first trip to Paris. Afterward he had several important encounters with the leading mathematicians in Germany. These meetings not only shed light on the contexts that motivated his work, they also reveal how he positioned him­self within the fast-changing German mathematical community.

Returning from Paris

After a rather uneventful and disap­pointing stay in Paris during the spring of 1886, Hilbert began the long journey

Fig. 1 . Hilbert in the days when he was re­

garded as merely one of many experts on the

theory of invariants.

home. Stopping first in Gottingen, he learned something about the contem­porary Berlin scene when he met with Hermann Amandus Schwarz, the se­nior mathematician on the faculty. Schwarz had long been one of the clos­est of Karl Weierstrass's many adoring pupils in Berlin; yet much had changed since the days when Charles Hermite advised young Gosta Mittag-Leffler to leave Paris and go to hear the lectures of Weierstrass, "the master of us all" (see [Rowe 1998]). During the 1860s and 1870s, the Berliners had dominated mathematics throughout Prussia, with the single exception of Konigsberg, which remained an enclave for those with close ties to the Clebsch school and its journal, Mathematische An­nalen (see [Rowe 2000]). However, af­ter E. E. Kummer's departure in 1883, the harmonious atmosphere he had cultivated as Berlin's senior mathe­matician gave way to acrimony. Weier­strass, old, frail, and decrepit, refused to retire for fear of losing all influence to Leopold Kronecker, who remained amazingly energetic and prolific de­spite his more than sixty years.

Presumably Hilbert heard about this situation from Schwarz, who would have conveyed the essence of the situ­ation from Weierstrass's perspective (see [Biermann 1988, 137-139]). If so, Hilbert would have heard how rela­tions between Weierstrass and Kro­necker had deteriorated mainly be­cause of the latter's dogmatic views, in particular his sharp criticism of Weier­strass's approach to the foundations of analysis. Only a few months after Hilbert passed through Berlin, Kro­necker delivered a speech in which he uttered his most famous phrase "God made the natural numbers; all else is the work of man" ("Die ganzen Zahlen hat der liebe Gott gemacht, alles an­dere ist Menschenwerk") [Weber 1893, 19]. Kronecker had never made a se­cret of his views on foundations, but by the mid-1880s he was propounding these with missionary zeal. No one was more taken aback by this than H. A. Schwarz, to whom Kronecker had writ­ten one year earlier:

If enough years and power remain, I will show the mathematical world

that not only geometry but also arithmetic can point the way for analysis-and certainly with more rigor. If l don't do it, then those who come after me will, and they will also recognize the invalidity of all the procedures with which the so­called analysis now operates [Bier­mann 1988, 138].

Weierstrass had written to Schwarz at around that time, claiming that Kro­necker had transferred his former an­tipathy for Georg Cantor's work to his own. And in another letter, written to Sofia Kovalevskaya, he characterized the issue dividing them as rooted in mathematical ontology: "whereas I as­sert that a so-called irrational number has a real existence like anything else in the world of thought, according to Kronecker it is an axiom that there are only equations between whole num­bers" (Weierstrass to Kovalevskaya, 24 March, 1885, quoted in [Biermann 1988, 137]). Whether or not Schwarz alluded to this rivalry when he spoke to Hilbert in 1886, he definitely did warn him to expect a cold reception when he pre­sented himself to Kronecker (Hilbert to Klein, 9 July 1886, in [Frei 1985, 15]). Instead, however, Hilbert was greeted in Berlin with open arms, and his ini­tial reaction to Kronecker was for the most part positive.

Back in his native Konigsberg, Hilbert reported to the ever-curious Klein about these and other recent events. He had just completed all re­quirements for the Habilitation except for the last, an inaugural lecture to be delivered in the main auditorium of the Albertina. His chosen theme was a propitious one: recent progress in the theory of invariants. Hilbert was pleased to be back in Konigsberg as a Privatdozent, even though this meant he was far removed from mathemati­cians at other German universities. To compensate for this isolation, he was planning to tour various mathematical centers the following year, when he hoped to meet Professors Gordan and N oether in Erlangen. Although he had to postpone that trip until the spring of 1888, it eventually proved far more fruitful than his earlier journey to Paris. What is more, it helped him so-

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc Volume 27, Number 1 2005 73

Fig. 2. Leopold Kronecker emerged as

Berlin's leading mathematician during the

1880s when his outspoken views caused con­

siderable controversy.

lidify his relationship with Klein, who always urged young mathematicians to cultivate personal contacts with fellow researchers both at home and abroad (see for example [Hashagen 2003, 105-116, 149-162]). Within Klein's net­work, the Erlangen mathematicians, Paul Gordan and Max Noether, played particularly important roles. The latter was Germany's foremost algebraic geometer in the tradition of Alfred Clebsch, and the former was an old­fashioned algorist who loved to talk mathematics.

Felix Klein knew from first-hand ex­perience how stimulating collabora­tion with Paul Gordan could be. Dur­ing the late 1870s, when Klein taught at the Technical University in Munich, he took advantage of every opportunity to meet with his erstwhile Erlangen col­league, who was himself enormously impressed by Klein's fertile geometric imagination. Gordan was widely re­garded as Germany's authority on al­gebraic invariant theory, the field that would dominate Hilbert's attention for the next five years. His principal claim to fame was Gordan's Theorem, which he proved in 1868. This states that the complete system of invariants of a bi­nary form can always be expressed in terms of a finite number of such in-

74 THE MATHEMATICAL INTELLIGENCER

variants. In 1856 Arthur Cayley had proved this for binary forms of degree 3 and 4, but Gordan was able to use the symbolical method introduced by Siegfried Aronhold to obtain the gen­eral result. During his stay in Paris, Hilbert had briefly reported to Klein about these matters (Hilbert to Klein, 21 April 1886, in [Frei 1985, 9]). There he learned from Charles Hermite about J. J. Sylvester's recent efforts to prove Gordan's Theorem using the original British techniques he and Cayley had developed. Hilbert thus became aware that the elderly Sylvester was still trying to get back into this race (see [Parshall 1989]). Presumably Hilbert thought that progress was unlikely to come from this old-fashioned line of at­tack, but neither had the symbolical methods employed by German alge­braists produced any substantial new results since Gordan first unveiled his theorem.

Over the next two years Hilbert had ample time to master the various com­peting techniques. Mter becoming a

Privatdozent in Konigsberg, he was free to develop his own research pro­gram, and his inaugural lecture on re­cent research in invariant theory clearly indicates the general direction in which he was moving. Still, there are no signs that he was on the path to­ward a major breakthrough. Indeed, tucked away in Konigsberg, it seems unlikely that he even saw the need to strike out in an entirely new direction in order to make progress beyond Gor­dan's original finiteness theorem. That goal, nevertheless, was clearly in the back of his mind when he set off in March 1888 on a tour of several lead­ing mathematical centers in Germany, including Berlin, Leipzig, and Gottin­gen. During the course of a month, he spoke with some twenty mathemati­cians from whom he gained a stimu­lating overview of current research interests throughout the country. Al­though we can only capture glimpses of these encounters, a number of im­pressions can be gained from Hilbert's letters to Klein, as well as from the

Felix Christian Klein

Hochzeitsbilder 1 875

Fig. 3. On leaving Erlangen for the Technische Hochschule in Munich in 1875, Felix Klein mar­

ried Anna Hegel, a granddaughter of the famous philosopher.

notes he took of his conversations with various colleagues. 2

A Second Encounter

with Kronecker

In Berlin, Hilbert met once again with Kronecker, who on two separate occa­sions afforded him a lengthy account of his general views on mathematics and much else related to Hilbert's own research. A gregarious, outspoken man, the elder Kronecker still exuded intensity, so Hilbert learned a great deal from him during the four hours they spent together. Reporting to Klein, he described the Berlin mathemati­cian's opinions as "original, if also somewhat derogatory" (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Hilbert told Kronecker about a paper he had just written on certain positive definite forms that cannot be repre­sented as a sum of squares. Kronecker replied that he, too, had encountered forms that cannot be so represented, but he admitted that he did not know Hilbert's main theorem, which dealt with the three cases in which a sum of squares representation is, indeed, al­ways possible ("Bericht iiber meine Reise," Hilbert Nachlass 741).

A noteworthy feature of this paper, [Hilbert 1888a], is that Hilbert actually credits Kronecker with having intro­duced the general principle behind his investigation. This work lies at the root of Hilbert's seventeenth Paris problem, which also played an important role in Hilbert's research on foundations of geometry. Interestingly, a second Hilbert problem, the sixteenth, also crept into his conversations with Kro­necker. It concerns the possible topo­logical configurations among the com­ponents of a real algebraic curve. The maximal number of such components had been established by Axel Harnack, a student of Klein's, in a celebrated the­orem from 1876 (for a summary of sub­sequent results, see [Yandell 2002, 276-278]). Kronecker assured Hilbert that his own theory of characteristics, as presented in a paper from 1878, enabled one to answer all questions of this type, clearly an overly optimistic assessment.

Whatever he may have thought about Kronecker's "priority claims," Hilbert stood up and took notice when his host voiced some sharp views about the significance of invariant the­ory. Kronecker dismissed the whole theory of formal invariants as a topic that would die out just as surely as had happened with Hindenburg's combina­torial school (which had flourished in Leipzig at the beginning of the nine­teenth century, but by the 1880s had entered the dustbin of history). The only true invariants, in Kronecker's view, were not the "literal" ones based on algebraic forms, but rather num­bers, such as the signature of a qua-

The on ly true

invariants , i n

Kronecker' s view,

were numbers ,

such as the

s ignature of a

quadratic form . dratic form (Sylvester's theorem, the algebraist's version of conservation of inertia). He then proceeded to wax forth over foundational issues, begin­ning with the assertion that "equality" only has meaning in relation to whole numbers and ratios of whole numbers. Everything beyond this, all irrational quantities, must be represented either implicitly by a finite formula (e.g., x2 = 5), or by means of approximations. Us­ing these notions, he told Hilbert, one can establish a firm foundation for analysis that avoided the Weierstrass­ian notions of equality and continuity. He further decried the confusion that so often resulted when mathemati­cians treated the implicitly given irra­tional quantity (say, x = v5) as equiv­alent to some sequence of rational numbers that serve as an approxima­tion for it.

Not surprisingly, Hilbert took fairly

extensive notes when Kronecker be­gan expounding these unorthodox views ("Bericht iiber meine Reise"). But he also jotted down a brief com­ment made by Weierstrass that sheds considerable light on the differences between these two mathematical per­sonalities. When Hilbert visited Weier­strass shortly afterward, he informed him of Kronecker's comments regard­ing invariant theory, including the pre­diction that the whole field would soon be forgotten, like the work of the Leipzig combinatorial school. Weier­strass responded by sounding a gentle warning to those who might wish to prophesy the future of a mathematical theory: "Not everything of the combi­natorial school has perished," he said, "and much of invariant theory will pass away, too, but not from it alone. For from everything the essential must first gradually crystallize, and it is neither possible nor is it our duty to decide in advance what is significant; nor should such considerations cause us to demur in investigating such invariant-theo­retic questions deeply" ("Bericht tiber meine Reise").

These words, with their almost fa­talistic ring, probably left little impres­sion on the young mathematician who recorded them. For Hilbert's intellec­tual outlook was filled with a buoyant optimism that left no room for resig­nation. He may not have enjoyed Kro­necker's braggadocio, but he was clearly far more receptive to his pas­sionate vision than to Weierstrass's more subdued outlook. Moreover, mathematically he was far closer to the algebraist than to the analyst. Even in his later work in analysis, Hilbert showed that his principal strength as a mathematician stemmed from his mas­tery of the techniques of higher algebra (see [Toeplitz 1922]). True, Klein and Hurwitz had drawn his attention to Weierstrass's theory of periodic com­plex-valued functions, about which he spoke in his Habilitationsvortrag shortly after returning to Konigsberg from Paris. Nevertheless, Kronecker's algebraic researches lay much closer to his heart. Soon after their encounter

2"Bericht uber meine Reise vom 9ten Marz bis ?ten April 1 888," Hilbert Nachlass 7 41 , Handschnftenabteilung, Niedersachs1sche Staats· und Umversitatsbibliothek Gbt·

tingen.

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1, 2005 75

in Berlin, Hilbert would enter Leopold theory important? Probably Hilbert Kronecker's principal research do­main, the theory of algebraic number fields. The latter's sudden death in 1891 may well have emboldened Hilbert to reconstruct this entire theory six years later in his "Zahlbericht." As Hermann Weyl later emphasized, Hilbert's am­bivalence with respect to Kronecker's legacy emerged as a major theme throughout his career [Weyl 1944, 613].

Like Hilbert, his two closest mathe­matical friends, Hurwitz and Minkowski, also held an1bivalent views when it came to Kronecker. No doubt these were colored by their mutual desire to step beyond the lengthy shadows that Kronecker and Richard Dedekind, the other leading algebraist of the older generation, had cast. Since Dedekind had long since withdrawn to his native Brunswick, a city well off the beaten path, it was only natural that the Konigsberg trio came to regard the powerful and opinionated Berlin math­ematician as their single most imposing rival. In later years, Hilbert developed a deep antipathy toward Kronecker's philosophical views, and he did not hes­itate to criticize these before public au­diences (see [Hilbert 1922]). Yet during the early stages of his career such mis­givings-if he had any-remained very much in the background. Indeed, all of Hilbert's work on invariant theory was deeply influenced by Kronecker's ap­proach to algebra.

Hilbert's encounters in the spring of 1888 with Berlin's two senior mathe­maticians left a deep and lasting im­pression. 3 Based on the notes he took of these conversations, he must have felt particularly aroused by Kro­necker's critical views with regard to invariant theory, for he surely found no solace in Weierstrass's stoic advice. Primed for action and out to conquer, Hilbert could never have contemplated devoting his whole life to a theory that might later be judged as having no in­trinsic significance. Whatever prob­lems he chose to work on-even those he merely thought about but never tried to solve-he always thought of them as constituting important mathe­matics. What makes a problem or a

carried that question within him for a long time, though anyone familiar with his career knows the answer he even­tually came up with; one need only reread his famous Paris address to see how compelling his views on the sig­nificance of mathematical thought could be. From the vantage point of these early, still formative years, we can begin to picture how his larger views about the character and signifi­cance of mathematical ideas fell into place. A few strands of the story emerge from the discussions he en­gaged in during this whirlwind 1888 tour through leading outposts of the German mathematical community.

Tackling Gordan's Problem

From Berlin, Hilbert went on to Leipzig, where he finally got the chance to meet face-to-face with Paul Gordan, who came from Erlangen. Despite their mathematical differences, the two hit

I t is neither

possi ble nor is

it our duty to

decide i n advance

what is s ign ificant . it off splendidly, as both loved nothing more than to talk about mathematics. Hermann Weyl once described Gordan as "a queer fellow, impulsive and one­sided," with "something of the air of the eternal 'Bursche' of the 1848 type about him-an air of dressing gown, beer and tobacco, relieved however by a keen sense of humor and a strong dash of wit . . . . A great walker and talker-he liked that kind of walk to which fre­quent stops at a beer-garden or a cafe belong" [Weyl 1935, 203]. Having heard about Hilbert's talents, Gordan longed to make the young man's acquaintance, so much so that he wished to remain incognito while in Leipzig to take full advantage of the opportunity (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]).

Although originally an expert on

3He recalled this trip when he spoke about his life on his seventieth birthday; see [Reid 1 970, 202].

76 THE MATHEMATICAL INTELLIGENCER

Fig. 4. Paul Gordan joined Klein on the Er­

langen faculty in 1874 and remained there un­

til his death in 191 2. His star student was

Emmy Noether, daughter of Gordan's col­

league, Max Noether.

Abelian functions, Gordan had long since focused his attention exclusively on the theory of algebraic invariants. This field traces back to a fundamen­tal paper published by George Boole in 1841, as it was this work that inspired young Arthur Cayley to take up the topic in earnest [Parshall 1989, 160-166]. Following an initial plunge into the field, Cayley joined forces with another professional lawyer who be­came his life-long friend, J. J. Sylvester. Together, they effectively launched in­variant theory as a specialized field of research. Much of its standard termi­nology was introduced by Sylvester in a major paper from 1853. Thus, for a given binary form f(x,y), a homoge­neous polynomial J in the coefficients off left fixed by all linear substitutions (up to a fixed power of the determinant of the substitution) is called an invari­ant of the form f In 1868 Gordan showed that for any binary form, one can always construct m invariants h, /z, . . . , Im such that every other in­variant can be expressed in terms of these m basis elements. Indeed, he proved that this held generally for ho­mogeneous polynomials J in the coef­ficients and variables ofj(x,y) with the same invariance property (Sylvester called such an expression J a con­comitant of the given form, but the term covariant soon became stan­dard).

In 1856 Cayley published the first

finiteness results for binary forms, but

in the course of doing so he committed

a major blunder by arguing that the

number of irreducible invariants was

necessarily infinite for forms of degree

five and higher [Parshall 1989, 167-179].

Paul Gordan was the first to show that

Cayley's conclusion was incorrect.

More importantly, in the course of do­

ing so he proved his finite basis theo­

rem for binary forms of arbitrary de­

gree by showing how to construct a

complete system of invariants and co­

variants. Two years later, he was able

to extend this result to any finite sys­

tem of binary forms. His proofs of

these key theorems, as later presented

in [Gordan 1885/1887], were purely al-

Fig. 5. Otto Blumenthal, Hilbert's first biog­

rapher, alluded to the critical meeting when

Hilbert and Gordan first met.

gebraic and constructive in nature.

They were also impressively compli­

cated, so that subsequent attempts, in­

cluding Gordan's own, to extend his

theorem to ternary forms had pro­

duced only rather meager results.

Little evidence has survived relating

to Hilbert's first encounter with Gor­

dan, but it is enough to reconstruct a

plausible picture of what occurred.

Gordan may have been a fairly old dog,

but this does not mean he was averse

to learning some new tricks. Even

though he and Hilbert had divergent

views about many things, they never­

theless understood each other well

(Hilbert to Klein, 16 March 1888 [Frei

1985, 38]). Their conversations soon fo­

cused on finiteness results, in particu­

lar a fairly recent proof of Gordan's

finiteness theorem for systems of bi­

nary forms published by Franz Mertens

in Grelle's Journal [Mertens 1887]. This

paper broke new ground. For unlike

Gordan's proof, which was based on

the symbolic calculus of Clebsch and

Aronhold, Mertens's proof was not strictly constructive. Gordan and

Hilbert apparently discussed it in con­

siderable detail, and Hilbert immedi­

ately set about trying to improve

Mertens's proof, which employed a

rather complicated induction argu­

ment on the degree of the forms. After

spending a good week with Gordan, he

was delighted to report to Klein that

"with the stimulating help of Prof. Gor­

dan an infinite series of thought vibra­

tions has been generated within me,

and in particular, so we believe, I have

a wonderfully short and pointed proof

for the finiteness of binary systems of

forms" (Hilbert to Klein, 2 1 March 1888

[Frei 1985, 39]) .

Hilbert had caught fire. A week

later, when he met with Klein in Got­

tingen, he had already put the finishing

touches on the new, streamlined proof.

This paper [Hilbert 1888b] was the first

in a landslide of contributions to alge­

braic invariant theory that would tum

the subject upside down. Between 1888

and 1890 Hilbert pursued this theme re­

lentlessly, but with a new methodolog­

ical twist which he combined with the

formal algorithmic techniques em­

ployed by Gordan. Beginning with three

short notes sent to Klein for publication

in the Gottinger Nachrichten [Hilbert

1888c] , [Hilbert 1889a] , [Hilbert 1889b] ,

he began to unveil general methods for

proving finiteness relations for general

systems of algebraic forms, invariants

being only a quite special case, though

the one of principal interest. With these

general methods, combined with the al­

gorithmic techniques developed by his

predecessors, Hilbert was able to ex­

tend Gordan's finiteness theorem from

systems of binary forms over the real

or complex numbers to forms in any

number of variables and with coeffi­

cients in an arbitrary field.

By the time this first flurry of activ­

ity came to an end, Hilbert had shown

how these finiteness theorems for in­

variant theory could be derived from

general properties of systems of alge­

braic forms. Writing to Klein in 1890,

he described his culminating paper

[Hilbert 1890] as a unified approach to

a whole series of algebraic problems

(Hilbert to Klein, 15 February 1890

[Frei 1985, 61]) . He might have added

that his techniques borrowed heavily

from Leopold Kronecker's work on al­

gebraic forms. Yet from a broader

methodological standpoint, Hilbert's

approach clearly broke with Kro­

necker's constructive principles. For

Hilbert's foray into the realm of alge­

braic forms revealed the power of pure

existence arguments: he showed that

out of sheer logical necessity a finite

basis must exist for the system of in­

variants associated with any algebraic

form or system of forms.

Hilbert found his way forward by

noticing the following general result,

known today as Hilbert's basis theo­

rem for polynomial ideals. It appears

as Theorem I in [Hilbert 1888c] . It

states that for any sequence of alge­

braic forms in n variables c{J1, c{Jz,

4>3, . . . there exists an index m such

that all the forms of the sequence can

be written in terms of the first m forms,

that is,

cP = CXlcPl + CX2cP2 + . . . + CXmcPm ,

where the ai are appropriate n-ary

forms. Thus, the forms cfJ1, c{Jz, . . . cPm serve as a basis for the entire system.

By appealing to Theorem I and draw­

ing on Mertens's procedure for gener­

ating systems of invariants, Hilbert

© 2005 Spnnger SCience+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005 77

proved that such systems were always finitely generated. There was, how­ever, a small snag.

Hilbert attempted to prove Theorem I by first noting that it held for small n. He then introduced a still more general Theorem II, from which he could prove Theorem I by induction on the number of variables. If this sounds confusing, a number of contemporary readers had a similar reaction, including a few who expressed their misgivings to Hilbert about the validity of his proof. Paul Gordan, however, was not one of them. According to Hilbert, to the best of his recollection, he and Gordan had only discussed the proof of Theorem II dur­ing their meeting in Leipzig (Hilbert to Klein, 3 March 1890 [Frei 1985, 64]). As

it turned out, Hilbert's Theorem II, as originally formulated in [Hilbert 1888c], is false. 4 Moreover, since it was conceived from the beginning as a lemma for the proof of Theorem I, Hilbert dropped Theorem II in his de­fmitive paper [Hilbert 1890] and gave a new proof of Theorem I. Nevertheless, the latter remained controversial, as we shall soon see.

Today we recognize in Hilbert's Theorem I a central fact of ideal theory, namely, that every ideal of a polynomial ring is finitely generated. Thirty years later, Emmy Noether in­corporated Hilbert's Theorem I (from [Hilbert 1890]) as well as his Nullstel­lensatz (from [Hilbert 1893]) into an abstract theory of ideals (see [Gilmer 1981]). Her classic paper "Idealtheorie in Ringbereichen" [Noether 1921 ] is nearly as readable today as it was when she wrote it. The same cannot be said, however, for Hilbert's papers (for En­glish translations, see [Hilbert 1978]). Not that these are badly written; they simply reflect a far less familiar math­ematical context.

In [Hilbert 1889a, 28] Hilbert hinted that much of the inspiration for both the terminology and techniques came from Kronecker's theory of module systems. When he wrote this, he knew very well that Kronecker held very neg­ative views about invariant theory, making it highly improbable that he

4See the editorial note in [Hilbert 1 933. 1 77).

would view Hilbert's adaptation of his ideas with approval. Indeed, Kro­necker had made it plain to Hilbert that, in his view, the only invariants of interest were the numerical invariants associated with systems of algebraic equations. Still, Hilbert quickly recog­nized the fertility of Kronecker's con­ceptions for invariant theory. Ac­knowledging his debt to the Berlin algebraist, he parted company with him by adopting a radically non-con­structive approach. Ironically, the ini­tial impulse to do so apparently came from his conversations with Gordan. Thus, with his early work on invariant theory Hilbert sowed some of the seeds that would eventually flower into his modernist vision for mathematics, thereby preparing the way for the dra­matic foundations debates of the 1920s (see [Hesseling 2003]).

Mathematics as Theology

Kronecker seems to have simply ig­nored Hilbert's dramatic break­through, but others closer to the field of invariant theory obviously could not afford to do so. Paul Gordan, who had initially supported Hilbert's work en­thusiastically, now began to express misgivings about this new and, for him, all too ethereal approach to invariant theory. His views soon made the rounds at the coffee tables and beer gardens, and more or less everyone heard what Gordan probably said on more than one occasion: Hilbert's ap­proach to invariant theory was "theol­ogy not mathematics" [Weyl 1944, 140].5

No doubt many mathematicians got a chuckle out of this epithet at the time, but a serious conflict briefly reared its head in February 1890 when Hilbert submitted his definitive paper [Hilbert 1890] for publication in Mathematis­che Annalen. Klein was overjoyed, and wrote back to Hilbert a day later: "I do not doubt that this is the most impor­tant work on general algebra that the Annalen has ever published" (Klein to Hilbert, 18 Feb. 1890, in [Frei 1985, p. 65]). He then sent the manuscript to Gordan, the Annalen's house expert on

invariant theory, asking him to report on it.

Klein, having already heard some of Gordan's misgivings about Hilbert's methods in private conversations, may well have anticipated a negative reac­tion. He certainly got one. The cantan­kerous Gordan forcefully voiced his objections, aiming directly at Hilbert's presentation of Theorem I, which Gor­dan claimed fell short of even the most modest standards for a mathematical proof. "The problem lies not with the form," he wrote Klein, " . . . but rather lies much deeper. Hilbert has scorned to present his thoughts following for­mal rules; he thinks it suffices that no one contradict his proof, then every­thing will be in order . . . he is content to think that the importance and cor­rectness of his propositions suffice. That might be the case for the first ver­sion, but for a comprehensive work for the Annalen this is insufficient." (Gor­dan to Klein, 24 Feb. 1890, in [Frei 1985, p 65]. Perhaps the misgivings come down to the non-constructivity in­volved in the [implicit] use of the Ax­iom of Choice. Concise modem proofs like [Caruth 1996] put the latter clearly in evidence.)

Klein forwarded Gordan's report to Hurwitz in Konigsberg, who then dis­cussed its contents with Hilbert. After that the sparks really began to fly. Clearly irked by Gordan's refusal to recognize the soundness of his argu­ments, Hilbert promptly dashed off a fierce rebuttal to Klein. He began by reminding him that Theorem I was by no means new; he had, in fact, come up with it some eighteen months ear­lier and had afterward published a first proof in the Gottinger Nachrichten [Hilbert 1888c]. He then proceeded to describe the events that had prompted him to give a new proof in the manu­script now under scrutiny.

This came about after he had spo­ken with numerous mathematicians about his key theorem; he had also car­ried on correspondence with Cayley and Eugen Netto, who wanted him to clarify certain points in the proof. Tak­ing these various reactions into ac-

5The earliest reference to Gordan's remark- ''Das ist keine Mathematik, das ist Theologie"-appears to be [Blumenthal 1 935, 394).

78 THE MATHEMATICAL INTELLIGENCER

count, Hilbert had prepared a revised

proof, which he had tested out in his

lecture course the previous semester.

Afterward he spoke with one of the au­

ditors in order to convince himself that

the argument as presented had actually

been understood. Having reassured

himself that this new proof was indeed

clear and understandable, he wrote it

up for [Hilbert 1890]. Hilbert then con­

cluded this recitation of the relevant

prehistory by saying that these facts

clearly refuted the ad hominem side of

Gordan's attack, namely his insinua­

tion that Hilbert's new proof of Theo­

rem I was not meant to be understood

and that he was content so long as no

one could contradict the argument.

Regarding what he took to be the

substantive part of Gordan's critique,

Hilbert stated that this consisted

mainly of "a series of very commend­

able, but completely general rules for

the composition of mathematical pa­

pers" (Hilbert to Klein, 3 March 1890

[Frei 1985, 64]). The only specific crit­

icisms Gordan made were, in Hilbert's

opinion, plainly incomprehensible: "If

Professor Gordan succeeds in proving

my Theorem I by means of an 'order­

ing of all forms' and by passing from

'simpler to more complicated forms,'

then this would just be another proof,

and I would be pleased if this proof

were simpler than mine, provided that

each individual step is as compelling

and as tightly fastened" (ibid.). Hilbert

then ended this remarkable repartee with an implied threat: either his paper

would be printed just as he wrote it or

he would withdraw his manuscript

from publication in the Annalen. "I am

not prepared," he intoned, "to alter or

delete anything, and regarding this pa­

per, I say with all modesty, that this is

my last word so long as no definite and

irrefutable objection against my rea­

soning is raised" (ibid.). Certainly Klein was not accustomed

to receiving letters like this one, and

especially from young Privatdozenten. Yet however impressed he may have

been by Hilbert's self-assurance and

pluck, he also wanted to preserve his

longstanding alliance with Gordan.

Moreover, in view of his older friend's

irascibility, Klein knew, he had to han­

dle the squabble delicately before it be-

came a full-blown crisis. Hilbert re­

ceived no immediate reply, as Klein

wanted to wait until he could confer

with Gordan personally. Over a month

passed, with no word from Gottingen

about the fate of a paper that Klein had

originally characterized as one of the

most important ever to appear in the

pages of Die Mathematische Annalen. Then, in early April, Gordan came to

Gottingen to "negotiate" with Klein

about these matters, which clearly

weighed heavily on the Erlangen math­

ematician's heart. To facilitate the

process, Klein asked Hurwitz to join

them, knowing that Hilbert's trusted

friend would do his best to help restore

harmony.

Gordan spent eight days in Gottin­

gen, following which Klein wrote

Hilbert a brief letter summarizing the

results of their "negotiations." He be­

gan by reassuring him that Gordan's

opinions were by no means as uni­

formly negative as Hilbert had as­

sumed. "His general opinion,'' Klein

noted, "is entirely respectful, and

would exceed your every wish" (Klein

to Hilbert, 14 April 1890, in [Frei 1985,

p. 66]) . To this he merely added that

Hurwitz would be able to tell him more

about the results of their meeting. But

then he attached a postscript that con­

tained the message Hilbert had been

waiting to hear: Gordan's criticisms

would have no bearing on the present

paper and should be construed merely

as guidelines for future work!

Thus, Hilbert got what he de­

manded; his decisive paper appeared in

the Annalen exactly as he had written

it. Gordan surely lost face, but at least

he had been given the opportunity to

vent his views. In short, Klein's diplo­

matic maneuvering carried the day.

Gordan knew, of course, that he was

dealing with someone who had little

patience for methodological nit-pick­

ing. He also knew that Klein consis­

tently valued youthful vitality over age

and experience. Hilbert represented

the wave of the future, and while this

conflict, in and of itself, had no imme­

diate ramifications, it foreshadowed a

highly significant restructuring of the

power constellations that had domi­

nated German mathematics since the

late 1860s.

A Final Tour de Force

If Hilbert was scornful of Gordan's ed­

itorial pronouncements, this does not

mean that he failed to see the larger is­

sue at stake. His general basis theorem

proved that for algebraic forms in any

number of variables there always ex­

ists a finite collection of irreducible in­

variants, but his methods of proof were

of no help when it actually came to

constructing such a basis. Hilbert ob­

viously realized that if he could de­

velop a new proof based on arguments

that were, in principle, constructive in

nature, then this would completely vi­

tiate Gordan's criticisms. Two years

later, he unveiled just such an argu­

ment, one that he had in fact been seek­

ing for a long time. In an elated letter

to Klein, he described this latest break­

through, which allowed him to bypass

the controversial Theorem I com­

pletely. He further noted that although

this route to his finiteness theorems

was more complicated, it carried a ma­

jor new payoff, namely "the determi­

nation of an upper bound for the de­

gree and weights of the invariants of a

basis system" (Hilbert to Klein, 5 Jan­

uary 1892 [Frei 1985, 77]) .

When Hermann Minkowski, who was

then in Bonn, heard about Hilbert's lat­

est triumph, he fired off a witty letter

congratulating his friend back in Konigs­

berg:

I had long ago thought that it could

only be a matter of time before you

finished off the old invariant theory

to the point where there would

hardly be an i left to dot. But it re­

ally gives me joy that it all went so

quickly and that everything was so

surprisingly simple, and I congratu­

late you on your success. Now that

you've even discovered smokeless

gunpowder with your last theorem,

after Theorem I caused only Gor­

dan's eyes to sting anymore, it really

is a good time to decimate the

fortresses of the robber-knights [i.e.,

specialists in invariant theory]­

[Georg Emil] Stroh, Gordan, [Ky­

parisos] Stephanos, and whoever

they all are-who held up the indi­

vidual traveling invariants and

locked them in their dungeons, as

there is a danger that new life will

© 2005 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 27, Number 1 . 2005 79

never sprout from these ruins again.

[Minkowski 1973, 45].

Minkowski's opinions were a constant

source of inspiration for Hilbert, so he

probably took these remarks to heart.

Indeed, this letter may well mark the

beginning of one of the most enduring

of all myths associated with Hilbert's

exploits, namely that he single-hand­

edly killed off the till then flourishing

field of invariant theory. As Hans

Freudenthal later put it: "never has a

blooming mathematical theory with­

ered away so suddenly" [Freudenthal

1971, 389].

Hilbert published his new results in

another triad of papers for the Gdt­tinger Nachrichten ([Hilbert 1891 ] ,

[Hilbert 1892a], [Hilbert 1892b ] ) . Nine

months later he completed the manu­

script of his second classic paper on in­

variant theory [Hilbert 1893].6 He sent

this along with a diplomatically worded

letter to Klein, noting that he had taken

pains to ensure that the presentation

followed the general guidelines Prof.

Gordan had recommended (Hilbert to

Klein, 29 September 1892 [Frei 1985,

85]). Then, in a short postscript, Hilbert

added: "I have read and thought through

the manuscript carefully again, and must

confess that I am very satisfied with this

paper" (ibid.). Klein reassured him that "Gordan

had made his peace with the newest

developments," and emphasized that

doing so "wasn't easy for him, and for

that reason should be seen as much to

his credit" (Klein to Hilbert, 7 January

1893, in [Frei 1985, p. 86]). As evidence

of Gordan's change of heart, Klein men­

tioned his forthcoming paper entitled

simply "Ober einen Satz von Hilbert"

[Gordan 1892]. The Satz in question was,

of course, Hilbert's Theorem I, which

really had caused Gordan's eyes to

sting, but not because he doubted its

validity. Nor did he ever doubt that

Hilbert's proof was correct; it was

simply incomprehensible in Gordan's

opinion. As he put it to Klein back in

1890: "I can only learn something that

is as clear to me as the rules of the mul­

tiplication table" (Gordan to Klein, 24

Feb. 1890, in [Frei 1985, p 65]).

Hilbert had claimed that he would

welcome a simpler proof of Theorem I

from Gordan, and here the elderly al­

gorist delivered in a gracious manner.

He began by characterizing Hilbert's

proof as "entirely correct" [Gordan

1892, 132] , but went on to say that he

had nevertheless noticed a gap, in that

Hilbert's argument merely proved the

existence of a finite basis without ex­

amining the properties of the basis el­

ements. He further noted that his own

proof relied essentially on Hilbert's

strategy of applying the ideas of Kro­

necker, Dedekind, and Weber to in­

variant theory [Gordan 1892, 133].

Probably only a few of those who saw

this conciliatory contribution by the

"King of Invariants" were aware of the

earlier maneuvering that had taken

place behind the scenes. Nor were

many likely to have anticipated that

Gordan's throne would soon resemble

a museum piece. 7 Not surprisingly, Hilbert put method­

ological issues at the very forefront of

[Hilbert 1893], his final contribution to

invariant theory. Here he called atten­

tion to the fact that his earlier results

failed to give any idea of how a finite

basis for a system of invariants could

actually be constructed. Moreover, he

noted that these methods could not

even help in finding an upper bound for

the number of such invariants for a

given form or system of forms [Hil­

bert 1933, 319]. To show how these

drawbacks could be overcome, Hilbert

adopted an even more general ap­

proach than the one he had taken be­

fore. He described the guiding idea of

this culminating paper as invariant the­

ory treated merely as a special case of

the general theory of algebraic func­

tion fields. This viewpoint was inspired

to a considerable extent by the earlier

work of Kronecker and Dedekind, al­

though Hilbert mentioned this connec­

tion only obliquely in the introduction,

where he underscored the close anal-

6For an English translation of this and other works by Hilbert. see [Hilbert 1 978].

ogy with algebraic number fields

[Hilbert 1933, 287] .

Hilbert's introduction also contains

other interesting features. In it, he set

down five fundamental principles

which could serve as the foundations

of invariant theory. The first four of

these he regarded as the "elementary

propositions of invariant theory,"

whereas the existence of a finite basis

(or in Hilbert's terminology a "full in­

variant system") constituted the fifth

principle. This highly abstract formu­

lation would, of course, later come to

typify much of Hilbert's work in nearly

all branches of mathematics. Indeed,

the only thing missing from what was

to become standard Hilbertian jargon

was an explicit appeal to the axiomatic

method. Immediately after presenting

these five propositions, he wrote that

they "prompt the question, which of

these properties are conditioned by the

others and which can stand apart from

one another in a function system." He

then mentioned an example that demon­

strated the independence of property 4

from properties 2, 3, and 5. These fmd­

ings were incidental to the main thrust

of Hilbert's paper, but they reveal how

axiomatic ideas had already entered into

his early work on algebra. 8 On September 1892, the day he sent

off the manuscript of [Hilbert 1893],

Hilbert wrote to Minkowski: "I shall

now definitely leave the field of invari­

ants and tum to number theory" [Blu­

menthal 1935, 395]. This transition was

a natural one, given that his final work

on invariant theory was essentially an

application of concepts from the the­

ory of algebraic number fields. One year

later, Hilbert and Minkowski were

charged with the task of writing a re­

port on number theory to be published

by the Deutsche Mathematiker-Vereini­

gung. Minkowski eventually dropped

out of the project, but he continued to

offer his friend advice as Hilbert strug­

gled with his most ambitious single

work, "Die Theorie der algebraischen

Zahlkorper," better known simply as the

Zahlbericht. 9

7Gordan later presented a streamlined proof of Hilbert's Theorem I in a lecture at the 1 899 meet1ng of the DMV in Munich. Hilbert was present on that occasion (see

Jahresbericht der Deutschen Mathematiker-Vereinigung 8(1 899), 1 80. Gordan wrote up this proof soon thereafter for [Gordan 1 899] .

8For a detailed examination of Hilbert's work on the axiomatization of physics, see [Corry 2004].

80 THE MATHEMATICAL INTELLIGENCER

Killing off a Mathematical Theory

Thus by 1893 Hilbert's active involve­ment with invariant theory had ended. In that year he wrote the survey article [Hilbert 1896] in response to a request from Felix Klein, who presented it along with several other papers at the Mathematical Congress held in Chicago in 1893 as part of the World's Columbian Exposition. Hilbert's ac­count offers an interesting partici­pant's history of the classical theory of invariants. At the time he wrote it, in­variant theory was a staple research field within the fledgling mathematical community in the United States, which first began to spread its wings under the tutelage of J. J. Sylvester at Johns Hopkins (see [Parshall and Rowe 1994]). Hilbert briefly alluded to the contributions of Cayley and Sylvester in his brief survey, describing these as characteristic for the "naive period" in the history of a special field like alge­braic invariant theory. This stage, he added, was soon superseded by a "for­mal period," whose leading figures were his own direct predecessors, Al­fred Clebsch and Paul Gordan. A ma­ture mathematical theory, Hilbert went on, typically culminates in a third, "crit­ical period," and his account made it clear that he alone was to be regarded as having inaugurated this stage.

What better time to quit the field? Hilbert realized very well that many as­pects of invariant theory had only be­gun to unfold, but after 1893 he was content to point others in possibly fruitful directions for further research, such as the one indicated in his four­teenth Paris problem. Although he did offer a one-semester course on invari­ant theory in 1897 (see [Hilbert 1993]), by this time his eyes were already on other fields and new challenges.

Mathematicians are constantly look­ing ahead, not backward, and by 1893 probably no one gave much thought to the events of five years earlier. Over time, Hilbert's decisive encounter with Gordan in Leipzig was reduced to a mi-

nor episode at the outset of his Siegeszug through invariant theory. Otto Blumenthal, Hilbert's first biogra­pher, even got the city wrong, claiming that Hilbert went to Erlangen to visit Gordan in the spring of 1888 [Blumen­thal 1935, 394]. By then forty years had passed, and presumably no one, not even Hilbert, remembered what had happened. Yet his own characteriza­tion of this encounter could not be more telling: it had been thanks to Gordan's "stimulating help" that he left Leipzig with "an infinite series of thought vibrations" running through his brain. Scant though the evidence may be, it strongly suggests that the week he spent with the "King of In­variants" gave Hilbert the initial im­pulse that put him on his way. Back in Konigsberg, he adopted several of Gor­dan's techniques in his subsequent work. Numerous citations reveal that he was thoroughly familiar with Gor­dan's opus, especially the two volumes of his lectures edited by Georg Ker­schensteiner [Gordan 1885/1887]. That work, the springboard for many of Hilbert's discoveries, was by 1893 prac­tically obsolete, though no comparable compendium would take its place.

His friend Hermann Minkowski saw that this presented a certain dilemma: it was all very well to blow up the cas­tles of those robber knights of invari­ant theory, so long as something more useful could be built on their now bar­ren terrain. Minkowski thus expressed the hope that Hilbert would some day show the mathematical world what the new buildings might look like. In the same vein, he kidded him that it would be best if Hilbert wrote his own mono­graph on the new modernized theory of invariants rather than waiting to find another Kerschensteiner, who would likely leave behind too many "cherry pits" (misspelled by Minkowski as "Ker­schensteine") [Minkowski 1973, 45].

Hilbert did neither, 10 leaving the theory of invariants to languish on its own while the Gordan-Kerschensteiner

volumes gathered dust in local li­braries. Invariant theory thus entered the annals of mathematics, its history already sketched by the man who wrote its epitaph. To the younger gen­eration, Paul Gordan would mainly be remembered for having once declared Hilbert's modem methods "theology." Now that he and his mathematical regime had been deposed, classical in­variant theory was declared a dead subject, one of those "dead ends" ("tote Strange") that Hilbert had decried in his letter to Klein from 1890.U Al­though Leopold Kronecker had pre­dicted this very outcome, he would hardly have approved of the execu­tioner's methods. Yet ironically, it was Hilbert's decision to move on to "greener pastures"-even more than the wealth of new perspectives his work had opened-that hastened the fulfillment of Kronecker's prophecy.

LITERATURE

[Biermann 1 988] Kurt-R. Biermann, Die Math­

ematik und ihre Dozenten an der Berliner Uni­

versitat, 1810-1933, Berlin: Akademie-Ver­

lag, 1 988.

[Blumenthal 1 935] Otto Blumenthal, "Lebens­

geschichte," in [Hilbert 1 935, pp. 388-429].

[Caruth 1 996] A Caruth, "A concise proof of

Hilbert's basistheorem." Amer. Math. Monthly

1 03 (1 996), 1 60-1 61 .

[Corry 2004] Leo Corry, Hilbert and the Ax­

iomatization of Physics (1898-1918): From

"Grundlagen der Geometrie" to "Grundlagen

der Physik", to appear in Archimedes: New

Studies in the History and Philosophy of Sci­

ence and Technology, Dordrecht: Kluwer

Academic, 2004.

[Fisher 1 966] Charles S. Fisher, "The Death of

a Mathematical theory: A Study in the Soci­

ology of Knowledge, " Archive for History of

exact Sciences, 3 (1 966), 1 37-1 59.

[Frei 1 985] Gunther Frei, Der Briefwechsel

David Hilbert-Felix Klein (1 886-1918) , Ar­

beiten aus der Niedersachsischen Staats­

und Universitatsbibliothek Gbttingen, Bd. 1 9 ,

Gbttingen: Vandenhoeck & Ruprecht, 1 985.

[Freudenthal 1 98 1 ] Hans Freudenthal, "David

Hilbert," Dictionary of Scientific Biography,

9For a brief account of the work and its historical reception, see the Introduction by Franz Lemmermeyer and Norbert Schappacher to the English edition [Hilbert 1 998,

xxiii-xxxvi].

1 0Perhaps the closest he came to fulfilling Minkowski's w1sh was the Ausarbeitung of his 1 897 lecture course. now available in English translation in [Hilbert 1 993]

1 1 Historical verdicts with regard to the sudden demise of invariant theory have varied considerably (see [Fisher 1 966] and [Parshall 1 989]). The merits of classical in­

variant theory were later debated in print by Eduard Study and Hermann Weyl. For an excellent account of this and other subsequent developments in algebra, see

[Hawkins 2000].

© 2005 Spnnger SC1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1 , 2005 81

I 6 vols. , ed. Charles C. Gillispie, vol. 6, pp.

388-395, New York: Charles Scribner's

Sons, 1 97 1 .

[Gilmer 1 981 ] Robert Gilmer, "Commutative

Ring Theory," in Emmy Noether. A Tribute to

her Life and Work, ed. James W. Brewer and

Martha K. Smith, New York: Marcel Dekker,

1 981 , pp. 1 31 -1 43.

[Gordan 1 885/1 887] Paul Gordan, Vorlesungen

uber lnvariantentheorie, 2 vols., ed. Georg

Kerschensteiner, Leipzig: Teubner, 1 885,

1 887.

[Gordan 1 892] -- , "Uber einen Satz von

Hilbert," Mathematische Annalen 42 (1 892),

1 32-1 42.

[Gordan 1 899] -- , "Neuer Beweis des

Hilbertschen Satzes uber homogene Funk­

tionen," Nachrichten der Gesellschaft der

Wissenschaften zu G6ttingen 1 899, 240-

242.

[Hashagen 2003] Ulf Hashagen, Walther von

Oyck (1856-1934). Mathematik, Technik und

Wissenschaftsorganisation an der TH

Munchen, Boethius, Band 47, Stuttgart:

Franz Steiner, 2003.

[Hawkins 2000] Thomas Hawkins, Emergence

of the Theory of Lie Groups. An Essay in the

History of Mathematics, 1 869-1926. New

York: Springer-Verlag, 2000.

[Hesseling 2003] Dennis E. Hesseling, Gnomes

in the Fog. The Reception of Brouwer's In­

tuitionism in the 1920's, Basel: Birkhauser,

2003.

[Hilbert 1 888a] David Hilbert, "Uber die Darstel­

lung definiter Formen als Summe von For­

menquadraten," Mathematische Annalen 32

(1 888), 342-350; reprinted in [Hilbert 1 933,

1 54-1 61 ] .

[Hilbert 1 888b] -- , "Uber die Endlichkeit des

lnvariantensystems fUr binare Grundformen,"

Mathematische Annalen 33 (1 889), 223-

226; reprinted in [Hilbert 1 933, 1 62-1 64].

[Hilbert 1 888c] -- , "Zur Theorie der alge­

braischen Gebilde 1 , " Nachrichten der

Gesellschaft der Wissenschaften zu G6ttin­

gen 1 888, 450-457; reprinted in (Hilbert

1 933, 1 76-183].

[Hilbert 1 889a] -- , "Zur Theorie der alge­

braischen Gebilde I I , " Nachrichten der

Gesellschaft der Wissenschaften zu G6ttin­

gen 1 889, 25-34; reprinted in [Hilbert 1 933,

1 84-1 91 ] .

[Hilbert 1 889b] -- , "Zur Theorie der alge­

braischen Gebilde I l l , " Nachrichten der

Gesellschaft der Wissenschaften zu G6ttin­

gen 1 889, 423-430; reprinted in [Hilbert

1 933, 1 92-1 98].

82 THE MATHEMATICAL INTELLIGENCER

[Hi lbert 1 890] --, "Uber die Theorie der al­

gebraischen Formen," Mathematische An­

nalen 36 (1 890), 473-534.

[Hilbert 1 891 ] -- , "Uber die Theorie der al­

gebraischen lnvarianten 1," Nachrichten der

Gesel/schaft der Wissenschaften zu G6ttin­

gen 1 891 , 232-242.

[Hilbert 1 892a] -- , "Uber die Theorie der al­

gebraischen lnvarianten I I , " Nachrichten der

Gesellschaft der Wissenschaften zu G6ttin­

gen 1 892, 6-1 6.

[Hilbert 1 892b] -- , "Uber die Theorie der al­

gebraischen lnvarianten I l l , " Nachrichten der

Gesel/schaft der Wissenschaften zu G6ttin­

gen 1 892, 439-449.

[Hilbert 1 893] -- , "Uber die vollen lnvari­

antensysteme," Mathematische Annalen 42

(1 893), 31 3-373; reprinted in [Hilbert 1 933,

287-344] .

[Hilbert 1 896] -- , "Uber die Theorie der al­

gebraischen lnvarianten," Mathematical Pa­

pers Read at the International Mathematical

Congress Chicago 1893, New York: Macmil­

lan, 1 896, 1 1 6-1 24; reprinted in [Hilbert

1 933, 376-383].

[Hi lbert 1 922] -- , "Neubegrundung der

Mathematik. Erste Mitteilung," Abhandlun­

gen aus dem Mathematischen Seminar der

Hamburgischen Universitat, 1 : 1 57-1 77;

reprinted in [Hilbert 1 935, 1 57-1 77].

[Hilbert 1 932/1 933/1 935] -- , Gesammelte

Abhandlungen, 3 vols. , Berlin: Springer, 1 932-

1 935.

[Hilbert 1 978] -- , Hilbert's Invariant Theory

Papers, trans. Michael Ackermann, in Lie

Groups: History, Frontiers, and Applications,

ed. Robert Hermann, Brookline, Mass . : Math

Sci Press, 1 978.

[Hilbert 1 993] --, Theory of Algebraic Invari­

ants, trans. Reinhard C. Lauenbacher, Cam­

bridge: Cambridge University Press, 1 993.

[Hilbert 1 998] -- , The Theory of Algebraic

Number Fields, trans. lain T. Adamson, New

York: Springer-Verlag, 1 998.

[Mertens 1 887] Franz Mertens, Journal fUr die

reine und angewandte Mathematik 1 00(1 887),

223-230.

[Minkowski 1 973] Hermann Minkowski, Briefe

an David Hilbert, eds. L. ROdenberg und H.

Zassenhaus, New York: Springer-Verlag,

1 973.

[Noether 1 92 1 ] Emmy Noether, "ldealtheorie in

Ringbereichen," Mathematische Annalen 83

(1 921 ) , 24-66.

[Parshall 1 989], Karen H. Parshall, "Toward a

History of Nineteenth-Century Invariant The­

ory," in The History of Modern Mathematics:

Volume 1 : Ideas and their Reception, ed.

David E . Rowe and John McCleary, Boston:

Academic Press, 1 989, pp. 1 57-206.

(Parshall and Rowe 1 994] Karen H. Parshall

and David E. Rowe, The Emergence of the

American Mathematical Research Commu­

nity, 1876-1900. J.J. Sylvester, Felix Klein,

and E.H. Moore, History of Mathematics,

vol. 8, Providence, Rhode Island: American

Mathematical Society, 1 994.

(Reid 1 970] Constance Reid, Hilbert. New York:

Springer Verlag, 1 970.

[Rowe 1 998] --, "Mathematics in Berlin,

1 81 0-1 933," in Mathematics in Berlin, ed.

H.G.W. Begehr, H. Koch, J . Kramer, N.

Schappacher, and E.-J. Thiele, Basel:

Birkhauser, 1 998, pp. 9-26.

(Rowe 2000] -- , "Episodes in the Berlin­

Gbttingen Rivalry, 1 870-1 930," Mathemati­

cal lntelligencer, 22(1 ) (2000), 60-69.

[Rowe 2003a] -- , "From Konigsberg to Gbt­

tingen: A Sketch of Hilbert's Early Career,"

Mathematical lntelligencer, 25(2) (2003),

44-50.

[Rowe 2003b] -- . "Mathematical Schools,

Communities, and Networks," in Cambridge

History of Science, val. 5, Modern Physical

and Mathematical Sciences, ed. Mary Jo

Nye, Cambridge: Cambridge University Press,

pp. 1 1 3-1 32.

[Rowe 2004] -- , "Making Mathematics in an

Oral Culture: Gbttingen in the Era of Klein

and Hilbert," to appear in Science in Con­

text, 2004.

(Toeplitz 1 922] Otto Toeplitz, "Der Alge­

braiker Hilbert," Die Naturwissenschaften 1 0 ,

(1 922) 73-77 .

(Weber 1 893] Heinrich Weber, "Leopold Kro­

necker," Jahresbericht der Deutschen Math­

ematiker-Vereinigung 2(1 893), 5-1 3.

[Weyl 1 932] Hermann Weyl, "Zu David Hilberts

siebzigsten Geburtstag, " Die Naturwis­

senschaften 20 (1 932), 57-58; reprinted in

[Weyl 1 968, vol. 3, 346-347].

(Weyl 1 935] --, "Emmy Noether," Scripta

Mathematica, 3 (1 935), 201-220; reprinted

in [Weyl 1 968, vol. 3, 435-444].

[Weyl 1 944] -- , "David Hi lbert and his Math­

ematical Work," Bulletin of the American

Mathematical Society, 50, 61 2-654; reprinted

in [Weyl 1 968, vol. 4, 1 30-1 72] .

(Weyl 1 968] -- , Gesammelte Abhandlun­

gen, 4 vols . , ed. K. Chandrasekharan, Berlin:

Springer-Verlag, 1 968.

(Yandell 2002] Ben H. Yandell, The Honors

Class. Hilbert's Problems and their Solvers,

Natick, Mass. : A K Peters, 2002.

i;i§lh§l,'tJ Osmo Pekonen , Ed itor I

Feel like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or, if

you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor: Osmo Pekonen, Agora

Center, University of Jyvaskyla, Jyvaskyla,

40351 Finland

e-mail: pekonen@mit.jyu.fi

Statistics on the Table: The History of Statistical Concepts and Methods Stephen M. Stigler

CAMBRIDGE, MASS , HARVARD UNIVERSITY PRESS

PAPERBACK 2002 (first pnnllng 1 999)

5 1 0 PAGES US $1 9.95 ISBN 0-674-00979-7

REVIEWED BY IVO SCHNEIDER

The book consists of 22 chapters, all of which except the first were pre­

viously printed as articles in reviewed journals or books. The chapters are distributed in five parts with the titles: I. Statistics and Social Science, II. Gal­tonian Ideas, III. Some Seventeenth­Century Explorers, IV. Questions of Discovery, and V. Questions of Stan­dards. One could argue about the choice of these titles, their order, or their interrelations. In a strictly histor­ical account one would begin with part III; part I is as much concerned with economic as with social questions, and Galtonian ideas are very much related to social science. In short, the selection arrangement of these 22 articles is less stringent than in a book which treats a topic systematically or strictly chrono­logically.

Trying to get the original publica­tions in order to find out about the changes made for the sake of this book (which according to the sample I could check are small), I learned that many of these articles are not easily avail­able. The book contains 21 of the 38 ar­ticles and books concerning the history of statistics that Stigler published be­tween 1973 and 1997, including his monograph from 1986, The History of Statistics-the Measurement of Un­certainty before 1900. So one advan­tage of the book is to make available some of Stigler's publications which are otherwise not easy to get.

From a historical point of view the most interesting question is: How does

Statistics on the Table relate to Stigler's History of Statistics, which appears to claim to cover the history of statistics, at least for the time before 1900?

First, Stigler justifies the new book with the argument that, because sta­tistical thinking and so statistical con­cepts permeate the whole range of hu­man thought, statistics in historical accounts is practically "never covered completely." Statistics on the Table represents, according to Stigler, "only a small selection of the possible themes and topics," but it treats, however in­completely, one of the most important aspects of statistical work: statistical evidence in the form of data and their interpretation for the solution of a problem. For many, this project in such a general formulation represents the whole of statistical science, so it is not surprising that Statistics on the Table and The History of Statistics have sev­eral things in common. Chapter 1 deals with the Karl Pearson of 1910/1 1 and so its material is not contained in The History of Statistics, which ends with 1900. It deals with the effect or non-effect of parental alcoholism on the alcoholism of the offspring. Pear­son's vote for non-effect in the light of data collected in the Galton Laboratory met with considerable resistance and criticism due to different factors and interests, one of them being the tem­perance movement of the time. What Pearson expected, or rather requested, from his critics was "statistics on the table," data which could confirm the position of his opponents and so dis­prove him. Interestingly, Pearson's re­quest was not seen by economists like Keynes, one of his opponents, as the appropriate method to deal with the controversy. Having shown in this way that "statistics on the table" was still far from being generally accepted as the method of handling questions of this kind, Stigler goes back in time in order to reconstruct the way in which collections of data were used before 1910.

© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number I , 2005 83

Starting with an evaluation of Quetelet's statistical work in chapter 2, which is less detailed than the chapter on Quetelet in his History, Stigler de­votes the next two chapters to the sta­tistical work of the economist Jevons, who is mentioned several times in the History but without any further evalu­ation of his statistics. Stigler's interest in Jevons is motivated by the effect of his statistical work in overcoming the typical mid-nineteenth-century separa­tion of statistics, understood as data collection, from the interpretation of the data, especially in the social and economic domain.

A chapter on the work of the econ­omist and statistician Francis Ysidro Edgeworth ends the first of the five parts of Statistics on the Table. Edge­worth does not figure prominently in the history of statistics, because his statistical ideas were dispersed over a great many not easily digestible arti­cles. So Stigler, like a Robin Hood of the history of statistics, takes away from the rich in reputation in order to give to the historically neglected Edge­worth, whom he had honoured already in his History with a full chapter, indi­cating that in his eyes Edgeworth was as important in the development of sta­tistics as Galton and Karl Pearson.

The five chapters devoted to "Gal­tonian ideas" in part II. differ from the Galton chapter in the History by em­phasizing different topics and the im­pact of Galton and his methods. So whereas Galton's work on fingerprints is only mentioned without any further analysis in the History, Stigler devotes the whole of chapter 6 of Statistics on the Table to it, including the accep­tance of fingerprints as evidence in court. Galton's and his contempo­raries' contribution to "stochastic sim­ulation," with a set of special dice used for the generation of half-normal vari­ants, plays no role whatsoever in the History. Regression is the topic of the next two chapters, of which only the second deals with Galton's contribu­tion to it, whereas chapter 8, "The his­tory of statistics in 1933," hints at the more subtle aspects of regression visi­ble in Hotelling's devastating review of Horace Secrit's book The Triumph of Mediocrity in Business from 1933, the

84 THE MATHEMATICAL INTELLIGENCER

year which Stigler likes to consider as the proper starting point of mathemat­ical statistics. Stigler's inclination to act occasionally as an agent provoca­teur might in this case be seen at odds with his History, which should be con­sequently "The history of non-mathe­matical statistics. " Chapter 10 then discusses the relatively early use (com­pared with other social sciences) of statistical techniques in psychology, which according to Stigler is due to ex­perimental design.

The following three chapters deal with publications of the second half of the 17th century. It is not clear to me why the first two of them belong in Sta­tistics on the Table. Nor do I under­stand the use of the word "probability" in the context of Huygens's tract on the treatment of games of chance or the correspondence between Pascal and Fermat common to these chapters. Pascal, Fermat, and Huygens were all well aware of contemporary concepts of probability, though none of these concepts appears in their treatment of games of chance. In chapter 13, how­ever, a contemporary concept of prob­ability is treated, as used by John Craig in order to determine the trustworthi­ness of statements concerning histori­cal events, or in Craig's term the "his­torical probability," which increases in proportion to the number of witnesses in favour of it and decreases in pro­portion to the time elapsed since the event. A function representing the de­pendence of Craig's historical proba­bility on time and distance is tested by putting on the table the data of Laplace's birth and death found in 65 books of the 19th century.

Of the six articles making up "Ques­tions of discovery," chapter 18, which treats the history of the so-called Cauchy distribution, has no relation to any concrete set of data put on the table, whereas the other five chapters do. The first chapter of this part is de­voted to eponymy, the practice in the scientific community of affixing the name of a scientist to a discovery, the­ory, etc. as a reward for scientific ex­cellence in the relevant field. Such a re­ward presupposes distance in time and place from the work honoured by eponymy. Accordingly, it cannot be ex-

pected that scientific discoveries are named after their original discoverer, or, to formulate it more aphoristically as a law of eponymy, "no scientific dis­covery is named after its original dis­coverer." Since Stigler sees the sociol­ogist Robert K. Merton as the originator of this so-called law, and since Stigler does not want to begin with a counterexample for the validity of this law, the title-giving eponymy of this law is "Stigler's law of eponymy." For the joke's sake it does not matter that this is not an eponymy proper, which would have demanded that the community of sociologists after an ap­propriate period of time ought to have accepted it. A proper eponymy­namely the affixing of the names of Gauss and sometimes Laplace, but not the name of de Moivre, the real "dis­coverer," to the normal distribution­is discussed on the basis of 80 books published between 1816 and 1976. The discussion showed that at least in this case, the eponymy was awarded only after considerable time by the scien­tific community. Seen with eyes ac­customed to eponymic practice, the next chapter (15), which answers the question "Who discovered Bayes's the­orem" with Nicholas Saunderson as the most probable candidate, appears as Stigler's next attempt to avoid a coun­terexample to the law of eponymy, be­cause most people believe that Bayes discovered Bayes's theorem. A prob­lem not touched so far when dealing with discoveries was treated by Thomas S. Kuhn, who pointed to si­multaneous discovery of the "same" thing by several people. In his discus­sion of Kuhn, Yehuda Elkana pointed to difficulties inherent in the concept of "sameness. " Difficulties of this kind are the topic of chapter 16, which de­scribes the first steps made by Daniel Bernoulli and Euler concerning the theme of maximum likelihood.

Sameness plays no important role in the next chapter, dealing with the claims of Legendre and Gauss con­cerning the method of least squares. The data of the French meridian arc measurements and their interpolation by Gauss in 1799 are interpreted as in­conclusive for Gauss's claim to have devised and applied the method of

least squares at that time or even be­fore. Again Stigler's social attitude as the Robin Hood of the history of sta­tistics becomes evident when he states that, despite Gauss's undisputed mer­its in developing algorithms for the computation of estimates, it was Le­gendre "who first put the method within the reach of the common man." However, Gauss's contribution to the method of least squares is seen much more positively in this article than in Stigler's History.

The last chapter (19) in this part is mainly concerned with a paper of Karl Pearson and his collaborators from 1913, in which Pearson fitted a quasi-in­dependent model to the data of incom­plete contingency tables, testing the fit by a chi-square test, which, as was rec­ognized by R. A Fisher, used the wrong number of degrees of freedom. But the concept of degrees of freedom had been introduced only in 1922 by Fisher, who had not seen that for the special class of tables considered by Pearson the use of the correct number of degrees of free­dom would not have changed Pearson's conclusions.

The last three chapters are sub­sumed under the title "Statistics and Standards." In the first the observation that many of the most powerful statis­tical methods, like the method of least squares, are originally connected with the determination of standards like the standard meter, is interpreted not as ac­cidental but as a consequence of the purpose of a standard to measure, count, or compare as accurately as pos­sible. In a second part Stigler describes how the fact that in experimental sci­ence no absolute accuracy can be achieved, that every measurement is in­evitably connected with error and so with uncertainty, eventually led to the creation of standards of uncertainty, standards in statistics like standard er­ror curves or standard deviations.

The last chapter (22), written to­gether with William H. Kruskal, is re­lated to standards in statistics in that the first part of it answers the question when and why the normal distribution was called "normal." The other parts of the chapter are concerned with the am­biguity of the words "normal" and "nor­mality," exemplified in paragraphs de-

voted to the terms "normal equations" in connection with the method of least squares, "normality in medicine," and "normal schools" in the educational system. Stigler maintains that there is a mutual dependence between the use of "normal" in science and in the realm of public discourse.

Before this last chapter I found my favourite "The trial of the Pyx," which is a test to control the quality and cor­rectness of the coin production at the Royal Mint for more than seven cen­turies. Stigler interprets the trial of the Pyx as "a marvellous example of a sam­pling inspection scheme for the main­tenance of quality." It is amusing to read his report of the most famous master of the Royal Mint, Isaac New­ton. Stigler amasses arguments for scepticism concerning Newton's hon­esty as master of the Mint, thus dis­qualifying Stigler forever as a member of the invisible college of Newtonians. However, perhaps concerned about his good relations to highly regarded British institutions, he finds on the ba­sis of the research work of others "no grounds for believing that he <Newton) took advantage of this knowledge for illegal personal gain."

Nearly all the 22 articles of this col­lection display Stigler's wit and hu­mour; they are good to read. They do not make up a coherent story or his­tory, but most of them contain, encap­sulated in some special question, the whole of statistics.

Muenchner Zentrum fUr Wissenschafts- und

Technikgeschichte

Deutsches Museum

80306 Muenchen, Germany

e-mail: IVo .Schneider@unibw-muenchen.de

Theory of Bergman Spaces Boris Korenblum, Haakan

Hedenmalm, and Kehe Zhu

HEIDELBERG, SPRINGER-VERLAG 2000. 286 PAGES, 2 ILLUS €59 50

REVIEWED BY DAVID BEKOLLE

Let D denote the unit disk of the complex plane. For 0 < p :s; oc and

- 1 < a :S oo, the Bergman space A€ of

D is the closed subspace of the Lebesgue space Lg : = LP(D, (1 - lzl2)"'dxdy (z = x + iy) consisting of holomorphic functions. For a = 0, we write AP = A{;. Intensive research on the theory of Bergman spaces has been carried on since the early 1970s. The start followed the "essential" completion of the theory of Hardy spaces on D. In particular, A� is a closed subspace of the Hilbert space a. We call Bergman projector and we denote by Pa the orthogonal projector of the Hilbert space L� onto its closed subspace A�. It is well known that Pa is the integral operator defined on L� by the Bergman kernal Ba(z,w) : = Ca(l - ZW)-(l+a).

One of the first fundamental results in the theory of Bergman spaces was established in 1984 independently by F. Forelli and W. Rudin [7] and E. M. Stein [10] . According to this result, for 1 :S p < oc and - 1 < a < oc, the Bergman projector Pa extends to a bounded pro­jector of L€ to A€ if and only if p > 1. For an account of the results obtained in the 1970s and 1980s, see the book of K. Zhu Operator Theory in Bergman Spaces [ 11 ] .

The aim of the present book by Korenblum, Hedenmalm, and Zhu is to present some deep results obtained in the 1990s in function theory and in op­erator theory in Bergman spaces on the unit disk, namely:

1. K. Seip's geometric characteriza­tions of interpolation and sampling sequences for A€;

2. the discovery by H. Hedenmalm of contractive zero divisors for A� and its implementation for A€ (p =t- 2) by P. Duren, D. Khavinson, H. S. Shapiro, and C. Sundberg;

3. outstanding results related to the "curiously resistant" characteriza­tion problem of zero sequences of A€ functions;

4. other striking results on the bihar­monic Green function due to H. Hedenmalm, P. Duren, D. Khavin­son, H. S. Shapiro, and C. Sundberg, and on invariant subspaces of A€ due to A Aleman, A. Borichev, H. Hedenmalm, S. Richter, S. M. Shimorin, and C. Sundberg.

The book under review is welcome and will be very useful-among many

© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1, 2005 85

reasons, because it includes a self-con­tained proof of K. Seip's geometric characterizations of interpolation and sampling sequences for Bergman spaces A g. Recall that a sequence r = {zj}j of distinct points of D is an inter­polation sequence for AR (0 < p < oo) if, for every sequence {w1)1 of complex numbers satisfying the condition

LCl - lzJI2?+ajwjf < 00, j

there exists a function! E Ag such that f(z1) = w1 for all j. A geometric char­acterization of interpolation sequences for Hardy spaces was obtained in 1958 by L. Carleson [5] for p = oo. For gen­eral p E (O,oo] see, e.g., the books [6] and [8].

Following the theorem of Forelli­Rudin and Stein stated above, interpo­lation sequences for Ag were first stud­ied by Eric Amar [ 1 ] , and it is unfair that his name is not quoted in this book regarding results of Chapter 4.

To prove his theorems, K. Seip re­lies heavily

• on a fundamental paper of B. Karen­blum [9] and

• on earlier results of A. Beurling [3] on interpolation for the Banach space of functions of exponential type :S a and bounded on the real line.

Seip's characterizations use notions of density inspired by Beurling and Karen­blum. One of these notions of density, denoted D1(f)), is defmed as follows. Let r = {z1)1 be a separated (with re­spect to the Bergman distance) se­quence in D, and let r E Cf, 1). We set

D(f,r) LllogjzJI : _!_ < lzJ I < r) . 2

J

log (-1 ) 1-r

For every z E D, we form a new se­quence

r z : = { _1Z..Li _-__ z_ }j· - ZjZ

The upper Seip density of r is defined by

D1(f) = limsupr-->1 SUPzErJJ(fz,r).

Finally, K. Seip's Theorem states the following: Let 0 < p < oo and - 1 < a <

86 THE MATHEMATICAL INTELLIGENCER

oo, let r be a sequence of distinct points of D. Then the following conditions are equivalent:

1. r is an interpolating sequence for A&'; 2. r is separated and D1(f) < " ; 1 •

The proof of this theorem sheds light on Korenblum's difficult paper [9].

The book under review is directed at graduate students and new re­searchers in the field. It will be very useful for senior researchers as well. At the end of each chapter, various ex­ercises are proposed to the reader. Many open problems are also stated. For a more recent report on open prob­lems on zero sequences, invariant sub­spaces, and factorization of functions in A2, the interested reader may con­sult the report by Aleman, Hedenmalm, and Richter [2].

As a minor point (compared to the quality and quantity of the book as a whole), the reviewer mentions the fal­sity of the proof of Theorem 1.21, page 21, which identifies the dual space of A&' (0 < p :S 1, - 1 < a < oo) with the Bloch space. The statement is correct, but its proof should be corrected in the second edition of the book. For a cor­rect proof, see [4] .

REFERENCES

[1 ] Amar, E. Suites d'interpolation pour les

classes de Bergman de Ia boule et du

polydisque de en. Can. J. Math. 30 (1 978),

71 1 -737.

[2] Aleman, A, H. Hedenmalm, and S.

Richter. Recent progress and open prob­

lems in the Bergman space (preprint).

[3] Beurling, A The Collected Works of Arne

Beurling by L. Carleson, P. Malliavin, J .

Neuberger, and J . Wermer. Vol. 2, Har­

monic Analysis, Boston, Birkhauser (1 989),

341 -365.

[4] Bekolle, D. Bergman spaces with small ex­

ponents. Indiana Univ. Math. J. 49 (3)

(2000), 973-993.

[5] Carleson, L. An interpolation problem for

bounded analytic functions. Amer. J. Math.

80 (1 958), 921 -930.

[6] Duren, P. Theory of HP spaces, Academic

Press, New York (1 970).

[7] Forelli, F. and W. Rudin. Projections on

spaces of holomorphic functions in balls,

Indiana Univ. Math. J. 24 (1 974), 593-602.

[8] Garnett, J. B. Bounded Analytic Functions,

Academic Press, New York (1 981) .

[9] Koren blum, B. An extension of the Nevan­

linna theory. Acta. Math. 1 35 (1 975), 1 87-

219 .

[1 0] Stein, E. M. Singular integrals and estimates

for the Cauchy-Riemann equations. Bull.

Amer. Math. Soc. 79 (1 973), 440-445.

[1 1 ] Zhu, K. Operator Theory in Function

Spaces. Marcel Dekker, New York (1 990).

Faculte des Sciences

Universite de Yaounde I

B.P. 81 2

Yaounde

Cameroon

e-mail: dbekolle@uycdc.uninet.cm

Gamma: Exploring Euler's Constant Julian Havil

PRINCETON, PRINCETON UNIVERSITY PRESS 2003

XXIII + 266 PAGES. US $29.95. ISBN 0-691 -09983-9

REVIEWED BY GERALD L. ALEXANDERSON

This is a Golden Age-well, at least it is for students and those of us

who love to read about mathematics outside our own area of expertise. In my youth we could choose from the books of E. T. Bell, What Is Mathe­matics? by Courant and Robbins, some of P6lya's books, and Rademacher and Toeplitz. There were others, but the list was short. Today the catalogues of Springer, Cambridge, Princeton, Ox­ford, the AMS, and the MAA overflow with general books, accessible to stu­dents and mathematical amateurs, and on a wide variety of subjects. Now we even see books coming out on specific numbers, notably Eli Maor's e: The Story of a Number (Princeton Univer­sity Press, 1998), Paul Nahin's An Imaginary Tale/The Story of v=I (Princeton University Press, 1998), David Blatner's The Joy of Pi (Walker, 1999), Charles Seife's Zero: The Biog­raphy of a Dangerous Idea (Penguin, 2000), Hans Walser's The Golden Sec­tion (MAA, 2002), and Mario Livia's The Golden Ratio: The Story of ¢, the World's Most Astonishing Number (Broadway, 2003). And here we have Havil's book on the Euler-Mascheroni constant. Given the plethora of inter-

esting numbers, this series could go on for some time.

Of course, numbers don't get very much more interesting than Euler's number 'Y· I was dubious when I read G. J. Chaitin's article, "Thoughts on the Riemann Hypothesis" in the Winter, 2004, issue of this magazine (vol. 26, no. 1) in which he included Havil's book in his list of recent books on the Riemann Hypothesis. I checked Harold Edwards's review of the RH books by Derbyshire, du Sautoy, and Sabbagh in the same issue and noted that he did not see fit to include Gamma in his re­view. At the time I was familiar with the three reviewed by Edwards and didn't see the relevance of Havil's book in this context. But I see that a case can clearly be made for its inclusion, as Chaitin does.

Havil is obviously enthusiastic about his subject and remarkably eru­dite. There are many references to de­velopments in mathematics that are re­lated to Euler's constant, often quite recent results. He obviously watches the literature. Most of the connections have to do with problems that at some level involve either natural logarithms or partial sums of the harmonic series.

He looks at ways of calculating 'Y to great accuracy. The problem is non­trivial, for y is defined as limn__,x (Hn -ln n), where Hn = 1 + t + f + i + . . . + _1_, Both Hn and ln n grow without n bound, but they grow very slowly, so just calculating the difference for larger and larger n is not efficient.

What do we know about y? First we learn what we don't know-whether it is irrational, for example. Unlike con­stants such as 1T and e, where questions of whether they are irrational and tran­scendental were settled before the 20th century, at the beginning of the 21st we still don't have this information about y. We do learn that from an Euler­Maclaurin expansion we get

n 1 1 1 r = I - - ln n - - + --

k � l k 2n 12n2

1 1 -- + -- + 120n4 252n6

One might ask how Lorenzo Mascheroni (better known for his proof that a geometric construction

possible with straightedge and com­pass can be carried out with the com­pass alone) got his name attached to y. He approximated it to 32 decimal places, only the first 19 of which were correct! In 1962 Donald Knuth com­puted 'Y to 1271 decimal places, and in 1999 it was calculated to 108,000,000 decimal places. Writing 'Y as a contin­ued fraction, we find that the conver­gent 323007/559595 differs from 'Y by 1.025 X 10- 12. Again using contin­ued fractions, Thomas Papanikolaou showed that if 'Y were to be a fraction, its denominator would be greater than 10242080, perhaps providing a hint that 'Y is irrational.

In demonstrating these and many other facts about y, we're led on an il­luminating tour of Bernoulli numbers; the Basel problem; f(x), Euler's gamma function; Stirling's approxima-

N u mbers don 't

get m uch more

interest ing than

Euler 's number y. tion formula; and much, much else. And if that weren't enough, in a series of appendices we find a concise intro­duction to complex function theory.

Sometimes the mathematical state­ments are sturming and leave one won­dering how things seemingly so dis­parate can be related. In the chapter about appearances of harmonic series, the author talks about musical tones, of course, then describes surprising re­sults on the infrequency of record rain­falls, for example. Next he provides an economical test for destruction of beams to check their strength and breaking points. Then there's a question of sending Jeeps across the desert, and problems of card sorting, Hoare's Quicksort algorithm, the maximum pos­sible overhang of playing cards placed on the edge of a table, and so on and so on. Some of these are fairly well known, though others were to me quite new and surprising. On logarithms, he describes clearly Benford's now well-known but surprisingly recent law on the lack of uniform distribution of digits in collec-

tions of data (like baseball statistics, ge­ographic areas, street addresses, death rates, and such).

Just as Benford's Law says that l's ap­pear more often than 2's as leading dig­its, 2's more often than 3's, and so on, for a descending curve of frequencies, for me the chapters of the book were rather the opposite, increasing in inter­est as I went along. I found the begin­ning material on the history of loga­rithms rather heavy going. Perhaps it's just too familiar, but there's also the problem that however valuable loga­rithms were in their infancy, the calcu­lations are not likely to be exciting to the modern reader. Havil quotes Laplace, however: " . . . by shortening the labors, [logarithms] doubled the life of the as­tronomer." The middle chapters have many cormections to interesting prob­lems; the latter ones make cormections to number theory and succeed in mak­ing these quite clear in spite of the de­tails being considerably more mathe­matically challenging to the reader.

The historical references are charm­ing, and many of the quotes are fasci­nating and, to me, unfamiliar. Here are a few examples:

• Leo Tolstoy: "A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the de­nominator the smaller the fraction."

• Heinrich Hertz: "One cannot escape the feeling that these mathematical formulas have an independent exis­tence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them."

• Krzysztof Maslanka: "We may­paraphrasing the famous sentence of George Orwell-say that 'all mathe­matics is beautiful, yet some is more beautiful than the other. ' But the most beautiful in all mathematics is the Zeta function. There is no doubt about it."

• Paul Erdos (paraphrasing Einstein): "God may not play dice with the Uni­verse, but there's something strange going on with the prime numbers!"

• David Hilbert (in response to a ques­tion of which mathematical problem

© 2005 Springer Science+Bus1ness Med1a, Inc., Volume 27, Number 1 , 2005 87

was the most important): "The prob­lem of the zeros of the Zeta function, not only in mathematics, but ab­solutely most important!"

• This is followed by the following from Morris Kline: "If I could come back after five hundred years and find that the Riemann Hypothesis or Fermat's last 'theorem' was proved, I would be disappointed, because I would be pretty sure, in view of the history of attempts to prove these conjectures, that an enormous amount of time had been spent on proving theorems that are unimpor­tant to the life of man."

The last is identified as the response in an interview. The book in which it appeared is identified correctly and the date is correct, but the interviewer and editors of the book are not identified anywhere. The interviewer happened to be this reviewer.

These and many other sections raise a question. There's a fine line between a book meant largely for entertaining and educating the generally informed but non-specialist reader, and a schol­arly book that carefully documents statements made in the text. The quo­tations above appear without specific references. Even more frustrating to this reader is the lack of specific ref­erences to theorems and articles cited. On page 1 13, the author cites a re­markable result of de la Vallee-Poussin of 1898, which he calls "baffling," that "if we divide an integer n by all inte­gers less than it and average the deficits of each quotient to the integer above it, the answer approaches 'Y as n ---> oo. " As an example he points out that _1_ Ig�gg cl lOOOOl - 10000) gives 10000 r� 1 I r r 0.577216 . . . . He then adds that the re-sult remains true if the divisors are those in any arithmetic sequence or if they are only the prime divisors. Some­one interested in pursuing this further is in for a literature search, for no spe­cific references are given, and de la Val­lee-Poussin, though listed in the index, is not listed in the set of references at the end. Because of the age of the work, Mathematical Reviews and MathSciNet will be of no help. And even Zentralblatt would fail if it hadn't

88 THE MATHEMATICAL INTELLIGENCER

recently incorporated the earlier mate­rial from the Jahrbuch. So tracking down something like this is not trivial. On page 1 15, Havil cites a 1990 result of "Maier and Pommerance," in a well­known paper on gaps between primes (Trans. Amer. Math. Soc. 322 (1) (1990) 201-237), but he does not give this reference, and a reader not know­ing something of the result would have a problem: there are lots of Maiers in mathematics-this one happens to be Helmut Maier-and Havil misspells the name of the second author. (The sec­ond author is, of course, Carl Pomer­ance.) The result, which Havil justifi­ably calls stupendous, is that

lim [ CPn + 1 - Pn) (log log log PnY 7 n->x [(log Pn)(log log Pn)(log log log log Pn)]

2: 4ey!c,

where c = 3 + e-c and Pn is the nth prime. (Incidentally, as stated in the book, the right-hand side is incorrect; it should be 4e�'/c.) I conclude that even in a general book of this sort, some endnotes pointing the reader to the sources would be helpful.

Now, at the risk of appearing petty I'll bring up something even less im­portant but still something of a prob­lem. Recently I reviewed another Princeton University Press book and commented that the use of both serif and sans serif type faces gives "the pages rather a strange look" Another reviewer was less generous and said it looked like an explosion in a font fac­tory. There's no such problem here. The type is dignified and carefully set, to the point where the typesetter has fastidi­ously made sure that in every differen­tial, the "d" is in roman and the "x" or "y" is in italics! But one could have hoped that the Press would bring on board a sufficient number of proof­readers and fact checkers that we would not find formulas misstated, mis­leading, or not corresponding to the ac­companying figure (see pages 44, 7 4, 94, 100). The errors cited are easily cor­rected, but when one fmds these slips from time to time, it makes a person wonder what might be wrong in the ma­terial that is not so easily checked and with which one is not familiar. Diacrit-

ical marks are only sometimes present (never, apparently, on Erdos and, in the case of the oft-cited de la Vallee­Poussin, almost always with a grave ac­cent instead of the correct acute ac­cent). Authors are not the best people for fmding these things; they have spent too much time with the manuscript. It is clear from the text that this author regards Euler as a mathematical hero, yet in citing Euler's Introductio, prob­ably his masterpiece, the title is mis­spelled and the date of publication is wrong (page 15). Other authors' names are misspelled-Montucla, Bernoulli (and why does Jacob Bernoulli have separate entries in the index?). There are the usual typographical errors and little slips in grammar. Those one ex­pects. But some of the errors I have cited are disturbing to the careful reader, and a press with the prestige of Princeton should not be placing some­thing like this in print without more careful checking.

For all that, this is a most enjoyable book-full of good historical asides, truly beautiful bits of mathematics, and clear exposition. I highly recommend it-but I hope that subsequent editions (including the paperback) can include some corrections and more references.

Department of Mathematics and Computer

Science

Santa Clara University

Santa Clara, CA 95053-0290

USA

e-mail: galexand@math.scu.edu

Schliisseltechnologie Mathematik Einbl icke in aktuelle Anwendungen der Mathematik Hans Josef Pesch

STUITGART/LEIPZIG/WIESBADEN: B.G. TEUBNER

VERLAG.

1 AUFLAGE 2002

1n the series Mathematik fur lngenieure und Naturwissenschaftler. 1 85 PAGES. € 22 90 ISBN 3-51 9-02389-X

REVIEWED BY GERHARD BETSCH

The title means what the author claims: Mathematics is a key

technology of our future. It permeates more and more our everyday life. Nevertheless, the number of students of mathematics and of subjects with a strong mathematical background is decreasing-at least in Germany. How can we change this develop­ment?

The intended readership consists of high school graduates, high school teachers, and freshmen in subjects with a strong mathematical "flavour" or "vein." But people from all profes­sions with a sufficient interest in math­ematics will profit considerably from this book

I give a sketch of the contents.

• John Bernoulli's problem of the brachistochrone (1696) and the Fer­mat Principle. The isoperimetric problem and its roots in antiquity. More generally: Calculus of varia­tions, its origin and development.

• Problems of optimal control. Pontr­jagin's Maximum Principle. Differen­tial Games. The key role of modem numerical mathematics in solving control problems.

• Problems and methods of Optimal Control and Numerical Analysis in connection with the development of the ISS (International Space Sta­tion). Return of a space shuttle into the at­mosphere. Experiments in space; the Genesis mission. Lagrange points in gravita­tional fields.

• Automatic steering of aircraft. Com­pensation, or avoiding of "micro­bursts."

• Optimal planning of robots. Optimal control of chemical processes. Con­trol problems in economics. Computation in real time.

Features of this book are an abun­dance of solid historical information, and very informative sections on tech­nical problems involved. The text is supported by very instructive high­quality illustrations.

For readers who do not skip the for­mulas, the author offers carefully se­lected (non-trivial) problems.

The author is a professor of engi­neering mathematics in the University of Bayreuth (Germany). The book be­longs to a series of textbooks for fu­ture engineers and scientists.

A translation of this work into Eng­lish would be desirable.

Furtbrunnen 1 7

7 1 093 Wei I im Sch6nbuch

Germany

e-mail: Gerhard . Betsch@t -online.de

Sync-How Order Emerges from Chaos in the Universe, Nature, and Daily L ife Steven Str-ogatz

QG, NEW YORK, HYPERION BOOKS, 2004, $US 1 4 95

ISBN 078688721 4

REVIEWED BY A . W . F . EDWARDS

''Being obliged to stay in my room for several days," wrote the

feverish Christiaan Huygens in Febru­ary 1665, "I have noticed an admirable effect which no-one could ever have imagined. It is that my two newly-made [pendulum] clocks hanging next to each other and separated by one or two feet keep an agreement so exact that the pendulums always oscillate to­gether without variation. After admir­ing this for a while I finally realised that it occurs through a kind of sympathy: mixing up the swings of the pendulums I found that within half an hour they al­ways return to consonance."

A third of a millennium later the de­signers of London's new Millennium Bridge for pedestrians-claimed as the world's flattest suspension bridge­were treated to another "admirable ef­fect" which perhaps they should have imagined. As the enthusiastic crowd crossed it on opening day its impercep­tible swaying motion caused walkers to adopt an unconscious sailor's roll so as to keep their balance. But of course they did so all together and, as with a child on a swing, positive feedback did the rest. The bridge was closed. Brian Josephson, Nobel Laureate in Physics in 1973, was first with the explanation.

Both these stories, and many more, are relayed in Sync, an eloquent grand tour of synchronous behaviour in physical, biological, and human sys­tems. Divided into three parts, Living Sync, Discovering Sync, and Explor­ing Sync, it would have been easier reading if Discovering Sync had come first. Sympathetic pendulums and swaying bridges are more accessible images than firing brain cells and cou­pled oscillators.

The physical chapters in Discover­ing Sync are particularly rewarding, partly because they are lighter on au­tobiographical detail. Strogatz almost makes quantum theory and Josephson junctions comprehensible.

Living Sync is burdened with a som­niferous chapter "Sleep and the daily struggle for sync" about circadian rhythms and experiments in which people spent long periods isolated from any information about the pas­sage of time (but no mention of the Po­lar Eskimos and their dayless winters).

In Exploring Sync the author dwells on the inappropriateness of linear models in human affairs-hardly a new thought, but one which needs constant repetition at a time when universities, at least in Europe, are increasingly af­flicted by intervention based on the bureaucratic assumption that each is merely the sum of its parts.

A penultimate chapter, "Small world networks," is an interesting account of the properties of communication net­works and how their structure-vary­ing between regular and random-in­fluences the path-lengths between typical nodes. One almost expects a further digression into population ge­nealogies and Bayesian probability net­works, but it is hard to fit network sto­ries into a book on synchrony.

There is no mathematics. "To con­vey the vitality of mathematics to a broad spectrum of readers, I've avoided equations altogether, and rely instead on metaphors and images from everyday life to illustrate the key ideas." With considerable success.

Gonville and Caius College

Cambridge CB2 1 TA, U .K.

e-mail: awfe@cam .ac.uk

© 2005 Spnnger Sc1ence+Bus1ness Medta, Inc . Volume 27, Number 1 , 2005 89

Mathematics Unl im ited-2001 and Beyond

dents to learn more and focus on what further away from the relevant prob­are really the difficult issues rather lems in the natural sciences and soci-

Bjorn Engquist and

Wilfried Schmid, editors

NEW YORK, SPRINGER-VERLAG, 2001 1 237 PAGES,

HARDCOVER US $59.95 ISBN: 3-540-6691 3-2

REVIEWED BY PETER W. MICHOR

Expectations were high at the turn of the century for publications that

would describe the state of mathemat­ics by looking into the future. Hilbert's problems at the Paris congress set the example. This book is the contribution from Springer-Verlag.

This is an anthology of 63 articles in­cluding five interviews, by a range of well-known mathematicians and other scientists. Among these articles one finds overviews over particular fields stressing open problems, descriptions of neglected themes, and essays on the relation between mathematics and soci­ety. Two themes that are interconnected in many ways in many of the articles are "applications" and "computing."

Let me start by quoting. 1

The authors remember from their high school days how they had to learn computing sines and cosines from ta­bles. The calculator was there, actu­ally we all had our own, but the edu­cational programs had not yet adapted to the new technology. Basi­cally, the same effect hits the univer­sities when we ignore the existence of Matlab, Mathematica, Maple, and similar software, which solves virtu­ally any exercise in basic calculus and linear algebra. Sometimes it ap­pears that many teachers in mathe­matics regard such software as a threat towards their profession. That is in our view a tragic misunder­standing; proper application of soft­ware would allow us to increase the level of calculus, by enabling the stu-

than wasting their time on repetitive trivialities. [ . . . j It is the author's opinion that the undergraduate edu­cation at most universities is out of phase with the modern professional application of mathematical models. [ . . . } Not only pure mathematicians seem to neglect the importance of do­ing computer based mathematics and the need to adapt the education ac­cordingly. Also in classical subjects, like physics and the geosciences, the role of mathematics and computers are kept at a moderate level with lit­tle impact on the culture or courses. Some trivial observations explain the slow progress at incorporating mod­ern computing tools. Students are sent like ping-pong balls between univer­sity buildings. Each building has its own traditional culture and theo­ries-and its own budget that must be protected. Each building gets its share of courses in a program, and the pro­fessors in the building put in much effort to preserve the traditions of their particular subject. The result is a set of 'pure' subjects and strong con­servatism-two characteristics that are not well correlated with a multi­disciplinary and rapidly developing technological world.

In basic mathematics, it seems that we really teach our students something that can be compared to doing astron­omy without telescopes, or doing biol­ogy without microscopes. Nobody should come out of basic mathematics instruction at universities without flu­ency in at least one general-purpose computer algebra program. The fol­lowing quotation reinforces this opin­ion2:

Do you have a message that you would like university mathematicians to hear? Of course, there is a very clear message. I am worried that mathe­maticians are moving further and

1 Langtanger, Tveito: How should we prepare the students of Science and Technology, p. 8 1 2 .

2Mathematics: From the outside looking i n . Achim Sachem interviewed b y V.A. Schmidt.

3After the "Golden Age": What next? Lennart Carleson interviewed by Bjorn Enquist.

4Bailey, Borwein: Experimental mathematics: recent developments and future outlook, p. 55.

5H. Cohen: Computational aspects of number theory, pp. 301-330.

90 THE MATHEMATICAL INTELLIGENCER

ety. In our core scientific engineering areas, we are continuously involved with mathematics, but there are very few mathematicians working at the labs. When we go to the universities, we find that even there the mathe­maticians are not dealing with the questions we confront. Why? A math­ematician's response to a very prac­tical problem is often that it is too complicated. A mathematician will often want to ignore a particular con­straint and limit the discussion to a specific problem that captures the essence. This approach is not helpful at all to the engineers. Furthermore, mathematicians seem to feel that there are enough beautiful problems within mathematics.

Also Lennart Carleson3 stresses the importance of maintaining "all the con­tacts with the neighboring applica­tions, not only with computer science but also with physics and scientific computation and chemistry and biol­ogy and so on."

Research in pure mathematics can benefit a lot from using computers. Consider the following remarkable for­mula4 "whose formal proof requires nothing more sophisticated that fresh­man calculus:

"' 1 ( 4 2 1T = :;?;0 16k 8k + 1

-8k + 4

1 1 ) --- ---8k + 5 8k + 6

This formula was found using months of PSQL computations, after corre­sponding but simpler n-th digit formu­las were identified for several other constants, including log(2)."

This theme is taken up in another article, 5 which describes in 22 gems the uses of computational techniques and of computer experiments in number theory, in particular in class field the­ory, and in arithmetic geometry, e.g.,

for the construction of tables of ellip­tic cuiVes of given conductors. This ar­ticle concludes with 12 challenges for the twenty-first century, including the Birch and Swinnerton-Dyer conjec­ture, which is also one of the seven mil­lennium problems.

Only 1 1 of the 63 articles are de­voted to pure mathematics alone. One of them6 describes an intriguing new countable class (ff of complex numbers called periods which contains all alge­braic numbers. The elementary defini­tion of a period says that its real and imaginary parts are given by the values of absolutely convergent integrals of rational functions with rational coeffi­cients, over domains in !Rn given by polynomial inequalities with rational coefficients. The number 7T, logarithms of algebraic numbers, and values of Riemann's zeta function at integers :::::2 are periods, whereas e, 117T, and Euler's constant y are conjectured not to be periods. It is still an open problem to exhibit at least one number which is not a period. The question is to find an algorithm whether or not two periods are equal, and the conjecture is that one may pass between two integral representations of a period by using only additivity of the integrals, the change of variables formula, and the fundamental theorem of calculus. Pe­riods have also an important role in the theory of £-functions and motives.

This book gives a wide overview of different aspects of the possible future development of pure mathematics, also by posing conjectures, on wide­open fields in applied mathematics and other sciences where mathematics plays or should play an important role, and on questions of education and the usefulness of mathematicians for an­swering questions in engineering and natural, economic, and social sciences. It is very interesting reading. I hope that it will have some impact on the way in which we educate young math­ematicians so that they will be able not only to push the frontiers in mathe­matics itself in the future (where the accomplishments and prospects are bright enough) but also to answer ex­pectations from outside mathematics.

6M. Kontsevich, D. Zagier: Periods, pp. 771-808.

Fakultat fUr Mathematik

Universitat Wien

Nordbergstrasse 1 5

1 090 Vienna

Austria

e-mail: peter.michor@esi .ac.at

The Mathematical Century by Piergiorgio Odifreddi

Arturo Sangalli, translator;

Foreword by Freeman Dyson

PRINCETON, PRINCETON UNIVERSITY PRESS 2004.

xx + 204 pages. US $27.95 ISBN 0-691 -09294-X

REVIEWED BY GERALD L. ALEXANDERSON

To choose and explain to a general audience the thirty most important

mathematical problems solved in the twentieth century takes courage. Pro­fessor Odifreddi has done it here with remarkable clarity and elegance. He recognizes the difficulty of the task, particularly if one tries to do it in 180 pages. The challenges are: (1) the ab­straction of modem mathematics and the difficulty of explaining the mean­ing of the theories to the non-special­ist; (2) the vast amount of mathe­matics produced, particularly in the second half of the century; and (3) the fragmentation of mathematics into subfields. These difficulties do not de­ter him, and he exhibits an amazing grasp of the various streams of modem mathematics. Further, he largely suc­ceeds in showing how the various problems are related.

The author is never without opin­ions. On computers: "As is often the case with technology, many changes are for the worse, and the mathemati­cal applications of the computer are no exception. Such is, for example, the case when the computer is used as an idiot savant, in the anxious and futile search for ever larger prime numbers. The record holder at the end of the twentieth century was 26,972.593 - 1, a number that is approximately 2 million digits long." On mathematics: " [A ma­jority of the sub fields of mathematics] are no more than dry and atrophied

twigs, of limited development in both time and space, and which die a nat­ural death." He claims that the disci­pline "has clearly adopted the typical features of the prevailing industrial so­ciety, in which the overproduction of low-quality goods at low cost often takes place by inertia, according to mechanisms that pollute and saturate, and which are harmful for the envi­ronment and the consumer. The main problem with any exposition of twen­tieth-century mathematics is, therefore . . . to separate the wheat from the chaff, burning up the latter and storing the former away in the barn."

When readers recover from trying to decide whether their own contribu­tions would be burned or stored, they can go on to Chapter 1, a brilliant es­say on the foundations of mathematics, before they explore the thirty prob­lems. Here the author shows his own mathematical predilection: mathemati­cal logic. In the seventeen pages of this introductory chapter the author takes us from Pythagoras to Leibniz, to Frege and Cantor, Russell, Zermelo, and Fraenkel, then on to Grothendieck, Godel, Bourbaki, Eilenberg and Mac Lane, Church and Rosser, Kleene and Scott. It's a fast tour but leaves the reader with a good idea of how to re­late sets, functions, categories, and lambda calculus, a treatment that even someone not much interested in foun­dations can enjoy. This chapter is something of a tour de force.

Now we come to his choice of prob­lems. With such a list one can never please everyone. He has relied strongly on Hilbert's famous list of twenty-three problems described at the 1900 Paris Congress, as well as those problems solved by Fields Medalists or by win­ners of the Wolf Prize. This is probably as rational a plan as any in searching for the "top 30" problems, but it also raises questions. Wiles's solution of the Fermat problem is included in spite of the fact that Wiles did not receive a Fields Medal because his age exceeded by a year the traditional cut-off age of 40. There is another question we could raise about using the Fields Medals as a criterion: some have suggested that

© 2005 Spnnger Science+Bus1ness Media, Inc , Volume 27, Number 1 , 2005 91

••• National Research Council Canada

CIST/'s

Conseil national de recherches Canada

C I ST I i s o n e of the com p reh e n s i ve sou rces for a l l S p r i n ger-Ver lag jou r n a l s , p rovid i n g wo r l d -wide doc u me n t de l i very serv ices .

You need the document just in time. We have it just in case.

Document Del ivery �

Contact u s :

1 -800-668- 1 222 c i sti . n rc . g c .ca info.c isti@nrc-cnrc .gc.ca

92 THE MATHEMATICAL INTELLIGENCER

certain branches of mathematics have been favored by Fields committees and some others have been ignored.

Choosing the most important thirty inevitably raises the "41st chair" ques­tion. L'Academie Fran<;aise chooses for membership the "40 immortals," and that makes people wonder who would occupy the 4 1st chair if there were one. From each of the past four centuries there were intellectual lumi­naries who did not make it: Descartes, Rousseau, Zola, and Proust are in that august company. So what are the prob­lems that might have occupied the 31st or 32nd slots? Or should others replace some of those already on the list? That could provide mathematical dinner party conversation for years.

Let's look at the problems that did make the author's list. They're divided into three categories:

Pure Mathematics

1. Analysis: Lebesgue Measure (1902); 2. Algebra: Steinitz Classification of

Fields (1910); 3. Topology: Brouwer's Fixed-Point

Theorem (1910); 4. Number Theory: Gelfond Trans­

cendental Numbers (1929); 5. Logic: Godel's Incompleteness

Theorem (1931); 6. The Calculus of Variations: Doug­

las's Minimal Surfaces (1931); 7. Analysis: Schwartz's Theory of Dis­

tributions (1945); 8. Differential Topology: Milnor's Ex­

otic Structures (1956); 9. Model Theory: Robinson's Hyper­

real Numbers (1961); 10. Set Theory: Cohen's Independence

Theorem (1963); 1 1 . Singularity Theory: Thorn's Classi­

fication of Catastrophes (1964); 12. Algebra: Gorenstein's Classifica­

tion of Finite Groups (1972); 13. Topology: Thurston's Classification

of 3-Dimensional Surfaces (1982); 14. Number Theory: Wiles's Proof of

Fermat's Last Theorem (1995); and 15. Discrete Geometry: Hales's Solu­

tion of Kepler's Problem (1998).

Applied Mathematics

1. Crystallography: Bieberbach's Sym­metry Groups (1910);

2 . Tensor Calculus: Einstein's Gen-

eral Theory of Relativity (1915) ; 3. Game Theory: Von Neumann's

Minimax Theorem (1928); 4. Functional Analysis: Von Neu­

mann's Axiomatization of Quan­tum Mechanics (1932);

5. Probability Theory: Kolmogorov's Axiomatization (1933);

6. Optimization Theory: Dantzig's Simplex Method (1947);

7. General Equilibrium Theory: the Arrow-Debreu Existence Theorem (1954);

8. The Theory of Formal Languages: Chomsky's Classification (1957);

9. Dynamical Systems Theory: The KAM Theorem (1962); and

10. Knot Theory: Jones Invariants (1984).

Mathematics and the Computer

1. The Theory of Algorithms: Turing's Characterization (1936);

2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950);

3. Chaos Theory: Lorenz's Strange At­tractor (1963);

4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976); and

5. Fractals: The Mandelbrot Set (1980).

One could easily argue about which problems belong where in the list (knot theory as part of applied mathematics?), but the author describes the distinction he makes between pure and applied mathematics thus: "Mathematics, like the Roman god Janus, has two faces. One is turned inward, toward the hu­man world of ideas and abstractions, while the other looks outward, at the physical world of objects and material things. The first face represents the pu­rity of mathematics, where the attention is unselfishly focused on the discipline's creations, seeking to know and under­stand them for what they are. The sec­ond face constitutes the applied side of mathematics, where the motives are in­terested, and the aim is to use those same creations for what they can do."

Overall the exposition is extraordi­narily fine. The context of a problem is set and the result is explained in terms as simple as the subject allows. Con­tributors at all levels of the solution are introduced, and if any were awarded a

Fields Medal or a Wolf Prize, that infor­mation is included. The language at times is almost poetic. Not having avail­able to me the original Italian edition, I cannot say whether the elegance of the language is primarily the contribution of the author or of the translator. Of course, occasionally one becomes aware that it is a translation. For exam­ple, Bishop Berkeley's classic descrip­tion of infinitesimals as "ghosts of de­parted quantities" when passed from English to Italian and back to English be­comes "ghosts of deceased quantities." It's not as good. Nor is it even correct!

There are a few hints that English is not the native language of the author or the translator, but they are rare: "a regular polyhedra" (page 72), "the best of the two" (page 88). The description of the Mobius strip with a top spinning on the surface (page 78) is a delightful device for explaining orientation, but it could be expressed with less ambigu­ity. It could have been made more clear by inserting parenthetically what is meant by "traveling once along the strip" or even by drawing a suitable figure. The Mordell Conjecture is cred­ited to Leo Mordell in various refer­ences and in the index when surely "Louis Mordell" was intended. The au­thor consistently attributes A. 0. Gel­fond's work to Gelfand. Hassler Whit­ney's name is usually spelled correctly but is misspelled on page 70. David Rodney Heath-Brown may be called "Roger" by his friends, but for the rest of us it looks odd. I would have re­ferred to I. R. Shafarevich, not "Igor." Perhaps the author is closer to some of these giants in mathematics than this reader. But this is quibbling when so much of the text is full of interesting insights and so eloquently expressed.

A person could ask about the in­tended audience for the book The au­thor divides the references into two parts: "for general readers" and "for advanced readers." When he says "gen­eral readers" he doesn't mean some­one without any mathematical back­ground. Though the author does a masterful job of describing difficult mathematics in accessible terms, still, sentences like the following are not for the faint of heart: " . . . W eil proposed his own conjecture, a version of the

© 2005 Spnnger Sc1ence+Business Med1a, Inc , Volume 27, Number 1 , 2005 93

Riemann hypothesis for multidimen­sional algebraic manifolds over finite fields, which became known as the Weil conjecture. It was proved in 1973 by Pierre Deligne [whose] proof was the first significant result obtained through the use of an arsenal of ex­tremely abstract techniques in algebraic geometry (such as schemas and l-actic cohomology) introduced in the 1960s by Alexandre Grothendieck. . . . "

As a lagniappe, the author includes four open problems for the 21st century:

(1) Arithmetic: The Perfect Numbers Problem (300 Be);

(2) Complex Analysis: The Riemann Hypothesis (1859);

(3) Algebraic Topology: The Poincare Conjecture (1904); and

llroled SWin Postai Servlce Statement of Ownership, Management, and Circulation · - -

Mathematical Intelligencer

( 4) Complexity Theory: The P = NP Problem (1972).

The last three are on the Clay Insti­tute's list of million dollar Millennium Prize Problems. In addition to the other Clay Institute problems, there are a few others one might have expected to see-the Goldbach Conjecture, the Twin Primes Conjecture, for example. They're famous and they're still at­tracting people to work on them.

In addition to the short list of sug­gested readings and a very helpful in­dex, the author includes chronological lists in a concluding chapter: Hilbert Problems, Fields Medalists, Wolf Prize winners, Turing Awardees, Nobel Lau­reates-but only those cited in the text. It would be interesting to see com-

plete lists, if only to see what the au­thor has not found to be worthy of in­clusion. For example, the first Fields Medalist on the list is Jesse Douglas, who won in 1936 and is cited earlier in the text as the first Fields Medalist, when Lars Ahlfors, who also won that year, was probably the first, at least al­phabetically. His contributions just didn't make it into the book

We should, however, be thankful for what we get, a truly gripping account of big problems of the twentieth century.

Department of Mathematics & Computer

Science

Santa Clara University

Santa Clara, CA 95053-0290

USA

e-mail: galexand@math.scu.edu

13 PubhcaiiOn

Tille Mathematical lntelli encer 1-4 luue Oale lor Cirw!mn � llelow Falll004

3 FllngO.bt

10/04

.. ExtentandNabnof�

A-. No. Copin &ell taue No. Copln of Single 11au1 Durlngf>rKedlngt2MontM PublllhedNMrfttloFiin9 Date

3500 3500 4 U.. FNQUa"'Cy

Quarterly 5 NurrtMtrof laSUN PuiJAshed Annually S Annuai S�Pnce

2687 4 $81�0

7 � Melling � cf KIIOWII � of �llo!l(Not pmter} (St1Ht r;ity, OOIItlty, stal8, arod ZJP.4) Springer Sc1ence + Business Media, Inc 233 Sprmg Street New YortZNY 10013-1522

8 Compleia u.lllngAdchaa of HNdquatterl. or General lluslneH OIIIca of Pubbhef (Notptlf!l8rj Springer Science + Business Media, Inc 233 Spring Street New Yor� NY 10013-1522 :.:���"':i::JofPubllsllw,€dilor,aodM!In!!!ln!!Edllor!'f?onotiN.,..b!!n:'sJ Springer Science+ Business Media, Inc. 1 0 1 Philip Dnve, Assin1pp1 Park

Norwell, MA 0206 1 - 1 6 1 5

E<!itor (Nll,., sn<f r:ampJets malllng arl<li'fl&} Chandler Davis, New College, University ofToronto, Dept. of Mathematics, Toronto, ON M5S 316

Man.gmg Ecil<>r (N- -<lt1trlpleie mll#lng 6/ddtu&}

Springer-Verlag New York, LLC, 233 Spring Street, New York, NY 10013-1 522

""""' "'� Carol Sabatino Telephone

2 1 2-460- 1500, X )05

IO <Mner (Do /IOI INw blllnk fltl>e � IS O'WFI«<bya cc:vpon�IJOfl, g<ve thenamellldaddr&ssofftleCOfPOiliiiOn ltllmf!d!IJ/81y loliowf!dbytt>e names Midaddte.s.ses ofaN stocl<holderS OW1llfiQ or /toldrng 1 percenl or moreofthetola!amountofs/ocl< lfnotoM�e<fby a cnrpora/Jon, g""" /h� names and 2ddresses of/he ftldovldualawn$r.S If owned by a patlnerslup orotherunmcorporated film !l'"" rl< nam;) �nd �ddress as well as /�ose ol eachftldov!dualawtMli'.Jfflle puiJ/icallOO IIl pub/Wl� by1{)0(1prot;/"'llamzaf/Ofl, !;ll""'ts namean<faddress )

Spnnger Sc1ence + Busmess Med1a, 8 V

t t Known Bondholderi, Mortgagees, aodOthe< Securrty Hoiden Ownongor Hokltfiii 1 Peroenl or More oi Tota1AmountoiBonds.�.or Other Secunbes W none, checl< llox

Springer Science + Business Media, B V

T1erga11enst• asse 17

D-69121 Heidelberg, GERMANY

0None

Com lete�lln Address T1ergartenstrasse 1 7

D-691 2 1 Heidelberg, GERMANY

12 TaxStatus(Forcnmplehon bynonprofito..pan<tatrorrs iJUihoriled tomaN atnonprofltrates} (Check one) The purpose runciiOO,and nonproft!status oflhtaorgarnzabonaodthe exempt status forrederaloncome tall purposes

� �= ��a��e:;;g��must&Ubmtl expl•nabM ol<:lrange willr lhis statemen/}

PSFonn J526, 0ctober 19£!9 (Seelnslnict•onson Re�erse}

94 THE MATHEMATICAL INTELLIGENCER

(2) Paklln-CtluntySut.cnp�on� St.tedon formlM1 (lnQude�af'IIOIMdrndl�r;op!IH)

{3) Sale$Through DealersanciCaniefs,StreeiVendotl,

(4) OtherCiusesMdedlhroughtheUSPS

c. Totai PaodarodlorRequested Cn•:.ulatlon {Sum oll.ib (IJ, (2},(3),afld(t))

.,_ Olst1bubon (1) outside-Countyas stated on form3541 ...... � (2) �asSialedonform)So41 -· ary,and (3) OtherCiasses Maded llvtlughlhe USPS """"''

e Fn•e O.stnb<tbon Outside llwl Mil� (C�prod>fti-/IINtl$}

I Tolalfr""DislrJbubon(Sum of l5d andl5e}

g TOiai Dislnbut!on (Sum of15c and15f}

o Tclal(Sumofl/ig -h} 1 Peft:en!Palllao<llor RequesledCn<:Uiallcn

(15c. <fMdedby 15g limes IOOJ

2687

2687 2687

157 157

53 53 210 2 1 0

2897 2897

603 603

3500 3500

92 75 92.75

� Pubhcallon req"'red I pooled onlhe-,t-__,-,

,---'""""oltnos pubhcal•on 1e Put>Ncabono! Stateme:jf"erslup Wmter 2005

O PuNocahcn not r"((uued

_jj - _ - , 17 Stgnature andz.v·;ublosher,�anager, o.-Owner

Instructions to Publishers CompleteaOOftle onecopyollll!Siormw.th yourpostmasterannualty on Of beforeOctobeo' I Keep a copyoflhecompletedftlnn lor your records

lncaseswherelhe stockhokler o•secun�yhok!erosa lruslee, .,cludeonrteiTI$10alld 1 1 thenarneoflhe�ono.-corpor.rllonfor whom the trustee os acting Also onclude the rl30mes and addre$11eS otondl...-.d\lllls vmo are S1ocllholders who own o.- hold 1 percent ormoreot the1o1al amount otllonds, IIXM1gages, orothar secunttesofthapvtllishongcorporatlon ln11em 1 1 dnone,cOOclt the bo• Usel>ank st>eets ofmore space os req"'red '"' 8esure lolumosh a�c"':ula110n onformab0ncalied foronotem15 Freearcutalk>nmuslbeshownon rtems 15d, e, aod f

ltem 15h Copoesnot Dostrlbuted, musltnctude (1) �stand copii!Songlnall,' atated on Form 35<11, and retumedtothe pv!>oshel (2)estomatedreturnsfromnewsagents,and(3),copresforol'liceuse, let'kwanl, spooled, andallother copiea nol doslribule<l

" the pvbhcabon had Penodocals aultlOnzabOn as a genenol Of raqvesterpublicahon, th•s Slalemenl ofOwnen;hop, Managemenl andCorculabon musl be publoshed, rtmuslbe pooted on any ossoo on Octobeo'or, d lt>e pvblocauonosool pu!>JSheddunng October lhefoi'Siossuepnntedal\erOctober

tnotem 16 >ndlcatatnedataol!he ossue onwhdlth•s StatementotOwner•h•pWIIIbe f>UbiJShed

ttem t7 mustbe stgned

Fallure toflle orpublrsh a statementofownenlrip maylndto suspenslon ofPerlo<llca/saurhorlzatlon

PSFo1111 3526, 0ctober 1999 (Reverse)

k1flrri.IQ.iij,i§i Rob i n W i l son I

The Ph i lamath' s A lphabet-G

Galois: Abel's work on the unsolv­ability of the general quintic equa­

tion was continued by the brilliant young French mathematician Evariste Galois (1811-1832), who determined (in terms of the so-called Galois group) which equations can be solved by radicals. Galois had a short and tur­bulent life, being sent to jail for politi­cal activism. He died tragically in a duel at the age of 20, having sat up the pre­vious night writing out his mathemati­cal achievements for posterity. Gauss: Carl Friedrich Gauss (1777-1855) presented the first satisfactory proof of the fundamental theorem of al-

Galois

Gauss

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6AA, England

e-mail: r.j.wilson@open.ac.uk

gebra (that every polynomial equation has a complex root) and made the first systematic study of the convergence of series. In number theory he initiated the study of congruences and proved the law of quadratic reciprocity. He also showed that a regular n-sided polygon can be constructed with straight-edge and compasses whenever n is a Fermat prime (such as 17, as shown on the stamp). Gazeta mathematica: The Roman­ian monthly Gazeta mathematica was first published in 1895. With its aim of developing the mathematical knowl­edge of high-school students, it has had an enormous influence on mathemati­cal life in Romania for many decades. Gerbert: Gerbert of Aurillac (938-1003) trained in Catalonia and was probably the first to introduce the Hindu-Arabic numerals to Christian Europe, using an abacus that he had designed for the purpose. He was crowned Pope Sylvester II in 999. Goldbach's conjecture: In 1742 Christian Goldbach wrote to Leonhard

Gazeta matematica

Gerbert

96 THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc

Euler conjecturing that every even in­teger (> 2) can be written as the sum of two prime numbers. Although this remains unresolved, a partial result of Chen Jing-Run (1966), shown on the stamp, implies that every sufficiently large even integer can be written as the sum of a prime number and a number with at most two factors. Gregorian calendar: The Julian cal­endar of 45 BC had 3651/4 days, which was eleven minutes too long. In 1582, Pope Gregory XIII issued an edict that corrected the over-long year by re­moving three leap days every 400 years, so that 2000 was a leap year but 2 100, 2200 and 2300 are not. The Gre­gorian calendar was quickly adopted by the Catholic world, and other coun­tries eventually followed suit: Germany in 1700, Britain and the American colonies in 1752, Russia in 1917, and China in 1949.

Goldbach's conjecture

Gregorian calendar