46 C O M P L E X I T Y © 1998 John Wiley & Sons, Inc.

Reviewsbook & software

D ehaene’s book addresses the math-ematical capabilities of the humanbrain. Its theme is that the ability

to do mathematics is engraved in the ar-chitecture of our brains and that this ar-chitecture is the result of biological evo-lution.

According to the author, the brain re-cruits cerebral areas that evolved forvarious purposes but that are more orless suitable for mathematics. Unlike acomputer, the brain works by associa-tion of ideas. This organization of men-tal representations is both a strength anda limitation. Dehaene believes that theseconstraints need to be recognized in theworld of mathematics education. At theend of the book, he also argues that theevidence from psychology and physiol-ogy supports a particular position in thephilosophy of mathematics, namely, amodern variant of intuitionism.

The “number sense” mentioned in thetitle is a brain function particularlyadapted to dealing with magnitude. Inspeaking of a number sense, there is thequestion of the number system used.Mathematicians usually distinguish be-tween the “natural number system,” usedfor counting, and the “real number sys-tem,” used for measurement. Eventhough each natural number may also beregarded as a real number, the functionsof the two number systems are quite dif-ferent. Which mathematical number sys-tem is closer to the system realized in thecircuitry of the brain? The original phi-losophy of intuitionism took for grantedthat the natural number system is pri-mary. It claimed that humans have a di-rect insight into the properties of the

natural numbers and that other math-ematical constructs should be derived asconsequences of this in-sight. The evidence re-viewed by Dehaene leadsto a somewhat modifiedview of this position. Itsuggests that the brains ofanimals and humanscontain an “accumulator”that allows the percep-tion and comparison ofnumerical magnitudes.These magnitudesrepresent naturalnumbers greaterthan zero but,when dealing withnumbers greaterthan three, the ac-cumulator is ananalog device. It al-lows us to countfairly large quanti-ties, but only in anapproximate man-ner. The fact thatthis is approximate gives the number sys-tem of the brain some of the character ofthe real number system, even though itsmain function is counting.

T he author speculates that thisprimitive mental representationnot only underlies our ability to

estimate, but also is the mechanism thatpermits our comprehension of symbolicnumerals, such as Arabic numerals. Ex-periments show that the speed withwhich we can compare Arabic numer-als depends on the size of the numbers

they represent and on the distance be-tween these numbers. This is as if they

were first translated intothe mental representa-tion and then comparedon that basis.

The first part of the bookconsiders the most basicnumerical abilities ofanimals, children, andadults. Some of the mostinteresting material in-volves children. The fa-

mous Swiss childp s y c h o l o g i s tPiaget and othersperformed experi-ments that weresaid to demon-strate that chil-dren lack an un-derstanding of theprinciple of con-servation of num-ber. In one suche x p e r i m e n t ,young children

were shown two rows of marbles withthe same number of elements. When themarbles in one row were spread out inspace, the children would report thatthere were more marbles in this row.Thus, according to Piaget and his school,the numerical sense in children arisesonly later, as a result of development.

The author of the book, however, de-scribes an experiment of McGarrigle andDonaldson that suggests that the resultcomes about because the young childrenmisunderstand the intentions of the ex-perimenter. They may have thought that

A Modern Variant of Intuitionism

THE NUMBER SENSE: HOWTHE MIND CREATES

MATHEMATICSby Stanislas Dehaene

Oxford University Press,New York, 1997, 274 pp., $25.00

(hard)

© 1998 John Wiley & Sons, Inc., Vol. 4, No. 1CCC 1076-2787/98/00000-00

C O M P L E X I T Y 47© 1998 John Wiley & Sons, Inc.

the experimenters would have had noreason to ask the question unless theyexpected a report of some kind of change.This was shown by varying the conditionsof the experiment. “While the experi-menter was conveniently looking else-where, a teddy bear lengthened one of thetwo rows. The experimenter then turnedand exclaimed, ‘Oh no! The silly teddybear has again mixed up everything.’Only then did the researcher again ask thequestion ‘Which is more?’ The underly-ing idea was that, in this situation, thisquery seemed sincere and could be in-terpreted in a literal sense. Since the bearhas messed up the two rows, the adult didnot know anymore how many objectsthere were and hence was asking thechild. In this situation, the vast majorityof children responded correctly on thebasis of number, without being influ-enced by row length.” This is in accordwith the Dehaene’s position that there isa primitive number sense even in veryyoung children.

The second part of the book con-cerns how the brain deals with morecomplicated mathematical tasks involv-ing symbolic systems and arithmeticalcalculation. As we grow up and as ourneed for mathematical skill moves be-yond approximation, we encounter newdifficulties. According to the author, ourbrain “evolved for millions of years in anenvironment where the advantages ofassociative memory largely compen-sated for its drawbacks in domains likearithmetic.” It now “has to tinker with al-ternate circuits in order to make up forthe lack of a cerebral organ specificallydesigned for calculation.”

It is no wonder that many of us reachhappily for a calculator. Dehaene findsnothing wrong in this. He notes that “di-vision and subtraction algorithms areendangered species quietly disappear-ing from our everyday lives—except inschools, where we still tolerate theirquiet oppression.” His recommendationis that mathematics education build onour natural capabilities. It should con-centrate on developing a rich repertoireof mental models of arithmetic based ona variety of concrete experiences.

The third and final topic in the book

is neuropsychology and its implicationsfor philosophy of mathematics. The ear-lier chapters drew a distinction betweentwo categories of arithmetic skills: el-ementary numerical abilities and ad-vanced symbolic and calculational skills.The evidence is that those categories relyon partially separate cerebral systems.The cerebral areas that handle numeralsare partially distinct from those that dealwith words. On the other hand, the areasthat support calculation are those in-volved in the memorization and repro-duction of automatic motor sequences,including verbal sequences. Thus whenwe, as children, struggle to learn the mul-tiplication tables, we say them over andover out loud, and this active repetitionbuilds our calculational ability.

How about more advanced math-ematical reasoning? Apparently littlework has been done on the neural basisof such abilities. However, one studysuggests that the circuits of neurons thatare responsible for algebraic knowledgeare largely independent of those in-volved in arithmetic calculation. In gen-eral, even simple functions of the braincall for the coordination of a number ofdifferent cerebral areas, each making acontribution.

The concluding chapter of the bookconsiders the implications of this pic-ture of brain function for the philosophyof mathematics. The subject of philoso-phy of mathematics is amazingly murky.Mathematics is said to be an exact sci-ence, but what kind of science? There areseveral contradictory answers. Here aresome of them:

1. Platonism: Mathematics exists as ab-stract reality.

2. Formalism: Mathematics is manipu-lation of symbols according to formalrules.

3. Intuitionism: Mathematical objectsare constructions of the humanmind.

Can there really be controversy aboutsuch basic issues concerning the natureof mathematics? Apparently yes.

Dehaene briefly reviews these issuesand then supports a modern form of in-

tuitionism. He states, “The structure ofa salt crystal is such that we cannot failto perceive it as having six facets. Itsstructure undeniably existed way beforehumans began to roam the earth. Yetonly human brains seem able to attendselectively to the set of facets, perceiveits numerosity as 6, and relate that num-ber to others in a coherent theory ofnumbers.” In his view, mathematics isthe result of an evolutionary process thatproduced a brain capable of mathemati-cal insight.

D ehaene argues vigorously againstformalism. Part of his case is basedon the existence of nonstandard

models of Peano arithmetic. Peanoarithmetic is a language for describingthe natural numbers. The standardmodel of Peano arithmetic is the systemof natural numbers. However, there arealso nonstandard models. These consistof the ordinary natural numbers, to-gether with a vast collection of othernumber-like objects that are regarded asbeing larger than every standard natu-ral number. The remarkable thing is thatevery true statement in Peano arith-metic about the natural numbers is alsoa true statement about the objects insuch a nonstandard model.

Thus, when we are using the languageto describe natural numbers, we couldunwittingly be describing a nonstandardmodel. The author regards this as a weak-ness of the formalist point of view. Heclaims, “Ironically, any five-year-old hasan intimate understanding of those verynumbers that the brightest logiciansstruggle to define. No need for a formaldefinition: We know intuitively what in-tegers are. Among the infinite number ofmodels that satisfy Peano’s axioms, wecan immediately distinguish genuine in-tegers from other meaningless and arti-ficial fantasies. Hence, our brain does notrely on axioms.”

This reviewer is not convinced thatthe evidence from neuropsychologypoints so clearly to a particular positionin the philosophy of mathematics. Theargument that our brains evolved to rec-ognize the six facets of a crystal is not adecisive refutation of the Platonist view

48 C O M P L E X I T Y © 1998 John Wiley & Sons, Inc.

For researchers interested in glo-bal biodiversity trends throughthe Phanerozoic Eon (the past

540 million years), a fundamentalquestion has persisted for the pastquarter century: Has natural selectionat the scale of individuals producedthe major features observed in thehistory of Phanerozoic diversity, in-cluding global biotic transitionsamong major faunal and floralgroups? It has been suggested bymany researchers that the answer isno, based on the view that any transi-tions accumulated through naturalselection among individuals would beswamped by more profound, rapidchanges associated with processes atthe species level and higher. This viewwas espoused formally in a temporalframework by Gould [1], on the heelsof more than a decade of research onpunctuated equilibrium. Gould sug-gested that biotic transitions reside inthree hierarchical tiers, the first ofwhich constitutes evolution by natu-ral selection operating through inter-vals of ecological time too brief to beresolved adequately in the fossilrecord. Gould argued that, on thebroader, geological time scales thatcan be observed in the record, evolu-tion at the first tier essentially leadsnowhere and is, instead, overriddenby allopatric speciation at the secondtier, producing punctuated equilibria.In addition, Gould proposed a thirdtier, characterized by major, globalperturbations that produce fairlyabrupt mass extinctions with an av-erage recurrence frequency on the or-der of tens of millions of years. In hisview, mass extinctions have been the

The Great Wall of Evolution

ultimate arbiters of Phanerozoic di-versification because of their roles inundoing and overriding the transi-tions that accrued during the “back-ground” intervalsbetween them.

One of themore conten-tious aspects ofthis hierarchyhas been the de-termination of amechanism(s)to account forthe lack of accu-mulated direc-tional changewithin species(i.e., stasis) atthe second tier,in the face of natural selection oper-ating at the first tier. Why should sta-sis be maintained within lineagesthrough extended intervals of geologi-cal time? Over the years, numerousexplanations for stasis have been pro-posed, but perhaps none are as com-pelling as that suggested by KeithBennett in Evolution and Ecology.Bennett expands on an argument hefirst presented in 1990 [2] to suggestthat stasis is a consequence of perva-sive climatic oscillations, related toperiodic variations in the Earth’s or-bit, that occur on time scales rangingfrom 20,000 to 100,000 years.

Bennett is a paleoecologist whohas conducted extensive analy-ses of distributional patterns

among terrestrial plants during theQuaternary Period (the most recent1.6 million years). Given its proximity

to the present day, it is possible to in-vestigate Quaternary biotic patternswith a high degree of temporal reso-lution compared to that attainable for

more ancient strata; much ofthe data for such studiescomes from soft sedimentcores collected from the bot-toms of present-day lakes.

Bennett points out that these dataprovide unique glimpses at patternsand processes operating at scales re-siding on the cusp of ecological andgeological time and, thus, allow directinvestigation of the relationship be-tween Gould’s first and second tiers.Moreover, given the body of evidencesuggesting that orbitally-forced cli-matic oscillations have been perva-sive throughout the Phanerozoic Eon,it is reasonable to assume, as Bennettsuggests, that such glimpses are rel-evant to the entire history of multicel-lular life, despite our inability to ob-serve these patterns directly throughmost of the record. The heart of thisbook is an extended description andcomparision of Quaternary distribu-tional patterns exhibited by a varietyof plants, animals, and protists thatlived in terrestr ial, marine, andaquatic settings through the Quater-

that there is notion of six in the universethat is independent of the imagination.Nor is the bald statement that we knowthe integers intuitively a conclusive blowagainst formalism. The existence of non-standard models suggests a principle ofrelativity in mathematics that could lendsupport to a formalist point of view.

Much of what is said in the book indi-cates that our intuition is not likely to getthings right the first time. The true na-ture of mathematics remains elusive.

One curious feature of the book mustbe mentioned: There is no picture of theauthor, but there is a picture of his brain.This was taken using magnetic reso-

nance imaging while he performed re-peated subtractions. He went to consid-erable trouble to write this book, and theresult is a valuable overview of recentcontributions of psychology to the un-derstanding of mathematical cognition.Reviewed by William G. Faris, Universityof Arizona.

EVOLUTION ANDECOLOGY:

THE PACE OF LIFEby K. D. Bennett

Cambridge UniversityPress, 1997, 241 pp.

C O M P L E X I T Y 49© 1998 John Wiley & Sons, Inc.

nary. While there is some variationamong patterns exhibited by differentgroups, several common themesemerge:

(1) In response to climatic oscillations,the geographic ranges of species areever-shifting.

(2) The geographic range shifts of spe-cies are generally independent of oneanother. Thus, there is little indica-tion of interdependence among dif-ferent species, and “community”compositions are dynamic throughtime, depending on the ephemeraloverlap in a given location of specieswith preferences for (or perhaps tol-erance of ) the local physical condi-tions.

(3) There has been remarkably little ex-tinction of species in the face of theseoscillations; most species seem quiteresilient to changing climatic condi-tions because of virtually continuousalterations to their geographic rangesthat Bennett likens to the kneadingof dough.

Bennett argues that these attributeseffectively explain stasis. Becausespecies ranges are in a state of con-tinual flux, there is little opportunityfor the development of species inter-actions and their corollaries that serveas hallmarks of natural selection.Rather, the Quaternary history of spe-cies appears to be one of individual-ism induced by the requirement tocontinually adjust one’s geographiclandscape. As Bennett notes, “I f

gradual changes are taking place, con-tinuously across fluctuating envi-ronmental conditions, then they areforced by some process other thannatural selection, because selectionpressures cannot be constantly in thesame direction for long enough toachieve this.”

The prospect of stasis in the faceof ever-changing conditions mayseem paradoxical. However, becauseclimatic conditions are continuouslychanging, species might be viewed asbeing well-accustomed to the flux.This is not to suggest that short-termperturbations cannot produce majorbiotic changes, including extinction.For example, a large-body impact, likethat implicated in the Late Cretaceousmass extinction, is a short-term per-turbation expected to have cataclys-mic effects on the biota. Perhaps thedistinction is that turnover-inducingperturbations must fall outside thepurview of those commonly experi-enced by a species during its evolu-tionary lifetime. Orbitally-inducedclimatic oscillations, however, are ap-parently so pervasive that they are es-sentially commonplace and, if any-thing, have added to the resilience ofspecies.

On this basis, Bennett argues for theaddition of a fourth tier to Gould’s hi-erarchy, between his first and secondtiers. This is the tier of stasis, andBennett suggests that it serves as virtualbarrier to the expression of microevo-lutionary processes on geological timescales.

My one concern with Bennett’s pre-sentation of supporting data is his con-sideration of major groups mainly in iso-lation from one another. For example,there is a subsection on terrestrialplants, another on terrestrial insects,another on terrestrial vertebrates, and soforth. However, it is well known thatsome of the most intricate interactionsamong species occur between speciesbelonging to rather different groups(e.g., insects and angiosperms). It is pos-sible that a more compelling array of in-terdependent temporal range variationswould be discovered by considering dis-tributional patterns of these differentgroups in conjunction with one another.

That concern aside, this is a suc-cinct, readable volume with astraightforward message. It toucheson a pivotal scale of paleontologicaldata that has rarely, if ever, been syn-thesized in this fashion, and meritsthe attention of anyone interested inrelationships among evolutionarypatterns and processes at differentspatio-temporal scales.Reviewed by Arnold I. Miller, Univer-sity of Cincinnati.

REFERENCES

1. S. J. Gould: The paradox of the first tier:An agenda for paleobiology. Paleobiology11: pp. 2–12, 1985.

2. K. D. Bennett: Milankovitch cycles andtheir effects on species in ecological andevolutionary time. Paleobiology. 16: pp.11–21, 1990.

Books Received

Evolutionary Computation: The Fossil Record, David Fogel, IEEE Press, Piscataway, NJ, 1998, pp. 641, $89.95

Impossibility: The Limits of Science and the Science of Limits, John D. Barrow, Oxford University Press, New York, 1998, pp. 256,$25.00 (hard)

Fundamentals of the Theory of Computation, Raymond Greenlaw and H. James Hoover, Morgan Kaufmann Publishers, Inc.,San Francisco, 1998, pp. 336, $42.50 (cloth)

Ecological Scale Theory and Applications, David L. Peterson and V. Thomas Parker, Eds. Columbia University Press, New York,1998, pp. 615, $60.00 (cloth), $35.00 (paper).

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Complexity is a bi-monthly, cross-disciplinary journal focusing on the rapidly expanding science of complex adaptive systems. The purpose of the journal is to advance the science of complexity. Articles may deal with suchmethodological themes as chaos, genetic algorithms, cellular automata, neural networks, and evolutionary game theory. Papers treating applications in any area of natural science or human endeavor are welcome, and especiallyencouraged are papers integrating conceptual themes and applications that cross traditional disciplinary boundaries.

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