I / i e w p o i n t

Poetic Metaphor and Mathematical Demonstration: A Shallow Analogy MIRIAM LIPSCHUTZ-YEVlCK

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chie f Chandler Davis.

J an Zwicky in "Mathematical Anal- ogy and Metaphoric Insight" [Zw] says that understanding a poetic

metaphor feels like understanding cer- tain mathematical demonstrations. She investigates the correspondences be- tween the notion of metaphor primar- ily as it is used in poetry and that of analogy in the development of mathe- matical demonstrations. Although she clearly states that metaphors and math- ematical analogies are not the same thing, she maintains that there are such f\mdamental similarities that they should both be considered as species of "analogical reasoning." She posits that the sense of understanding, the "flash of insight" (the "I get it!", the "Eu- reka moment") on grasping a metaphor or a demonstration, is closely related in the two domains.

Analogy, the drawing on associa- tions, is all-pervasive in our thinking, in our language, and in our creative en- deavors, be they artistic, scholarly, or everyday: concocting a new Italian recipe; racial stereotyping; formulating a legal opinion; making a medical di- agnosis; the "coup de foudre" estab- lishing a romantic link; and so on. As- sociation appears as an essential concept in the Fourier logic represen- tation of brain function proposed by Karl Pribram [Pr], [LY1]. Similes are imbedded in our language, carrying much of our meaning (as proverbs, be- fore they were so displaced by techni-

cal jargon, used to do). Zwicky's article abounds in them: "the field of reso- nance", "lift off the page", "has no pur- chase on", "cede pride of p l a c e " . . . . Hannah Arendt [A] wrote that "all con- ceptual or metaphysical language is ac- tually and strictly metaphorical."

Zwicky argues that metaphor and mathematical demonstration have spe- cial kinship, in that in both, the new in- sights derive from discovery of unsus- pected analogies between facts long known but wrongly believed to be strangers to each other. But this kinship extends to all creative thinking!

I maintain that analogical reasoning, being a generally present feature of thought, can not prove mathematical reasoning any closer to poetry than say- ing that both are thought.

Going beyond this commonality, we follow the divergent aspects in the further use of analogy in the two domains--"Points of Non-Correspon- dence", as Zwicky calls them--and find two different "Languages of the Brain". To me, they look complemen- tary (as the word is used in physics). Mathematical thinking analyzes; it is modelled, perhaps, by digital logic of networks [vN]. Poetical thinking embellishes; it more resembles holo- graphic pattern recognition. Let us look at the dichotomy. Though the two use analogy differently, their symbiosis may suggest a more insightful mode of thought.

Metaphorically Valid? While my husband and I were graduate students at M,I.T. during WWII,

the young Walter Pitts, a brilliant prot6ge of the great mathematician Nor- bert Wiener, offered to deliver a lecture on "Sinkiewicz's Theorem" to an eager audience of graduate students. Pitts gave, as usual, a dazzling per- formance. He proved the theorem moving seamlessly through a maze of lemmas and analogies, with frequent hand-waving to bypass the "obvious." His (almost poetic) presentation was received with applause and admiring comments. The lemmas were profound, the theorem still more so. Even though the lecture had the fornl of---and felt like--a proof, unbeknownst to us it was fiction. (Sinkiewicz was in fact a Polish novelist.)

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Proof vs. Gestalt: Two Modes of Creative Endeavor Poincar~ [P] defined Discovery as "ap- pearances of sudden illuminations, ob- vious indications of a long course of pre- vious unconscious work. All that one can hope for from these inspirations which are the fruit of unconscious work, is the point of departure for such calculations. They must be done in the second pe- riod of conscious work: results must be verified and consequences deduced."

This is the dichotomy: on the one hand Zwicky's visual intuition, the "see- ing as"; on the other, a r igorous proof, which requires analytic validation, a de- rivation from axioms, or an algebraic computation.

Creating a mathematical proof Professor Norbert Wiener's lectures dur- ing my graduate studies at M.I.T. (1943-1947) were revelations of the re- searcher at work. The classroom had blackboards on three sides. Wiener, starting on the left wall, wou ld write the theorem he in tended to prove. He pro- ceeded to accomplish this by assem- bling a chain of valid deduct ions from various lemmas he had previously given in class. Talking to himself as much as to u s - - t h o u g h with an occasional "just a minute, just a m i n u t e " - - h e would pro- ceed from blackboard to blackboard, never losing his thread, though perhaps leaving us well behind. After much trial and error he might exclaim, "Let us do a Ces~ro job on this" or the like. At the end (often close to the last space on the right wall b lackboard, near the door) the proof would be complete.

To the students he appeared to be pulling the proper tools from his toolkit,

as an experimental scientist would reach for instruments. If he thought of his tools in an analogic way, like the "Ces~ro job", this was not the defining feature of his work. It is hardly necessary to insist on the magnitude of Wiener 's discoveries [M]. No doubt he had to perceive un- suspected analogies between facts long known but wrongly bel ieved to be strangers to each other, as Zwicky says; but then came the checking; his lectures both displayed the checks and displayed his conviction of their importance.

Creating a poem Allow me to give some impressions of the exper ience of writing a poem.

This poem was inspired by what I had learned in 1948 about the fate of my Jew- ish classmates at a genteel girls' school in Antwerp. The school had a hateful dis- ciplinarian Principal with a strict rule for the wearing of gloves. I had not thought of her for years�9 Only on recently en- countering an acquaintance with similar background, and seeing her gloves and how she removed them to shake hands, as we were taught, did a chain of asso- ciations start up which led to the poem [LY2]. Some lines from the end:

"This one and that one," They p icked out my former class-

mates. One by one they ga thered them�9 Even the b londe, b lue-eyed Berthe

Perelman Whose name bet rayed her. "A Jewess!"

The Principal s tood by her post at the head of the passageway,

As the girls wa lked by to the trucks waiting outside�9

Did she check to see if the girls wore their gloves?

The writ ing of the p o e m seemed to arise from a consciousness distinct in character from mathematics. After read- ing Jan Zwicky's article I subjected my poem to critical scrutiny and noted nu- merous metaphors . I might have started with a problem: my feelings of guilt for having escaped the fate of my friends; my bur ied wish for revenge on my Principal; the contrast be tween the en- forcement of good manners and the brutality of the N a z i s . . . but t h e n - - n o poem. Yet the p o e m I did write holds all of this "compact in one."

A poem, even a long one, may some- times be grasped as an emotional whole; a lengthy proof can be recon- structed only s tep by step, even if one has first grasped the general thrust.

The thought processes in the writing of a p o e m do not take the conscious form "this reminds me of this reminds me of this . . . ", though they may do so in an at tempt to implement an intu- itive mathematical perception�9 Rather the search for "the right word" for use as poet ic metaphor is often a tip-of-the- tongue phenomenon , and in all cases feels almost the reverse of mathemati- cal puzzling.

Conclusion Associations, "atoms" of analogies, guide our discoveries, be they poetic, mathe- matical, judicial, culinary, amorous, or whatever. Sometimes the "right" analogy is d iscovered by a chain of steps�9 On other occasions it pops up sponta- neously as though by "ghosting" the ob- ject with another stored jointly in an as- sociative hologram, or by the emergence

MIRIAM LIPSCHUTZ-YEVICK was bom in Scheveningen, Holland, and arrived in the U.S. in 1940 after a three-months-long flight from the Nazis. She eamed her doctorate at M.I.T. in 1947 (one of the few women in mathematics up to then). She was at Rutgers (University Col- lege) from 1964 until retirement She has published on probability, on her invention holographic logic, and on other areas, including a text Mathematics for the Billions for her remedial students. She is a deeply devoted grandmother.

Miriam Lipschutz-Yevick 22 Pelham Street Princeton, NY 08540 USA e-mail: gandmyevicl<@rcn.com

12 THE MATHEMATICAL INTELUGENCER

of a cluster of unconnected associations which together recreate the object.

Comparing the domains on which analogy acts, I find another contrast: po- etic analogy casts a wide net (the search for the telling metaphor runs through a wide web of relevant associations); mathematical analogy deals with con- cepts appropriate to a particular theory, and the validation is deductive and se- quential in character. When mathemat- ical research does widen its scope, it is by generalization, which "by condens- ing compresses into one concept of wide scope several ideas which seemed widely scattered before" (P61ya [Po], p. 30). Alas, this sometimes relies on for- realism which obscures the ideas re- lated, thereby impoverishing the mean- ing and insight.

"Mathematics does not f i t all. "Cloth- ing humanistic and social sciences in

mathematical garb is a technique fre- quently used by scholars in the non- quantitative fields (The Phillips Curve, The Bell Curve, and so on) to over-awe a quantitatively uneducated population. Perhaps we agree that the technique can be pernicious. I have tried to argue that, likewise, poetry has nothing to gain from overstating its resemblance to mathematics--and vice versa. Rather our aim should be to teach the general public to appreciate the insights of the two domains. And beyond that, to un- derstand and act upon the problems of our world with rational thought and em- pathic feeling.

[A] Hannah Arendt, quoted in Dwight Bolinger, Language the Loaded Weapon, Longman, London, New York, 1980; p. 143.

[LY1 ] Miriam Lipschutz-Yevick, "Holographic or

Fourier Logic," Pattem Recognition 7 (1975), 172-213.

[LY2] Miriam Lipschutz-Yevick, "Gloves," The

Kelsey Review, 2003. [M] Michael Marcus, review of "Dark Hero of

the Information Age," Notices of the Amer.

Math. Soc. 53 (2006), 574-579. [P] Henri Poincare, The Foundations of Sci-

ence, The Science Press, New York, 1929. Chap. Ill, "Science and Hypothesis."

[Po] George P61ya, Mathematics and Plausible

Reasoning, Vol. h Introduction and Analogy

in Mathematics, Princeton University Press, Princeton, 1954.

[Pr] Karl Pribram, Languages of the Brain, Pren- tice-Hall, Englewood Cliffs N J, 1971.

[vN] John von Neumann, The Computer and

the Brain, Yale, 1958. [Zw] Jan Zwicky, "Mathematical Analogy and

Metaphorical Insight," Mathematical Intelli-

gencer 28 (2006), no. 2, 4-9.

Jan Zwicky Responds:

I n her helpful reply to my essay, Miriam Lipschutz-Yevick argues that it is not surprising that we should ex-

perience some kinship between mathe- matical analogies and metaphors since "both are thought" (p. 11); but there the resemblance ends. "Mathematical think- ing analyzes," she asserts, but "poetical thinking embellishes" (p. 11 ). A proof in mathematics is the result of "a deriva- tion from axioms, or an algebraic com- putation" (p. 12) while, for Lipschutz- Yevick, "the search for 'the right word' " when writing a poem "feels almost the reverse of mathematical puzzling" (p. 12).

Except for her claim that 'poetical thinking' embellishes, I couldn't agree more. Lipschutz-Yevick claims that I ar- gue that "metaphor and mathematical demonstration have special kinship" (p. 11), but in fact, I do not. I am at pains to distinguish between mathematical in- sight (or 'invention') and mathematical demonstration or proof, and wish to ar- gue that metaphorical thought shares structural features--and, most impor-

tantly, its relationship to truth---only with the former. I suggest that if we adopt Hardy's notion of proof as a rhetorical flourish designed to get other people to see what we ourselves see, then there is a surprising parallel between mathemat- ical proofs and the analogies they con- firm, on the one hand, and poems and the metaphors to which they give ex- pression, on the other. Poems, too, of- ten serve as rhetorical flourishes which position an auditor or reader to grasp the insight the poem's composer wishes us to see. This argument--whether or not it is sound-- is conceptually impos- sible to mount if one fails to distinguish between insight and demonstration.

It is also impossible to mount if one moves, as Lipschutz-Yevick does, seamlessly between "metaphorical" and "poetical", the paired distinction on the literary side of the equivalence. "Poet- ical thinking" (p. 11)--whether we mean by this effusive Victorian rhetoric or, simply, poetry-- is not co-extensive with metaphor. Poetry (the reading of "poetical thinking" that interests me) can contain metaphors, and often does, but needn't (witness the excerpt from her own poem that Lipschutz-Yevick quotes). Metaphors can, and often do, live inside poems; but they also inhabit well-written prose, as well as garden-variety oral con- versation. The distinction is as crucial to

my argument as the distinction between insight and demonstration.

Why, then, does Lipschutz-Yevick re- spond as though I had conflated both? I believe two things may be contribut- ing to her misreading. The first is that it seems utterly obvious to her that both metaphorical insight and mathematical insight are, as she says, "thought" (p. 11). This is actually al l--or nearly a l l - - I wish to establish! (I do also wish to draw out a few consequences of so con- ceiving metaphor.) I am delighted that my central claim is something she thinks we should take for granted, but I sup- pose I have spent too long among the skeptics to rest easy. To many of my colleagues in the humanities and to sev- eral in the sciences (vide P61ya, quoted in my essay), it is anything but obvious that metaphorical contemplation consti- tutes a way of thinking. (Even some po- e t s - a s Shakespeare's and Wordsworth's characterizations of metaphors under- line---can, on occasion, claim not to take metaphorical discernment seriously as a form of thought.) It is, many suggest, mere 'play', a gesture without signifi- cant meaning, or, worst of all, a rhetor- ical embellishment of ideas that might be rendered more clearly, if less attrac- tively, in plain prose. (I will return to Lipschutz-Yevick's characterization of 'poetical thinking' as embellishment in

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