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VielNpoint

Mathematical Analogy and Metaphorical Insight JAN ZWICKY

The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the Editors-in-Chief endorse or accept responsibility for them. A Viewpoint should be submitted to either qf the Editors-in-Chief, Matjorie Senechal or Chandler Davis.

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THE MATHEMATICAL INTELLIGENCER

y interest in this topic grows out of a long-standing, hands­ on engagement with the mak­ ing of metaphors. As a poet, and as an editor and reader of poetry, I have of­ ten been struck by the power of good metaphors to change my stance in the world, to alter in a profound and, it seems, permanent way how I look at things. Whence this power? And further, how is it that we are able to distinguish such 'good', world-altering, metaphors from metaphors that are merely outre or arcane-surprising linguistic con­ structions that lack, or seem to lack, genuine ontological depth? It is difficult to offer examples with­ out quoting whole poems or para­ graphs. A metaphor is like a depth charge: if you know nothing about the material in which it is embedded, it can be difficult to evaluate its force. Some very powerful metaphors can speak to us sans context; but often metaphors in the shallow and mid-ranges won't yield up their full meaning on their own. As a consequence, almost any candidate for a shallow metaphor that I might offer without context can appear as a challenge to invent a context in which it would appear effective. With this caveat, let me offer with­ out further commentary the following: "the eyes are the windows of the soul" (a good or strong metaphor); "the table fizzed like a platypus" (a weak or shal­ low metaphor); "the river/Is a strong brown god" (good); "the luggage re­ sembled/godly" (weak); "the road was a ribbon of highway, perfect for Pekinese" (weak); "If I have exhausted the justifications I have reached bed­ rock, and my spade is turned" (good). My interest, as I say, is in what appears to be an intuitive capacity many of us have for being struck by certain metaphors, and for being left cold by others. What is involved in the compre­ hension of good metaphors? How do metaphors mean? (It will be clear from the preceding that I am using the term 'metaphor' broadly to cover any lin­ guistic expression of focussed analogi­ cal thinking. Thus what we would

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Inc.

strictly regard as a simile is also 'metaphorical' in the sense I am con­ cerned with.) In reflecting on these issues, three things struck me more or less simulta­ neously. The first was that metaphors involve what Wittgenstein called "seeing­ as" [1], a seeing of one thing in terms of another. The second was that, as a working poet, I find that understanding a metaphor feels like understanding cer­ tain kinds of mathematical demonstra­ tions: I am aware of features of various figures or expressions, or various im­ ages or ideas, being pulled into reveal­ ing alignment with one another by the demonstration or the metaphor. The fi­ nal observation-or, in this case, idea­ was that seeing-as involves a kind of re-cognition, and, as such, is what we mean when we say we understand something. This last claim is essentially empirical in nature. If we ask "When do people say that they 'understand' or 'get' or 'see' or 'grasp' something?", it turns out that the experience of 'getting it' seems to involve a reconfiguration of an initially problematic array or scenario-a redirection of emphasis that somehow affects the overall shape of the problem. And the emergence of this new way of looking at things is often accompanied by a feeling of astonish­ ment, or of things falling into place, of their coming home. In sum, I began my investigations with the intuition that both metaphors and certain kinds of mathematical demonstrations are species of analogi­ cal reasoning: both say, in effect, "Look at things like this, if you want to un­ derstand them". But how close is the connexion between mathematical anal­ ogy and metaphor? (And here I should perhaps emphasize that my use of the phrase 'mathematical analogy' is in­ tended to be at least as broad as my use of 'metaphor'-it embraces every­ thing from certain visual proofs of the Pythagorean theorem to Euler's result about the sum of the reciprocals of the squares.) Mathematics clearly involves reasoning-but poetry? Aren't literary metaphors simply inventions-airy noth-

ings, loose types of things, fond and idle names? Can understanding in mathe­ matics really be compared with under­ standing in literature? These are the questions l wish to explore in what fol­ lows. First, l will offer more detailed testimony from mathematicians and po­ ets to support the claim that there is an important correspondence between metaphors and analogies in mathemat­ ics. Then I will look brietly at two points of apparent non-correspondence. I will conclude by suggesting that the answer to our initial questions about the power and recognizability of good metaphors lies with the phenomenon of what we might call metaphorical insight: to grasp a good metaphor is, like understanding a fruitful mathematical analogy, to ex­ perience the significance of a newly seen alignment for what the figure, con­ cept, or thing actually is. Or, to put this another way, a good metaphor changes the way we see the world because it is not a mere linguistic fiction, but is in some sense-a sense analogous to that which attaches to mathematical demon­ stration-true.

Evidence for a Correspondence Because of the third idea mentioned above-that seeing-as is at the root of our experience of understandinr-I was led to the work of Max Wertheimer, one of the leading figures in the de­ velopment of gestalt psychology [2]. There I found elegant and thoroughly researched descriptions of the phenom­ enon of 'getting it,' couched in terms of re-arranging 'internal' structural rela­ tions [3]-in effect, re-seeing an initial configuration in a different way. For example: In this square with a parallelogram strip across it (Fig. 1) the lines a and hare given. Find the sum of the con­ tents of the two areas. One can pro­ ceed thus: The area of the square is a2, in addition that of the strip is . . . ? But suppose that one hits upon the idea:

[Sci=] (square + strip) = (2 triangles, base a, altitude b) [= Sc2] [Sc2= ) . . . . . . . . . . . . . . . . . . . . . . =

(2 a2b)

=

ab

[ =P l . The solution has thus been attained, so to speak, at a single stroke [4).

Figure I .

Indeed, most of Wertheimer's examples were drawn from elementary geometry or arithmetic, a few from music-and none from poetry. But his summary characterization precisely captured cen­ tral features of the experience of grasp­ ing a metaphor: In general we see that in [trying to dis­ cern whether S is [S'll, the object (S) . . . is given as [something defined by a certain set of characteristicsl-but there is no direct route from S to [S') . . . It frequently occurs that [seeing) the required relationship to [S') is only possible when [S) has been re-formed, re-grasped, ro-centred in a specific way. And it is nor less frequently the case that to effect this process a deeper penetration into the nature and structure of S is required [5] . Subsequently, in Poincare, I found descriptions of the process of mathematical creation that appeared to echo the process of actually making metaphors. To create consists precisely in not making useless combinations . . . , [in choosing to study facts] which re­ veal to us unsuspected kinship be-

is a philosopher and award-winning poet Another passion of hers is for chamber music (as comes through very strongly in some of her poems).

JAN ZWICKY

Department of Philosophy University of Victoria Victoria, BC V8W 3P4, Canada e-mail: [email protected]

tween other facts, long known, but wrongly believed to be strangers to one another. Among chosen combinations the most fertile will often be those formed of elements drawn from do­ mains which are far apart. Not that I mean as sufficing for invention the bringing together of objects as dis­ parate as possible; most combina­ tions so formed would be entirely sterile. But certain among them, very rare, are the most fruitful of all [6). This could easily be Robert Hass, talk­ ing about metaphor: "Metaphor, in gen­ eral, lays one linguistic pattern against another. It can do so with a sudden­ ness and force that rearrange categories of thought" [7). But it was ultimately Kepler's re­ marks on analogy that seemed to me most suggestive of all: The geometrical voices of analogy must help us. For I love analogies most of all, my most reliable masters who know in particular all secrets of nature. We have to look at them es­ pecially in geometry, when, though by means of very absurd designa­ tions, they unify infinitely many cases in the middle between two ex­ tremes, and place the total essence of a thing splendidly before the eyes [8). The phrase "by means of very absurd designations" seems designed to invoke the image of metaphor, which, of course, in its strict sense proceeds via an 'ab­ surd' designation. Kepler's claims that mathematical analogies "unify infinitely many cases" and "place the total essence of a thing splendidly before the eyes" both underscore the connexion with gestalt thinking [9] and are echoed in the remark of poet Anne Michaels that "metaphor unifies separate components into a complex whole, creating some­ thing greater than a sum of parts" [10] . (Kepler does not say that the "total essence" that is placed "splendidly [lu­ culenter] before the eyes" is greater than a sum of parts, but the rhetorical con­ struction here suggests that he experi­ ences it as something other than a com­ puted arithmetical average or simple mean.) Also I believe Kepler's sense of what analogy does is surprisingly close to poet jane Hirshfield's sense of what metaphor does: both are, in Hirshfield's words, "central devices for ordering the

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plenitude of being" [11] . Finally, Kep­ ler's suggestion that these mathemati­ cal analogies "know all secrets of na­ ture" could be a paraphrase of poet Charles Simic's claim that, surprising though it sounds, metaphor is "the supreme way of searching for tmth" [12] . And in this connexion, we should note that Kepler is not alone in think­ ing analogy is of vital importance to discovery in mathematics. Eberhard Knobloch points out that both Bernoulli and Leibniz made similar claims [13] . More recently, George P6lya has argued that analogical thought is fundamental to both mathematical insight and ped­ agogy [14]. A final point of correspondence between metaphors and mathematical analogies concerns an awareness on the part of practitioners that they can lead us astray, but a refusal to cede pride of place either to analytic description or logicist investigation. Both Kepler and Leibniz explicitly acknowledged the po­ tential of analogies to mislead, yet they remained advocates of analogical rea­ soning. P6lya, though one of its most vigorous champions, notes that it is "hazardous, controversial, and provi­ sional" [15]. (Their attitude is nicely captured in Butler's observation that "though analogy is often misleading, it is the least misleading thing we have" [16].) The poet Charles Wright is speak­ ing of a similar difficulty in literary com­ position when he discusses the disci­ pline of learning to distinguish between true and false images [17]; Simic also al­ ludes to it in discussions of the episte­ mology of poetic composition [18].

Apparent Points of Non-correspondence No one is seriously going to maintain that mathematical analogies and meta­ phors are essentially the same thing. There are several obvious points of non­ correspondence, of which I will discuss two. My aim here is to concede differ­ ences, while arguing that the most sig­ nificant among them need not damage our impression of fundamental similar­ ity. Indeed, I wish to suggest that these apparent differences point to deeper connexions we have not considered, and that they will help us begin to for­ mulate answers to our original ques­ tions about metaphor.

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THE MATHEMATICAL INTELLIGENCER

The first dissimilarity arises as a di­ rect consequence of consideration of the last point of correspondence men­ tioned above. Precisely because ana­ logical reasoning can be misleading, in mathematics one often does constmct "analytic" or ''logical" proofs to back up one's analogical or visual intuitions. One's reasoning remains suspect until one can produce the four-lane ax­ iomatic deduction that leads to the same

Kepler is not alone in th inking analogy is of vital importance to discovery in math e matics.

destination as the leap of gestalt imag­ ination. There seems no comparable procedure or demand in the case of metaphor. Books of poetry do not usu­ ally come organized like Euclid's Ele­ ments or with appendices that parse and defend every metaphor in the body of the text. Indeed, while we can imag­ ine providing such explanations or elab­ orations, they would seem to be anti­ requisite if the metaphor is to remain literarily pleasing or effective. In this, metaphors closely resemble jokes, as Ted Cohen has pointed out [19l. The second point of non-correspon­ dence focuses on the relations between metaphors and mathematical analogies and the world. Mathematical analogies are in some robust, though perhaps in­ tuitive, sense true, and are perceived to be so, even by members of the general public. This is an honour rarely, if ever, accorded to metaphors except by poets themselves. P6lya, for example, sug­ gests that poets "feel some similarity [when they compare a young woman to a flower], . . . but they do not con­ template analogy. In fact. they scarce­ ly intend to leave the emotional level or reduce that comparison to some­ thing measurable or conceptually definable. " [20]. I agree that most poets would resist attempts to quantify or schematize their metaphors, but I think P6lya is just wrong to suppose that poets don't "con­ template'' analogy. The points of corre­ spondence, and the cited testimony, point to a genuine concern with tmth on the part of many poets. The key

question, I think, is how we can best make sense of this concern-and how our intuitions about tmth in mathemat­ ics can assist us in making sense of it. To this end, let me first suggest ways of accounting for the points of appar­ ent non-correspondence in which the disanalogy with metaphor does not ap­ pear so severe. Then I will return briefly to the issue of the nature of necessary truth which is, I believe, at their root. With respect to Apparent Disanalogy No. 1-the absence of linear or 'ana­ lytic' proof in metaphoric contexts-it is important to reflect a moment on how analogies function in mathematical con­ texts. Guldin, in his discussion of Kep­ ler, comments, "I consider [his] analo­ gies to be useful for the invention of things more than for their demonstra­ tion" [21]. That is: the analogy expresses the insight; the proof, by contrast, es­ tablishes the incontrovertibility of the insight. It is a sentiment one finds echoed in implicit and explicit forms throughout the literature on analogy and proof. Interestingly, it appears to repeat the first point of non-correspon­ dence and then collapse it into the sec­ ond: no proof, no tmth. But in so do­ ing it points once again to the deep similarity between mathematical analo­ gies and metaphors. True, there is noth­ ing corresponding to linear or algebraic proof in metaphoric, i.e., literary, con­ texts-but this absence (and its corre­ sponding presence in mathematical con­ texts) is a feature of the context, not of metaphor itself. And what Guldin's (and P6lya's, and, arguably, Kepler's own) view underlines is that the same holds for mathematical analogies: they are ve­ hicles of insight, not proofs themselves. This leads directly to a question about the nature of proof: how is it that a proof 'establishes the incontrovertibil­ ity' of an insight? Isn't its 'incontrovert­ ibility' precisely what makes us call something an insight in the first place, rather than a guess or a hypothesis? (Even when we're wrong?-And note that we can be wrong about proofs as well as analogies.) What, exactly, is a proof anyway? As with most fundamental notions in most disciplines, the answer is unclear. I am inclined to follow Hardy: I have myself always thought of a mathematician as in the first instance

an observer, a man who gazes at a distant range of mountains and notes down his observations . . . when he sees a peak he believes that it is there simply because he sees it. If he wishes someone else to see it, he points to it, either directly or through the chain of summits which led him to recognize it himself. When his pupil also sees it, the research, the argument, the proof is finished. The analogy is a rough one, hut I am sure that it is not altogether misleading. If we were to push it to its extreme we should be led to a rather paradoxi­ cal conclusion; that there is, strictly, no such thing as mathematical proof; that we can, in the last analysis, do nothing hut point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychol­ ogy; pictures on the board in the lec­ ture, devices to stimulate the imagi­ nation of pupils [22]. But if Hardy is right-if a proof is a "rhetorical flourish" designed to get other people to see what you see-then there is a clear parallel with poetry: as proof is to mathematical analogy, so the poem is to metaphorical insight. The poem it­ self is a 'rhetorical flourish' that positions its reader or auditor in such a way that she or he sees what the poet saw. An example may be helpful. Con­ sider the claim "The human heart is a red pepper". Yeah. OK, we say: it's about the right size, is in roughly the right colour range (assuming the heart is alive or relatively fresh, and has been exposed to the air) . But at first blush, this looks like one of those mid-range or even overtly shallow observations­ it's not that interesting. Here, however,

is its full context, a poem by Sue Sin­ clair [23]:

Red Pepper Forming in globular convolutions, as though growth were a disease, a patient evolution toward even greater deformity. It emerges from under the leaves thick and warped as melted plastic, its whole body apologetic: the sun is hot. Put your hand on it. The size of your heart. Which may look like this. abashed perhaps, growing in ways you never predicted. It is almost painful to touch, hut you can't help yourself. It's so familiar. The dents. The twisted symmetry. You can see how hard it has tried. The analogy becomes more and more resonant as we proceed through the second and third stanzas, the final line 'explaining' the deformities observed in the first stanza and clinching the be­ wilderment, the almost-pain, and the odd familiarity of the second and third. (That the claim, as articulated above, is not fully explicit in the poem, is a fea­ ture of what I'd like to call the gestalt­ rhetoric of much lyric poetry. An ex­ ploration of this phenomenon would take us too far afield; but it is worth noting that we also see such implicit­ ness from time to time in mathematical proofs, as for example in Bhaskara's vi-

sua! demonstration of the Pythagorean theorem [24].) "But still," one may wish to protest, "isn't it the case that Kepler's analogies, or P6lya's analogical presentation of the Pythagorean theorem, express TRUTHS, in a way that metaphors never do? Aren't metaphors creatures of the imag­ ination rather than delineations of real­ ity?" Here, I think a couple of assump­ tions, one about literature and one about mathematics, dovetail to produce questionable prejudice. I'll address the literary side of the matter first. Note that the phrasing of Apparent Disanalogy No. 2 includes the phrase "are perceived to he [robustly true] even by members of the general public''. This is one clue that we are dealing with a phenomenon conditioned by, perhaps even expressive of, culturally deter­ mined levels of literacy and numeracy. My suspicion is that if, as members of either the general or expert public, we were all equally and highly literate and numerate, we would be less inclined to imagine all mathematical demonstra­ tions were transparently true and more inclined to be struck by the profundity of certain metaphorical claims. The poem as 'proof'-that is, rhetorical flourish that positions its reader or au­ ditor to see what the poet has seen­ works by a subtle interplay of rhythm, assonance, denotation, and connota­ tion, much of which our schooling does not prepare us to pick up. Just as we are rarely informed in grade school of the existence of contested proofs, or that mathematicians themselves debate the nature of mathematical truth. I do not wish to deny, however, that most of us at some time or another have

--

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been struck by what I called earlier the incontrovertibility of some mathemati­ cal demonstrations. Consider the visual proof that Plato offers in Meno that a square double in size is built on the di­ agonal of a given square (Fig. 2). I have taught this proof countless times, mostly to students who are less interested in lifting it off the page of Plato's prose than they are in the socio-political drama of Socrates's interrogation of a slave. But the proof, once drawn on the board, is easily grasped: It is also elegant and very powerful. And time after time, there is someone in the class who experiences it with a physical shock-an audible gasp or an involuntary "Oh!" as the light dawns. What underlies this experience, I think, is not only 'getting' that the square dou­ ble in size is built on the diagonal, but that this has to be the case. What im­ presses is not simply the claim's truth, but its necessity. This-necessary truth-would seem to be something a metaphor cannot possess. Metaphors, we think, are crea­ tures of linguistic play, not deductive logic; surely there is nothing conceptu­ ally necessary about a claim like the heart is a red pepper. But let us think hard about this for a moment. Good poems-including poems like Sinclair's "Red Pepper," which elaborates a sin­ gle metaphorical insight-are notori­ ously difficult to teach: ask any poet or sensitive English professor. You can build a tolerable lecture around a mediocre poem, which often requires lots of external information to make it comprehensible; but one often has the sense with a good poem that everything that can be said has been said, and per­ fectly, in the poem itself. Either you get it or you don't. In this, it seems to me, good poems resemble the simple visual proofs we try to teach students in an­ cient-philosophy classes. Yes, there are some who grasp the Meno proof with a gasp; but there are others who don't see it the first, or even the second, time. If they don't get it, there's little I can do but say the same thing-walk through the demonstration, read the poem­ again. And when they get the poem, grasp its central metaphorical insight, there is often an expression of aston­ ishment just as there is with the theo­ rem: a sudden stillness in the room, oc­ casionally tears. These are not in all

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THE MATHEMATICAL INTELLIGENCER

cases the same acknowledgements of necessity as we find in mathematics, but that is nonetheless what I believe they are. Their differences stem from the na­ ture of the necessities compassed by the two domains: mathematics, I believe, shows us necessary truths unconstrained by time's gravity; poetry, on the other hand, articulates the necessary truths of mortality.

What the Correspondence Suggests It is time now to return to our initial questions: Whence the vision-altering power of some metaphors? And how is it that we are able to distinguish between such metaphors and arresting, but mere,

Metaphors can he insighfful in just the way that mathematical a nalogies can .

linguistic confections? Metaphors, I sug­ gest, can be insightful in just the way that mathematical analogies can: they reveal to us "unsuspected kinships" be­ tween "facts long known, but wrongly believed to be strangers to one an­ other. " And, as in mathematics, "the most fertile will often be those formed of elements drawn from domains which ., are far apart. This does not mean, as Poincare notes, that we simply bring to­ gether the most disparate objects we can think of-such a tactic can produce surprise, but it is surprise without depth, "sterile," is Poincare's word. It means that, as in the case of mathematical analogies, metaphorical power is a product of discernment, to borrow again from Poincare. But discernment of what? I propose that the correspondence between meta­ phors and mathematical analogies sug­ gests that we distinguish between pro­ found and shallow metaphors along the lines of a mathematical distinction be­ tween important or fruitful mathemati­ cal conceptions and unimportant ones [251. Here, it is helpful to note that even a realist like Kepler, who believes that good mathematical analogies reveal truths about the actual universe, argues that the value of analogy lies in the "most spacious" field of invention that it opens [26]. In other words, discern­ ment in mathematics, and, as part of

this, the development of 'true' analo­ gies, consists in perceiving connexions that point the way to yet other con­ nexions. The power, the value, of an analogy lies not in a definitive mapping of some territory, but, paradoxically enough, in its freeing of the imagina­ tion for further discovery. To put this yet another way: a fruitful or important analogy is one that establishes a deep field of resonance. It might be objected that this pro­ posal clouds the issue more than it clar­ ifies it. The notion of a fruitful or im­ portant conception is so contested in mathematics that it cannot usefully form the basis of a parallel account of good and weak metaphors. For underlying questions about which conceptions will prove fruitful or important is a debate about the nature of mathematical cre­ ativity: is it a form of discovery, or a type of invention? In metaphysical terms, the question is whether mathe­ matical entities and truths have an ex­ istence independent of the human minds that eventually discover them, or whether they are-as metaphors are of­ ten supposed to be-simply constructs of human discourse and imagination. Neither argument nor experience has yet been able to settle the issue. As Demidov notes, the experience of the working mathematician supports both claims [27]. It is not my intention to settle the de­ bate here. My aim is to suggest that, rather than rendering the comparison between powerful metaphors and fruit­ ful mathematical analogies problematic, it may point yet again to a fundamen­ tal similarity, obscured yet again by lit­ erary prejudice we are disinclined to ex­ amine. For what the existence of the debate between realism and constructivism in mathematics should suggest to us, given the correspondences we have noted, is that a similar debate might be joined with respect to metaphorical insight. It may be that the world does exist independently of human activity and discourse, and that writers whose metaphors are consistently strong are not just good at manipulating language, they are good at perceiving the way that world actually is. Yet, as noted earlier, we tend to think it is ob­ vious that metaphor has no purchase on what I have previously called a 'robust' conception of truth-that (to

paraphrase Nietzsche) metaphor's self­ conscious use marks a liberation of the human understanding from the stultify­ ing effects of naive (or even sophisti­ cated) realism. If, however, there is rea­ son to construe the power of metaphors along the lines of the importance or fruitfulness of mathematical concep­ tions, there is reason to think the real­ ism-constructivism debate might be a live one for metaphorical insight. In that case, we must accept that we have been given grounds for a radical reconstrual of the role of metaphor in our lives: we must take seriously the possibility that metaphors are not invented but are per­ ceived, and that the true ones among them limn the structure of a resonant, mind-independent universe. ACKNOWLEDGEMENTS

[6] Henri Poincare, "Mathematical Creation",

Samuel Butler, London: A. C. Fifield, p. 94.

Ch. Ill, Science and Method in The Foun­

1 91 9.

dations of Science, trans. George Bruce

[1 7] Charles Wright, "Narrative of the Image:

Halsted, New York: The Science Press,

A Correspondence with Charles Sirnic",

p. 386. 1 929.

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[7] Robert Hass, "Images", Twentieth Century

views, Ann Arbor: The University of Michi­ gan Press, p. 59. 1 995.

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[1 8] Charles Sirnic, "Narrative of the Image:

p. 287. 1 984. [8] Johannes Kepler, Ad Vitellionem par­

A Correspondence with Charles Sirnic",

alipomena, quibus astronomiae pars op­

ibid., p. 72. See also "Notes on Poetry and

tica traditur, etc. in Franz Hammer, ed. ,

Philosophy", p. 64, and "Visionaries and

Johannes Kepler: Gesammelte Werke,

Anti-Visionaries", p. 78, both in Wonderful Words, Silent Truth [1 2].

Vol. I I , trans. E. Knobloch, rev. G. Shrimp­ ton, Munich: C . H . Beck, p. 92. 1 604/

[1 9] Ted Cohen, "Metaphor and the Cultivation

1939.

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[9] Cf. Wertheimer's description of the "fun­

pp. 3-12. 1 978.

damental 'formula' " of Gestalt theory

[20] George P61ya, op. cit. Note 1 4, p. 1 3.

in "Gestalt Theory" in Willis D. Ellis, op.

[21 ] Paul Guldin, De Centro Gravitas, Book 4,

cit. [2].

Ch. 4, Proposition 5, trans. G. Shrimpton, Vienna: Gregor Gel bar, p. 327. 1641 .

[1 0] Anne Michaels, "Cleopatra's Love" in Tim

My thanks to my colleague, Colin Macleod, for insightful comments on an early draft of this essay, and to jenny Petch for assistance with the final ver­ sion.

Lilburn, ed. Poetry and Knowing: Specu­

[22] G. H. Hardy, "Mathematical Proof " , Mind,

lative Essays and Interviews, Kingston, ON: Quarry Press, p. 1 79. 1 995.

Vol. 38, p. 1 8. 1 929. [23] Sue Sinclair, "Red Pepper", Secrets of Weather & Hope, London, ON: Brick

[1 1 ] Jane Hirshfield, Nine Gates: Entering the Mind of Poetry, New York: HarperCollins,

Books, p. 24. 2001 . Reprinted by per­

p. 1 1 1 . 1 997. REFERENCES

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382.

[25],

p.

379,

© 2006 Springer Sc1ence+Bus1ness Media, Inc., Volume 28, Number 2, 2006

p.

9

e: The

Master of All BRIAN J. McCARTIN

Read Euler, read Euler. He is the master of us all.

-P. S. de Laplace

L

aplace's dictum may rightfully be transcribed as: "Study e, study e. It is the master of all . " just a s Gauss earned the moniker Princeps mathemati­ corum amongst his contemporaries [1], e may be dubbed Princeps constantium symbolum. Certainly, Euler would concur, or why would he have endowed it with his own initial [2]? The historical roots of e have been exhaustively traced and are readily available [3, 4, 5]. Likewise, certain funda­ mental properties of e such as its limit definition, its series representation, its close association with the rectangular hy­ perbola, and its relation to compound interest, radioactive decay, and the trigonometric and hyperbolic functions are too well known to warrant treatment here [6, 7l. Rather, the focus of the present article is on matters re­ lated to e which are not so widely appreciated or at least have never been housed under one roof. When I do in­ dulge in reviewing well-known facts concerning e, it is to forge a link to results of a more exotic variety. Occurrences of e throughout pure and applied mathe­ matics are considered; exhaustiveness is not the goal. Nay, the breadth and depth of our treatment of e has been cho­ sen to convey the versatility of this remarkable number and to whet the appetite of the reader for further investigation.

The Cast Unlike its elder sibling 7T', e cannot be traced back through the mists of time to some prehistoric era [8]. Rather, e burst into existence in the early seventeenth century in the con­ text of commercial transactions involving compound inter­ est [5] . Unnamed usurers observed that the profit from in­ terest increased with increasing frequency of compounding, but with diminishing returns.

10

THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+Business Med1a, Inc.

Thus, e was first conceived as the limit e= lim (1 + 1/n)11 --""' 11

= 2.718281828459045 · · · ,

(1)

although its baptism awaited Euler in the eighteenth cen­ tury [2] . One might naively expect that all that could be gleaned from equation (1) would have been mined long ago. Yet, it was only very recently that the asymptotic development

ev= e

!

S1(v + k, k)

k�O

(v + k)!

k ( - 1)1

I

/!

1�0

'

(2)

with 51 denoting the Stirling numbers of the first kind [9], was discovered [10]. Although equation (1) is traditionally taken as the defin­ ition of e, it is much better approximated by the limit _

. e- IJ-'>X ltm

[

(n + 2)n+2 ( n + 1)n+t

_

(n + l) n+l n"

]

(3)

discovered by Brothers and Knox in 1998 [ 1 1 ] . Figure 1 dis­ plays the sequences involved in equations (1) and (3). The superior convergence of equation (3) is apparent. In light of the fact that both equations (1) and (3) pro­ vide rational approximations to e, it is interesting to note that 878 = 2.71826 · · ·provides the best rational approxima323 tion to e, with a numerator and denominator of fewer than four decimal digits [ 1 2] . (Note the palindromes!) Consider­ ing that the fundamental constants of nature (speed of light in vacuo, mass of the electron, Planck's constant, and so on) are known reliably to only six decimal digits, this is re­ markable accuracy indeed. Mystically, if we simply delete

2. 7

0

t

n

n_ �

" "

r.

.r.

,...,

"

e

2.6

2.5

2.4 Eq.1 0 = Eq. 3

2.3

2.2

2.1

2

2

Figure I.

4

6

8

10

Exponential sequences.

the last digit of both numerator and denominator then we obtain the best rational approximation to e using fewer than three digits (13]. Is this to be regarded as a singular property of e or of base 10 numeration? In 1669, Newton published the famous series represen­ tation for e [14],

12

n

14

16

*·

e=

Lk=O k! X

1+ 1+

1

=

-

-

1

-

1 1 1 . (1 + . . . )))), . (1 + . (1 + - . (1 + 4 5 2 3

(2 k)!

(5)

.

Figure 2 displays the partial sums of equations (4) and (5) and clearly reveals the enhanced rate of convergence. A va­ riety of series-based approximations to e are offered in [15). Euler discovered a number of representations of e by con­ tinued fractions. There is the simple continued fraction (16] 1 e = 2 + -------.1-1 + 2+� 1+ � 1+

�����

(4)

4+

1+

(6)

� 1+ �

or the more visually alluring [5]

;

BRIAN J. McCARTIN studied Applied Mathe­ matics at the University of Rhode Island and � .. ......� . ,R Jil went on to a Ph.D. at the Courant Institute, � . l. . • New York University. He was Senior Research Mathematician for Unrted Technologies Re­ . search Center and Chair of Computer Science at RPI/Hartford before joining Kettering University in 1993. He received the Distinguished Teaching Award from the Michigan Section of the MAA in 2004. Along the way, he also earned a degree in music from the University of Hartford ( 1994). To fill in the details see http://www.kettering.edu/-bmccarti.

1 e = 2 + -------.1---1 + 2=2+ f== =1:====

lm�

'

20

2k+ 2

established by application of the binomial expansion to equation (1). Many more rapidly convergent series repre­ sentations have been devised by Brothers [14] such as a: 0

18

i

'·.J.�, L._. .. .. ,,

'

Applied Mathematics Kettering University Flint, MI.' 48504-4808 USA

(7)

3+4+-;���=:::4+

5+

6 6+ � 7+�

In 1655, John Wallis published the exhilarating infinite product �

2 2 4 4 6 6 8 8 10 10 12 12 14 14 16

- = --------------- .. ' 2

1 3 3 5 5 7 7 9

9 11 11 13 13 15 15

(8)

However, the world had to wait until 1980 for the "Pip­ penger product" (17] e

2

(�)1/2 (� 1

) (

i l/l i 5.

3 3

��

�)1/H (� � � E_ E_ � � li)ti16 . . . .

5 7 7

9 9 11 11 13 13 15 15

© 2006 Springer Science+Business Media, Inc.. Volume 28, Number 2, 2006

(9)

11

2. 8

i

2.6

0

e

2. 4

2.2 2!

Eq. (4) o = Eq. (5)

1. 8

1.6

1. 4

1.2

1

2

0

Figure 2.

3

Exponential series.

In spite of their beauty, equations (8) and (9) converge very slowly. A product representation for e which converges at the same rate as equation (4) is given by [18] U1

e

= 1; =

Un+1

fi

n-1

= (n + 1)( un +

Un + 1

=

Un

1) =>

3._ 2_ ..!§.. i2_ 326 . . (lO) 1 4 1 5 64 325 .

An Interesting lnequalit-e In 1744, Euler showed that e is irrational by considering the simple infinite continued fraction (6) [19] . In 1840, Liouville showed that e was not a quadratic irrational. Finally, in 1873, Hermite showed that e is in fact transcendental. Since then, Gelfand has shown that e71' is also transcen­ dental. Although now known as Gelfand's constant, this number had previously attracted the attention of the influ­ ential nineteenth-century American mathematician Benjamin Peirce, who was wont to write on the blackboard the fol­ lowing alteration of Euler's identity [2, 5]:

4

n

(12) he also established the more general inequalities {3> a 2: e

=>

af3> {3"'; e 2: {3> a> 0

=>

{3"'> af3,

Barel-e Transcendental

·

·

·

·

·

12

THE MATHEMATICAL INTELLIGENCER

·

p

1

O 0) exponential integral: Ei(x) = Jx -dt - t

Gamma function: f(z) = J�

"'

e1

orthogonal polynomials: e- x2 on ( - oo ,oc) Hermite: weight w(x) Laguerre : weight w(x) xa e- x on (O,oo) , a > - 1 integral transforms: Fourier: f'"x e1xv f( y)dy Laplace: fo e-xv f( y) dy Gauss: J::x e-<x-yl2 .fC y)dy =

=

•

© 2006 Springer Science+Bus1ness Media. Inc., Volume 28, Number 2, 2006

19

Top ln(e 10) Reasons Why e Is Better than

first 1 0-digit prime found in consecutive digits of e

Figure I 0.

. com

Google-plexed.

I t should be clear from the above that one would be challenged to find a branch of pure or applied mathemat­ ics which has not felt the enriching presence of e. Readers are invited to add their own favorite branches to this al­ ready robust family tree.

Conclusion The time has come to end our stroll through the Garden of e-den, even though we have yet to taste many of its most succulent fruits, such as exponential splines [62]. However, I would like to conclude my p-e-an to e in a lighter vein. The initial decimal digits of e are quite easy to remem­ ber: 2.7 1828 1828 45 90 45. In spite of this, many mnemon­ ics based on the number of characters of each word have been developed. A particular favorite is: "It enables a num­ skull to memorize a quantity of numerals" [63]. For the more sophisticated, I offer the pleasingly self-referential: "I'm form­ ing a mnemonic to remember a function in analysis" [13] . O f course, anything can b e taken to excess, a s i s evidenced by the gargantuan 40-digit mnemonic of [64]. Internet giant Google holds the distinction of having transformed a mathematical noun (the googol [65]) into a verb. As further evidence of their mathematical pedigree, when filing the registration for their 2004 initial public of­ fering of stock (IPO) with the Securities and Exchange Com­ mission (SEC), they chose not to state the number of shares to be offered. Instead, they declared $2,718,28 1 ,828 as their estimate of the money that would be so raised. Later that same year, from deep within their corporate headquarters in Mountain View, California (the Googleplex), a most curious recruitment strategy was hatched. Billboards and banners, such as that in Figure 10, appeared anonymously in Silicon Valley and Cambridge, Massachusetts. After deci­ phering the puzzle (whose solution begins strangely enough at the 101st digit of e) and visiting 7427466391 .com, one was eventually led to a solicitation of employment for Google. Even though e finally has its own biography [5), it has traditionally had to dwell in the shadow cast by the head­ line-grabbing 7r. Witness the fact that in [12), 7r has 8 pages devoted to it, while e is allocated a paltry 1/2 page! How­ ever, the situation has improved to the point where e now has its own Web page [66), from which Figure 1 1 is bor­ rowed.

10) 9)

8) 7) 6) 5)

e

7T

spell than 7r. e 2.718281828459045 The character for e ca n b found on a keyboa rd but 7r sure can't. Ev tybody fights for a pi ce of the pie ln(7r) is a really na ty number, but ln(e) 1. e is used in calculu whil 7r is used in baby geom­ is easier to

7r =

3 . 1 4 while

=

,

=

etry.

'e' is the most commonly picked vowel in Wheel of Fortune. 3) e sta nds for Euler's umber; 7r doesn t stand for

4)

'

2)

squat.

don't need to know Greek to be able to u se e. e with a food product.

You

1 ) You can't confus Figure I I .

e-gotism.

REFERENCES

[ 1 ] C. C. Gillispie, Biographical Dictionary of Mathematics, Charles Scribner's Sons, New York , 1 991 .

[2] W. Dunham, Euler: The Master of Us All, Mathematical Associa­ tion of America, Washington, DC, 1 999.

[3] U. G. Mitchell and M. Strain , The Number e, Osiris 1 (1 936), 476-496. [4] J. L. Coolidge, The Number e, American Mathematical Monthly 57 (1 950), 591 -602. (5] E. Maor, e: The Story of a Number, Princeton University Press, Princeton, NJ, 1 994. [6] A. H. Bell, The Exponential and Hyperbolic Functions and Their Ap­ plications, Pitman & Sons, London, 1 932.

[7] M. E. J. Gheury de Bray, Exponentials Made Easy, Macmillan & Co. , London, 1 92 1 .

[8] P. Beckmann, A History of 1r, Barnes & Noble, New York, 1 993.

[9] J. H. Conway and R. H. Guy, The Book of Numbers, Springer­ Verlag, New York, 1 996. [1 0] M. Brede, On the Convergence of the Sequence Defining Euler's Number, Mathematical lntelligencer 27 (2005), 6-7. [1 1 ] H. J. Brothers and J. A. Knox, New Closed-Form Approximations to the Logarithmic Constant e, Mathematical lntelligencer 20 (1 998), 25-29. [1 2] D. Wells, The Penguin Dictionary of Curious and Interesting Num­ bers, Penguin Books Ltd. , Middlesex, England, 1 986. [1 3] M. Gardner, The Unexpected Hanging and Other Mathematical Di­ versions, Simon and Schuster, New York, 1 969. (1 4] H. J. Brothers, Improving the Convergence of Newton's Series Approximation for e, College Mathematics Journal 35(1 ) (2004), 34-39. [1 5] J. A. Knox and H. J. Brothers, Novel Series-Based Approxima­ tions to e, College Mathematics Journal 30(4) (1 999), 269-275 . (1 6] C. D. Olds, The Simple Continued Fraction Expression for e, Amer­

ACKNOWLEDGMENTS

The author thanks Mrs. Barbara McCartin for her dedicated assistance in the production of this paper. Figure 7 is cour­ tesy of http://static .filter. com while Figure 9 is courtesy of http://FreeNaturePictures.com.

20

THE MATHEMATICAL INTELLIGENCER

ican Mathematical Monthly 77 (1 970), 968-974. [1 7] N. Pippenger, An Infinite Product for e, American Mathematical Monthly 87(5) (1 980), 391 . (1 8] X. Gourdon and P. Sebah, The Constant e and Its Computation, http://numbers.computation. free. fr/Constants/Eie. html.

[1 9] A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 1 975. [20] I. Niven, Which is Larger, e" or 1r8? in A Century of Calculus, Part II: 1969- 1 99 1 , Mathematical Association of America, Washington,

DC, 1 992, 445-447.

[21 ] J . M . Borwein and P. B. Borwein, Pi and the AGM, Wiley­ lnterscience, New York, 1 987. [22] S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003. [23] R. Courant and H. Robbins, What Is Mathematics?, Oxford Uni­ versity Press, New York, 1 941 . [24] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Third Edition, Oxford University Press, London, 1 954. [25] P. Ribenboim, The New Book of Prime Number Records, Springer, New York, 1 996. [26] S. M . Ruiz, A Result on Prime Numbers, Mathematical Gazette 81 (1 997), 269-270. [27] E. T. Bell, Men of Mathematics, Simon and Schuster, New York, 1 965. [28] H. V. Baravalle, The Number e- The Base of the Natural Loga­ rithms, The Mathematics Teacher 38 (1 945), 350-355. [29] B. Friedman, Lectures on Applications-Oriented Mathematics, Holden-Day, San Francisco, 1 969. [30] G. H. Hardy, Ramanujan, AMS Chelsea, Providence, Rl, 1 999. [31 ] J . Aczel, On Applications and Theory of Functional Equations, Academic Press, New York, 1 969. [32] H. Bateman, Differential Equations, Chelsea, New York, 1 966. [33] R. E. O'Malley, Thinking About Ordinary Differential Equations, Cambridge University Press, New York, 1 997. [34] G. Boole, Calculus of Finite Differences, Chelsea, New York, 1 970. [35] S. lyanaga and Y. Kawada, Encyclopedic Dictionary of Mathe­ matics, MIT Press, Cambridge, MA, 1 980. [36] J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, New York, 1 987. [37] K. W. Morton, Numerical Solution of Convection-Diffusion Prob­

[44] J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1 980. [45] F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, New York, 1 973. [46] E. M . Weisstein, Exponential Transform, http://mathworld.wolfram. com/ExponentiaiTransform. html. [47] F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1 969. [48] N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Se­ quences, Academic, San Diego, 1 995. [49] R. J. Riddell, Jr. , Contributions to the Theory of Condensation, Ph. D. Dissertation, University of Michigan, UMI Dissertation Ser­ vices, Ann Arbor, Ml, 1 951 . [50] A. S. Householder, The Numerical Treatment of a Single Nonlin­ ear Equation, McGraw-Hill, New York, 1 970. [51 ] K. Holmstrom, A. Ahnesjo, and J. Petersson, Algorithms for Ex­ ponential Sum Fitting in Radiotherapy Planning, Technical Report /Ma-TOM-2001 -02, Department of Mathematics and Physics, Malarden University, Sweden (2001 ). [52] G. R. de Prony, Essai experimental et analytique, Journal de /'Ecole Polytechnique, 1 (22) (1 795), 24-76. [53] J. J. O'Connor and E. F. Robertson, Gaspard Clair Frangois Marie Riche de Prony, http://www.groups.dcs.st-and.ac.uklhistory/ Mathematicians/De_Prony.html. [54] H. S. Schultz and B. Leonard, Unexpected Occurrences of the Number e, Mathematics Magazine 62(4) (1 989), 269-271 . [55] N. MacKinnon, Another Surprising Appearance of e, Mathemati­ cal Gazette 74(468) (1 990), 1 67-1 69. [56] D. J. Colwell, J . R. Gillett and B. C. Jones, That Number e Again, Mathematical Gazette 75(473) (1 991 ), 329-330.

[57] J. Hornberger and J-H. Ahn, Deciding eligibility for transplantation when a kidney donor is available, Medical Decision Making 1 7 (1 997), 1 60-1 70. [58] H. Cramer, Mathematical Methods of Statistics, Princeton Univer­ sity Press, Princeton, NJ, 1 946.

lems, Chapman & Hall, London, 1 996.

[59] S. M. Stigler, The History of Statistics, Harvard University Belknap

merical Methods for Problems with Initial and Boundary Layers,

[60] A. Papoulis, Probability, Random Variables, and Stochastic

[38] E. P. Doolan, J . J . H. Miller, and W. H. A. Schilders, Uniform Nu­ Boole Press, Dublin, 1 980. [39] F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1 956. [40] B. J. McCartin, Exponential Fitting of the Delayed Recruitment/ Renewal Equation, Journal of Computational and Applied Mathe­ matics 1 36 (2001 ), 343-356. [41 ] G. A. Baker, Jr. and P. Graves-Morris, Pade Approximants, 2nd Edition, Cambridge University Press, Cambridge, 1 996. [42] D. Mortimer, Research on Asymptotics, http://www.maths.ox.ac. uk/ � mortimer/research/david research .shtml. [43] A. C. Fowler, Mathematical Models in the Applied Sciences, Cam­ bridge University Press, Cambridge, 1 997.

Press, Cambridge, MA, 1 986. Processes, Second Edition, McGraw-Hill, New York, 1 984. [61 ] H. Bateman, Higher Transcendental Functions, Krieger, Malabar, FL, 1 981 . [62] B. J. McCartin, Theory of Exponential Splines, Journal of Approx­ imation Theory 66( 1 ) (1 99 1 ), 1 -23. [63] M. Gardner, The Scientific American Book of Mathematical Puz­ zles and Diversions, Simon and Schuster, New York, 1 959. [64] Z. Barel, A Mnemonic for e, Mathematics Magazine 68 (1 995), 253. [65] E. Kasner and J . Newman, Mathematics and the Imagination, Si­ mon and Schuster, New York, 1 967. [66] D. Sapp, The e Home Page, http://www.rnu.org/�doug/exp/ etop1 O.html.

© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 2, 2006

21

M at h e iT1 a t i c a l l y Bent

Col i n Ad a m s , Editor

Class Reunion COLIN ADAMS

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, W i l l i am s College, Wil l iamstown, MA

01267 USA

e-mail: C o l i n . C .Adams@wi l l iams.edu

22

ell, if it isn't cosine! I haven't seen you in 20 years. You look great. Still smooth as a baby's bottom. How do you do it? Hey, I heard your wife left you. Sorry about that. So ex is your ex. I hope you don't mind my saying, but she's an amazing function. I had a crush on her in high school. And I have to tell you, she hasn't aged a day. No one in her family has. I always had trouble differ­ entiating between her and her mother. I know you really loved her. Hey, you might want to take it easy on that punch. You know, there's rum in there. Me? I'm not married. Never have been. Back in high school, with all the girlfriends I had, who would have guessed that natural log would still be single? But it's not like I'm incapable of a monogamous relationship. I am a one­ to-one kind of guy. I just haven't met the right function. But I'm doing my best to meet as many as possible, if you know what I mean. Hey, speaking of break-ups, did you hear about step function? Secant left him. Said she needed some continuity in her life. And get this. Now she's see­ ing absolute value. So the guy's con­ tinuous, but what about his derivative? Well, I guess it could be worse. At least he's Lipschitz. Hey, look who's standing over there, talking to the Gauss map. It's logistic growth. Oh, is she hot. Did you hear? She's a model. Made it to the big time. Does all kinds of modeling. Drug dis­ sipation, blood alcohol content, con­ strained population growth, disease spread. You name an important mod­ eling assignment, she's there. Saw her on the cover of the New England jour­ nal of Medicine last month. And over there by the hors d'ouevres is Riemann zeta. Remember, he was class president? You know, after high school he did the lawyer thing. But he

THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media. Inc.

got tired of it pretty quickly. Found himself representing all kinds of sleazy functions, some nowhere differentiable, some not even well defined. Decided he wanted to give something back to society, so he went into politics. But you know, his first election was a tough one. He went up against the incumbent, some polynomial who was a functional illiterate as far as I could tell. There was all kinds of mudslinging going on. Then the incumbent brings up the values issue. Challenges Rie­ mann to divulge where his zeros occur. Makes a big deal out of it. Personally, I didn't know what to think. How much privacy should a politician be allowed? Then Riemann gives this incredibly persuasive speech about how your zeroes are your busi­ ness, and it is the right of every func­ tion to decide whether or not to divulge where they occur. It was truly moving. So the electorate buys Riemann's hy­ pothesis and he's elected. Now every­ body wants to be like him; L-functions, Dedekind zeta, you name it. Hey, remember arctan? I think she was in your homeroom. Did you hear she went to Wall Street? Works in de­ rivatives. She's done extremely well for herself. Uh-oh, speaking of arctan, there's her old flame 1/(x2 + 1). Look the other way. Poor guy. Had some substance abuse is­ sues. He was completely dysfunctional. Now he's trying to integrate himself back into society, but you can guess the chances of that. He's no arctan. And of course there was his brother, .x-2 + 1 . Did you hear about him? So na'ive. One night, he fell in with some polynomials defined over IQ[i]. They got him toasted, lead him down a dark al­ ley and in the morning, all the police found were his factors. Hey, maybe you should switch to soda water. You are starting to list. You know, life is always full of ups and downs. You'll find someone else. Sheesh, listen to me, telling you about ups and downs. Like you have never had time as the variable on your hori­ zontal axis.

Look, over by the registration table. There's gamma. What an athlete he was, huh? Captain of the football and the wrestling teams. You know he went to college. Got a full scholarship to West­ ern Harmonic. Everyone assumed he would pledge Gamma Gamma Gamma, but instead he went Tri Delta with his buddies Dirac and the Laplacian. Now, that Laplacian; he was an operator. Hey, how do you like the band? I booked them. They're called Dr. Dedekind and the Minor Functionaries. I always liked func. Now, look who's stuffing himself with

pigs in a blanket. None other than phi, or as he insists, Euler's totient function. Talk about pretentious. Totient's not even a word. What an asymptote. I heard he was dating your ex, but I find it hard to believe. She's way to classy for him. Whoops. Dropped your glass. Not that anybody cares. What's a broken glass here or there? So, you keeping busy? I'm still get­ ting lots of work. Some physics, some engineering, a little pure math. I don't know if you saw it, but I even had a cameo appearance in A Beautiful Mind. Hey, if you get a chance, you should

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go next door. The series are having their reunion. Talk about wild. Taylor series was doing his imitation of Maclaurin. "Look at me. a =0. Dumdedumdedum." I laughed so hard I squirted Sprite out my nose. Those guys know how to party. It would do you good. Put a smile on your face. Hey, there's cube root. I sat behind her in algebra. Always had a thing for her. I think I'll go say hello, see what she's up to now. With any luck, she's available. Anyway, it's been great talk­ ing with you. Good luck with the al­ cohol thing. Catch you at the next one.

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The Gold Standard for Mathematical Publishing Anlm�ted Tube Plol • To mu.e 1n IWiirn.atfd t1.1b• plot 1 Type ., expreulon in ont or two variabln 2 WICh the insertion point in the exprenion,. 3 Optn the Plol Propertjes tUJog 11nd,. on the SIIIT111 VIIriabltl The next onimation 'hows 11 knot being (Qwn, Plot 30 A.nknated +- Tube

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JMacKichan to,wur,rli'c

www.mackichan.com/mi • Emai: info@maclfa1 � jUID �lf aa*,

The author has been on the University since 1946, working now on one branch of mathematics, now on another. He is author of a book on ring theory and co­ author of books on categorial logic and on history and philosophy of mathematics. The present arti­ cle forms part of an ongoing study of nonlinear logic and its application to linguistics. He in­ forms The lntelfigencer (somewhat cryptically: try to make sense of this!) that he has canred out a detailed investigation of the number two. JOACHIM LAMBEK

staff of McGill

Department of Mathematics and Statistics McGill University Montreal, H3A 2K6 Canada e-mail: [email protected]

42

THE MATHEMATICAL INTELLIGENCER

then a*

=

a*l --" ct'(aa e )

= ( a*a)a c --" l ac = a t .

Thus a* --" a e , and similarly a C --'> a* . Here are some ele­ mentary properties of pregroups, the proofs of which will be left to the reader: le

= 1 , (ab)e

=

h e a e , if a -'> h then be --"

ae,

and similarly for right adjoints. Moreover

cae y = a = ( a r)f

Free Pregroups The linguistic application I have in mind leans primarily on pregroups freely generated by partially ordered sets. We begin with a partially ordered set (poset) of basic types, which may differ from one language to another, and which is meant to express certain elementary grammatical con­ cepts and their features. From the basic types one forms simple types by repeated adjunction. Thus, a simple type has one of the following forms:

where a is a basic type. Finally, we define a (compound) (vpe to be a string of simple types. The types form a monoid under concatenation (1 being the empty string), which is easily seen to be partially ordered, provided we stipulate that, for any simple type x, X-'>

y implies y e --"

xe,

hence X -'>

y implies xee --" y ce,

and similarly for right adjoints. The partial order may be extended to compound types by stipulating that X -'>

x' and Y --" y ' imply

.X:Y --"

x'y ' .

Moreover, the partially ordered monoid of types i s seen to be a pregroup , with adjunctions defined i n du c t ive ly

thus:

l e = 1 = 1 ", (xy)e = y ex e , (xyY

=

y rx ".

The resulting pregroup is the free pregroup generated by the given poser of basic types. (In the technical language of category theory, this means that we have constructed a functor left adjoint to the so-called "forgetful functor" from pregroups to posers. )

Contractions Suffice What makes free pregroups particularly amenable for com­ putation is the following observation. LEMMA When showing that

for simple zypes X; and YJ· one may assume without loss of generality that all contractions a e a --'> 1 and aar --'> 1 pre­ cede all expansions 1 --'> aa C and 1 --'> a ra. This was proved in [ 1 5] , where pregroups were first in­ troduced (under this name) to facilitate language process-

ing .3 The following argument will give an idea of the proof. A calculation in which an expansion immediately precedes a contraction may look like this: suppose a ----'> h and h ----'> c, then

q = question (including yes-or-no questions and wh-ques­ tions) and to postulate

A Small Fragment of English This calculation can be replaced by simply citing the tran­ sitivity of the partial order a ----'> b ----'>

c,

using neither expansions nor contractions. Why is this lemma useful to grammarians? Among other things, they are interested in checking whether a given string of words is a grammatically well-formed sentence. To do this, one may look at the corresponding string of simple types, say x1 • • • Xm, and check whether it reduces to a simple, or even basic, type y, for example the type of a declarative sentence or question. To check whether

We begin by assigning some types to a few English words: he has type 1r5 ( = third person subject), her has type o (= direct object), sees has type 1r3s1oe to indicate that we require a third person subject on the left and a direct object on the right. (This idea may have been anticipated in a chemical analogy by Charles Sanders Peirce [27], who would have said that sees resembles a mol­ ecule with two unsaturated chemical bonds. ) Now look at the sentence

X1 • · • Xm ----'> Y

when y is simple, one may assume that all contractions pre­ cede all expansions. But, if the right-hand side is simple, there will be no expansions at all! Thus, for sentence ver­ ification, we may confine attention to contractions only. This is not to say that expansions are useless; they play a role in proving the mathematical properties of pregroups discussed above.

1T3

=

=

·

'----J

�

We calculate in two steps: 1r3(1r3s 1 of ) = (7r.o7r3)sloe ----'> 1 s1oe = s 1oe, (s1oe)o s1 (ofo) -----"' s1 1 = s 1 . =

It is convenient to indicate contractions by underlinks 4 Sim­ ilarly, we have

Some Basic Types In what follows, we will study the pregroup of types freely generated by a poset of basic types for some very small frag­ ments of three modern European languages. The following common set of basic types will work in all three examples. , 6 1r; = jth personal subject pronoun, where .f 1 , · denotes the three persons singular followed by the three persons plural. Actually, in modern English, the original second person singular has disappeared and has been replaced by the sec­ ond person plural. Moreover, there is no morphological dis­ tinction between the three plural verb forms. hence in Eng­ lish we may put 1r2 1r4 1r'i = 1T(, . sk = declarative sentence in the kth simple tense ( k = 1 ,2, . . . ) . Here k = 1 and k = 2 stand for the present and past in­ dicative, respectively. English and German also have two subjunctives, but express the future as a compound tense. Literary French has altogether seven simple tenses, but two of these are in the process of disappearing. qk = yes-or-no questions in the kth simple tense, o = direct object, p2 = past participle of intransitive verb, i = infinitive of intransitive verb. Both of the last-mentioned types may also apply to com­ pound verb phrases. It is convenient to introduce also the types: 1r = subject when the person is irrelevant, q = yes-or-no question when the tense is irrelevant,

he sees her (7r3SJO£ ) 0 ----'> S1

I saw her, 1r1 (7rr s2oe ) 0 ----'> s2, where the first underlink represents the generalized con­ traction

·

In our next example, we make use of two further type as­ signments: has has type 1r3s 1 p�, seen has type p2oe.

=

3Buszkowski [5] has shown that this lemma is essentially a cut elimination

4Such linkages go back to Zellig Harns [1 OJ, as I learned from A K. Joshi.

theorem

The former requires one complement on each side, the lat­ ter only a single complement on the right, to give he has seen her 1r3 (7r3s1 p�)(pzof)o ----'> SJ. '-----'

�

L.._,J

Note in contrast I have seen her 1r1 (7r rs � pi)(pzoe)o -----"' s1 '-----'

'--'

L.._,J

you had seen her 7rz (7rrSzpi)(pzof)o ----'> Sz. '-----'

�

L.._,J

Unfortunately, has must be assigned a different type in direct questions, namely

for a certain logical system, here called

compact bilinear logic.

© 2006 Springer Science+Business Medta, Inc., Volume 28, Number 2, 2006

43

to obtain

Note that this analysis requires the type assignment with : qoeecqe ,

has he seen her?

(ql p�7T�) 7T3(pzoe)o ......,. q1 I

'------'

I '---'

Not wishing to overload the mental dictionary with multiple type listing, one may adopt certain metantles. In the present case, such a metarule would convert the type 1rjskxe into qkxe7TJ. In English, this metarule is restricted to auxiliary verbs, here with x p2; but, in German, it applies to all verbs. I will assign the following type to the object question word =

whom : qoeeqe,

where o ......,. o . The reader will notice that, following the late Inspector Morse, I distinguish between whom and the sub­ ject question word who, which must be assigned a differ­ ent type. 5 Thus we have whom has he seen-?

(qoecqe) (ql p�7T�) 7T3 Cpzoe) I

I

L...-...1

I

'----'

so far the only example of a triple left adjoint. However, people who normally avoid whom would probably avoid this construction and replace it by whom have I seen her with-? which I will not analyze here further (see [20]).

A Very Short Glance at French In this section, which Francophobes may wish to skip, we will show how double left adjoints can help to analyze clitic (i.e . , unstressed) pronouns in French. To start with, con­ sider the clitic pronoun Here we have introduced a new basic type i, strictly larger than i and postulate -

since qfql ......,. qfq ......,. 1 and offof ......,. oecoe ......,. 1 , the latter in view of the contravariant adjunction. The dash here repre­ sents what Chomsky used to call a trace. It turns out that double adjoints will always appear in the presence of traces. It may be of interest to see how the last calculation may be performed step by step when the four words are heard in succession. Here are four stages of the calculation: whom: qoeeqe

(whom) has: (qoecqe) Cq1p�7T�) ......,. qoecp�7T� (whom has) he: (qoeep�7T�) 1r3 ......,. qoeep� '----'

The purpose of the overbar will be made clear later. We begin with some sample sentences: je veux dormir

7Tl (7Trslie) i

je veux voir jean

1..-..J

been : p 2oeep� . Notice another double adjoint anticipating a Chomskyan trace in the following example: seen

'-----'

with

whom

have

I

I

I

THE MATHEMATICAL INTELLIGENCER

= o,

n ___,.

n ......,. o,

je

7T3·

veux

le

voir

'---...J I

L....__..J

7T1 (7T�s1le) cioee le) (ioe)

......,. s1 .

I

To explain the purpose of the bar, we introduce another clitic pronoun

where

w

=

indirect object

is a new basic type. We assign two types to the verb don­ ner, namely

je veux donner

?

Cpzoe)o ......,. q .

l -===----'1 '---' '-----' L

5The reader may wonder why I did not take 6

44

seen her

......,. s1

The name jean here appears as a direct object, but it could also be the subject of a sentence. Had we assigned the type n to names, we should have postulated

I

Apparenty, most Americans, including some prominent linguists, use who in place of whom, except when whom is governed by a preposition, as in

(qomqe)(qoeeqe)(ql p� 7Tl ) 7Tl

L-...-J

to justify the sentences

1T3 (7T rSzp� ) (pzoeepz) (pzoe) ......,. Sz. '------' I

L....__..J

We are now able to handle

When hearing the question, one must hold successively 3, 6, 4, 5, 3, 5, 1 simple types in temporary storage. It is tempt­ ing to identify these simple types with G. A. Miller's chunks of information. In an influential paper [23], Miller suggested that humans can hold a maximum of seven (plus or minus two) chunks of information in their short-term memory. The auxiliary verb be may be employed for construct­ ing passives in English. In particular, its past participle then has the following type:

been

......,. sl

1r1 (7rrs1le) Cio)e) o

'-----'

(whom has he) seen: (qoeep� ) (p2oc) ......,. q I '------' I

she had

-

i -f> i ___,. i.

I

1r1 (7rrs1le) (iweoe) '-----' I

(un livre) (a jean)

o

'-----'

w I

......,. s1.

and je veux donner

1r1 (7rrs1le) (ioewe)

but the explanation would take us too far afield.

(a jean) (un livre)

w

0

The hat here guards against the following, which is correct in English, but not in German:

Now consider je

veux

le

(a jean)

donner

7T1 ( 7Tis 1 ie) cioeeie) (iotwe) L..--J

L.___.J I

L..--J I

je veux lui donner

7TJ ( 7T rsl ie) (iwUiC) (iweoe) L..--J I

je

L..--J I

veux

le lui

w

( un livre) 0

� s1

I

'------'

� S1

donner

lui

I

'------'

we can re-analyze

I

he

Note that linguists usually put an asterisk on the left of an incorrect sentence. For a fuller treatment of French, the in­ terested reader may wish to consult [1).

Not wishing to assume that readers of this article are famil­ iar with German, I will avail myself of a trick due to Mark Twain [29]. He employed it to illustrate the vagaries of Ger­ man by using English words with German word order. It is my contention that this strange word order will be triggered by the types assigned to the verbs. Here are some examples: you see him

7Tz C1TS s1 oe) o

� s1

'------'

see you

him?

(ql oe7T� ) 7Tz o I

�

'------' I

ql

The metarule for forming questions, which applies to En­ glish auxiliary verbs, applies to all verbs in German. I

have

him

seen

7TJ (7Tis lp� ) o (orpz) I can

�

'-----' I him see

I

7T1 ( 7T \"s1ie) o (o ri) I

he

s1

� s1

'----' I

can seen

become

r rr1•) r .e) (orp2) (p2o 1T3 ( 7T 3s1t L_______j

can

he

I

L.._..-_.1

seen become

I

'---'

I

I

'------'

I

II

'----'

'-----'

Some Limitations to Our Approach Other languages for which a preliminary pregroup analysis has been applied are Italian [7], Polish [12] , Japanese [6] and Arabic [2] . Work on Latin and Turkish is in progress. How­ ever, evidence for double adjoints has so far been uncov­ ered only in modern European languages. We have been working with the free pregroup gener­ ated by partially ordered sets. Thus we admit postulates of the form a � f3 only when a and f3 are basic types. As long as this restriction is borne in mind, the Lemma offers a decision procedure for sentence verification. However, it is doubtful whether all linguistic phenomena can be han­ dled successfully using free pregroups only. As Buszkowski [4] formally proved, grammars based on free pregroups are context-free. It is known [26] that certain languages, most familiarly Dutch, are not context-free. The usual proof of this relies on the well-known argument in formal language theory that the intersection of two context-free languages need not be context-free. This suggests that one should in­ corporate the intersection symbol into the pregroup, that is, to work with free lattice pregroups. As far as I know, Even in English, there are some problems with our ap­ proach. Consider the noun phrase

?

II

people whom I know -

Here the English become is used to translate the German passive auxiliary. The above analysis follows [16] in its use of double right adjoints. However, a different approach [22] shows that double right adjoints can be avoided in this con­ text, by assigning a second type to the past participles of transitive verbs. Say, with the help of an appropriate metarule we obtain where o is a new basic type, strictly smaller than o, that is, such that

are pregroups endowed with a binary operation

p (p rpoeese) 7TJ ( 7T\" sl oe ) � PI · �

I

I

'------'

I I

Here I have used the types

p plural noun phrase, s = declarative sentence when the tense is irrelevant and =

the postulate

The relative pronoun, in this context whom : prpoeese ,

o + o � o. 6Lattice pregroups

become

such a project has not yet been attempted 6

(ql ie7T� ) 1T3 (orpz) Cp So rri) � ql I

can seen

1T3 ( 7T 3si:) (pzoe) (op 2i) � s 1

without using double right adjoints.

German a Ia Mark Twain

'------'

= 0.

le donner

'-+--' I

I

'------'

+ s1 .

'------' L......,L-J

By assigning the following revised type to the passive aux­ iliary

7TJ ( 7TIS!ie) (iwUit) (ioeeie) (ioew 0 one has g(n + p) g(n) for all n. =

As simple examples, consider g(n) : = s, where s is a fixed integer (this is called the s-ball cascade). For s = 1 , this pattern can be juggled most easily: simply throw the ball from the right hand to the left and vice versa. The case s 2 is also simple: at even time units catch ball number one and throw it immediately again with the left hand so that it can be caught two time units later with the same hand, similarly the right hand deals with ball number two at odd time units. A more de­ manding juggling pattern is the three­ ball cascade. At every time unit a ball is caught and immediately thrown again, so that it is in the air for three time units. (This seems to be the simplest juggling pattern where three balls are involved.) If g is periodic with period p it suf­ fices to know the values of g for p con­ secutive values: g can be reconstructed uniquely up to a translation. For exam­ ple, "312" stands for a juggling pattern where the g-sequence looks like . . . 312312312 . . . , and the three-ball cas­ cade can be abbreviated by "3". Finite sequences which arise in this way (like 312 and 3 in the preceding example) are called juggling sequences. =

Theorems and Corollaries The theory starts with simple results ("If a0, . . . , ap- 1 is a juggling sequence then so is the sequence a0 + d, . . . , ap- 1 + d, where d is a fixed positive in­ teger."), but rather soon the assertions are more interesting and the proofs are trickier. For example, let a juggling se­ quence a0, . . . , ap- 1 be given. One can show that the number of balls needed to juggle it is precisely (ao + + ap-1)/p this is intuitively clear since this number is the number of balls in the air "in the average" . It follows that the s-cas­ cade "s" needs s balls (which is clear) ·

·

·

and that one must have 2 balls to jug­ gle "312". This "average theorem" has a nice corollary: A finite sequence of inte­ gers will give rise to a juggling sequence here p. is the usual Mdbiusfunction, and only if the average is an integer. How­ juggling sequences which are identical ever, this condition is not sufficient. As up to cyclic permutations are identified. an example, consider 321 . The average is 2, but the associated function n ,...,. n + More Sophisticated Definitions Juggling sequences are the basic objects g(n) is not one-to-one. Surprisingly, it can be shown that the of interest; their study covers the first condition is nearly sufficient: If the av­ chapters of Polster's book. More com­ erage of a finite sequence a of non­ plex notation is needed when weaken­ negative integers is an integer, then ing the assumptions of the first ap­ there exists a permutation of a which proach. How can one deal with the is a juggling sequence. The proof is possibility of throwing and catching rather involved, but nevertheless-as it more than one hall in a given time unit (multiplex juggling)? What modifications is to be expected-elementary. Now suppose that someone is jug­ are necessary if not one but several jug­ gling a certain juggling sequence a0 . . . glers are involved (multihand juggling)? Formally, a multiplex juggling se­ ap- 1 of length p. After some time it might be desirable to pass without in­ quence is a finite sequence of finite terruption to another pattern. Fix an i nonempty ordered sets of non-negative and an integer d such that 0 < d :s: ai. integers. These sets encode what kinds If ai is replaced by ai+ d + d and ai+ d of throws are made on every beat. For b y a i - d, then this will b e again an example, ! 1 ,4}{ 1 } has to be realized as admissible sequence. For example, the follows: On the first beat, two balls are ball thrown at time unit i will be caught caught, they are immediately thrown at that moment when in the original pat­ again, the first one to height one, the tern the ball thrown at time unit i + d second to height four; on the second is going to land. (For example, in the beat, one ball is caught and thrown to case p = 3, i 0, and d 1 one will height one; the actions on beat one and two are repeated again and again. pass from 642 to 552.) Multihand juggling needs an even The operation of generating a new sequence in this way is called a site swap. more elaborate notation, one has to As another, more elementary operation, pass to matrices, the columns of which one can consider the cyclic shift where prescribe what has to be done at a cer­ one replaces ao ap- 1 by tain time unit. For example, one can a1 az ap-l ao. It is rather surprising learn from the second entry of such a that site swaps and cyclic shifts suffice to column what action "hand" number two pass from any juggling sequence to any is assumed to perform at the corre­ other provided that length p and the sponding time: catch 5 balls, throw two of them such that they arrive at "hand" number ofballs (a0 + + ap 1)/p co incide. For example, one could start with number one two units later and throw hh b (which is identical with the cas­ the remaining balls such that they are cade b) and arrive at C pb)O 0 (this at "hand" four at the next time. The emphasis is similar to that in the corresponds to the pattern where at every pth time unit a ball is thrown so strongly case of the above juggling sequences of a single player. One can prove average that it is in the air for ph time units). The book also contains many result� cen­ theorems. it is possible to count the tering on counting: What is the number of number of essentially different patterns, juggling sequences with a prescribed prop­ the mutual dependence of these pat­ erty? A� a sample theorem, consider the case terns can be visualized by graphs, etc. With the notations of multihand jug­ of minimal juggling sequences. (A juggling sequence CkJ • • • ap- l is called minimal if gling at hand it is also possible to it cannot be written as a repetition of change the point of view. If h balls are smaller juggling sequences; so 441 is juggled by h hands in a certain way, minimal, but 44444 is not.) It can be one may interchange the roles of balls shown that the number of minimal jug­ and hands: Fix the position of the balls gling sequences of period p with aver­ and let the hands move! Now the balls juggle the hands using a pattern which age b is precisely =

·

·

=

·

Fachbereich Mathematik und lnformatik Freie Universitat Berlin Arnimallee 2-6 D-1 41 95 Berlin Germany [email protected]

Flatter land hy Ian Stewart

·

·

·

is in a sense dual to the original one. Claude Shannon, the famous informa­ tion theorist, was one of the first to in­ vestigate this "duality theory" . The book contains much more. A survey of the history of juggling, hints for jugglers, the connections with bell ringing, to mention a few. For me it was very stimulating. I tried to juggle some of the simpler juggling sequences, and I also wrote a computer program to visualize even very complicated ao, . . . , ap- 1 (which no human being will ever manage). The mathematics involved are very interesting: I had not expected to see so many connections with algebra, graph theory, and combinatorics. In a review of this book in the No­ tices of the AMS (Tanuary 2004) Allen Knutson argued that the historical part is not free of errors. I do not have the background information to decide whether this criticism is justified. For me, it is a fascinating book from which I learned a lot.

·

·

-

·

·

·

·

­

CAMBRIDGE, MASSACH USETIS, PERSEUS PUBLISHING, 2001, 301 PP. $25.00 ISBN 0-7382-0442-0

The Annotated Flatland A romance of many dimensions by Edwin A. Ahhott Introduction and notes by Ian Stewart CAMBRIDGE, MASSACHUSETIS, PERSEUS PUBLISHING, 2002, 239 PP. $31.00 ISBN 0-7382-0541-9 REVIEWED BY VAGN LUNDSGAARD HANSEN

I

n 1884, Edwin Abbott Abbott (18381926) published his renowned book "Flatland" , which has amused read­ ers now for more than a century. In 2002 Perseus Publishing reissued the

© 2006 Spnnger Science+Business Media. Inc. . Volume 28, Number 2. 2006

89

book under the title "The Annotated Flatland" with carefully written com­ ments and notes by Ian Stewart, in­ cluding a comprehensive biography of Abbott. Although he showed great tal­ ent for intellectual work during his years of study at St. John's College, Univer­ sity of Cambridge, England, 1857-1862, Abbott chose another career than uni­ versity and became a priest in the An­ glican Church in 1 863. He spent most of his life as an enthusiastic and capa­ ble educator and became headmaster of the City of London School in 1865, stay­ ing in this position until his retirement in 1889. In mathematical circles, Abbott is known only for his book "Flatland". In Flatter/and, the distinguished math­ ematician and popular science writer Ian Stewart takes on the almost impos­ sible and daring task of writing a con­ temporary sequel to Abbott's classic. In his charming joking style of exposition, Stewart draws on the contemporary fan­ tasy world and phenomena that in many cases would not have made any sense in Abbott's times. In other cases, the jokes are slightly strained, such as the language joke on page 238: "the nat­ ural logs in the hearth were burning brightly". In general, though, Stewart is entertaining and, admittedly, the re­ viewer laughed even in the case men­ tioned. A review of Tbe Annotated Flatland seems improper at this place since it will amount to a review of Abbott's clas­ sic, for which time has already revealed its great merits. There is, however, good reason for an appraisal of Stewart's scholarly and elegantly written com­ mentaries, many of which are pearls. This holds in particular for his comments on nine quotations in Flatland from Shakespeare (the only author quoted by Abbott), and for several short biogra­ phies of mathematicians and physicists of importance for understanding the profundity in Abbott's book. ()()()

90

THE MATHEMATICAL INTELLIGENCER

Stewart's book Flatter/and is a de­ manding one. The reader has to be ac­ quainted with much of the language of modern physical theories to appreciate the elegance with which Stewart writes about the continuing attempts by sci­ entists to understand the shape and ori­ gins of the universe and the nature of space, time, and matter. Ever since Ein­ stein formulated his revolutionary ideas of general relativity in 1915, these ef­ forts have been intimately related to de­ velopments in differential geometry and topology, and contemporary mathemat­ ics altogether. It is difficult to tell how Stewart's exposition will appeal to the lay reader, but an educated reader with some background in science can gain insights into many recent concepts from cosmology that regularly pop up in the media, such as "big bang", "gravitational lenses", "antimatter" , and "wormholes" . A slightly annoying thing is Stewart's way of naming the persons in his nar­ rative by concatenations like 'Haus­ dorffbesicovich' (page 69); it is easier with 'Alberteinstein' (page 45); or is it? Flatter/and is a science-fiction story about Victoria Line, who comes upon her great-great-grandfather A. Square's diary, hidden in the attic. The diary helps her to contact the Space Hopper, who tempts her away from her home and family in Flatland and becomes her guide and mentor through 10 dimen­ sions-the manifold dimension of the extension of space-time describing the universe. During the journey with Space Hopper, we experience large parts of contemporary mathematics through the virtually enhanced eyes of the truly amazed Victoria Line and registered in her 'Diary Dear'. It is impossible to mention all the things Victoria Line discovers during her journey, but here are a few highlights. To interpret the diary of her great-great­ grandfather, Victoria must first under­ stand some basic elements of cryptog­ raphy (Chapter 2). When she first meets the Space Hopper he gives her a com-

prehensive introduction to seeing 3-di­ mensional things through 2-dimensional sections and to Kepler's sphere-stacking problem (Chapter 3). Stewart has earlier written about fractals, and lovely de­ scriptions are given of fractal dimension and Mandelbrot sets in Chapter 5. In "The topologists' tea-party" (Chapter 6) we meet the Mobius cow and milk in Klein bottles, and also the Alexander horned sphere. The projective plane is discussed as a one-sided surface, and there is a good introduction to finite projective planes (Chapter 8). The var­ ious types of geometries as described by Felix Klein in terms of their trans­ formation groups are beautifully ex­ plained in Chapter 9. In "Cat Country" (Chapter 1 1), there is a well-thought­ out exposition of the theory of ele­ mentary particles and the duality be­ tween waves and particles. On it goes with a description of general relativity in "The domain of the Hawk King" (Chapter 1 3) and a beautiful introduc­ tion to Hawking's universe and the Penrose map in "Down the Wormhole" (Chapter 1 4) . Through the remaining part of her journey, Victoria Line gets a stimulating introduction to theories about the shape of the universe, in­ cluding an excellent explanation of the redshift and presentation of the argu­ ments that the universe expands. Ian Stewart has written many excel­ lent books, mathematics textbooks as well as popular writings. Flatter/and is not his best book, but it is still an en­ joyable and stimulating book from which to capture the spirit of some ba­ sic trends in contemporary mathemat­ ics and science and to be entertained as well. Ian Stewart is indeed a master of exposition. Department of Mathematics Technical University of Denmark Matematiktorvet, Building 303 DK-2800 Kgs. Lyngby Denmark e-mail: [email protected]

$j£1

1.19.h•i§i

..

Robin Wilson

1005-World Year of Physics ANIL NAWLAKHE AND ROBIN WILSON

Since the eighteenth century it had been assumed that all physical motion throughout the universe satisfies New­ ton's laws of motion. Moreover, these laws still hold for systems moving at a constant speed relative to each other. In 1905 Albert Einstein (1879-1955) published his special theory of relativ­ ity, proposing an entirely new point of view based on the notion that one can­ not determine the absolute motion of

I

a moving object. He made three as­ sumptions: (i) the maximum velocity in the universe is that of light; (ii) the mea­ sured value of the speed of light is con­ stant, no matter how fast the observer or light source is moving; (iii) absolute speed cannot be measured--only speed relative to another object. It fol­ lows that objects foreshorten in the di­ rection in which they move, and the faster they move the shorter they be­ come. It also follows that mass is a form of energy, and that the energy E and the mass m are related by the relativ­ ity equation E= mc 2 , where c is the speed of light. Until that time it had been assumed that Maxwell's wave equations are valid only in a particular frame of reference (that of the "ether" that carries the waves), and are thus unlike Newton's laws, which hold for all observers in uniform motion. Einstein reconciled this apparent discrepancy by starting

REPUBLIOUE DU MAU

from the simple postulate that the ba­ sic laws of physics (including Maxwell's equations) are the same for all ob­ servers in uniform motion relative to one another. In 1 905 Einstein also explained the 'photoelectric effect', that light behaves like particles and can liberate electrons on impact with a metal surface. Ein­ stein's paper, which led to his 1921 No­ bel Prize in Physics, showed that Planck's equation E hv is a funda­ mental feature of light itself, rather than of the atoms. Despite the many exper­ iments showing light to consist of waves, it also behaves as though it con­ sists of particles, each carrying one quantum ( h v) of energy. =

Anil M. Nawlakhe 41 , Gayatree Nagar Ganishpur Bhandara-441 904 M.S. India

ALBERT EI:-<STEI.

1879 19:;5

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics.

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