The Mathematical Intelligencer

Note

Well,Papa,Can You Multiply Triplets? SOPHIE MORIER-GENOUD AND VALENTIN OVSIENKO

We show that the classical algebra of quaternions is a commutative Z2
Z2
Z2 - graded algebra. A similar interpretation of the algebra of octonions is impossible.

his note is our ‘‘private investigation’’ of what really happened on the 16th of October, 1843 on the Brougham Bridge when Sir William Rowan Hamilton engraved on a stone his fundamental relations:

T

2

2

where h , i is the scalar product of 3vectors. Indeed, hr(i), r(j)i = 1 and similarly for k, so that i, j and k anticommute with each other, but hr(i), r(i)i = 2. The product i  i of i with itself is commutative and similarly for j and k, without any contradiction. The degree (1) viewed as an element of the abelian group Z2
Z2
Z2 satisfies the following linearity condition rðx  yÞ ¼ rðxÞ þ rðyÞ;

ð3Þ

2

i ¼ j ¼ k ¼ i  j  k ¼ 1: Since then, the elements i, j and k, together with the unit, 1, have denoted the canonical basis of the celebrated four-dimensional associative algebra of quaternions H: Of course, the algebra H is not commutative: The relations above imply that the elements i, j, k anticommute with each other, for instance i  j ¼ j  i ¼ k: Yes, but...

The Algebra of Quaternions Is a Graded Commutative Algebra Our starting point is the following observation. The algebra H indeed satisfies a graded commutativity condition. Let us introduce the following ‘‘triple degree’’: rð1Þ ¼ ð0; 0; 0Þ; rðiÞ ¼ ð0; 1; 1Þ; rðjÞ ¼ ð1; 0; 1Þ;

ð1Þ

for all homogeneous x; y 2 H: The relations (2) and (3) together mean that H is a Z2
Z2
Z2 - graded commutative algebra. We did not find the above observation in the literature (see however [1] for a different ‘‘abelianization’’ of H in terms of a twisted Z2
Z2 group algebra; see also [2, 3, 4]). Its main consequence is a systematic procedure of quaternionization (similar to complexification). Indeed, many classes of algebras allow tensor product with commutative algebras. Let us give an example. Given an arbitrary real Lie algebra g, the tensor product gH :¼ H R g is a Z2
Z2
Z2 - graded Lie algebra. If furthermore g is a real form of a simple complex Lie algebra, then gH is again simple. The above observation gives a general idea of studying graded commutative algebras over the abelian group C ¼ Z2
  
Z2 : |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} n times

rðkÞ ¼ ð1; 1; 0Þ: Then, quite remarkably, the usual product of quaternions satisfies the graded commutativity condition: p  q ¼ ð1ÞhrðpÞ;rðqÞi q  p;

ð2Þ

provided each of p; q 2 H is proportional to one of the basis vectors and

One can show that, in some sense, this is the most general grading, in the graded-commutative-algebra context, but we will not provide the details here. Let us mention that graded commutative algebras are essentially studied in the case C ¼ Z2 (or Z). Almost nothing is known in the general case.

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in Z2
Z2
Z2 ; i.e., the componentwise addition (modulo 2), for instance,

...But Not the Algebra of Octonions After the quaternions, the next ‘‘natural candidate’’ for commutativity is, of course, the algebra of octonions O: However, let us show that: The algebra O cannot be realized as a graded commutative algebra. Indeed, recall that O contains 7 mutually anticommuting elements e1, ...,e7 such that ðe‘ Þ2 ¼ 1 for ‘ ¼ 1; . . .; 7 that form several copies of H (see [2, 3] for a beautiful introduction to the octonions). Assume there is a grading r : e‘ 7! C with values in an abelian group C, satisfying (2) and (3). Then, for three elements e‘1 ; e‘2 ; e‘3 2 O; such that e‘1  e‘2 ¼ e‘3 ; one has rðe‘3 Þ ¼ rðe‘1 Þ þ rðe‘2 Þ: If now e‘4 anticommutes with e‘1 and e‘2 ; then e‘4 has to commute with e‘3 because of the linearity of the scalar product. This readily leads to a contradiction.

Let us now take another look at the grading (1). It turns out that there is a simple way to reconstitute the whole structure of H directly from this formula. First of all, we rewrite the grading as follows:

ð4Þ

k $ ð11 ; 12 ; 0Þ: Second of all, we define the rule for multiplication of triplets. This multiplication is nothing but the usual operation

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but with an important additional sign rule. Whenever we have to exchange ‘‘left-to-right’’ two units, 1n and 1m with n [ m, we put the ‘‘-’’ sign, for instance ð0; 12 ; 0Þ  ð11 ; 0; 0Þ ¼ ð11 ; 12 ; 0Þ; since we exchanged 12 and 11. One then has for the triplets in (4): i  j $ ð0; 12 ; 13 Þ  ð11 ; 0; 13 Þ ¼ ð11 ; 12 ; 0Þ $ k; since the total number of exchanges is even (12 and 13 were exchanged with 11) and j  i $ ð11 ; 0; 13 Þ  ð0; 12 ; 13 Þ ¼ ð11 ; 12 ; 0Þ $ k; since the total number of exchanges is odd (13 was exchanged with 12). In this way, one immediately recovers the complete multiplication table of H:

ACKNOWLEDGMENTS

We are grateful to S. Tabachnikov for helpful suggestions.

REFERENCES

1. H. Albuquerque, S. Majid, Quasialgebra structure of the octonions, J. Algebra 220 (1999), 188–224. 2. J. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145–205. 3. J. Conway, D. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, Ltd., Natick, MA, 2003. 4. T. Y. Lam, Hamilton’s quaternions, Handbook of Algebra, Vol. 3, 429–454, NorthHolland, Amsterdam, 2003.

Multiplying the Triplets

1 $ ð0; 0; 0Þ; i $ ð0; 12 ; 13 Þ; j $ ð11 ; 0; 13 Þ;

ð11 ; 0; 0Þ  ð11 ; 0; 0Þ ¼ ð0; 0; 0Þ; ð11 ; 0; 0Þ  ð0; 12 ; 0Þ ¼ ð11 ; 12 ; 0Þ;

morning: ‘‘Well, Papa, can you multiply triplets?’’ and always getting the same answer: ‘‘No, I can only add and subtract them’’, with a sad shake of the head. This story now has a happy ending. As we have just seen, Hamilton did nothing but multiply the triplets. Or should we rather say added and subtracted them?

REMARK 3.1 The above realization is, of course, related to the embedding of H into the associative algebra with 3 generators e1 ; e2 ; e3 subject to the relations e2n ¼ 1; en em ¼ em en ; n 6¼ m:

for

This embedding is given by i 7! e2 e3 ;

j 7! e1 e3 ;

k 7! e1 e2

and is well known. Everybody knows the famous story of Hamilton and his son asking his father the same question every

Universite´ Paris Diderot Paris 7 UFR de mathe´matiques case 7012 75205 Paris Cedex 13 France e-mail: [email protected] CNRS, Institut Camille Jordan Universite´ Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex France e-mail: [email protected]

Viewpoint

Mathematics Ho! Which Modern Mathematics Was Modernist? I. GRATTAN-GUINNESS

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 editor-in-chief endorses or accepts responsibility for them. Viewpoint should be submitted to the editor-inchief, Chandler Davis.

n fact, if A accuses B of being reactionary, B can always reply ‘On the contrary, I am merely reacting against your reaction against reaction’. Constant Lambert, Music Ho! [1934, 201]

I

A variety of philosophies of mathematics has been developed over the centuries. Some positions have been allied to philosophies of science or of knowledge in general (empiricism, positivism, conventionalism, fallibilism); a few have stressed psychological aspects (phenomenology, psychologism); some have been largely tied to mathematics itself (formalism, logicism—of which more anon); a few carry the names of leading philosophers (Plato, Aristotle, Kant). Occasionally, a philosophy has drawn upon broader cultural contexts for inspiration; so far the prime example has been modernism. In this article I discuss the specification and the range of the modernist thesis, suggest several new examples and further contexts for it, and question some parts of the pertinent historical record. I also raise several general issues involved, especially the relationship between mathematics and symbolic logics and between pure and applied mathematics. The breadth and depth of the thesis relative to mathematics as a whole is assessed at the end. To ground this discussion, I consider two important books. One is a study by the German historian Herbert Mehrtens that seeks to reinterpret the concerns with the foundations of mathematics, especially from the 1890s to the 1930s and largely in Germanspeaking countries, as a dispute between modernist and antimodernist mathematicians [Mehrtens 1990]. The

second is Jeremy Gray’s [Gray 2008], which treats the place in research-level mathematics of modernism in roughly the same period; this book comes soon after a relevant collection of papers on the philosophy of mostly pure modern mathematics that Gray co-edited [Ferreiro´s and Gray 2006]. To save much space I cite the books by Mehrtens and Gray respectively as ‘M’ and ‘G’; occasionally I note the co-edited volume, which is cited as ‘FG’.

Modernism in the Arts and in Mathematics Modernism has received most attention in the context of the arts. The word suggests disagreement with the current situation and opposition to conservative traditions, and looks forward to a different future. Modernism is also time dependent. For example, from the late nineteenth century to the Great War, manifestations included cubist and expressionist art, Sigmund Freud (especially as an analyst of anxiety) and Arnold Schoenberg’s launch of serialist music; manifestations after the War included Igor Stravinsky’s neoclassical works,1 and T. S. Eliot’s poem ‘The Waste Land’, which was hailed as a modernist work on its publication in 1922. A prominent feature of modernism was pluralities of various kinds, such as multiple viewpoints in cubism, bitonality in music, and ambiguity in language (such as with small words like ‘us’ with Eliot) [Butler 1994]. Another feature was extremism—’deliberate silliness’ for Lambert [1934, 64]—such as nonsense words and language, examples of the initiatives that a modernist might take. Mehrtens approaches modernism contextually in his early chapters, in which he stresses the freedom of the

1

The rise in the status of modernism in music after the Great War was especially marked, and one of the casualties was Edward Elgar [Lambert 1934, 205]. My great-uncle, a talented amateur string player, knew Elgar and played in orchestras under his direction. He told me in June 1953 that on one occasion in the 1920s they performed one of the oratorios, and there were more people on the stage than in the audience.


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mathematician to create theories and the need for rigour and axioms, and regards pure mathematics as more certain than applications [M, pts. 1–2]. Gray begins with a compatible definition of modernism in mathematics, given in a jumbo sentence [G, 1] as […] an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious—rather than a naı¨ve relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve. So modernism is predicated on both a mathematical theory and the mathematicians who develop it, who are not working in isolation. However, while Gray relates his definition to a text by Guillaume Apollinaire on cubist art, it is surely incomplete and so too wide; for example, there is no reference to mathematics at all, and no time dependence. In some further exegesis broadly similar to Mehrtens’s, modernist mathematicians were concerned with, and even became anxious about, rigour and foundational questions, emphasized pure much more than applied mathematics, stressed the axiomatisation and structure of theories, and became concerned with language. Modernism seems to cover both the state of a mathematical theory in being modern (or not) and the process of modernizing a theory (or not): the modernist mathematician is not necessarily contributing to the axiomatisation or structuring of a theory but prefers to work in one that is already in such a form. Desiderata for rigorous theories included hopefully complete and consistent axiomatisations, carefully formed definitions and detailed

proofs; I would add a high status given to existence theorems. Mehrtens gives prominent places to intuition and perception, especially in connection with the standpoints of antimodernist mathematicians in which they play a significant role [M, pt. 3]. These include those involved in the so-called ‘crisis’ in foundations in the 1920s,2 and later some of the Nazi mathematicians; these two contexts were among the initial motives for his study. In addition, he draws upon the Bourbaki collective (but not Saunders Mac Lane) for its structuralism [M, 315–326]. He also considers, as does Gray in a rather more systematic way, several German philosophers and psychologists from Immanuel Kant onwards, who often acted under his influence.3 Gray treats both epistemologically oriented forms such as Kant’s pure intuition that pertain to modernist mathematics, and more naı¨ve manifestations such as those associated with sensible intuition and heuristics that normally belong to the pre-axiomatic creation and development of a mathematical theory [Po´lya 1954]. The discussion of these important differences should have come much earlier than G [197–202]. An important part of the modernist creed is anxiety over errors in mathematical theories, which must be detected and eliminated. Errors can come in various forms: For example, logical issues such as muddling necessary and sufficient conditions, unintentionally assuming the theorem to be proved, or filling in missing gaps in proofs; ambiguities of expression or definition, requiring new distinctions of sense; and errors in calculation arithmetical or algebraic. Maurice Lecat compiled an impressive catalogue [1935] of around 500 errors and their (partial) correction from across the history of mathematics. Later, Rene´ Dugas [1940] and Rostand [1960, 1962] discoursed at length on inexactitude and unclarity in mathematics, and con-

sidered means of production of mathematics more rigorous, even maybe axiomatic or modernist (a word that they did not use); Rostand focussed upon mathematical analysis and related subjects and drew largely upon French authors. This substantial body of work anticipated some of the considerations offered in all three books; it is a pity that our authors do not consider it, especially in M [474]. In my own philosophical work on structuresimilarity between mathematical theories [Grattan-Guinness 1992] I found their contributions very useful. Three principal mathematicians featured in both books are Felix Klein, Hermann von Helmholtz, and Henri Poincare´. They might qualify as modernists for their advocacy of some kind of important role for Kantian philosophy in mathematics, and also for other reasons; for example, Klein on account of his innovative use of group theory in geometry. However, all three are non- or even antimodernists for their use of intuition, and maybe also for their prosecution and encouragement of applied mathematics. The position of Poincare´ is obscure: Apparently he was modernist when separating mathematics from physics and invoking conventionalist philosophy [G, 34]; on the other hand, he does not come readily to mind when thinking of rigorous mathematics. There seems to be some improvised historiography here; Mehrtens classifies Klein and Poincare´ as antimodernists tout court [M, pt. 3]. Two further major figures are the thoroughly modern Richard Dedekind and, above all, David Hilbert. The importance of Hilbert’s modernising axiomatic and metamathematics4 is well brought out by both authors. One obvious clarion call for modern mathematics is his famous speech of 1900 in Paris on mathematical problems to be tackled in the new century [Hilbert

2 The accounts of ‘crisis’ talk among modernist German-speaking mathematicians in the 1920s and 1930s in these books do not mention the economic crises of that time, though they must have helped to stimulate the attitude. 3 One of them was Carl Stumpf; Gray attributes his major book to Edmund Husserl [G, 206]. Another is Wilhelm Wundt [G, 393-400]; Cantor’s letters to him, dealing with the relationship between set theory and space and time, are published in [Kreiser 1979]. On the context of Kant and his influence upon 19th-century science see, for example, [Friedman and Nordmann 2006], to which Gray contributed an article. 4 In support of G [29, 412], Hilbert used ‘formalism’ only in the context of metamathematics and never to characterise his philosophical position as a whole [M, 292]. This step was taken by L. E. J. Brouwer in 1927, as an insult, and was taken as such by its recipient. Sadly, the use of this word to name Hilbert’s position became standard even among his followers and commentators, including [M, 121, 577].

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1901]. This item is duly noted, and Gray has written on it at length elsewhere; here they are suitably temperate about it, for it was much more a personal selection of topics made by Hilbert in some haste than a grand considered overview [Grattan-Guinness 2000b].

Geometries Both authors deal at some length with the history of projective and non-Euclidean geometries and the relationship of all geometries to space and thereby to physics. The modernism arises especially in the separation of these categories and the use of other pieces of modern mathematics: For example, group theory, especially with Klein, the legitimacy of studying n-dimensional spaces when n[ 3, and the application of kinematics to geometry. An early historical and philosophical commentator on some of these developments up to Poincare´ was Rougier [1920], published when modernism itself was developing; an assessment of his work would have been welcome. Among earlier figures, C. F. Gauss and Bernhard Riemann are stressed, especially for their contributions to intrinsic geometry; and Gauss again and also N. I. Lobachevsky and Ja´nos Bolyai for their pioneering advocacies of non-Euclidean geometries. However, this last trio worked in substantial isolation (Gauss by choice), which renders doubtful their status as modernists. I missed a connected account of this early consequence of Hilbert’s axiomatisation of 1899. E. H. Moore’s student Oswald Veblen, helped by his supervisor, introduced the notion of categoricity in his doctoral dissertation of 1904, an important step forward for a theory that was then still young. Mehrtens ignores this story, while Gray splits it up [G, 183, 205] — unfortunately, for model theory was surely modernist in allowing for different interpretations of a given axiom system. As emerges from [FG, 209–248 (P.

Mancosu)], a summary would have enhanced the discussion of modelling in general that both authors offer.

Mathematical Analysis and Set Theory When someone will speak to us for the first time of, for example, a continuous function in Cauchy’s sense, we shall enter into the symbolic game of inequalities that attend this definition, without asking ourselves if, in making it more precise, he can help us to translate the vague feeling that we may have of continuity. Rene´ Dugas [1940, 7] In his handling of aspects of the history of mathematical analysis, Gray starts out from A. L. Cauchy’s emphasis on limit theory during the 1820s and later,5 and its extension by Karl Weierstrass and his followers from the late 1850s onwards (when Mehrtens joins in). I see much of this development as based upon a single limit theory being refined into a multiple limit theory [Grattan-Guinness 1970]; Gray does not put it this way, though some of his comments are consistent with it [G, 275]. Further, in the emphasis on rigour of proof, care over definitions, and detachment from experience, both Cauchy and Weierstrass surely perform quite modernistically relative to their respective periods: Gray demurs over Cauchy for in effect not being sufficiently Weierstrassian [G, 62–67]. Gray could have strengthened his remarks on the status of integration by noting the great rise in prominence, not due to Cauchy, of line and surface integrals, especially with the growth of potential theory [G, 71–75, 267]. Among later manifestations of modernism he notes the rise of summability theory, where the mathematician chooses the method by which an infinite series may be summed [G, 234]; again, this correct claim would have gained greater

weight if he had emphasised the general conception put forward by E´mile Borel around 1900 rather than the important pioneering case studied in the 1870s by Ernesto Cesa`ro [Tucciarone 1973]. It is natural to move from mathematical analysis to set theory, for when Georg Cantor founded the latter theory in the early 1870s, his basic concept was that of the ‘derived’ set, the set of limit points of a set. After 25 years of some opposition and much indifference, his theory rapidly gained attention from the mid 1890s on. An important part of the reception at this time was its application to mathematical analysis and related topics. Unlike Mehrtens, Gray describes the development of measure theory [G, 217–223]; however, neither author has much on functional analysis [G, 224–225; FG, 381–382] or on integral equations [M, 133–135; G, 254, 454], although Hilbert devoted nearly a decade to the latter topic (after not mentioning it at all in his 1900 speech!). Surely both topics were prime cases of modernism in the 1900s and later, and deserve some exegesis. So, too, do at least parts of the (pure) theory of differential equations; but nothing is said about the contributions of figures such as Jacques Hadamard, Emile Picard, Edouard Goursat or Ludwig Schlesinger, and very little on Klein’s friend Sophus Lie. Indeed, this massive branch of pure and applied mathematics does not even gain an index entry in any of the three books. Another sign of modernism is the axiomatisation of set theory, encouraged by Hilbert, achieved by Ernst Zermelo in 1908, and developed in various ways in the 1920s by Abraham Fraenkel, Johann von Neumann and others.6 Their approach contrasted with the traditional intuitive conception of sets, which was followed by Cantor himself (modern in the 1870s, rather old-fashioned 30 years later) and maintained by figures such as Felix Hausdorff. Yet Hausdorff, too, counts as

5 Although I did not use the language of modernism, my account [Grattan-Guinness 1990] of French mathematics in the period 1800–1840 can be seen as including the modernising of parts of both pure and applied mathematics by men such as J. B. J. Fourier, Cauchy, A. M. Ampe`re, and A. J. Fresnel (Fourier analysis, a theory of limits, real- and complex-variable mathematical analysis, the inauguration of classical mathematical physics) against the traditionalists such as S. D. Poisson and J. B. Biot (power series, molecular physics). 6 Part of Zermelo’s contribution was to introduce the axiom of choice in 1904. Contrary to his supposed opposition to it [G, 256], Russell (‘not a mathematician’ [G, 455]) came to a form of the axiom slightly earlier than Zermelo, and gave it special attention, partly because of the difficulty of expressing it within mathematical logic [GrattanGuinness 1977, esp. pp. 80–81].


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a modernist, because of his affirmation of the freedom to create mathematics [M, 174–177], or his adhesion to Kantian philosophy [G, 222–223]. But now the breadth of conception of modernism arises again; was anybody not modernist? Maybe Artur Scho¨nflies is a candidate; [G, 264–266] provides some evidence. One major source for anxiety in both set theory and logic was Bertrand Russell’s paradox. He found it in 1901 from taking a special case of Cantor’s power-set theorem.7 In characterising this result as a paradox or antimony, our authors follow many previous commentators [M, 151–157; G, 164, 251] in not emphasising that it is a double contradiction, since the set of all sets that do not belong to themselves belongs to itself if and only if it does not do so. Russell realized at once that this was much more serious then an ordinary contradiction. Mehrtens claims as a new reading of the history of the foundations of mathematics a reduction in importance of the paradoxes [M, 149– 152]; but at least one outsider had tried already to give that impression [GrattanGuinness 1977, 1988].

Algebraic Logic and Mathematical Logics Both authors rightly devote some space to symbolic logics. But two traditions developed [G, 21–22] that were so different from each other that the use of symbols is about the only common factor [Grattan-Guinness 2004a]; hence two quite different relationships to mathematics are obtained. Briefly, the algebraic logicians (George Boole, Augustus de Morgan, 7

C. S. Peirce, Ernst Schroeder) applied mathematics (namely, their favourite algebras) to logic,8 handled collections in the traditional manner of part-whole theory, viewed quantification extensionally as an extension to infinitude of conjunction and disjunction, and within language stressed nouns and adjectives [Brady 2004]. In addition, Schro¨der proposed a passport-like version of logicism in which a mathematical theory was tested on its own to see if it satisfied (or not) five logical tests [Peckhaus 1991]. In great contrast, the mathematical logicians either kept the logic in mathematics separate (Giuseppe Peano and his followers) or applied this logic to mathematics by developing an integrated form of logicism where mathematical theories were built up in a nesting sequence of key definitions (the rather isolated Gottlob Frege for arithmetic and a somewhat unclear amount of real-variable mathematical analysis, A. N. Whitehead and Russell for ‘‘everything’’9), took their prime influence from mathematical analysis, handled collections by means of Cantorian set theory, (usually) regarded quantification as a binder across some Cantorian set of values, and in language also emphasized particles such as ‘the’ and ‘some’ [Grattan-Guinness 2000a, chaps. 5–9]. Despite the prime place given to axiomatisation and rigour and the major role assigned to set theory, logicism was antimodernist [G, 29], presumably because philosophically Russell invoked Leibniz rather than Kant: More improvisation? A striking manifestation of the difference between algebraic logic and mathematical logic occurred in August

1900 at the International Congress of Philosophy, held in Paris. Peano spoke on the correct ways of framing definitions in mathematical theories, including the need to individuate ‘the’; Schro¨der, in the audience, objected; Peano held his ground. The historical importance of this occasion lies in the fact that Whitehead and Russell were also in the audience, experiencing Peano’s approach for the first time [Grattan-Guinness 2000a, 290–291]. Pace [G, 240], especially from the 1910s onwards mathematical logic came to eclipse its older brother, perhaps to excess;10 Mehrtens ignores algebraic logic almost entirely. Its revival dates only from the 1940s onwards, by the Polish logician Alfred Tarski, among others, and involved further algebras such as group theory; it also used set theory rather than part-whole theory.11 But Peirce had been involved in an extraordinary sequence of influences and reinterpretations, especially in the 1880s. It starts out from a graphical treatment of quantics by W. K. Clifford, on to quantics and then multisets with A. B. Kempe [Vercelloni 1989, 1–46], then the first version of Peirce’s existential graphs (that provide a topology of syntax), and finally the development of a particular kind of relation by the philosopher Josiah Royce [GrattanGuinness 2002a]. We have here cases of modernism even to the cliche´ of being ahead of its time by decades, twice over; in particular, Peirce’s later versions of the existential graphs have become a major tool in semiotics.12 Mehrtens has nothing on this story, while Gray records only a few parts of it, leaving out the multisets and the graphs [G, 243–244].13

It should be made clear that the indeed ‘nice explanation’ of this fact given in [Coffa 1991] (and better in the paper [Coffa 1979]) was based on his discovery of a Russell manuscript. The same explanation was found independently in [Grattan-Guinness 1978], using a different manuscript; it is acknowledged in [M, 160]. 8 That is to say, the algebraic logicians were applied mathematicians. This feature is especially clear in, for example, the title of Boole’s first book, The Mathematical Analysis of Logic (1847). Gray does not pick this up and even calls their work ‘mathematical logic’ [FG, 384]. 9 Contrary to [M, 160, 291], Frege did not attempt to ground all mathematics in his logic. Contrary to G [240], Russell did not take over logicism from Frege; he formulated it around February 1901, probably from reading some Peanist literature, and his serious reading of Frege was still at least 18 months away [GrattanGuinness 2009a]. Logicism states that (some branches of) mathematics are part of mathematical logic; carelessly, Russell sometimes stated it as an identity thesis, and Gray follows suit [G, 280, 285]. 10 The most noticeable manifestations of algebraic logic between the wars occurred around the early 1920s: The influence of Schro¨der upon Leopold Lo¨wenheim and Thoralf Skolem, the replacement of quantification with functional abstraction by Moses Scho¨nfinkel, and a study of some metaproperties of the propositional calculus and its extension to many-valued logic by Emil Post. All these authors, especially the first two, were informed by model theory. Several of their main texts are available in English in van Heijenoort [1967]. 11 On aspects of this development see, for example, Maddux 1991, by a former student of Tarski. 12 The fine but little known Nachlass of Peirce’s friend in semiotics, Lady Welby [G, 384], is held at York University in Canada [Grattan-Guinness 2002b]. 13 In a review of recent Peirce scholarship [G, 239–240] Gray fails to mention the mammoth that Writings of Charles S. Peirce, Chronological Edition, the Harvard Collected Papers edition in course of publication since 1982 by the Indiana University Press. It is intended to replace the Harvard edition, which came out in eight volumes (not the four of [G, 239]) between 1931 and 1958.

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Post-War Logic A prime case of modernism is Kurt Go¨del’s incompletability theorems of 1931.14 Not only did they open up a new epoch in foundational studies by refuting both logicism and metamathematics, but the first theorem (which is misstated in G [433] though not elsewhere) also exhibited a plurality of theories in that for any axiomatised mathematical theory M to which it applies, an undecidable proposition P pertains, leading to the bifurcation of two further incompletable axiomatised theories (M and P) and (M and not-P). The theorems are duly recognized in both books, more in Gray’s; in referring to consistent theories, the accounts relate to the modified version of the first theorem proved in 1936 by Barkley Rosser rather than to Go¨del’s own. But another feature of the theorems, equally redolent of pluralities, is rarely noticed: I was alerted to it by Rosser. It is that the distinction between logic and its metalogic was of central importance and must be rigidly observed (for otherwise the proof would not work). Prior to the theorems, logic was viewed as some kind of all-embracing theory: Even the stratification built into Hilbert’s distinction between mathematics and metamathematics did not fully contravene it. But now metalogic (Carnap’s word, 1930, quickly after hearing of the theorems from Go¨del) was properly established; similarly, ‘metalanguage’ soon came in from Tarski, who was working in similar territory [Wolenski 1989, chap. 8]. Some other developments in logics during the 1920s and 1930s involve pluralities and thereby exhibit modernism, especially the introduction of many-valued and modal logics as alternatives to the classical two-valued

logic. The contributions of Royce’s former doctoral student C. I. Lewis in the 1930s are particularly noteworthy: Partly under the influence of model theory, he not only expounded modal logic, in which the notions of necessity and possibility were central, but even compared several different systems.15

Mathematics = Logic Fruitful contacts between mathematics and logic are exceptional: Normal is the disdain in which mathematicians hold logic as a subject for explicit study, even if they work in branches where rigour is important. (Poincare´ explicitly derided mathematical logic—antimodernism, therefore?) From the ancient days when Euclid and Aristotle were not talking to each other,16 through Boole’s and Peirce’s algebras of logic being regarded only as curiosities, to mathematicians ‘find[ing] so little mathematics’ in Whitehead and Russell’s Principia Mathematica (1910–1913) [G, 284] although it litters hundreds of the latter pages, mathematicians have regarded the logic that they need as ‘‘obvious’’, be it to operate the proof method by contradiction, say, or to state the problem of the independence of the parallel axiom in Euclidean geometry. It is to the credit of some modernists that they took logic less unseriously than usual, and actually studied it. However, even then there were limitations. Go¨del’s theorems of 1931 show them well: Logicians, and mathematicians and philosophers interested in foundational questions, came fairly quickly to realize their fundamental significance [M, 297–299; G, 408–409, 427–430], but their impact on the mathematical community in general dates only from the mid 1950s, 25 years later [Grattan-Guinness 2009b]. One reason is that Go¨del worked with a far tighter

conception of proof than mathematicians normally entertain; but another was this continuing uninterest. Nevertheless, logic will not go away. In taking his initiatives the modernist mathematician may have wished to be autonomous of applications [M, 184– 186; G, 454–455], but he must have wanted to have a consistent theory, which shows him not to be autonomous of logic. As the modernist Hilbert put it in his 1900 speech in Paris, ‘If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist’ [Hilbert 1901, problem 2].17

Modernism Elsewhere Some other features of mathematical practice conform to modernism. One aspect of the story that Gray exhibits is the significance of modernism in several parts of American mathematics (for example, model theory); the country grew rapidly in importance in mathematics, from a modest status, from the 1890s onwards. Of special note is the Open Court Publishing Company [G, 350–352], which was much devoted to the diffusion and translation of German cultural texts in English. They published translations of books and of articles in the journals Open Court and especially The Monist, which was edited by Paul Carus, a former student of Hermann Grassmann.18 As Mehrtens realises [M, 331–350], another modernizing tendency from the late nineteenth century onwards was international collaboration of various kinds, including in academic contexts [Stump 1997]. Both authors note the series of International Congresses of Mathematicians from 1897, and in other disciplines from 1900 in connection with the ‘universal exhibition’ in Paris that year (the occasion for

14 Contrary to G [427], the Vienna Circle of philosophers did not collectively advocate any one philosophy, although, thanks to enthusiasts such as Carnap, logical positivism received the most publicity. Member Go¨del was a Platonist, while other members followed Wittgenstein, or practiced phenomenology, or antifoundationalism, and so on [Stadler 2001]. 15 This development still requires a history to match its significance; in particular, Lewis in the 1930s has a different context from Lewis in the 1910s (which is noted in G [387–388]). Meanwhile use [Parry 1968]. 16 The first serious attempt in the West to grasp the logic of Euclid was made in [de Morgan 1839]; and he found it hard to grasp. 17 In today’s climate of logical pluralism [Beall and Restall 2005] the postmodernist mathematician can seek a suitable ground from a vast range of nonclassical logics; in the period studied in this book, prior to Lewis there was some anticipation of modal logics by Hugh MacColl [Astroh and Read 1999], a few Polish forays into manyvalued logics [Wolenski 1989, chap. 6], and most noticeably the intuitionism of antimodernist Brouwer [M, esp. 257–287; G, 301–306], which did not gain many followers although Weyl was one of them [M, 289–294; G, 413–423]. 18 Carus is ‘worthy of a full-length study, as are the workings of his Open Court Press [sic]’ [G, 350]. Well, in fact we do have such a study of each: Respectively, [Henderson 1993] and [McCoy 1987]. ‘I am told there is an abundance of source material’; yes indeed, a remarkable collection held in the Archives of the Southern Illinois University at Carbondale, that much enriched, for example, [Grattan-Guinness 2000a, esp. chaps. 8 and 9].


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Hilbert’s famous speech). Other cases of international mathematics include the remarkable rise to importance of the Circolo matematico di Palermo [Brigaglia and Masotto 1982]; this is an example of the growing professionalisation of mathematicians, which itself may have motivated some of them to go pure; we are mathematicians, distinct from physicists, chemists, philosophers [M, 329-350; G, 33-37]. As part of his discussion of linguistics and semiotics, Gray notes the penchant at that time for international languages by Peano and others [G, 374–378].19 Mehrtens highlights another manifestation of modernism: ‘Progressive’ reforms in mathematics education in several countries [M, 350–377]. Examples include the seminar run in Berlin by Karl Schellbach, which influenced several of Weierstrass’s students [Mu¨ller 1905]; the revolt in Britain from the 1870s onwards against teaching Euclidian geometry only Euclid’s way [Price 1994, chaps. 1–3]; and rather more interesting contemporary handling of the teaching of geometries in Italy [Giacardi 2006]. Some other enterprises of Klein manifested social modernism: The International Commission on Mathematics Education [Coray et al. 2003, 19–66; Menghini et al. 2008, passim]; the Encyklopa¨die der mathematischen Wissenschaften (duly noted by both authors); and his collaboration in the mid 1890s with the culture minister Friedrich Althoff in a doctoral programme for women [M, 577]. The first graduate was Grace Chisholm, who married W. H. Young soon afterwards; their collaboration in the first husbandand-wife partnership in mathematics [Grattan-Guinness 1972] played an important part in the tardy but rapid modernisation by the British (including Russell) in mathematical analysis and set theory [Grattan-Guinness 2009d]. Klein is also one of the considerable numbers of native German speakers who wrote on the history of mathe-

matics. Describing this activity, Gray tries to capture it for modernism on the grounds that their historical writing often reflected their own times rather than those of their historical periods [G, 365–373]. I share his own misgivings about this claim, on the grounds that this kind of modernisation is and always has been extremely common in the field, including among historians of no apparent modernist intent; the main reason is the capacity of many mathematical theories to be reformulable in (various) more modem forms.20

Where Is Applied Mathematics? The foundations of mechanics have often been written about in recent years. One has said many intellectually rich things, much critically valuable, admittedly also many a superficiality about general concepts such as force, mass, cause, atoms, mechanical worldviews, etc., concepts that had the fortune or misfortune to have the critical glance directed at them because of metaphysical unclarity and a certain popularity adhering to them for the moment. But the serious attempt to build up mechanics really strongly for once has been made only very rarely […] What we need above all is a strong grounding of classical mechanics. The following essay should be seen as an attempt in this direction. Georg Hamel [1909, 350] In order to protect the autonomy of the modernist mathematician from the input of information from science and physical experience, our authors stress pure mathematics. For Mehrtens, ‘the modernism of mathematics has drastically extended the potential for applications of mathematics and at the same time made it invisible’ [M, 184], so that applied mathematics is discussed only in connection with its role in the development of the German

mathematical community [M, 364–369, 390–392]. The editors of FG regard applications as ‘well represented’ in their pages [FG, 20], an opinion that I find somewhat optimistic. In his own book Gray admits several applications. For example, he attends to some aspects of the development of relativity theory [G, 321–328] as an historical successor to an extensive discussion of non-Euclidean geometries. It is worth stressing that the name ‘relativity’ itself implies pluralities: While an unfortunate name for that theory, Einstein’s choice of it, around 1912, belongs to its time [Feuer 1971]. The issue of measurement in mathematics and in physics is also nicely registered [G, 328–333], and Roche [1998] shows that it can be developed considerably. On the general issue of applications, those who reject Eugene Wigner’s influential claim that the applicability of mathematics to the sciences is a mystery [Grattan-Guinness 2008a] see instead applications as a vast resource for initiatives to be taken in the creation of new mathematics, even if it cannot be called ‘modernist’. For both the sciences and applied mathematics often use notions of great generality and/or ubiquity: For example, among many, optimisation, convexity, invariance, symmetry, linearity, equilibrium, continuity and randomness. Thus the constraints imposed upon mathematics by applications are far less stringent than they appear. Indeed, the consideration of applied mathematics in general would encourage the study of apparent modernism in the other sciences such as physics, and links to mathematics [Garber 1999]. In this connexion is worth noting the category of applicable mathematics, when no application is explicitly made but such potential is clearly evident: Differential equations and their solutions provide obvious examples. A major branch of applied mathematics is mechanics. Its axiomatisation shows all the characteristics of those for

19 As is noted in G [376–378], another enthusiast for international languages was the logician and historian Louis Couturat; Russell was a frequent and polite listener on the matter in their enormous correspondence [Schmid 2001]. Also mentioned in this context is C. K. Ogden, whose superb but little-known Nachlass is held at McMaster University in Canada, alongside the Bertrand Russell Archives. 20 See [Grattan-Guinness 2004b] on the distinction between history and heritage. A major claim of many historians of mathematics from the late nineteenth century onwards, especially some German writers, was that Euclid’s geometry was actually closet common algebra. The grounds given were that his theorems could be restated in terms of it and that they played a role in its creation by the Arabs. But these facts relate to heritage from Euclid, not to his own history, as historians have understood now for some years. On the metahistorical issues involved here see [Grattan-Guinness 1996].

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arithmetic, set theory and geometries [Grattan-Guinness 2006]: Different formulations (Newtonian, analytic, workenergy), missing axioms detected (for example, the couple), the intuitive plausibility of axioms, the possible reductions of statics to dynamics or vice versa, even modernist axiomatisations in 1905 by Hilbert himself [G, 319–320] and four years later by his former student Hamel (not in G [320]). The background to these last initiatives is the same as the modernising of branches of pure mathematics just mentioned, namely, a change, again inspired by Kantian philosophy, of viewing axioms as chosen by the mathematician instead of taken to be self-evident truths [Pulte 2005, chaps. 7–8]. It is also a rich source of inexactitude and unclarity [Dugas 1940, pt. 2]. So what about, for example, quantum mechanics, with its use of differential equations and parts of linear algebra and abstract algebras? Do we not see modernism there also, for instance in the philosophy of complementarity advocated by Niels Bohr and others, which is only noted in passing in G [364]? Were von Helmholtz and Hilbert not (seeing themselves as being) modern(ist) when studying physics, or Hamel when presenting a strong grounding for classical mechanics?

Verdict Methodology does not account better for mathematical intuition than literary criticism accounts for poetic intuition. Analysis can be good to explain a text to someone who does not understand it; but, even to the extent that one will understand this text, one will understand a thing quite other than that that can be explained about it. Franc¸ois Rostand [1962, 149] Modernism manifested itself in various parts of research-level pure mathematics: In addition to arithmetic, analysis, set theory, logic and geometries, evidence is also provided for parts of algebraic number theory and

certain abstract algebras in both books, and in some of the articles in FG (whose authors do not emphasise modernism as such). Mehrtens shows that the thesis is not restricted to research-level mathematics but includes also social aspects, such as the growth of the mathematics profession and educational movements. What were the motivations for mathematicians to adopt modernism at all? Mehrtens shows that mathematics and the arts enjoyed some common backgrounds [M, esp. 538–561], but one can agree with Gray that there is little direct influence of ‘cultural modernisms’ upon mathematics and not much more vice versa [G, 7]; however, the case of mathematician Hausdorff influencing philosopher novelist Hausdorff, or vice versa, is rather nice.21 It seems likely that incentives came from closer to home; not only freedom and creativity, but also the desire to justify and even explain the generality to which some mathematical theories were susceptible, the certainty that mathematical knowledge was held to possess. Concerning the specification of the thesis, Mehrtens more clearly than Gray identifies antimodernist mathematicians; but the modernist thesis needs an indicative list of topics that were modernist, nonmodernist, and antimodernist, and of the major relevant mathematicians, in order to clarify its limitations as well as its scope. As it is, the qualifications for (anti-)modernism seem to be complicated and even occasionally improvised: When, for example, Klein and Riemann seem to be modernist and anti-modernist at once, presumably from different points of view [M, 57–62], one feels the axiom of choice coming on. Further, the status of logic, whether symbolic or not, is unclear, since logical pluralism arises only in the context of L. E. J. Brouwer’s intuitionism. Also worth considering here is the preference for linear theories, which is quite noticeable throughout the nineteenth century and had become quite a fixation for some mathematicians by its

end and later [Grattan-Guinness 2008b]: Were the advocates of nonlinear theories [West 1985] for or against modernism, or neutral about it? Further, in a thesis that is much concerned with changes in theories and differences in standpoints, I would have welcomed some consideration of the difference between innovations in theories where new notions preponderate, and revolutions where replacement of theory predominates. For example, Cantor largely innovated with his point-set topology but revolutionised our understanding of the actual infinite (for his nondissenters, anyway). One may wonder about the depth of the impact of modernism, the readiness of mathematicians to receive it, and the size of the attentive audience. Geographically, the effects of modernism are rightly seen to have fallen most strongly upon Germany and Italy: Gray might have discussed Berlin more, where ‘pure maths good, applied maths bad’ was a prominent theme in their talk [Biermann 1988, chaps. 5–6], in great contrast to the long-term and systematic promotion of applied mathematics by Klein at Go¨ttingen [M, 200–209].22 The place of Kantian philosophy and its offshoots influenced all the major figures mentioned above in some way, and also others; for example, in Hilbert’s circle, his favoured philosopher Leonard Nelson [G, 209–212] and Hermann Weyl [M, 289–295; G, 413– 423]. However, I doubt that these considerations permeated among modernist mathematicians in general, especially in non-German-speaking countries (the American Moore, for example, or the modernising Britons); the authors do not prove that they did. A text that is consistent with modernism does not necessarily exemplify modernist intent on the part of its author. On the place of modernism in mathematics as a whole, the claim that ‘Modernism and Antimodernism make between them the success history of mathematics in the twentieth century’ [M, dustjacket] requires us to rejudge as failures or footnotes the developments

21

See [M, 165–176], [G, 221–222] and especially [FG, 263–289 (M. Epple)]. [M, 208] cites this gorgeous passage from the faculty discussion at Berlin University on appointing a successor to antimodernist Leopold Kronecker in 1892 [Biermann 1988, 305–306]: Weierstrass: Schwarz sticks to the point. Good lecture. Klein nibbles more. Hoodwinker. […] von Helmholtz: Kronecker gave a very unfavourable opinion about Klein. He considered him a charlatan. 22


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was for the most part unintelligible, and at the same time apparently so exhaustive that it excited the absurdest expectations’, to quote J. H. Stirling on the philosophy of G. W. F. Hegel [Schwegler 1879, 441]? Am I just reacting against the reaction of modernist historians to traditional reactionary historiography?

theoretical physics in Europe, 1750– 1914, Basel: Birkha¨user. Giacardi, L. 2006. (Ed.), Da Casati a Gentile. Momenti di storia dell’insegnamento secondario

della matematica in Italia, Lugano: Agora` Publishing. Grattan-Guinness, I. 1970. The development of the foundations of mathematical analysis from Euler to Riemann, Cambridge, Mass.: M.I.T. Press. Grattan-Guinness, I. 1972. ‘A mathematical

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of, for example, topology, mathematical statistics, and numerical methods! Similarly, Gray’s claim to uncover rich connexions between foundations and research ‘across every branch of mathematics’ [G, 3, italics inserted] and assertion that ‘by far the major part of mathematics was transformed in a modernist way’ [G, 14] are unproven: Several major parts of mathematics do not seem to have been affected by modernism. A reasonably general impression may be gained from the large collection of articles on landmark published writings in mathematics that appeared under my editorship a few years ago: 66 writings from the period 1800–1940 were described [GrattanGuinness 2005, chaps. 22–77], and only about a half of them, at most, seem to be modernist. So the subtitle of Gray’s book, ‘the modernist transformation of mathematics’, is a considerable exaggeration in both extent and depth, in danger of reducing to the tautology that modernism influenced those branches of mathematics that were influenced by modernism. But that would be unintended silliness, no? In several other respects the thesis is very problematic, both in content and presentation. Pure mathematics is extolled, yet Gray treats several applications (not all discussed above). His examination of the distinction between different kinds of intuition should form a central part of the thesis. The requirement of the modernist mathematician to belong to a group need not be mandatory. The lack of time dependence in the specification is surely an oversight: Is modernism not applicable also to, for example, the efforts of the Analytical Society at Cambridge in the 1810s to replace Isaac Newton’s version of the calculus with Euler’s and Lagrange’s, or even the context of Euclid’s Elements [M 47–49, 473]? The contributions of Lecat, Dugas and Rostand deserve appraisal; and a few of the other items cited here might be worth adding to the bibliographies of these two books, rich though they already are. As Saint Paul put it in a somewhat different context, ‘I count not myself to have apprehended’ (Philippians 3:13). Are aspects of the modernist thesis that seem to me unclear, exaggerated, or improvised actually evidence of ‘a scrutiny of thought so profound that it

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Viewpoint

What Poetry Is Found in Mathematics? What Possibilities Exist for Its Translation? JOANNE GROWNEY

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 editor-in-chief endorses or accepts responsibility for them. Viewpoint should be submitted to the editor-inchief, Chandler Davis.

ne question leads to another. And so it was that when I was invited to be part of a panel 1 presentation on the challenges of translating poetry into English—invited because of my activity in translating work by Romanian poets—I began to think about the poetry of mathematics and to ask, What possibilities exist for its translation? The remarks that follow include some of the ideas presented in the conference panel—and others that came as a consequence.

O

Good Mathematics Is Poetry Writers of stature and wisdom have, without reservation, linked mathematics to poetry. Albert Einstein (1879– 1955) said, ‘‘Pure mathematics is, in its way, the poetry of logical ideas.’’ Ralph Waldo Emerson (1803–1882) observed, ‘‘If a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic musical.’’ From within the mathematical community, we have words from Karl Weierstrass (1815–1897): ‘‘It is true that a mathematician who is not somewhat of a poet will never be a perfect mathematician.’’ More recently, from Lipman Bers (1914– 1993): ‘‘…Mathematics is very much like poetry… what makes a good poem—a great poem—is that there is a large amount of thought expressed in very few words. In this sense formulas like e ip + 1 = 0 or $exp(-x2)dx = Hp are poems’’ [1]. My own view is that good mathematics—whether a formula or an elegant definition, a theorem and its proof, or a counterexample, whether verbal or symbolic—is poetry.

Some Poetry Uses Mathematics Without Being Mathematics In the blogosphere, poetry and mathematics often are connected—sometimes

1

by elementary-school teachers making interdisciplinary assignments, sometimes by writers from India, where many who make their living in technical industries use the Internet for aesthetic expression and do not hesitate to link mathematics with poetry and the spirit. In the 2008 film, Harold and Kumar Escape from Guantanamo Bay, Kumar Patel ends the tale with a recitation of a mathematical love poem by David Feinberg, ‘‘The Square Root of Three’’ [4]. In current terminology, the adjective ‘‘mathematical’’ is applied to poems that fall into one of two categories: First, to poems whose structure involves particular mathematical ideas—perhaps geometric shape, or counting or substitution into a form or formula; next, to poems that make careful and deliberate use of mathematical imagery—such as circles, vectors, or parallels—to vivify the work. It cannot be said that ‘‘mathematical’’ poems are mathematics. Rather, they may be considered as applications of mathematics or as translations—for they can connect nonmathematicians to the power and beauty of mathematical ideas.

Mathematics May Structure a Poem In every age, some of the poets have shaped their work by counting. Long traditions embrace the fourteen-line sonnet with its ten-syllable lines. Fiveline limericks and seventeen-syllable haiku also are familiar forms. Moreover, patterns of accent and rhyme overlay the line and syllable counts for even more intricacy. In 2006 a new syllable-count form emerged when a blogger, Gregory K. Pincus [12], began to promote the Fib. In a six-line Fib, the syllable counts are based on the first six nonzero Fibonacci numbers. Pincus offers this example [13]:

Note: This panel presentation took place at the annual conference of the AWP (Association of Writers and Writing Programs, www.awpwriter.org) in Chicago, February 11–14, 2009.

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One Small, Precise, Poetic, Spiraling mixture: Math plus poetry yields the Fib. Danish poet Inger Christensen (1935–2009) wrote a book-length sequence based on the nonzero members of the Fibonacci sequence: The first poem has one line, the second poem two, the third three, the fourth five, each number in the sequence being the sum of the previous two (1, 1, 2, 3, 5, 8, 13,…). The work stops with the letter n, which, as the 14th letter of the alphabet, generates a poem of 610 lines [2]. A triangle poem may be formed by lines whose syllable count increases from 1 to some fixed positive integer. A square poem has the number of syllables per line equal to the number of lines. ‘‘Elevens,’’ a recent poem by Stanley Plumly [14], consists of eleven square stanzas with each stanza having eleven lines of eleven syllables each. Online, in the UbuWeb’s collection of historical visual poetry, we find a 10 9 10 square

by Henry Lok [10]. The text of Lok’s poem is presented in the table. Here are two additional squares and a triangle [8]:

If everything is seven times as large, does the seven matter?

All over the world, fashionable shoes— trendy, hazardous, uncomfortable— keep women in place.

A group of French intellectuals, the OULIPO (OUvroir de LItte´rature POtentielle/Workshop of Potential Literature), have invented new ways to use mathematics in poetry [11]. A follower of OULIPO applies mathematical or other algorithms to help the mind escape the unconscious rules and limitations that stifle the new or creative and to discover the unlikely. A popular OULIPO algorithm is called S + 7 (for ‘‘substantif plus 7’’), in English N + 7, and is a procedure that replaces each noun in a familiar passage (perhaps a poem) with the seventh noun that follows it in a specified dictionary. For example, we can apply N + 7 to the first two lines of Robert Frost’s poem, ‘‘Fire and Ice’’ [5]: Some say the world will end in fire, Some say in ice. With the help of my desktop Webster’s dictionary, the website www. rhymezone.com, and side rules restricting the words to fit the form, the N + 7 algorithm gave me

One added forever joined by zero, paired to opposites— we build the integers.

Some say the wound will end in ire, Some say in lice.

‘‘Square Poem in Honor of Elizabeth I’’ by Henry Lok (1597) For an image of this poem with its borders and embellishments, see [10].

The new word choices are, I think, thought-provoking, and might lead the way to a new poem.

For analysis of cross designs and other sub-poems within Lok’s square, see [15].

God

Hath

pourd

forth

Rare

Grace

On

This

Isle—

And

Makes

Cround

your

rule

Queene

In

the

same

So

Still

Kings

Lawd

This

saint

Faire

that

with

truth

doth

Stand

Rule

so

long

time

milde

Prince

ioy

land

it

Will

For

proofe

you

shows

wise

of

earths

race

whome

There

Heauens

haue

vp

held

Iust

choice

whome

God

thus

Shields

Your

stocke

of

Kings

worlds

rich

of

spring

and

Feare

States

fame

Knows

farre

Praise

Isle

which

ALl

blisse

Yields

Hold

God

there

fore

sure

stay

of

all

the

Best

Blest

is

your

raigne

Here

Builds

sweet

Peace

true

Rest

Poems with Imagery from Mathematics Beyond use of numbers and shapes and substitutions to aid construction of poetry, a second variety of ‘‘mathematical’’ poem draws on the language of mathematics as one might draw from the classics of literature, importing imagery to create or expand meaning. Here are samples of what is possible:

AUTHOR

......................................................................................................................................................... JOANNE GROWNEY began in abstract algebra at the University

of Oklahoma, but soon widened her view to embrace many other arts, verbal and visual. Retired now after a teaching career at Bloomsburg University in Pennsylvania, she is a poet, still crossing disciplinary lines. It is all set forth at http://joannegrowney.com. 7981 Eastern Avenue, #207 Silver Spring, MD 20910, USA e-mail: [email protected]


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Source Poem

Sample of ‘‘Mathematical’’ Imagery

‘‘July 18, 2005’’ by Deanna Nikaido [6]

this light bearing equation of love

‘‘3.141592 …’’ by Peter Meinke [6]

In school I was attracted

explain, but a way to share the magic— using music to interpret particular joys and beauties of mathematics for the nonmathematician.

to irrational numbers … ‘‘Several Hypotheses and a Proposition’’ by Jacqueline Lapidus [6] ‘‘Sex and Mathematics’’ by Jonathan Holden [6]

one of these days we’ll intersect again. Making love we assume may be divined by the equation for the hyperbola y = 1/x, …

REFERENCES

1. Albers, D.J., G.L. Alexanderson, and C. Reid, editors, More Mathematical People, Academic Press, New York, 1993, p. 16. 2. Christensen, Inger, alphabet, translated by Susanna Nied, New Directions, New

‘‘The Shape of Desire’’ by Emily Grosholz [6]

the frail parabolas of love

‘‘Tales from a Sonnetarium’’ by Diane Ackerman [6]

the world is all subtraction in the end

‘‘Geometry’’ by Rita Dove [6, 7]

I prove a theorem and the house expands

‘‘The One Girl at the Boys’ Party’’ by Sharon Olds [7]

her body hard and indivisible as a prime number

York, 2000. 3. Dalglish, Cass, Humming the Blues, Calyx Books, Corvallis, Oregon, 2008.

From Nikaido to Olds, each of these poets has employed a bon mot from mathematics to offer a vivid image— and, in turn, these ‘‘mathematical’’ poems interpret or translate a thin slice of esoteric mathematics into accessible poetry.

‘‘The Poetry of Mathematics’’ I belong to a listserve of poets who pose questions and respond on matters of interest to writers. ‘‘How will they respond to a poem from mathematics?’’ I wondered. And so, with curiosity, I wrote and posted a version of Euclid’s proof that there are infinitely many primes (Book IX, Proposition 20). I— to whom many proofs are poems— selected this as a somewhat accessible sample of ‘‘the poetry of mathematics.’’ However, not one of the listserve’s readers, a few of whom are mathematically literate, replied that she/he saw it as a poem. It seems, then, that a theorem-proof is—not unlike verses written in Portuguese or Arabic—a poem that requires translation. But how may we do that? How to communicate—to those not fluent in the language of mathematics—the poetic beauty of an elegant proof is a question for which I do not have good answers. A suitable test case is Euler’s identity, eip + 1 = 0, one of the equation-poems suggested by Lipman Bers. Is it translatable? My first answer came easily—an emphatic NO: One must learn the

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language of mathematics to gain access to the poem-formula, to understand its beauty and the disparate entities that it connects. After setting down my initial response and looking for my next sentence, inexplicably, my mind did some leaping: The Euler identity caused me to think of a crossing—and then a train— and then a poem by Kenneth Koch, ‘‘One Train May Hide Another’’ [9]. Much as Euler’s identity focuses a cluster of meaning around a concise statement, Koch’s title resonates throughout his sixty-seven-line poem and through the life of each reader with awareness of many and diverse experiences also linked to a single statement. I invite you to visit Koch’s poem and see whether this connection of Koch’s title and Euler’s identity as interpretive-equivalent works for you. At the AWP conference which prodded my questioning about translation of mathematical poetry, one of my fellow panelists was Cass Dalglish, a poet who has learned Sumerian and set for herself the task of translating texts written in cuneiform by the poet-prince-priest Enheduanna around 2300 BCE [3]. Among Dalglish’s explorations as a translator has been interpretation with the cuneiform signs using improvisation in the manner of a jazz musician. I don’t yet know how to apply this idea to translation of mathematics; perhaps it can be done by a mathematician who, unlike me, is a musician. There may be an opportunity here—not a way to

4. Feinberg, David, ‘‘The Square Root of Three,’’ http://www.herecomesthescience. com/2008/04/26/square-root-of-three/. 5. Frost, Robert, ‘‘Fire and Ice,’’ http://rpo. library.utoronto.ca/poem/844.html. 6. Glaz, Sarah and JoAnne Growney, editors, Strange Attractors: Poems of Love and Mathematics, A.K. Peters, Ltd, Wellesley, MA, 2008. 7. Growney, JoAnne, editor, Numbers and Faces: A Collection of Poems with Mathematical Imagery, Humanistic Mathematics Network, 2001. Out of print; electronic version available from the editor, japoet@ msn.com. 8. Growney, JoAnne, ‘‘Square Poems,’’ http://joannegrowney.com; triangle: http:// mathdl.maa.org/images/upload_library/4/ vol6/Growney/MathPoetry.html#Triangle. 9. Koch, Kenneth, ‘‘One Train May Hide Another,’’ http://www.poetsorg/viewmedia. php/prmMID/15592. 10. Lok, Henry, ‘‘Square Poem in Honor of Elizabeth I,’’ from Sundry Christian Passions, http://www.ubu.com/historical/ early/early07.html (accessed May 2009). 11. Motte, Warren F., Jr., translator and editor, OULIPO: A Primer of Potential Literature, Dalkey Archive Press, Normal, Illinois, 1998. 12. Pincus, Gregory K., http://gottabook. blogspot.com/. 13. Pincus, Gregory K., http://gottabook. blogspot.com/2006/04/fib.html. 14. Plumly, Stanley, ‘‘Elevens,’’ Kenyon Review XXVI, no.2 (Spring 2004), http://www. kenyonreview.org/issues/spring04/plumly. php. 15. Roche, Thomas P, Jr., Petrarch and the English Sonnet Sequences, AMS Press, New York, 1989.

Mathematically Bent

The proof is in the pudding.

Colin Adams, Editor

The Adventures of Robin Caruso COLIN ADAMS

B 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 disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.

Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected]

eing the Adventures of one Robin Caruso, a Mathematician who was wash’d up on an Island where he was forc’d to spend many a Year.

This Tale that I tell you is the true Story of the Adventures of my Life, including a long Sojourn upon an Island far in the Caribbean, well beyond the reach of Civilization as we know it. Altho’ the Events that I relate occurr’d over many a Year, the Tale itself is not nearly so long. But let me begin my Story at the Beginning. I was born in a small Town in the midwestern Province of a large and industrious Nation. My Childhood was uneventful, and my Schoolwork uninspir’d. I found most Subjects quite dull, and did not experience the suppos’d thrill of interacting with Children of my Age, either of my Gender or of the Other. But one day, when I was Twelve, our Mathematics Instructor happen’d to explain to us how as a Child, Gauss had summ’d all the Numbers from one to one hundred in just a few Seconds. I alone found this Story fascinating. Over the next few Months, I taught myself how to add longer and longer Strings of consecutive Numbers in extremely short Periods of Time, and in the Process, manag’d to alienate all the other Children in my Class. Soon, my Interests spread beyond Addition to other aspects of Mathematics, and the day-to-day World began to appear drab in comparison. My Soul thirst’d for more Mathematics. Finally, at the age of Seventeen, I cou’d stand it no longer, and I left my

Hometown and travel’d many miles to a large University in a neighboring State. I had heard they did amazing Mathematics there. But once I had sign’d on, I found myself requir’d to take Courses in which I had no Interest. And they bill’d my Parents for the Privilege of it. Where was the aesthetic Beauty of Mathematics? Where were the brilliant Insights and the awe-inspiring Theorems? After an insufferable Four Years, I finally managed to escape with a Piece of Paper expounding my Abilities in what I consider’d to be Elementary Mathematics, and with a Partner who had herself sign’d on to become my Spouse. I subsequently utiliz’d the Diploma to gain Entrance to a Graduate Program in Mathematics at an even better-known University of Higher Learning in an even more-distant State. There I finally did see some of the Mathematics that I had long’d for; Theorems that requir’d true Insight, and abstract Mathematics that took substantial Effort to conquer. And I was finally given the Chance to make my own Contribution. I craft’d a beautiful Thesis that prov’d Facts no one had previously realiz’d in the entyre History of Humankind. But much to my Chagrin, on the Basis of that Thesis, I was informed that it was Time for me to leave. I was deemed ready to begin my Research Career. I land’d a Job at a small liberal arts College where I arriv’d the next Fall with my growing Family, three Boxes of Books and boundless amounts of Energy and Enthusiasm for the Mathematics I wou’d shortly be discovering. But I soon learn’d that my idealiz’d view of Academia was far from the Reality. I was sign’d up to teach Six Courses a Year, with hundreds of Students, many of whom were taking the wrong Course for worse Reasons, with little Background and even less Motivation. I cou’d no more find Time to do Research than I might juggle Spivak’s Collection of Differential Geometry Books. But at least during the Summers, I cou’d carve a small amount of Time for Research. I try’d to keep active, and in my spare Time, I took to carrying around Whitehead’s thick Tome, ‘‘Homotopy

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Theory.’’ When I was waiting for my Children at the Dentist’s Office, or bound by Traffick, I cou’d read a Theorem or Two, and then cogitate upon it. After Five Years, and a hard-won Handful of Publications, Frustration was looming large. Tho’ Tenure seem’d likely, it was not clear to me that this was the Path I desir’d for my Life. Nearing the End of the subsequent Fall Semester, I looked forward to having Time for Mathematics during the much too short Winter Break. But over Thanksgiving, we visit’d my In-Laws, and my Father-in-Law announc’d to all present that he and his Wife were taking the entyre extend’d Family on a Caribbean Cruise over the Holiday Break. For me, the Idea of such a cruise was Anathema, but my Wife and Children made it clear that saying no was not an Option. This was how I found myself aboard a Festival Cruise Lines Ship known as the Sensation. And Sensation it was, but of the unpleasant Variety. There were a Multitude of Activities, including Water Slides, Pools, Sauna, Bingo and Floor Shows. Everywhere you went there was pounding Music, gyrating Lights and Wait Staff attempting to convince you to buy multi-color’d Drinks. It was a Nightmare on Water. Needless to say, I often claim’d Seasickness and slipp’d away to peruse Mathematics. One evening, half way through the Voyage, I manag’d to escape to the Stern of the Ship, which was one of the few quiet Spots I had discover’d on board. Enjoying a soft Breeze, I lean’d against the Rail, cogitating over Whitehead. Unfortunately, a Boy of perhaps Fifteen Years appear’d, and much to my Annoyance, began performing Tricks about the Deck on his Skateboard. He was wearing Earphones plugg’d into some Kind of Musical Device. ‘‘Go away,’’ I snapp’d, but he did not heed me. I try’d to ignore him and return’d my Attention to my Book. But then, as he was attempting to land on the spinning Board, he lost his Footing and tripp’d, slamming into me and smashing me hard up against the Railing. The Book was knock’d from my Hand, and land’d just on the Edge of the Deck on the Far Side of the Rail. ‘‘You Miscreant,’’ I yell’d, as I lean’d over to grab the teetering Book. Just as I had stretch’d far enough to grasp it in my Hand, the Teen push’d off me as he skateboard’d away. My Foot slipp’d on the 16

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wet Deck, and I found myself balanc’d precariously on the Rail for just an Instant before tumbling over the Side, down, down, down toward the black Water churning behind the Ship. The Shock of my Impact with the Water took my Breath away. When I resurfac’d, spluttering but with the Book still gripp’d in my Hand, the Boat was already steaming away. ‘‘Help me,’’ I scream’d, waving the Book, but to no avail. Surely, I thought, the Boy wou’d report the Incident. But as the Ship reced’d into the Distance, I realiz’d it was not to be. Either fearing the Repercussions, or more likely oblivious to the Results of the Collision, the boy wou’d not be reporting my Disappearance over the Side. This is so unfair, I thought to myself, as I tread’d Water. My mathematical Career had hardly begun, and it is about to end. Above me shone the Moon, and for lack of a better Plan, I began to swim in its Direction, using the Book to pull myself along. After Fifteen Minutes, I stopp’d and paddl’d in Place, worrying whether this particular Part of the Ocean was home to giant Squids or other Monsters of the Deep that might grab me by the Legs and pull me into the Depths. Near the Horizon, straight ahead of me, I notic’d that the Stars were not visible, block’d out by Something, perhaps an Island rising out of the Ocean. I continu’d to swim in that Direction ‘till I cou’d hear Waves breaking on a Shore. With rising hope, I forc’d myself to push on. Eventually, as I was nearing total Exhaustion, a Wave caught me up, pulling me forward as it suck’d me under. I desperately try’d to deliver myself from its Grasp as I gulped down mouthfuls of salty Water in futile attempts to obtain Air. Suddenly, it spit me out on a sandy Beach. Crawling from the Water, I collaps’d on the Sand and fell into an exhaust’d Slumber. The next Morning, I awoke, my Face encrust’d with Salt and Sand. In my Hand, I still had a firm grip on a soggy Whitehead. I pull’d myself to my Feet. The Beach stretch’d for a half mile in either Direction before disappearing around the Edge of the Island. It was border’d by dense Jungle and was apparently untouched by human Hands. I realized that I was completely alone here. I had only the wet Clothes upon my Back, one very damp Book, and neither Food nor Water to Drink. I wou’d shortly perish of Thirst or Hunger.

‘‘Oh, Whitehead,’’ I exclaim’d, ‘‘It appears we have survived the Ocean only to be dealt a lonely Death upon this wretched Island.’’ As I confront’d my dismal Circumstances, I began to plod slowly down the Beach along the Water’s Edge. I shortly came upon a Stream crossing the Sand and emptying into the Ocean. Leaning down, I cupp’d a bit of Water to taste. It was fresh. This was a very good Sign. I drank greedily. Once sated, I look’d up at the nearby Palm Trees and saw a profusion of Coconuts dangling from them. Perhaps I cou’d survive here. Not all was lost. But wou’d I go crazy with Loneliness? Wou’d I miss Human Companionship? And then, slowly, it dawn’d on me that my Dream had come true. Here I was with plentiful Food and fresh Water, good Weather, and nothing but Time on my Hands. I cou’d do Mathematics from Dawn to Dusk every Day. It was an incredible Realization. Whitehead wou’d dry out, and there was enough Mathematics in him to last me many a Year. I cou’d expand upon it, develop the Field, prove new Theorem after new Theorem, ad infinitum. ‘‘Whitehead,’’ I said enthusiastically, ‘‘this might work out after all!’’ Whitehead grinn’d back at me with his bright yellow Cover. But I wou’d need Something upon which to write. As I consider’d this, I pok’d at the Sand with my Foot. ‘‘That’s it, Whitehead!’’, I exclaim’d. ‘‘I can write with a Stick in the Sand. It is the perfect Pad. Temporary Work can be done below the Tideline, and the permanent Work I can place up higher. There must be Miles and Miles of Sand here. I can work for Years!’’ I was ecstatic. And so it began. Each Morning, after a breakfast of Bananas and Coconuts, wash’d down with cold fresh Water, Whitehead and I wou’d head down to the Water’s Edge where we wou’d spend an Hour or two discussing Mathematics. Then I wou’d think how I might push the Arguments further. I wou’d tinker and make Notes down nearer the Water as the Tide went out and then I wou’d save my Nuggets, my interesting Facts, and my clever Observations up by the Tree-line. Eventually, I wou’d box off the very good Stuff, the Stuff I knew was publishable. Then every four Days, I wou’d go over it with my Stick, just to make sure the Wind did not erase any of the Results, a potential Loss not just to me, but to Humanity.

I took breaks to spear Fish, to hunt and to collect Fruit. Evenings were spent with Whitehead enjoying a Bonfire by the Beach, discussing the day’s Results. And so the Pattern of my Life was establish’d. I rarely consider’d Rescue, for I think I was as happy as I had ever been in my Life. I was fulfill’d, productive and all of my physical Needs were met. The Years pass’d, and the body of my Work grew. It now took almost a Day to retrace all of the best Results. But as I retrac’d them, I wou’d think more about them, and consider means to push them yet further. It gave me tremendous Pleasure to see my Writings extend down the beach as far as I cou’d see. But then one Morning, Five Years into my time on the Island, I awoke to signs of a Storm brewing off to the North. Climbing a Tree to get a better View of the impending Weather, I was surpriz’d to spy a Ship dock’d in the Bay. I slid down the Tree, grabb’d up Whitehead, and sprint’d down the Path to find a Dinghy beach’d on the Sand. Before it stood a Collection of Sailors and one who must be the Captain. I strode up to him. ‘‘Sir, you are standing on my best Theorem,’’ I said. ‘‘I know not of what you speak,’’ reply’d the Captain. ‘‘I am standing on Sand.’’ ‘‘But do you not see my writing in the Sand? Do you not see my Theorem Egregious?’’ ‘‘I see only Marks in the Sand that look as if a Seagull has strutt’d about.’’ ‘‘These are the great Theorems that I have prov’d. You must bring Paper and Pen from the Ship so that I may record them.’’

‘‘Have you gone batty?’’ he reply’d. ‘‘There is no Time for your Theorems. Do you see that approaching Storm? It will dash our Ship upon the Reef, and we will all be trapp’d here with your infernal Theorems.’’ ‘‘I will not go without my Theorems.’’ ‘‘Hold him fast then, Men, and throw him in the Dinghy. We must make haste.’’ With that, the Sailors grabb’d me, and toss’d me like Luggage into their Craft. As they push’d the Boat in to the Surf, I struggl’d to escape, and in the process lost hold of Whitehead, and he fell into the Water. ‘‘No, no, save Whitehead!’’ I wail’d, but the Sailors merely held me down, ignoring my Pleas. I watch’d helplessly as Whitehead bobbed away. The Wind was picking up now, and we crash’d through the Waves. ‘‘Please, let me go,’’ I cry’d. ‘‘I must save Whitehead.’’ I cou’d still see his yellow head appearing once and again between the swells. The Oarsmen ignor’d me, rowing as fast as possible, as the Storm descend’d upon us. We reach’d the Ship just as Lightning creas’d the Sky and Thunder roar’d. I was dragg’d up onto the Deck. The Captain yell’d to hoist Anchor, and the ship turn’d toward the open Ocean, as the Rain began to fall in Sheets and the Wind began to howl. The Ship clear’d the Reef just as the Gale hit full Force. The Captain turn’d the Ship into the Wind and we spent the next Twelve Hours wondering if our final resting Place wou’d be the Ocean Floor beneath us. But presently, the Storm abat’d, and I begg’d the Captain to return to the Island

so I might copy down my Theorems. He just laugh’d. ‘‘Do you think your Chicken Scratchings will have surviv’d that Storm? It has scour’d the Beaches as clean as the Day you arriv’d on the Island. Your Mathematics as been return’d to from whence it came.’’ I do not know what he meant by that last Part, but I knew he was correct. The Beaches wou’d be clean now. And so I return’d to Civilization. I again took up my Position at the College. And I did manage to dredge a Theorem or Two from Memory, and ultimately wrote up a Paper that was duly publish’d in the Journal of Existential Mathematics. But the Truth was that I cou’d not recreate the vast Majority of what I had discover’d on that Island. Most of it seem’d as a Dream, and when I went to figure out the Details, they had wash’d away just as my Scribblings on the Beach had done. But perhaps anyone can ask what Permanence there is in Mathematics. More often than not, the next Generation overwrites the Mathematics of the Former. Results that seemed so fundamental at the time are generalized and then subsumed in subsequent Work. A few Theorems are sustain’d. Some of the work of Gauss, Euler, Riemann. But in due time, my Theorems wou’d have disappear’d below the Surface of the sea that is Mathematics anyway. Does it matter so much that they did so before anyone else had the Opportunity to appreciate them? This is a Question I ask myself much too often, I am afraid. Much too often.

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Further Stellations of the Uniform Polyhedra JOHN LAWRENCE HUDSON

he process of stellation has a long and chequered history. Kepler (c. 1619) described the stellated octahedron (stella octangula) and two stellations of the dodecahedron—the ‘‘small’’ and ‘‘great’’ stellated dodecahedra. In 1809 the French geometer Poinsot discovered the reciprocals of Kepler’s two forms, the great dodecahedron and the great icosahedron. These four, socalled Kepler-Poinsot solids, are regular, but unlike the other regular solids (the five Platonic forms), these polyhedra have either five-pointed star polygons as faces or similarly star-shaped vertex figures. In 1811 Cauchy showed that these nine solids were the complete set of finite, regular polyhedra. In 1938 Coxeter, et al. [1] described the complete set of stellated icosahedra. It was this set of fifty-nine solids which, fifty years later, I made for the Science Museum in London. Wenninger [2] presented some further stellated forms, including the two semiregular solids (the cuboctahedron and the icosidodecahedron).

T

Origins of Present Work It was this work that prompted me to apply the process of stellation to the large class (43 members) of nonconvex uniform polyhedra (n-cUP), and in 1988 John Kingston and I published an article, in this journal, in which we presented computer graphics and card models of various stellated n-cUP [3]. As our exemplar, we used the small cubi-cuboctahedron (SCCO) and gave a detailed explanation of how its fourteen planes formed complex, external cells around the solid to produce five main-line stellations (Fig. 3). We made no attempt to generate every possible stellation in the way

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Coxeter had done—the possibilities would have been manifold and our (primitive) computing power was already stretched to its limits! This article forms the basis for the brief description of the stellation process described in the following text. In the years after the publication of our article I developed the software to take advantage of the more powerful home-computers that were coming onto the market. Their increased memory enabled me to work with more complex solids and to produce more realistic graphics. By 1996 I had produced computer images of the main-line stellations of all forty-three members of the nonconvex uniform polyhedra. Table 2 lists the number of such stellations for each of the nc-UP. Figure 18 is one example. While working on these images I realised that the same procedure that had previously been used to produce stellations (external to the solid) could also be used to create what I term internal stellations—that is to say, to reveal forms that are already contained within the solid by an exactly analogous (internal) stellation process that removes, layer by layer, cells to reveal internal main-line stellations. In the following text, I briefly describe the derivation of (external) stellations with reference to one of the n-cUP, introducing the technical terms relating to it. I then detail the related production of internal stellations and illustrate it in the main section of this article using two further members of the n-cUP as exemplars: the quasi-truncated hexahedron and the quasi-truncated great stellated dodecahedron. In conclusion, further, notable, internal stellated forms are presented with some brief comments. Table 2 lists the numbers of main-line, internal stellations for each of the 43 n-cUP.

A stellation cell Triangle face Line in stellation diagram contributed by facial plane of triangle

Part of octagon face Square face Line in stellation diagram contributed by facial plane of octagon

Line in stellation diagram contributed by facial plane of square

Figure 1. Small cubi-cuboctahedron sitting on the stellation diagram for its square face.

The Stellation Process: An Overview A polyhedron, P, has a surface composed of subsets of planes; the subset for any particular plane, which need not be a simply connected region, is called a face. The plane itself is referred to as the facial plane. The full set of facial planes of P divide space into finite and infinite 3-D regions. The finite regions may be referred to as 3-D cells, each cell itself being a convex polyhedron (v. [1], pp 15ff). The 2-D surface polygons of each cell will be referred to as facets (a traditional term used to describe the convex surface polygons of cut gemstones such as diamonds). It is just these 2-D facets that appear as the finite, convex, 2-D regions making up the facial planes. It can thus be noted that each 2-D facet may show itself:

AUTHOR

......................................................................... JOHN LAWRENCE HUDSON In the best

traditions of mathematical procrastination, John has eventually got round to submitting an update on his original article from 1988. This, he claims, is due to the following factors: 1) waiting for the technology to catch up with his demands for RAM and colour facilities, 2) work on extensions to his house since 1996, 3) appointment (in absentia) as chief gardener, landscaper, compost-heap processor, etc., etc. In between times John teaches IT skills and Math at two of Her Majesty’s penal establishments. He recently reached publication with an article aimed at teachers using a different method for subtraction. West Nottingham College, Nottingham, Nottinghamshire, UK e-mail: [email protected]

1. in a plane stellation diagram (see text that follows). 2. as the surface polygon of a cell. 3. as part of a planar face of a new polygon constructed with a subset of cells. Further, each facet either separates exactly two cells, fitting together perfectly at the facet, or separates a cell from an infinite region. In order to stellate P we first extend all its faces until they intersect with other facial planes, creating cells as defined previously. A stellation of P is then a polyhedron comprising any set of these cells. Because this generally admits a large number of possible stellated forms, we define a restricted stellation (henceforth referred to simply as a ‘‘stellation’’) S as having the following properties: 1. S is a subset of the finite cells defined by the facial planes of P. 2. The complement of S is a connected region. 3. S maintains the original symmetries of P. A fuller discussion of these restrictions can be found in [1].

Stellating a Nonconvex Uniform Polyhedra Figure 1 shows the small cubi-cuboctahedron (SCCO). This is one member of a set of 43 polyhedra collectively known as the nonconvex uniform polyhedra (n-cUP). In this context, uniform means that all the faces are regular polygons (including the star polygons) and all the vertices are congruent to each other; nonconvex means that the polyhedron is self-intersecting. The SCCO has three types of face, an example of each being shown: square (in red), triangle (blue), and octagon (green). This last is not so readily apparent because only four (rectangular) facets can be seen in the diagram. The remaining four triangular facets lie between two of the rectangular parts, but face in the opposite direction (we can say they have a negative sense—the outward normal to the interior of the SCCO Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009

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actually points towards its centre). In total there are six facial planes containing a square face (lying in the planes of a cube), eight facial planes containing a triangle (lying in the planes of an octahedron), and six more facial planes containing an octagon (again in the planes of a cube). It is these component forms (Platonic polyhedra) that are reflected in the name of the solid; the ‘‘small’’ is appended in contradistinction to the ‘‘great’’ cubicuboctahedron.

Stellation Cells The lines of these diagrams divide the facial plane into a set of polygons (facets). Facets from the various facial planes are used to construct cells around the solid, and any desired stellation is built up from a set of these cells. One such cell is shown in Fig. 1, adjacent to the square face of the SCCO. The blue triangular face is covered by another (different) cell. Finally the four facets of the rectangular ‘‘dimple’’ are covered by a third type of cell. These three cells are shown in Fig. 2.

The Stellation Diagram If we now consider any facial plane F of P, then every other nonparallel facial plane of P will intersect F in a straight line. The set of lines thus generated is called the stellation diagram for that particular facial plane type. The diagrams for facial planes containing similar faces will themselves be similar. Referring again to Fig. 1, the stellation diagram for the SCCO’s square face is shown beneath the solid. The arrows indicate the lines contributed by the corresponding faces. Since the SCCO has three types of face, it has a total of three stellation diagrams. It is left to the reader to sketch out the other two.

Main-Line Stellations Bearing this illustration in mind, the creation of main-line stellations of (polyhedron) P can be defined thus: use all the boundary facets (from the full list of finite cell facets) of P to select all finite cells (if any), which are not part of P, and which contain any of these boundary facets. Adding all these cells to P will produce the (unique) first main-line stellation, S1. Repeating this selection process (where possible) on S1 leads to S2, and so on. The sequence of mainline stellations of P is complete when the final stellated form is bounded only by infinite cells. The SCCO has five main-line stellations, as shown in Fig. 3. A fuller presentation of the stellation process and its detailed application to the small-cubi-cuboctahedron can be found in the article referred to earlier [3].

Internal Stellation

Figure 2. Cells for SCCO main-line stellation 1.

So far, main-line stellations have been presented as the addition of external cells to the original solid—the stellation diagrams and the cells they form surround the outside of the polyhedral surface. And this has indeed been the focus of previous research—for example, Coxeter, et al. and

Figure 3. The small cubi-cuboctahedron and its five main-line external stellations (back row 1,2,3; front row 4, SCCO, 5). 20

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A

B

E

D

C F G I

H

Figure 4. Quasi-truncated Hexahedron. Figure 7. The internal cells of the Quasi-truncated hexahedron.

Table 1. Number of internal cells in QtrH

Figure 5. Octahedral facial plane.

Figure 6. Hexahedral facial plane.

Cell A B C D E F G H I

No. 24 6 24 24 24 8 12 8 1

Figure 8. 1st internal stellation. Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009

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Figure 9. Two further internal stellations—non mainline. Internal Stellation 1 2 3 4 5

Figure 10. Dodecahedral Facial plane.

Internal Stellation 1 2 3 4 5

Figure 13. Internal stellation 1.

Figure 11. Icosahedral Facial plane.

Figure 12. Quasi-truncated Great Stellated Dodecahedron. 22

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Figure 14. Internal stellation 2.

Figure 16. Internal stellation 4. Figure 15. Internal stellation 3.

Figure 17. Non main-line internal stellations.

Wenninger present only external stellations of (respectively) the icosahedron and the two semiregular forms, the cuboctahedron and the icosi-dodecahedron. The reason for this is readily apparent—these forms are convex and have no internal structure (e.g., vertices, planes, or cells). However, n-cUP, by definition, are self-intersecting, as noted previously. As a consequence they have interior structure—their facial planes intersect internally, inside the solid, and will generally create vertices, edges, facets, and cells interior to the polyhedral boundary surface. Indeed, the SCCO itself has a simple internal cell, namely a cube, three of whose sides are shown by the heavy lines in Fig. 1. This may be thought of as its single, internal stellation, revealed when cells adjacent to the SCCO’s surface (eight tri-hedral pyramids and six cuboids) are removed. In the more complex members of the n-cUP that I now describe, this process (removal of internal cells) may generally admit

a large number of possibilities. We therefore restrict the removal of internal cells so that the resulting internal stellation retains the properties 1, 2, and 3 of the restricted stellation defined earlier. The unique sequence of main-line internal stellations of a polyhedron P can be defined in an analogous manner to that of external stellation provided earlier: use all the boundary facets (from the full list of finite cell facets) of P to select all cells that are part of P and that contain any of these boundary facets. Removing all these cells from P will produce the first (unique) main-line, internal stellation S1i. Repeating this process on S1i leads to the complete sequence of main-line, internal stellations. Note that such stellations may not be simply connected, nor even connected. This process is now illustrated, in detail, with reference to two n-cUP. Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009

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Table 2. Inventory of stellated forms—nonconvex Uniform Polyhedra

Wenninger’s Reference Number

External

Internal

Tetrahemihexahedron

1

68

Octahemioctahedron

2

0

69

Small cubicuboctahedron

5

1

70

Small ditirgonal icosidodecahedron

12

0

71

Small icosidodecahedron

22

0

72

Small dodecicosidodecahedron

16

1

73

Dodecicosidodecahedron

7

1

74

Small rhombidodecahedron

75

Truncated great dodecahedron

76

Rhombidodecadodecahedron

22

1

77

Great cubicuboctahedron

4

1

78

Cubohemioctahedron

2

0

79

Cubotruncated cuboctahedron

4

1

80

Ditrigonal dodecahedron

4

3

81

Great ditrigonal dodecicosidodecahedron

16

1

82

Small ditrigonal dodecicosidodecahedron

14

3

83

Icosidodecadodecahedron

13

3

84

Icosidodecatruncated icosidodecahedron

16

1

85

Quasirhombicuboctahedron

5

4

86

Small rhombihexahedron

4

0

87

Great ditrigonal icosidodecahedron

9

5

88

Great icosicosidodecahedron

14

5

89

Small icosihemidodecahedron

8

0

90

Small dodecicosahedron

9

3

91

Small dodecahemidodecahedron

4

0

92

Quasitruncated hexahedron

0

3

93

Quasitruncated cuboctahedron

6

0

94

Great icosidodecahedron

5

5

95

Truncated great icosahedron

9

5

96

Rhombicosahedron

16

2

97

Quasitruncated small stellated dodecahedron

0

98

16

0

7

1

3

3

Quasitruncated dodecahedron

17

2

99

Great dodecicosidodecahedron

12

3

100

Small dodecahemicosahedron

5

0

101

Great dodecicosahedron

7

3

102

Great dodecahemicosahedron

4

2

103

Great rhombihexahedron

3

2

104

Quasitruncated great stellated dodecahedron

105

Quasirhombicosidodecahedron

106

Great icosihemidodecahedron

3

1

107

Great dodecahemidodecahedron

2

1

108

Quasitruncated icosidodecahedron

109

Great rhombidodecahedron

The Quasi-Truncated Hexahedron The Quasi-truncated Hexahedron (QtrH) is one of the simpler nonconvex Uniform Polyhedra and is shown in Wenninger, (op. cit., p 144). A line drawing of it is shown in Fig. 4. As can readily be seen, it has two types of facial planes. The octahedral plane (Fig. 5) is dominated by the large facets in each corner of the triangle (striped horizontally).

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Main-line stellations

67

Totals

24

Name

5

5

11

12

17

10

5

10

366

104

The cubic plane (Fig. 6) features an octagram with a small square at its centre (striped vertically). The relation between the solid and its facial planes is readily apparent. The fourteen planes of the QtrH divide its interior into cells of eight kinds, as shown in Fig. 7, where they are labelled A-I. Table 1 lists the numbers of each cell present. Facets of the QtrH are shaded. It is the cells (A-D) that contain these facets that have to be removed, revealing

Figure 18. Small Dodecicosa-dodecahedron. Stellation 3.

Figure 20. Quasi Rhombi-cosi-dodecahedron Internal stellation 5.

Figure 19. Great Dodec-icosi-dodecahedron Internal stellation 3.

Figure 21. Great Rhombi-dodecahedron. Internal stellation 2.

cells E, F, and G, which comprise the first internal stellation (Fig. 8). The facets required to make this solid are shown shaded medium grey in Figs. 5 and 6. A second internal stellation has no facets from the hexahedral plane, simply the three triangles shaded black in Fig. 5. These form Kepler’s ‘‘Stella Octangula’’ (compound of two tetrahedra). Its cells are labelled H in Fig. 7. Nestling inside it is the final internal stellation, namely a simple octahedron (pale grey in Fig. 5), and cell I in Fig. 7. Although I have generally concentrated on main-line stellations, I show in Fig. 9 two non main-line forms to give an idea of the variety possible. The first of these solids is formed by the cells labelled G in and the second from cells E. Note that although there are facets common to both (i.e.,

the large, unshaded parallelograms in the octahedral plane; Fig. 5), in the first form the interior contains the centre of the solid and so these facets have the usual positive sense. In the second form, the in-centre (of the solid) is not contained by the cells, and so these facets have a negative sense. In fact there are twelve hollow tetrahedral shafts penetrating from side to side, one of which can just be seen above the forward facing tetrahedral dimple. It is just the same ‘‘shafts’’ that comprise the first form in Fig. 9!

The Quasi-Truncated Great Stellated Dodecahedron The Quasi-truncated Great Stellated Dodecahedron (Wenninger, op. cit., Ref 104) is shown in Fig. 12. Like the QtrH, it Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009

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Tabular Summary

Figure 22. Rhombi-cosahedron Internal stellation (non mainline).

also has two facial planes (Figs. 10 and 11). The facets of the triangular (icosahedral) plane are shaded vertically and those of the decagram in the dodecahedral plane are shown shaded horizontally. As will be seen from Table 2, this solid has five (external) main-line stellations; it also has five internal (mainline) stellations, computer images of which are shown in Figs. 13–16. The first internal stellation (Fig. 13) is a ribbed icosahedron with the vertices of another icosahedral form peeping up from between the ribs. The second has a similar form (Fig. 14), but the ridged ribs have been replaced by grooved, di-hedral ribs, the vertices of which are impressive pentahedral pyramids. The third stellation (Fig. 15) is more complex in its details. The ribs are now ridges again, but the icosahedral vertices of the first form are replaced here with shallow, trihedral dimples, whereas the vertices of the ribs take on an even more complex form. The fourth internal stellation (Fig. 16) sees the start of the usual simplification process; it is, not unsurprisingly, a truncated form of the Stellated Dodecahedron. The final internal stellation (not illustrated) is the dodecahedron itself. Again I show two internal stellations that are not found in the main-line sequence (Fig. 17). They are hollow versions of the internal stellations 1 and 2 previously shown. Note that the facets underlying the ‘‘ribs’’ will have a negative sense as described previously.

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Table 2 presents a numerical summary of the stellations, external and internal, of the nonconvex Uniform Polyhedra. The following computer images show some of the more notable examples. The Small Dodecicosa-dodecahedron (Wenninger’s Ref. 72) is unique in having a main-line, external, stellation (No. 3) where the vertices of two cells actually touch, so that the solid is no longer topologically equivalent to a sphere. This can be seen in Fig. 18, where the vertices of the (blue) pentagon planes touch. This solid has 16 external stellations. This is easily eclipsed by the Quasi-rhomb-icosidodecahedron (Ref. 105) which has 23 main-line stellations, shared between 11 external and 12 internal, making it unique in having more internal than external stellations. However, the record must go to the Quasi-truncated icosidodecahedron (Ref. 108) which, although ‘‘only’’ having 17 external stellations, has 10 internal stellations, giving it the grand total of 27! As will be seen from the table, where it is listed with Wenninger’s Reference number (92), the Quasi-truncated Hexahedron is the only member of its class that has no external stellations. There are, however, several members that have no internal stellations, either because their planes form no (internal) cells that are closed or because they are ‘‘hemi’’ forms and have no central, interior space apart from the cells composing the solid itself. The Great Dodec-icosi-dodecahedron is unique in that it is the only n-cUP having a main-line stellation, internal or external, that falls into disjoint cells (Fig. 19). Its third (and final) internal stellation is composed of a truncated dodecahedron with pentagonal pyramids ‘‘floating’’ above the centre points of the former’s decagonal faces. The final three images (Figs. 20, 21, 22) are included to give some idea of the variety and complexity of forms that these beautiful polyhedra give rise to. Table 2 gives an inventory of the main-line stellations (external and internal) for all the n-cUP. Computer images of all the itemised solids have been produced. I would like to thank the referee for his comments and J. G. Kingston for his support and suggestions.

REFERENCES

1. H S M Coxeter, et al., (1938), The 59 Icosahedra. University of Toronto. 2. M J Wenninger (1971), Polyhedron Models, CUP. 3. J Hudson and J Kingston (1988), Stellating Polyhedra, Mathematical Intelligencer 10(3), 50–61.

Mnemosyne Emily Grosholz A musical black hole, like some enormous bellows Breathing animus into the pipe organ of space Unlike Bach’s polyphonic organ in the Leipzig Kapelle, Sings in the oval rills of the Perseus Galaxy Cluster Two hundred fifty million light years away from us, Two hundred fifty million years ago, but whether Past or present hangs on how the reminiscent soul Decides to reckon time. At fifty octaves deeper Than our true middle C, it plays only B flat, a singlet Theme for a B minor mass that no one ever hears, But we still hear that mass performed in Bach’s Kapelle Days after nine-eleven-one, not to applause but tears.

Department of Philosophy, Pennsylvania State University University Park, PA 16802, USA e-mail: [email protected] Ó 2009 The Author, Volume 31, Number 4, 2009

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Years Ago

David E. Rowe, Editor

Brouwer’s Intuitionism Vis a` Vis Kant’s Intuition and Imagination BERNARD FREYDBERG

Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

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Editor’s Note: Brouwer as Philosopher and Critic Within the last two decades there has been considerable interest in the life and work of L. E. J. Brouwer, one of the twentieth century’s most colorful and controversial figures (Stigt 1990; Dalen 1999, 2005; Kuiper 2004). Brouwer’s name has long been associated with two spheres of activity. As a pioneering actor in topology, he published a series of papers between 1909 and 1913 that were to exert a major impact on this fledgling field of research. Two famous results from this period were his proof of the invariance of dimension and his elegant fixed-point theorem, both of which were nonconstructive in nature (these results can be found in the classic text of Hurewicz and Wallman 1948). Soon afterward, however, Brouwer began an earnest program for a truly constructive mathematics. He took up a critical analysis of set theory and the concept of the continuum, activities that led him to pursue a radically new research orientation that came to be known as intuitionism. With it Brouwer aimed to sweep away standard methods that had long been used for proving existence theorems in analysis, techniques which had become increasingly controversial in the wake of the new foundational research undertaken by Cantor, Hilbert, Zermelo, Schoenflies and others. It has been customary to cordon off these two spheres of Brouwer’s activity: His fertile contributions to mainstream topology from his early career and the later (in many ways less successful) work on intuitionist foundations that made him such a controversial figure. This division was adopted, for example, by the editors of his Collected Works (Brouwer 1975, 1976). Moreover, Brouwer’s biography would seem to justify this separation, as he himself described his turn to foundations circa 1914 with the dramatic phrase ‘‘the second act of

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intuitionism.’’ Still, Brouwer’s creative vision was, from the start, fundamentally geometrical and, as a topologist, the continuum remained his natural domain of inquiry. Thus, viewed in larger terms, the thrust of his intuitionist program should be seen as standing in firm opposition to the longstanding trend associated with the ‘‘arithmetization of analysis.’’ Already in the early 1870s, much effort had been expended on freeing the real numbers from all vestiges of ‘‘geometrical intuition.’’ For Weierstrass, Cantor and Dedekind, the real number system was a purely arithmetical construct, whereas Kronecker spoke out vociferously against this conception. By 1900, Hilbert famously proclaimed that Kronecker had been dead wrong; Hilbert maintained that the very existence of the ‘‘arithmetized continuum’’ could be proved, once and for all, by demonstrating the consistency of his axioms for a complete ordered Archimedean field. The algebraist Kronecker could only turn over in his grave, but the topologist Brouwer mounted a strong counterattack. Thus, behind Brouwer’s intuitionism stood his fierce opposition to the general notion that the continuum could be reduced to an arithmetical object; for him the intuitive continuum was far richer than Dedekind’s or Cantor’s atomistic conceptions of an infinite point set whose individual elements were captured in their entirety by means of arithmetical properties alone. He was also steadfastly opposed to Hilbert’s formalist methods. Brouwer was ferociously committed to his vision and knew full well that this placed him on a collision course with Hilbert, the most influential mathematician of the era. His intuitionist conception of the continuum aimed at nothing less than overturning Hilbert’s formalized axiomatic methods, an approach Brouwer regarded as contentless. Brouwer’s intuitionist ideas gained widespread currency after the First

World War, especially after 1921 when Hermann Weyl published his essay on the ‘‘New Foundations Crisis in Mathematics.’’ In it, Weyl explicitly cast aside his own earlier attempt to rebuild the continuum on the basis of a refined set theory, noting as he did so that he had never conceived of his ‘‘Weylean number system’’ as identical with the’’ intuitively given continuum.’’ Weyl’s provocative text, strongly colored by the political events of the era, was openly partisan: So I now abandon my own attempt and join Brouwer. I tried to find solid ground in the impending dissolution of the State of Analysis (which is in preparation, even though still only recognized by few) without forsaking the order upon which it is founded, by carrying out its fundamental principle purely and honestly. And I believe I was successful—as far as this is possible. For this order is in itself untenable, as I have now convinced myself, and Brouwer—that is the revolution! . . . It is Brouwer to whom we owe the new solution of the continuum problem. (Weyl 1921, pp. 98–99) Not to be outdone, Hilbert took up Weyl’s political metaphor and defended the State of Analysis: What Weyl and Brouwer do amounts in principle to following the erstwhile path of Kronecker: They seek to ground mathematics by throwing overboard all phenomena that make them uneasy by establishing a dictatorship of prohibitions a` la Kronecker. But this means to dismember and mutilate our science, and if we follow such reformers, we run the danger of losing a large number of our most valuable treasures. . . . I believe that, just as Kronecker in his day was unable to get rid of the irrational numbers (Weyl and Brouwer, incidentally, allow the preservation of a torso) so today Weyl and Brouwer will be unable to push their program through. No: Brouwer is not, as Weyl believes, the revolution, but only a repetition, with the old tools, of an attempted coup that, in its day, was undertaken with more dash, but nevertheless failed completely; and now that the power of the state has been armed and strengthened by Frege, Dedekind, and Cantor, this

coup is doomed to fail. (Hilbert 1922, p. 200) Six years later, Brouwer was pushing his ideas harder than ever, prompting Hilbert to resort to more mundane power politics. This conflict culminated in 1928 when Hilbert, despite strong misgivings within his own camp of loyalists, ousted Brouwer from the editorial board of Mathematische Annalen (Dalen 1990) Soon thereafter, Brouwer largely withdrew from the arena of active research; while continuing to dabble with foundations issues, he failed to complete any of the three drafts he made for a monograph on intuitionist mathematics (Stigt 1990). Needless to say, Hilbert and Brouwer never reconciled after this dramatic rupture. The 1920s represented the ideological phase in the foundations debates; after Brouwer retired from the scene, the polemical language died down quickly. Perhaps not coincidentally, in 1930 leading representatives of the three leading ‘‘philosophical schools’’ convened at a conference in Ko¨nigsberg that was devoid of the customary fireworks. Rudolf Carnap spoke on behalf of logicism, John von Neumann represented formalism, whereas Arend Heyting served as spokesman for intuitionism. In the meantime, Heyting had given a formalization of intuitionist logic, a development Brouwer had approved, albeit with the understanding that this formal language contained nothing new or even important for intuitionist mathematics. At any rate, this conference successfully canonized the three approaches that would for the next several decades come to dominate discussions in the philosophy of mathematics. It was thus both fitting and ironic that a diminutive young logician slipped into the conference hall to make an announcement of a new result about formal systems, one that apparently only von Neumann grasped at the time. His name was Kurt Go¨del, and the result was the first of his two famous incompleteness theorems. Soon thereafter, Hilbert’s program for establishing the consistency of the axioms of arithmetic for the natural numbers (his first step toward salvaging the continuum) was in a shambles. It has been customary to look back on the battle between Brouwer and Hilbert from the vantage point of

subsequent events: Go¨del’s theorems, Heyting’s intuitionist logic, and Paul Bernays’s subsequent work on formal systems and the axiomatization of set theory. Yet these developments tell us relatively little about the ideas that originally motivated the principal actors. To understand these, one must go back in time to recall Hilbert’s encounters with figures like Kronecker and Paul Gordan (Rowe 2003b, 2005), his use of formal axiomatics to ‘‘arithmetize geometry,’’ and his insistence on using the axiomatic method to refute Kronecker’s skepticism regarding the ontological status of the continuum as an arithmetical object. Hilbert’s views were already made highly visible in his famous Paris lecture on ‘‘Mathematical Problems’’ from 1900, an address that profoundly influenced the young Brouwer. But unlike Hilbert, Brouwer was deeply engaged in the broader philosophical discourse of the times, and so he quietly distanced himself from the buoyant optimism that Hilbert had made his own. When Hilbert pronounced that there is no ignorabimus in mathematics—that each and every wellformulated mathematical problem can either be solved or not, hence its truth value was certain—Brouwer saw this as unwarranted hubris. Indeed, he identified this claim with a fundamental methodological weakness that would later become a cornerstone of his intuitionist philosophy: The fallacy of assuming the unrestricted validity of the logical tertium non datur, the Law of the Excluded Middle. Brouwer, probably more than any other contemporary mathematician, tried to weld his mathematics to fit deeply held philosophical views. This alone helps to explain the strong attraction he had for Hermann Weyl, a mathematician with similar philosophical sensibilities (Rowe 2003a). For Brouwer, the pursuit and attainment of mathematical knowledge had to be understood in human terms and not by appealing to some transcendental sphere of eternal, never-changing harmonies. Mathematics is thus created in time, a notion that already played a key part in the epistemology of Immanuel Kant. Experts on Brouwer’s thought have long noted the role of Kant’s ideas in his writings (Stigt 1990, 127129), especially in connection with


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Brouwer’s primordial conception of time. Nevertheless, the last word on this topic has surely yet to be spoken. In the essay that follows, the philosopher Bernard Freydberg offers a new assessment of the intellectual affinities between Brouwer’s philosophy of mathematics and the epistemology of Kant. This reading derives in part from Freydberg’s original analysis of the role of productive imagination in Kant’s philosophy (for which the reader should consult Freydberg 1994, 2005). His findings lead to a very different interpretation of Brouwer’s intuitionism

than current views of this philosophy which have been strongly colored by subsequent developments in intuitionistic logic. Freydberg’s reading offers a view of Brouwer’s thought that suggests there was no radical break that led to intuitionism, and that it is a mistake to separate Brouwer’s early career as a topologist from his endeavors to reform set theory and the foundations of analysis. Rather, following this account, Brouwer’s intellectual journey, culminating with the ‘‘second act of intuitionism,’’ was guided by a coherent vision that he held already from his

student days. As Freydberg argues, the thrust of Brouwer’s thought resonated on a deep level with Kantian motifs, a conclusion that coincidentally shatters another standard picture long associated with Hilbert’s philosophical views. For the ironic upshot of the analysis below is that it was not Hilbert—the native of Kant’s Ko¨nigsberg who cited the famous philosopher approvingly on numerous occasions—but rather his arch-nemesis, Brouwer, who carried the torch of Kantian philosophy forward among mathematicians a century ago. D.E.R.

hat follows is an apodeixis in the original Platonic sense as opposed to modern or contemporary philosophical usage. Kant used the word apodiktisch to mean ‘‘absolutely certain’’ or ‘‘incontrovertible.’’ So, too, did a philosopher who was at the center of the early twentiethcentury discussions of the foundations of mathematics, Edmund Husserl. This meaning has become the lexicographically respectable one, to the degree that its earlier Platonic sense has been all but completely obscured. This more original sense, in fact, carries a very different connotation, drawing on the literal Greek meaning as ‘‘showing forth’’: deiksis from deiknumi—to show, apo = forth, from. In Euclid’s Elements the apodeixis constitutes the actual proof, being the fifth among six steps in the formal enunciation of a proposition. In Plato’s Phaedrus, Socrates applies the word apodeixis to the great myth of Eros that comprises the heart of the dialogue. The myth speaks of the immortality of the soul and of the soul’s repeated journeys. It speaks of the worthy human soul ascending to where it can get a glimpse of the procession of the gods and of the divine banquet, where the divinities—standing on the earth’s outer crust—feast on ‘‘being beingly being.’’ It speaks of beauty as providing the recollection of this vision, and of love of beauty as the way humanity most fully realizes itself on

earth. Most provocative are the words spoken by Socrates immediately before beginning this remarkable apodeixis: ‘‘The apodeixis [often translated as ‘‘proof’’] shall be one that the wise will trust, but the clever will disbelieve.’’ (Plato 2003, 245 b.c.) I risk an analogous claim for this apodeixis of the Brouwer-Kant relationship. The connection between Brouwer’s intuitionism and Kantian philosophy is on some level certainly plain, and there is no doubt that Brouwer read Kant’s Critique of Pure Reason (Stigt 1990, 127–132). However, he was certainly not a Kant scholar. Moreover, it would be a mistake (and would undoubtedly overstate the case) to claim that Kant influenced him. Rather, Brouwer found a kindred spirit in Kant; one might better think of Kant as his dialogical partner, a fellow member of Schopenhauer’s ‘‘Republic of Geniuses’’ who speak to one another from mountain peaks across centuries. There is little or no evidence that Brouwer read anything beyond Kant’s first critique and, as I will suggest from the evidence, his most attentive and conscientious reading of the first critique seemed to be concentrated upon one section alone, albeit a crucial one. But like many eager tellers of tales, I have gotten a bit ahead of myself. First, a few words about the remarkable philosophical inclinations of Luitzen Egbertus Jan Brouwer (1881– 1966) are surely in order.

Brouwer’s Philosophical Outlook

W

1

A recent study that deals with this theme is Kuiper 2004.

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Although several of Brouwer’s works have been absorbed into the mathematical mainstream, his larger philosophical outlook has received rather little detailed attention.1 Furthermore, much of that attention has been dismissive, and one can easily understand why. In his early work, Life, Art, and Mysticism (Brouwer 1996), written when he was 24, Brouwer argued that both science and the human intellect are sources of sin and evil. Pessimism, misanthropy and arrogance set the dominant tone. His otherwise admiring chronicler, Dirk van Dalen, apparently found this work at least somewhat embarrassing (Dalen 1999, pp. 66–77). However, his other major biographer, W. J. van Stigt, emphasized that much in this radical philosophical tract foreshadowed his future work in mathematics. Van Stigt also noted the significant historical fact that Brouwer not only never disowned this work but actually affirmed it throughout his life. ‘‘It proudly features in every one of his entries for various biographical dictionaries as the first of his two books’’ (Stigt 1990, p. 35). One can understand why this is so as well. For in Life, Art, and Mysticism, Brouwer asserts an underlying belief in a natural—intuitive— access to reality that antedates the intrusion of these ‘‘evils,’’ a view that he not only never gave up, but that

Figure 1. Immanuel Kant (1724–1804) was baptized Emanuel, but changed the spelling of his name after learning Hebrew.

informed many of his singular mathematical insights. Brouwer’s philosophical thought was unsystematic, and he disdained the mundane practice of offering arguments for his views. Taking his ideas out of context, they may appear merely bizarre and irrational, providing easy prey for critics. This does not mean, however, that his thought contains little of value, nor does it lack inner consistency and cogency. One who is trained in the history of philosophy can surely find echoes of thinkers from Heraclitus to

Merleau-Ponty. Perhaps an intrepid philosopher will one day write a commentary that plumbs the depths of Brouwer’s thought and presents a unified account of it. However, the Brouwer-Kant affinity with respect to the history of mathematics is the concern here, and those insights of Brouwer most under consideration are the following: 1) The First Act of Intuitionism In several of his later lectures, Brouwer described the historical evolution of his thought, culminating in what he called

its first and second acts of intuitionism. These rested on two insights, the first of which he had already reached as a student, namely ‘‘the uncompromising separation of mathematics from mathematical language and thereby from the linguistic phenomena as described by theoretical logic’’ (unpublished Berlin lectures, 1927, quoted in Stigt 1990, p. 96). Brouwer’s second insight came around 1916 and led him to his notion of free-choice sequences, the key to his understanding of the continuum (see Stigt 1990, pp. 71–77). In his Berlin lectures, this second act of intuitionism was described as ‘‘a form of self-unfolding of the Primordial Intuition’’ that allowed not only for lawlike sequences but also those that were freely chosen, yet subject to certain constructive restrictions.2 These latter constructs were notoriously complicated, but luckily we need not be concerned with them for present purposes, as we are only concerned with Brouwer’s first act of intuitionism, which he characterized as follows in his postwar Cambridge lectures: Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics (Dalen 1981, p. 7).

2) Rejection of the Principle of Excluded Middle From early on Brouwer denied the legitimacy of applying Aristotle’s

2 (Stigt 1990, p. 96). Brouwer’s later description of the Second Act of Intuitionism in his Cambridge lectures is more revealing: Admitting two ways of creating new mathematical entities: Firstly in the shape of more or less freely proceeding infinite sequences of mathematical entities previously acquired (so that, for decimal fractions having neither exact values, not any guarantee of ever getting exact values admitted); secondly in the shape of mathematical species, i. e., properties supposable for mathematical entities previously acquired, satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be ’equal’ to it, definitions of equality having to satisfy the conditions of symmetry, reflexivity and transitivity… (Dalen 1981, p. 8).


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Finally it must be remarked that the intention of the work under discussion is not to propose considerations under which different persons can establish different conclusions, but to establish truths that…anybody who has once understood them will forever affirm (cited in Dalen 1978, p. 31). These passages suggest an integrated picture of the nature of mathematical knowledge: Mathematics is an act of the human mind, which fashions the images surrounding it into particular patterns that, once understood, have universal validity.

Brouwer’s Kantian Affinity

Figure 2. L. E. J. Brouwer lecturing on a favorite topic: The intuitionistic continuum.

tertium non datur to propositions that concern infinite sets or infinitely many possibilities. He also came to identify the notion that this principle was universally applicable with Hilbert’s sweeping claim that every well-posed problem in mathematics has a solution, that in the realm of mathematical knowledge there can be no ignorabimus. This article of faith was propounded by Hilbert in his famous Paris lecture on ‘‘Mathematical Problems,’’ delivered in 1900; Brouwer referred to it as ‘‘Hilbert’s dogma.’’ For him, all human knowledge was historically conditioned and subject to change. Thus, the faulty belief in the universal validity of the principle of the excluded third was a phenomenon of the history of civilization; Brouwer likened it with the former belief in the rationality of p, or in the rotation of the firmament around the earth. 3) On the Nature of Mathematical Knowledge Brouwer once wrote that ‘‘properly speaking the building of intuitive mathematics per se is an action

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[een daad]… and not a science (een theorie),’’ a remark that Dirk van Dalen compared with Hermann Weyl’s later dictum: ‘‘Mathematics is more an action than a theory [mehr ein Tun als eine Lehre]’’ (Dalen 1999, p. 105). Calling mathematics more an action than a science requires that Brouwer give some account of what belongs to mathematical truth. He seems, at one point, to restrict not only mathematical truth but all truth to an entirely subjective criterion: For the only truth for me is my own ego [ikheid] of this moment, surrounded by a wealth of images in which the ego believes and which make the ego live. There is no sense in the question whether those images are ‘true,’ for my ego only the images exist and are real as such; a second corresponding reality, independent of my ego, is out of the question (cited in Dalen 1978, p. 293). However, when he speaks of mathematical truth more specifically, a merely subjective criterion is clearly insufficient:

I will now consider the three topics listed in the previous section serially. 1) The First Act of Intuitionism: It is not difficult to locate certain textual connections between Brouwer and Kant, especially between the First Act of Intuitionism and Kant’s Transcendental Aesthetic in the Critique of Pure Reason, the section that deals with space and time as pure intuitions. However, the nature of the connection is not altogether straightforward. First of all, there is a conundrum in Kant’s notion of pure intuition. At the outset, Kant claims that for us humans, intuition—sensation in general—must be given. They are distinguished from concepts, which must be thought. In the same paragraph, he also claims that in order for a representation to be pure, it must have no empirical (given) content (Kant 2003). Therefore, at the very least there is a tension in the notion of pure intuition. Decades ago I took a graduate seminar with the great American Kant scholar Lewis White Beck, who rejected the notion of ‘‘pure intuition’’ and actually had us substitute ‘‘pure form of intuition’’ whenever the formulation ‘‘pure intuition’’ occurred. This phrase meant merely the pure element in every (empirical) intuition. I have since come to have serious doubts about Beck’s interpretation, and here I can attempt to shed some light upon Brouwer’s powerfully creative affinity for Kantian thought. Pure intuition: What does it mean? ‘‘Pure,’’ as we have seen, means ‘‘independent of sensation.’’ Human intuition, as we have also seen, must be given.

Taking the two together, we can make sense of them only if we interpret pure intuition as intuition that we give ourselves. Here, in The First Act of Intuitionism, Brouwer instinctively understood Kant’s point, namely that pure intuition—more specifically, time as the form of all intuition—is essentially nonconceptual, that is, nonlinguistic (for Brouwer, there is no pure intuition of space). One can say preliminarily that language limps along afterwards, that the function of language is second-order and secondary. The function of language—mathematical language and any other—is to serve the intuition, to point to the intuition that resides at a more fundamental level than any language. Consider the very first sentence of the Transcendental Aesthetic, which reads: ‘‘In whatever manner and by whatever means a mode of knowledge [Erkenntnisse] may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought as a means is directed’’ (Kant 2003, pp. A19, B33). Thought (language) is in service to intuition. Though the need humans have for language cannot (and, of course, should not) be denied, there is no doubt that for Kant intuition is primary. Brouwer’s claim that mathematics has lost its way because it has confounded itself with mathematical language echoes loudly from the Kantian analysis that I have just proposed. However, many critics have noted what they see as a disconnect, or perhaps even a contradiction, in Brouwer’s philosophy of mathematics with respect to language and the role of logical notation. After all, Brouwer employs both in order to formulate intuitionist mathematics. I might add that it is indeed insufficient merely to assert the nonlinguistic character of intuition based upon its separability in reflection from language. Much more is required to account for this apparent paradox. I suggest strongly that other texts from the Critique of Pure Reason amply supply this account, although Brouwer does not refer to them and may indeed have breezed past them. With the preparation in place, I can explore a deeper connection between the Kantian philosophy and Brouwer’s

intuitionism. In my view, the crux of this connection is not located in the ‘‘Transcendental Aesthetic’’ but in the ‘‘Transcendental Analytic.’’ There, one must come to terms with a Kantian element that, to my knowledge, Brouwer never mentions: Productive imagination. I will argue that Kant’s productive imagination and matters connected with it—according to a particular interpretation which, in my view, is governed by a rigorous textual reading—tacitly undergirds The First Act of Intuitionism and, a fortiori, all that follows from it. Early in the ‘‘Transcendental Analytic,’’ Kant writes: Synthesis in general (u¨berhaupt), as we shall hereafter see, is the mere result (Wirkung) of the power of imagination (Einbildungskraft), a blind but indispensible function of the soul, without which we should have no knowledge whatsoever, but of which we are scarcely ever conscious. To bring this synthesis to concepts is the sole function of the understanding, and it is through this function that we first gain knowledge properly so called. (Kant 2003 pp. A78, B103) How can the Brouwerian twoity be seen in light of productive imagination? The synthesizing work of imagination is sharply separated from the understanding’s work of bringing the synthesis to concepts. Synthesis, an apparently technical word, also has a literal Greek meaning: Placing together. For Kant, synthesis is defined as the act of running through (a manifold of possibilities) and holding it together. Productive imagination, of which Brouwerian ‘‘memory’’ can be considered an implicit offshoot, holds together the two moments of time. Time, the pure form of all intuition, is successive by nature, and a twoity is the succession of two moments emptied of whatever content these moments might contain. A biographical-historical note that deserves attention here: Brouwer’s unorthodoxy was clear early on, so he had difficulty finding someone to direct his dissertation. D. J. Korteweg, though an applied mathematician, was very open-minded and agreed to serve as Brouwer’s Ph.D. advisor. Both of Brouwer’s biographers note that this led to some lively exchanges, like this one:

Korteweg: Can one mention the name of Kant in a mathematical article? Brouwer: Yes, Russell and Couturat did so, and the subject forces me to do so. Korteweg: did you study Kant sufficiently enough to form an opinion? Brouwer: I cannot prove that I did, but I have repeatedly and seriously studied the passages I need (Stigt 1990, p. 128). Brouwer goes on to beg Korteweg for the understanding of his basic ideas, but concludes his plea with these less than humble words: ‘‘[E]ven though you find [my thoughts] absurd, [this is] because I am a child of another time than you are’’ (ibid). So much for gratitude for Korteweg’s willingness to direct a dissertation that was outside his field; more than once, he had to put up with Brouwer’s arrogance and inelegant social skills. But gratitude and social tact, while admirable qualities, have no relevance for judging philosophical and mathematical excellence, as Brouwer surely knew. He did indeed belong to a generation of mathematicians who would criticize conventional wisdom while raising new questions about the foundations of mathematical research. 2) Rejection of the Principle of Excluded Middle: The tertium non datur had been considered fundamental to philosophy since Aristotle, who was credited with first formulating it. Kant certainly knew Aristotle’s logic, and he believed that Aristotle had essentially completed the subject. However, though he wrote and lectured a great deal on logic, to my knowledge Kant never mentioned the Principle of the Excluded Middle or any of its equivalents. Not only is it absent in the Critique of Pure Reason, where one might expect to find it, but it never appears in his many lecture courses on logic. This absence is striking: A theological analogy might be to imagine a rabbi or priest failing to mention the creation of light among God’s works at the outset of the Old Testament. One might certainly conjecture that in his reading of the Critique of Pure Reason, the young Brouwer took notice of and perhaps heart in the absence of this so-called ‘‘fundamental’’ principle.


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A brief consideration of Kant’s Mathematical Antinomies of Pure Reason should be enough to illustrate Brouwer’s affinity with the Kantian ‘‘absence:’’ First Antinomy Thesis: The world has a beginning in time, and is also limited as regards space. Antithesis: The world has no beginning, and no limits in space; it is infinite as regards both time and space (Kant, 2003 A426–427, B454–455). Second Antinomy Thesis: Every composite substance in the world is made up of simple parts, and nothing anywhere exists save the simple or what is composed of the simple. Antithesis: No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple (Kant, 2003 pp. A434– 435, B 462–463). In the late eighteenth century, only Aristotelian logic was available, according to which each term in a proposition was assumed to have members, or existential import.3 However, for Kant, ‘‘world’’ is a concept of the unconditioned, that is, it is an idea of reason that transgresses the sensible condition to which all concepts are bound if they are to have meaning and significance—or in other words, world is a concept with no object—‘‘world’’ does not exist. As it is independent of time (Brouwer’s language), ‘‘world’’ cannot belong to any life moment—a notion that is crucial to his First Act of Intuitionism, as we have seen. Since the concept ‘‘world’’ refers to nothing at all, all propositions that predicate some quality to it are false. Thus, the theses of both the First and Second Antinomies, the mathematical antinomies, are false. According to the Principle of the Excluded Middle, their antitheses must be true. However, the antitheses are both also false for the same reason that the theses are false: They predicate some quality to ‘‘world’’! According to the Principle of the Excluded Middle, p _ *p; if *p is false, then

3

p must be true. Therefore, the Logical Law of Excluded Middle does not hold in these important cases. One can surmise that in his reading of the Critique of Pure Reason, the young Brouwer would have grasped these key arguments, and it seems likely that he later took heart in the absence in Kant’s philosophy of this so-called ‘‘fundamental’’ principle. Simmering beneath the surface of Kant’s analysis, one can discern both the development of certain kinds of philosophical positivism and of Boolean logic. But both he and Brouwer resisted these tendencies. On the other hand, Kant does pay much attention to the principle of contradiction, which is merely a principle of formal (i.e., analytic) logic—according to which any proposition in which the predicate contradicts the subject must be false. It is a conditio sine qua non of truth, a necessary condition, but emphatically not a sufficient condition. There is another philosophical antecedent of the Principle of Contradiction, which Brouwer accepts. It is found in Plato’s Republic, and in a manner most suitable to Brouwer because of its connection to time: ‘‘It is obvious that the same thing will not be willing to do or undergo opposites in the same part of itself, in relation to the same thing, at the same time.’’ (Plato 1968, p. 436b, emphasis mine). I might mention also that for Kant, the real logic—what might be called his positive and original logic—is Transcendental Logic, which includes a relation to time and the movement that time implies. Formal logic merely abstracts from transcendental logic, leaving behind time and the movement that time implies—elements that also belong to what Brouwer called ‘‘life moments.’’ 3) On the Nature of Mathematical Knowledge: The affinity on this matter is clear in this essential respect: Mathematics, for Brouwer as for Kant, is an act of construction. The difference, accidental in this context in light of Brouwer’s study of Kant’s Transcendental Aesthetic, is that, for Brouwer, mathematics is more art than science, while it is science alone for Kant, who sought to establish philosophy as an

a priori science and to include mathematics under its sway. Philosophical knowledge is knowledge by means of concepts; mathematical knowledge is knowledge by means of the construction of concepts. In geometry, his most often used example, the concept ‘‘triangle’’ serves as a rule for its exhibition in time and space. Of course, the drawn triangle upon which proofs are performed is not the Euclidean triangle, properly speaking. However, the Euclidean triangle is plainly intended and the proofs issuing from its use are incontrovertible. For Brouwer, mathematics is also a free creation of the human mind, but it is art more than science. From his early philosophical discourses in Life, Art, and Mysticism, Brouwer expressly exempted certain kinds of art from the stigma of sinfulness to which he consigned science. Such art, which transcends individuality and history, can provide ‘‘eruptions of truth’’ that are sensed but not articulated. Accordingly, the Brouwerian view of mathematical truth finds an affinity in Kant’s aesthetic philosophy. In the ‘‘Analytic of the Beautiful’’ in the Critique of Judgment (Kant 1970), beauty is characterized as disinterested and as commanding universal assent, that is, as transcending both the conditions of time and space and the vagaries of individual opinion. Kant calls this subjective necessity. In order to make this claim, he assumes that there is a sensus communis of which only some avail themselves. Thus, it is a conditioned necessity, conditioned namely by the contingency of its exercise. Such a sensus communis accords entirely with Brouwer’s view of mathematical truth, just at it harmonizes with his view of mathematics as art. However, just as in the case of the actual construction of mathematics, a difference between the two thinkers obtains in the determination of the nature and limits of human receptivity. For Kant, this subjective necessity pertains to beauty, not truth; his view of truth is conventionally scientific. But Brouwer’s requirement of universal assent to established mathematical truths, once they are under-

In Boolean logic, where there is no such requirement, there would be a middle, but this does not affect the issue.

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stood, allows for more than their strong kinship with Kant’s sensus communis— it allows for their virtual interchangeability. The experience of Thomas Hobbes with the Pythagorean theorem, as recorded in John Aubrey’s Brief Lives, provides a wonderful instance of Brouwer’s claim that when a mathematical truth is established it claims universal assent to all who understand it: He was 40 years old before he looked in on Geometry; which happened accidentally. Being in a Gentleman’s Library, Euclid’s Elements lay open, and ‘twas the 47 El. libri I. He read the Proposition. By God, sayd he (he would now and then swear an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which Proposition he read. That referred him back to another, which he also read. Et sic deinceps that at last he was demonstratively convinced of that trueth. This made him in love with Geometry (Aubrey 1957, p. 150).

Conclusion In entirely different ways, both Brouwer and David Hilbert found inspiration in the work of Kant. In fact, Hilbert’s Foundations of Geometry begins with this citation from the Critique of Pure Reason: ‘‘All human knowledge begins with intuitions, proceeds from there to concepts, and ends with ideas’’ (Kant 2003 pp. A702, B730). One reads this as foreshadowing Hilbert’s formalism, in which the foundations of mathematics rest on axioms that reflect the deeper principles embodied in knowledge that began in the realm of Kantian intuitions. Hilbert’s success with Euclidean geometry led him to hope that all of mathematics could be grounded in abstract axiomatics. His proof theory was a direct result of this vision. It aimed to demonstrate that the axioms for arithmetic had to be consistent, a proposition Hilbert believed was apodiktisch in the sense of Kant. Ironically, these hopes were dashed by another Kantian-oriented mathematician, Kurt Go¨del, in 1931. Yet one who had thoroughly

understood Hilbert’s Kantian Leitwort might have realized that it was out of place here. For Kant’s principal message was that the ascent to unconditioned ideas leads reason into areas beyond its scope and competence. On the other hand, Brouwer’s rigorous attention to intuition in Kant’s writings makes him a more faithful Kantian, and his creative appropriation of Kant’s notion of pure intuition plays a direct and positive role in his mathematical work. I hope that this brief apodeixis is useful for appreciating the instinctive, implicit, but acute affinity of Brouwer for the Kantian philosophy, and that this appreciation may be helpful in promoting a deeper understanding of Brouwer’s ideas on the foundations of mathematics. In my research, I found a most intriguing appraisal of Brouwer’s influence in a London Times obituary written by the logician and mathematician G. T. Kneebone, who likened Brouwer to another philosopher whose outlook and intellectual demeanor seems far removed from Kant’s: Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic. He in fact rejected the universally accepted logic of deductive reasoning which had been codified initially by Aristotle, handed down with very little change into modern times, and very recently extended and generalized out of all recognition with the aid of mathematical symbolism (Kneebone 1966). In terms of his willingness to call long held assumptions into radical question, this comparison seems most appropriate. Nietzsche’s influence is felt throughout much of twentieth and twenty-first century philosophy, as well as in literature, criticism, psychology and philology; and one can certainly make a suggestive analogy to Brouwer’s influence upon mathematics. However, one would be hard pressed to make a comment upon Brouwer like the one Heidegger made of his great predecessor: ‘‘Nietzsche hat mich verru¨ckt ge-

macht.’’ [‘‘Nietzsche has made me crazy.’’] Brouwer sustained the essence of his vision throughout his life, as did Kant after 1781—the fundamental insights were set, and any problems of exegesis occurred in terms of them. Brouwer proceeded more like a creative artist than the much more prosaic Kant, but, like Kant, he was guided by a singular vision. Theirs is an affinity of kindred minds and of purposeful seekers. The two cultures here are a synthesis: A twoity that prefigures philosophy and mathematics—a twoity that can only be the product of imagination.

REFERENCES

Aubrey, John. 1957. In Aubrey’s Brief Lives, ed. Oliver Lawson Dick, Ann Arbor: University of Michigan Press. Brouwer, L. E. J. 1975. Collected Works, Vol. 1: Philosophy and Foundations of Mathematics, ed. A. Heyting, Amsterdam: North-Holland. Brouwer, L. E. J. 1976. Collected Works, Vol. 2: Geometry, Analysis, Topology and Mechanics, ed. H. Freudenthal, Amsterdam: North-Holland. Brouwer, L. E. J. 1996. Life, Art, and Mysticism. Ed. W. P. van Stigt, Notre Dame Journal of Formal Logic, 37: 381–429. Dalen, Dirk van. 1978. Brouwer: The Genesis of his Intuitionism. Dialectica, 32(3–4): 291–303. Dalen, Dirk van, ed. 1981. Brouwer’s Cambridge Lectures on Intuitionism, Cambridge: Cambridge University Press. Dalen, Dirk van. 1990. The War of the Frogs and the Mice, or the Crisis of the Mathematische Annalen. Mathematical Intelligencer, 12(4): 17–31. Dalen, Dirk van. 1999. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Volume 1: The Dawning Revolution. Oxford: Clarendon Press. Dalen, Dirk van. 2005. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Volume 2: Hope and Disillusion. Oxford: Clarendon Press. Freydberg, Bernard. 1994. Imagination and Depth in Kant’s Critique of Pure Reason. New York: Peter Lang. Freydberg, Bernard. 2005. Imagination in Kant’s Critique of Practical Reason. Bloomington, IN: University of Indiana Press.


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Hilbert, David. 1922. The New Grounding of

Kuiper, J. J. C. 2004. Ideas and Explorations:

Mathematics: First Report. Trans. W. Ewald, in Mancosu, Paolo, ed. 1998.

Brouwer’s Road to Intuitionism. Ph.D Dissertation. Utrecht: Utrecht University

From Brouwer to Hilbert. The Debate on

Department of Philosophy.

the Foundations of Mathematics in the

Mancosu, Paolo, ed. 1998. From Brouwer to

1920s, New York: Oxford University

Hilbert. The Debate on the Foundations

Press, pp. 198–214.

of Mathematics in the 1920s. New York:

Hurewicz, Witold and Wallman, Henry. 1948.

Oxford University Press.

Rowe, David. 2005. Hilbert’s Early Career: Encounters with Allies and Mathematical Intelligencer,

Rivals. 27(1):

72–82. Stigt, W. P. van. 1990. Brouwer’s Intuitionism, Amsterdam: North-Holland. Weyl, Hermann. 1921. On the New Foundational Crisis of Mathematics. Trans. B.

Dimension Theory, Princeton: Van No-

Plato. 1968. The Republic of Plato. Trans.

strand. Kant, Immanuel. 1970. Critique of Judgement.

Allan Bloom, New York: Basic Books. Plato. 2003. Plato’s Phaedrus. Trans. Stephen

Mu¨ller, in Mancosu, Paolo, ed. 1998. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the

Scully, Newburyport: Focus Books.

1920s. New York: Oxford University

Trans. J.H. Bernard, New York: Free Press. Kant, Immanuel. 2003. Critique of Pure Reason. Trans. Norman Kemp Smith, New York: Palgrave Macmillan. Kneebone, G. T. Obituary for L. E. J. Brouwer. In London Times, December 17, 1966.

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Rowe, David. 2003a. Hermann Weyl, the

Press, pp. 86–118.

Reluctant Revolutionary. Mathematical Intelligencer, 25(1): 61–70. Rowe, David. 2003b. From Ko¨nigsberg to Go¨ttingen: A Sketch of Hilbert’s Early Career. Mathematical Intelligencer, 25(2): 44–50.

Koc¸ University Sariyer-Istanbul Turkey e-mail: [email protected]

Mathematical Entertainments

Bidding Chess JAY BHAT AND SAM PAYNE

I

Michael Kleber and Ravi Vakil, Editors

t all started with a chessboard

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This column is a place for those bits

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of contagious mathematics that travel

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from person to person in the

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community, because they are so

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elegant, surprising, or appealing that

2

one has an urge to pass them on. Contributions are most welcome.

1 a

b

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d

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and a bottle of raki

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]

at an otherwise respectable educational institution. SP’s friend Ed had just returned from Turkey with a bottle of the good stuff. We were two and a half sheets to the wind, feeling oppressed by the freshman dormroom, its cinderblock walls and our mediocre chess skills. The details of what transpired are clouded with alcohol and time, but Ed was enrolled in a seminar on auction theory and a site called eBay had recently emerged on the web. Somehow we decided that alternating moves

was boring—too predictable. Some moves are clearly worth more than others, and the game would be much more interesting if you had to pay for the right to move. Enter Bidding Chess. How to play. We both start with one hundred chips. Before each move, we write down our bids, and the player who bids more gives that many chips to the other player and makes a move on the chessboard. For example, if I bid 19 for the first move and you bid 24, then you give me 24 chips and make a move on the chessboard. Now I have 124 chips and you have 76 and we bid for the second move. By the way, you bid way too much and now you’re toast! How to win. You win by capturing your opponent’s king. Or, rather, I win by capturing your king. None of this woo-woo checkmate stuff. I don’t care if you have me in checkmate when I have enough chips to make seven moves in a row. What if the bids are tied? Quibblers. The bids are never tied. There is an extra chip, called *. If you have * in your pile, then you can include it with your bid and win any tie. And if you win, then * goes to your opponent along with the rest of your bid. But if you wuss out and save * for later, then you lose the tie. Bidding Chess is meant to be played, so set up the board and grab a friend, maybe one you never really liked much anyway, and try it out. Think carefully—the game is never won on the first move, but it is often lost there. After you have played a few times, take a look at the following transcript from a game played in 1015 Evans Hall, the common room of the UC Berkeley math department, in Fall 2006. Names have been changed for reasons you may imagine. We write N* to denote a pile or bid with N chips plus the * chip. A sample game. Alice and Bob both start with 100 chips. Alice offers Bob the * chip, but he refuses. Alice shrugs, takes the black pieces, and bids 12 for

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the first move. Bob bids 13 and moves his knight to c3.

8 7 6

would like to counter, Bob bids fifteen for the next move, which seems fair. Alice bids 22 and takes the pawn at f2. Bob realizes with some dismay that he must win the next move to prevent Alice from taking his king. She bids all 65 of her chips, and he bids 65*. King takes bishop.

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Now Alice has 113* chips, and Bob has 87 as they ponder the value of the next move. Alice figures that the second move cannot possibly be worth more than the first, because it would be silly to bid more than 13 and end up in a symmetric position with fewer chips than Bob. So she bids 11 plus *, which feels about right. Bob reasons similarly, and also bids 11. Alice wins the tie with * and moves her king’s pawn forward one to e6. Bob, who played competitive chess in high school, is puzzled by this conservative opening move. Feeling comfortable with his board position, he decides to bid only nine of his 98* chips for the third move and is mildly surprised when Alice bids 15. She moves her bishop to c5.

8 7 6 5 4 3 2 1 a

b

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Now Alice has 87 chips, while Bob has 113*. Since Alice bid 15 for the last move and started an attack that he 1

and let R(G) be the critical threshold between zero and one such that Alice wins if her proportion of the bidding chips is more than R(G) and Bob wins if she has less than R(G), with realvalued bidding.

R ICHMAN ’ S T HEOREM Let G be a finite combinatorial game. Then RðGÞ ¼ 1
PðGÞ:

Furthermore, a move is optimal for random-turn play if and only if it is an optimal move with real-valued bidding. The proof of Richman’s Theorem is disarmingly simple: One shows that R(G) and 1 - P(G) satisfy the same recursion with the same initial conditions, as follows. For any position v in the game G, let Gv be the game played starting from v.

4 3 2 1 a

b

c

d

e

f

g

h

Now Bob has a material advantage, but Alice has 130* chips. Pondering the board, Bob realizes that if Alice wins the bid for less than 30, then she can move her queen out to f6 to threaten his king and then bid everything to win the next move and take his king. So Bob bids 30, winning over Alice’s bid of 25. He moves his knight to f3. But Alice can still threaten Bob’s king by moving her queen to h5. Since she has 160* chips, she can now win in the next two moves, regardless of what Bob bids, and capture his king. Alice suppresses a smile as Bob realizes he has been defeated. Head in his hands, he mumbles, ‘‘That was a total mindf**k.’’ Richman’s Theory. David Richman invented and studied a class of similar bidding games in the late 1980s in which bids are allowed to be arbitrary nonnegative real numbers, not just integers. One of Richman’s discoveries is a surprising connection between such bidding games and random-turn games in which, instead of alternating moves, players flip a fair coin to determine who moves next.1 For simplicity, suppose Alice and Bob are playing a finite, loop-free combinatorial game G. Let P(G) be the probability that Alice wins, assuming optimal play,

P ROOF . Suppose v is an ending position, so it is a winning position for either Alice or Bob. If v is a winning position for Alice, then R(G) and 1P(G) are both equal to zero; if v is a winning position for Bob, then both are equal to one. Suppose v is not an ending position. By induction on the length of the game, we may assume that R(Gw) is equal to 1 - P(Gw) for every position w that can be moved to from v. Let R+(v) be the maximum value of R(Gw), over all positions w that Bob can move to from v, and let R-(v) be the minimum over all positions that Alice can move to. Then it is straightforward to check that RðGv Þ ¼

Rþ ðvÞ þ R
ðvÞ ; 2

and an optimal bid for both players is |R+(v)-R-(v)|/2, since Alice will always move to a position that minimizes R, and Bob will move to a position that maximizes it. Similarly, Alice will always move from v to a position that minimizes 1 - P, and Bob will move to a position that maximizes it. These probabilities are equal to R+(v) and R-(v), by induction, and 1 - P(Gv) is the average of the two, since we flip a fair coin to determine who moves next. Real versus discrete bidding. Realvalued bids are convenient for

The details of this result, and many other related facts, were presented by Richman’s friends and collaborators in [LLPU96, LLP+99].

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theoretical purposes and are essential to Richman’s elegant theory. They are, however, disastrous for recreational play, as becomes obvious when pone ffiffi player bids something like e
p þ log 17: Whole-number bids are playable and fun, but the general theory and practical computation of optimal strategies become much more subtle. For instance, the set of optimal first moves for Tic-Tac-Toe with discrete bidding depends on the number of chips in play [DP08]. Nevertheless, when the number of chips is large, discrete bidding approximates continuous bidding well enough for many purposes, and Richman’s theory gives deep insight into discrete-bidding game play. Bidding Hex. Richman’s Theorem is especially exciting in light of recent developments in the theory of randomturn games. The probabilists Peres, Schramm, Sheffield and Wilson found an elegant solution to Random-Turn Hex, along with a Monte Carlo algorithm that quickly produces optimal or near-optimal moves [PSSW07]. This algorithm has been implemented by David Wilson [Wil], and the computer

beats a skilled human opponent more than half the time, though anyone can beat it sometimes, by winning enough coin flips. Elina Robeva has implemented a similar algorithm for Bidding Hex that is overwhelmingly effective— undefeated against human opponents. Online. See the Secret Blogging Seminar post by Noah Snyder [Sny08] for an excellent online introduction to bidding games and links to further resources. JB and Deyan Simeonov have developed Bidding Tic-Tac-Toe and Bidding Hex for online play. Visit http://bttt.bidding-games.com/online and http://hex.bidding-games.com/ online to play against the computer at a range of difficulty settings. You can also play against the computer or challange your friends to a game of skill and honor through Facebook. Goto http://apps.facebook.com/biddingttt and http://apps.facebook.com/biddinghex and try it out!

REFERENCES

[DP08] M. Develin and S. Payne, Discrete bidding games, preprint, 2008.

[LLP+99] A. Lazarus, D. Loeb, J. Propp, W. Stromquist and D. Ullman, Combinatorial games under auction play, Games Econom. Behav. 27 (1999), no. 2, 229–264. [LLPU96] A. Lazarus, D. Loeb, J. Propp and D. Ullman, Richman games, Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol. 29, Cambridge Univ. Press, Cambridge, 1996, pp. 439–449. [PSSW07] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Random-turn hex and other selection games, Amer. Math. Monthly 114 (2007), no. 5, 373–387. [Sny08]

N. Snyder, Bidding hex, http:// sbseminar.wordpress.com/2008/11/

[Wil]

28/bidding-hex/, November 2008. D. Wilson, Hexamania, http:// research.microsoft.com/*dbwilson/ hex/.

Stanford University Stanford, CA 94305 USA e-mail: [email protected], [email protected]

Erratum: In The Mathematical Intelligencer, vol. 31, no. 2, the Viewpoint column: Kronecker’s Algorithmic Mathematics by Harold M. Edwards, p.13, lines 10 and 11 of the first column should read: See Page 39 of the critical edition (1962) of Galois’s works. Galois’s Mathematics,...

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The Mathematical Tourist

New York Mathematical Tourists JOE HAMMER

Dirk Huylebrouck, Editor

ourists sometimes claim that whatever they seek in any city will be found in New York. The city is known for its museums, arts centers, finance and corporate headquarters. There are numerous ‘‘must see’’ sites, including the Statue of Liberty, Empire State Building, Times Square and the Broadway theater scene. In this article, the mathematical tourist will virtually visit two science museums and an aviary, all featuring interesting architectural geometry. Their exhibits are complementary, providing a vista of the sciences of today.

T

The Rose Center Does your hometown have any mathematical tourists attractions such as statues, plaques, graves, the cafe´ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

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First we visit the Rose Center, a wing of the American Museum of Natural History on the west side of Central Park between 77th and 81st Streets. It is the largest institution of its kind, boasting more than 36 million artifacts and specimens. Twenty-five interconnected buildings in an area of 1.25 million sq ft (about 116,000 m2) include 46 exhibition halls, several research labs and an invaluable library. In its 140-year history, it has undergone numerous developments, extensions, alterations and modernizations, so it is not surprising that the buildings exhibit hybrid styles. The Rose Center for Earth and Space is a recent addition to the museum, opened in 2002. Its main purpose is to make astronomy, astrophysics and earth sciences accessible to the public. Architects James S. Polshek and Todd H. Schliemann succeeded in designing a landmark complex, adding a 350,000 sq ft space to the museum, with exhibition areas designed by Ralph Appelbaum and Associates. The centerpiece of the complex is the 87 ft diameter sphere, the Hayden Sphere, clad with spherical trapezoidshaped aluminum panels (Fig. 1). The fabrication of these panels required precision to achieve seamless joints along both longitudinal and latitudinal division lines on the sphere (Fig. 2). Because of spherical symmetry, only a

THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

small number of different sized panels were needed, making mass production possible. The skeleton of the sphere is a longitudinal and latitudinal steel frame grid on which are laid the panels. The skeleton is held together by a massive ring truss built along its ‘‘equator.’’ The giant sphere is encased in a 120 ft glass cube. Its huge walls are made from totally transparent clear waterwhite glass that creates the illusion that the sphere is floating weightlessly in space. That look is further enhanced by placing the supporting stainless steel truss structure of the walls at a distance from them so as not to obstruct the transparency of the glass. Moreover there are no visible supporting columns or suspending cables to the sphere at any level of the complex. Suspended from the ceiling of the cube are 13 smaller colored spheres surrounding the Sphere. One depicts Saturn, easily recognized by its system of rings. Another is apparently Jupiter with its colored bands. The impression is that the Sphere (the ‘‘sun’’) with its surrounding ‘‘planets’’ represents our solar system or perhaps some other part of the universe, or it might be a monumental geometric sculpture with celestial themes. The Sphere is lit at strategic points with fluctuating blue fluorescent lights creating the illusion that it rotates in the company of the rest of the stars in the sky. It is a nighttime excursion site not to be missed. All this is best seen from the adjacent Terrace Garden, itself a cosmic-related exhibition area. However, the silver sheen Sphere with its surrounding planets and the enveloping cube are not just architectural bravado. They have multitudinous functions to facilitate the exhibitions of the Center. Along the above mentioned ring truss, the Hayden Sphere is divided into two hemispheres. The upper and bigger portion houses a Space Theater incorporating the Hayden Planetarium (named after the founder of the

Figure. 1. The Hayden Sphere encased in the glass cube showing a V column and some of the smaller spheres (photo: Tania Hammer).

previous planetarium). It has a 68 ft spherical dome screen. The planetarium is equipped with a one-of-a-kind custom-made Zeiss Star Projector, capable of projecting over 9,000 stars in a night sky. The lower hemisphere houses the 46 ft diameter Big Bang Theater. At the bottom of this theater is an 8 ft deep, 36 ft diameter concave spherical screen on which the birth of the universe is simulated. Emanating from the Big Bang Theater is a 364 ft long, 8 ft wide spiral ramp called the Cosmic Pathway, on which a 13 billion year linearly scaled history of the universe is chronicled, from the time of the Big Bang to the present. The 200 computer generated illustrations are accompanied by explanatory text and diagram panels. The fascinating finale of this journey is the length of human history, which is just about the breadth of a human hair, as one inch corresponds to about 3 million years. Fabricating the geometry of the ramp was a challenging engineering problem. Its surface bends in two directions: Winding one and a half times around the lower half of the Hayden Sphere, and simultaneously descending gradually with about a 16 ft fall. An antecedent of the Spiral Ramp, just across Central Park, on the corner of Fifth Avenue and 89th Street, is Frank Lloyd Wright’s Guggenheim Museum. The geometry of its main exhibition gallery is a spiral ramp too, but differing structurally. It is a 1,320 ft long ramp spiraling around six times, creating a

Figure. 2. Close-up view of a part of the sphere showing Saturn and Jupiter, a supporting column entering the sphere, and some aluminum panels (photo:Tania Hammer).

rotunda atrium with a circular skylight dome on the top. The ramp is cantilevered from massive concrete walls that are also the walls of the building itself. The second antecedent originates from genetics. In 1953, when the double helical structure of the DNA was discovered, the ‘‘helix’’ became a buzz word and subsequently an inspiration for art and architecture. In the Hall of Universe, the bottom part of the Hayden Sphere can be seen 23 ft above the ground—its illusion of floating appearing here most vividly. Its secret lies in the clever placement of the attachment points of the supporting columns to the Sphere (Fig. 2). These columns consist of three V-shaped steel tubes planted on the ground of the Hall. Remarkably, from no point can the three columns be seen together. Their six branches enter the aluminum cladding of the Sphere through six holes: There is no direct contact with the aluminum surface. The supporting tubes are attached to the ring-truss inside the Sphere. Additionally, branching from the V-tubes, several supplementary tubes support the torsion tube of the spiral pathway. These columns themselves avert the attention from the main role of the V-tubes; it appears as if their only role is to support the spiral pathway. On the corner of Columbus Avenue and 79th Street is the newest addition to the museum, the Weston Pavilion, opened in 2002. It is an attractive 43 ft high, 1,800 sq ft glass box; a

mini-version of the Rose Cube. It holds a collection of items related to astronomy. The centerpiece is an 18 ft diameter galactic armillary sphere—a stainless steel and aluminum sculpture suspended from the ceiling of the glass cube (Fig. 3). This is a modern version of its ancestors—the Chinese and the Greek armillaries of antiquity where the Earth was placed at the center. At the entry plaza of the Pavilion is a stainless steel 5 9 5 9 5 ft sculpture— the New York Times’s Capsule. It is a time capsule in which late twentiethcentury artifacts have been stored, collected from all over the world. The planned opening date is at the end of this millennium. The sculpture was designed by the Spanish architect Santiago Calactrava. Its symmetric concave surface is meant to be a topological image of a sphere and is an emblem of the museum. Bystanders might liken it to a budding flower (Fig. 4). Usually, a time capsule is buried for safety, but this capsule is visible, touchable and even climbable.

The New York Hall of Science The second museum in this visit is the New York Hall of Science, on the grounds of Flushing Meadows Corona Park. The architectural icon of this museum is the Great Hall (Fig. 5), originally built as a science pavilion for the 1964 World’s Fair. Architect Wallace K. Harrison is known for designing the United Nations Headquarters in New

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Figure 3. The armiliary inside the Weston Cube (photo: Tania Hammer).

Figure 4. The Times’s capsule (photo: Gael Hammer).

Figure 5. Aerial view of New York Hall of Science. Note in the left at mid distance the geodesic dome (photo: courtesy and ÓSteven Turner).

York, as well as the Metropolitan Opera House at Lincoln Center. The Hall is an 80 ft high 7,000 sq ft building with a free-form asymmetric floor plan. This column-free structure consists of a mere one-foot thick single concrete wall. The concrete is formed into 20 vertically sinuous undulations with the two ends overlapping, the space between providing an unusual entrance doorway. The geometry of the undulations is composed by arcs of circles of 12.5 ft

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radius. The entire wall is formed into a grid of an amazing 5,400 mini-windows all fitted with cobalt blue glass. Inside the Hall, the daylight filters through this billowing, multifaceted, blue glassfilled windows creating a feeling of being enveloped by a starry night sky (Fig. 6). Exhibition galleries were built underneath the Hall in a hexagonal shaped area. Years later, a cylindrical entry atrium and an oval shaped

auditorium were added. The decisive breakthrough in the development of the museum was in 2004 when it was extended to provide 55,000 sq ft more exhibition space and teaching facilities. Architecturally, the key building—the Hall of Light—does not seek to compete with the revered Great Hall, instead affording geometric counterpoints (Fig. 5). The new building is a long, slightly folded, angular, horizontal, low-lying form, contrasting with the towering, vertical curved form of the old building. The galleries house over 400 constantly updated computer aided do-it-yourself exhibits in biology, chemistry, physics and aerodynamics, and an entire section devoted to mathematics. Features include topics in probability theory, projective geometry, topology, celestial mechanics and the history of mathematics. The exhibition, called ‘‘Mathematica: A World of Numbers and Beyond,’’ was designed by the educator and artist couple Charles and Ray Eames in 1961. Some of their work can be seen online. One of the numerous interactive exhibits is in the probability section: The formation of a bell-shaped (or normal) curve is shown on a device where hundreds of small, identically sized plastic balls cascade through a maze of steel pegs. At any stage you can stop the cascading to see the shape of the developing bell. Another highlight is in the graph theory station, called Ropes and Pulleys, designed by Kayle

Figure 6. Inside the Great Hall showing the blue glass-filled wall (photo: Gael Hammer).

Subsequently, Buckminster Fuller designed several domes, of different types and for different purposes, across North America. Modified versions have now been built all over the world, a recent one being the New Mariinsky Theater in St Petersburg, Russia, by French architect Dominique Perrault. A surprising analogue of Fuller’s domes was discovered in 1985 by Harold Croto in Sussex, England, who found a new form of the element carbon: Since it was a cluster of 60 carbon atoms making a spherical cage resembling a geodesic dome, he consequently named it ‘‘Fullerene 60.’’ Subsequently, more fullerenes and other new molecules from clusters of atoms with highly symmetrical geometries were produced in ‘‘molecular architecture.’’ Our last stop is Flushing Meadows Corona Park, the site of the 1964 World’s Fair. A 120 ft-diameter steel terrestrial globe, the ‘‘Unisphere,’’ was erected in the Park (Fig. 7) as a symbol for world peace. Its dedication reads, ‘‘Peace Through Understanding.’’ ACKNOWLEDGMENTS

Thanks are due to V. Guy Maxwell, AIA, of Polshek Partners, and Ed Murray of New York Hall of Science for providing information.

LITERATURE

Ball, P. 2005, A New Kind of Alchemy. New Scientist 495(2), pp. 30–33. Buckminster Fuller, R. 1977, Synergetics. New York: Macmillan. Castelvecchi, D. 2008, Small but Super. Science News 173, pp. 14–19. Futter, E. 2001, Rose Center for Earth and Space: A Museum for the Twenty-First Century. A Collection of essays. Laiserin, J. 2000, Many Models Make the Rose Center. Architectural Record 118, pp. 171–179.

Figure 7. Unisphere (photo: Gael Hammer).

Lyal, S. 2002, Masters of Structure. London:

Dries. This is a network of over 100 pulleys linked by cables. It responds dynamically to force by rotating wheels situated in front of the network. Movement in one place ripples across the entire network, and you can never see the same pattern twice. It demonstrates how sensitive networks can be. With just small adjustments, the shapes can change much.

The Surroundings

Laurence King Publishing.

At a five-minute’s walk from the museum is a ‘‘geodesic dome’’ (Fig. 5) designed by R. Buckminster Fuller for the 1967 World’s Fair. Today it is an aviary with 50 species of American birds. The dome, 85 ft high and 135 ft in diameter, is an aluminum network of spherical triangles, the inside space being free of supporting columns.

Stephens, S. 2005, Polshek Partnerships’ Hall of Science. Architectural Record 193, pp. 224–229. School of Mathematics University of Sydney Sydney, NSW 2006 Australia e-mail: [email protected]

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Prime Simplicity MICHAEL HARDY

AND

CATHERINE WOODGOLD

O

nce upon a time a learned professor explained to a class of bright students how Euclid proved the existence of infinitely many prime numbers in the third century BC. He said that Euclid began by supposing only finitely many prime numbers exist and ultimately deduced a contradiction. He also remarked on the admirable simplicity of what he said was Euclid’s proof by contradiction. He explained that the proof began by letting p1 ; . . .; pn be all the prime numbers that exist, in increasing order. He directed the students’ attention to the number p1 . . .pn þ 1: He showed that that number cannot be divisible by any of the primes p1 ; . . .; pn : Then he said, ‘‘By our assumption, no other primes than those exist. This number is therefore not divisible by any primes. Since it is not divisible by any primes, it must itself be prime. But that contradicts our initial assumption that no other primes than p1 ; . . .; pn exist.’’ We shall see that this account of what Euclid did in his famous proof of the infinitude of primes is commonplace among some (not all) of the best number-theorists and among a broad cross-section of mathematicians and others, and that it is historically wrong. The learned professor in our story misled his students into several confusions and errors about prime numbers, about good ways to write proofs, and about history. The proof that Euclid actually wrote is simpler and better than that. Reductio ad absurdum is not the best way to prove this proposition, and Euclid did not do it that way. One of the students knew something of history. She later became an eminent number-theorist and wrote a book in which she asserted that this was among the earliest proofs by contradiction in mathematics. That is an error. Another student had been taught that any proof in which the existence of something is inferred by deriving a contradiction from the assumption of its nonexistence is nonconstructive. He concluded that this was among the earliest nonconstructive proofs. That is an error. The proof that Euclid wrote is constructive. After the professor stated that the number p1 . . .pn þ 1 must itself be prime, one alert student filed permanently in her mind the ‘‘fact’’ that if p1 ; . . .; pn are the first n primes, then p1 . . .pn þ 1 is also prime. She delighted in her new

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understanding of how to prove that, and in her new knowledge that Euclid had already proved it circa 300 BC. That is an error: observe that 2
3
5
7
11
13 þ 1 ¼ 59
509: And Euclid never concluded that p1 . . .pn þ 1 is prime, even contingent on an assumption later proved false. Another bright student realized that the same argument that works with p1 . . .pn þ 1 would also work with p1 . . . pn  1: He concluded that if p1 ; . . .; pn are the first n primes, then p1 . . .pn  1 is also prime. Since p1 . . .pn þ 1 and p1 . . .pn  1 had both been proved to be prime, there must exist infinitely many ‘‘twin primes’’—pairs of prime numbers that differ by 2. But observe that 2
3
5
7  1 ¼ 11
19: Another student saw at once that if p1 . . .pn þ 1 is prime whenever p1 ; . . .; pn are the first n primes, then that obviously entails that there are infinitely many primes. She felt her understanding was complete and she tuned out the rest of the professor’s presentation, which she knew her weaker classmates needed whereas she did not. Another student, alertly following the whole argument, remembered for the rest of his life that Euclid’s proof of the infinitude of primes considers the number p1 . . .pn þ 1; where p1 ; . . .; pn are the n smallest primes. That is an error. The proof that Euclid wrote never explicitly considered that number. Another student realized that the conclusion that p1 . . .pn þ 1 is prime relied on an assumption later proved false, and therefore could not be considered validly proved. One thing she always remembered after that is that the prime factors of p1 . . .pn þ 1 are bigger than the primes p1 ; . . .; pn initially considered in Euclid’s proof. That is an error. We will see that in some cases the prime factors of p1 . . .pn þ 1 are smaller than some of p1 ; . . .; pn , and in some cases smaller than all of them. Any proof that is not by contradiction can be rewritten as a proof by contradiction in a way that superficially seems trivial, but that can have quite unexpected consequences: just prepend to the proof the assumption that the theorem is false, then follow the proof by the words ‘‘We have arrived at a contradiction; thus our initial assumption must be false.’’ That is done to Euclid’s proof of the infinitude of primes when it is called a proof by contradiction. Even if that way of proving this proposition were not the cause (as

we will further argue below) of the students’ misunderstandings enumerated previously, this frequently presented proof by contradiction is needlessly complicated. Simplicity is good. To add to a proof an extra complication that serves no purpose is to make the problem appear more complicated than it really is. We have nothing against proof by contradiction in general. Neither did Euclid, for within his proof of the infinitude of primes he has a lemma that he proves by contradiction. He also proved other pffiffiffithings by contradiction, for example the irrationality of 2.

How Not to Write Proofs by Contradiction

This very clear and very close paraphrase of Euclid’s proof in modern terminology and notation is on page 65 of Øystein Ore’s book Number Theory and Its History [104]: Euclid’s proof runs as follows. Let a; b; c; . . .;k be any family of prime numbers. Take their product P ¼ ab. . .k and add 1. Then P + 1 is either a prime or not a prime. If it is, we have added another prime to those given. If it is not, it must be divisible by some prime p. But p cannot be identical with any of the given prime numbers a; b; . . .;k because then it would divide P and also P + 1; hence it would divide their difference, which is 1, and this is impossible. Therefore a new prime can always be found to any given (finite) set of primes. Within this argument, we do find one small lemma proved by contradiction. That is where it says, ‘‘But p cannot be identical with any of the given prime numbers a; b; . . .;k because then it would divide P and also P + 1; hence it would divide their difference, which is 1, and this is impossible.’’ But the proof as a whole is not by contradiction: Ore says ‘‘Let a; b; c; . . .;k be any family of prime numbers’’; he does not say, as so many books do, ‘‘Suppose only finitely many primes exist, and call them p1 ; p2 ; . . .; pn ’’, and he does not end the proof by showing that that assumption leads us into contradiction.

Only the premise that a set contains all prime numbers could make one conclude that if a number is not divisible by any primes in that set, then it is not divisible by any primes. Only the statement that p1 . . .pn þ 1 is not divisible by any primes makes anyone conclude that that number ‘‘is therefore itself prime’’, to quote no less a number-theorist than G. H. Hardy in [55], where he actually attributed that conclusion to Euclid! (Euclid’s statement ‘‘Certainly [that number] is prime, or not’’, to be examined at greater length below, clearly shows that Euclid’s reasoning did not follow that path.) The mistake of thinking that p1 . . .pn þ 1 has been proved to be prime is made all the more tempting by the very obvious fact that that would entail the result to be proved. The proposal to prove the twin prime conjecture by saying that p1 . . .pn þ 1 and p1 . . .pn  1 are both prime came from a student with whom one of us (M. Hardy) spoke, who a few months later entered a graduate program in mathematics. The seemingly trivial rearrangement of the proof into a reductio ad absurdum has thus led us quickly into territory that the straightforward proof could not have hinted at, wherein lie substantial mathematical errors that would not have been approached otherwise. In any proof by contradiction, once the contradiction is reached, one can wonder which of the statements asserted to have been proved along the way can really be proved in just the manner given (since the argument supporting them does not rely on the initial assumption later proved false), which ones are correct but must be proved in some other way (since the argument supporting them does rely on the initial assumption), and which ones are false. It is easy to neglect that task. One’s consequent ignorance of the answers to those questions can lead to confusion: after all, when one remembers reading a proof of a proposition,

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dropped out of the graduate program in mathematics, then went back and got a Ph.D. in statistics. He has taught mathematics and/or statistics here and there over the years, at (for instance) the University of North Carolina, MIT, the Woods Hole Oceanographic Institution, and the University of Minnesota. He would like to be more multilingual, but the only language other than English in which he feels really comfortable is Esperanto.

CATHERINE WOODGOLD, daughter of

AUTHORS

Euclid’s Proof

MICHAEL HARDY

School of Mathematics University of Minnesota Minneapolis, MN 55455 USA e-mail: [email protected]

the mathematician G.F.D. Duff, was born and raised in Toronto, and graduated in Engineering Science from the University of Toronto in 1982. Since then she has lived and worked as a seismologist in Ottawa where she and her husband have enjoyed raising their two children. 424 Cambridge Street South Ottawa, ON K1S 4H5 Canada e-mail: [email protected]


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might one not think the proposition has been proved and is therefore known to be true? G. H. Hardy probably was aware that because the conclusion that p1 . . .pn þ 1 ‘‘is therefore itself prime’’ was contingent on a hypothesis later proved false, it could not be taken to be proved. But he did not say that explicitly. It seems hard to justify a similar confidence that all of his readers avoided the error into which he inadvertently invited them. Euclid’s proof as presented by Øystein Ore earlier spares us that task by limiting the use of proof by contradiction to the narrowest possible scope.

The Historical Evidence By some accounts (see [4] and [54]), James Williamson’s translation [144] of Euclid’s Elements into English is the best. Williamson’s translation of Euclid’s Proposition 20 in Book IX appears in the text that follows. Italics are Williamson’s and indicate additional words not in the original Greek. Euclid often identified numbers with line segments. That is why when he adds DF to DE, he gets EF. Where we would say 6 is divisible by 2 or 2 divides 6, Euclid’s translators, including Williamson, say 6 is measured by 2 or 2 measures 6. PROP. XX. There are more prime numbers than any proposed multitude of prime numbers. Let A, B, C be the proposed prime numbers; I say that there are more prime numbers than A, B, C. For let the least number (by 38.7) measured by A, B, C be taken; and let it be DE; and let unity DF be added to DE; certainly EF is prime, or not: first let it be prime; therefore prime numbers A, B, C, EF are found more in number than A, B, C. But now let EF not be prime; therefore it is measured by some prime number (by 33.7); let it be measured by the prime number G; I say that G is the same with neither of the numbers A, B, C: for if G be the same with one of the numbers A, B, C, the numbers A, B, C measure DE; therefore also G will measure DE; and it also measures EF; and therefore G, being a number, will measure the remainder, the unity DF; which is absurd: therefore G is not the same with one of the numbers A, B, C; and it is supposed prime; therefore prime numbers A, B, C, G have been found more than the proposed multitude A, B, C. Which was to be demonstrated. (For the original Greek and an English translation side by side see [43] or the translation on the web at hhttp:// farside.ph.utexas.edu/euclid.htmli.)

Proof by Contradiction? • In order that Euclid’s proof be a proof by contradiction, he would have had to intend the words ‘‘Let A, B, C be the proposed prime numbers’’ to mean a finite set containing all prime numbers. But they immediately follow the words ‘‘any proposed multitude of prime numbers’’,

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so Euclid appears to intend ‘‘A, B, C’’ to be any proposed multitude, i.e., any finite set, of prime numbers. Nowhere does he state that it contains all primes, nor does he need such an assumption. Those words give us no reason to impute it to him. • The words ‘‘I say that there are more prime numbers than A, B, C’’ fit into the context perfectly if ‘‘A, B, C’’ means any arbitrary finite set of prime numbers, but one would not write such a thing if ‘‘A, B, C’’ means ‘‘the set of all prime numbers, which we have just assumed to be finite’’. • If Euclid had meant a finite set containing all prime numbers, rather than any arbitrary finite set of prime numbers, then he would have written ‘‘certainly EF is not prime’’ rather than ‘‘certainly EF is prime, or not’’. • The proof concludes by saying ‘‘Which was to be demonstrated’’, not by saying a contradiction or absurdity has been reached. One part of Euclid’s argument is by contradiction. That is the part that begins by saying ‘‘for if G be the same with one of the numbers A, B, C’’ and concludes with ‘‘which is absurd’’. In this too, Ore follows Euclid closely in his modern paraphrase. To say that Euclid’s proof is by contradiction is to attribute more words to Euclid than we find in the text. If we do that, we must say that he added a gratuitous complication and two infelicitous phrasings to his proof, despite the lack of evidence and despite the fact that his proof is very well and very efficiently phrased if we adopt the alternative interpretation.

Prevailing and Persisting Perceptions of this Proof The erroneous view that Euclid’s proof of the infinitude of primes is by contradiction is not only widespread among the best mathematicians (and consequently among a broad public). It is also regarded by some of the best mathematicians as of some importance. Indeed, some authors regard Euclid’s proof as interesting largely because it is (allegedly) a proof by contradiction. Kac and Ulam [77], said Euclid’s proof is ‘‘one of the earliest known proofs by Contradiction’’ (perhaps that is true of the small lemma in the middle, but it is not true of the proof as a whole). Averbach and Chein [6] wrote, ‘‘His proof is interesting to us because’’ it is a proof by contradiction. Fletcher [44] wrote that it is ‘‘another classical proof by contradiction’’. Goodfriend [50] called it ‘‘a classic example of proof by contradiction.’’ Gullberg [53], writing on the history of mathematics, has a section on reductio ad absurdum in which he uses Euclid’s proof as his most prominent example, following it immepffiffiffi diately with a proof of the irrationality of 2, a negative result that can only be proved by contradiction. Henle and Tymoczko [60] also have a section titled Proof by Contradiction in which Euclid’s proof is the main example, and they write ‘‘Notice that the (false) assumption that there is a greatest prime was a powerful tool, since it gave us an extra assumption that we could use.’’ That extra assumption is completely superfluous. Johansen and Johansen [74], writing on ancient philosophy, mention Euclid’s proof in order

to exemplify reductio ad absurdum. Leveque [86], writing on number theory, presents the proof primarily for the same purpose. Liebeck [87] begins the proof by saying, ‘‘This is one of the classic proofs by contradiction.’’ Paulos [105] begins his account of the argument by calling it ‘‘a beautiful example of proof by reductio ad absurdum.’’ Similarly G. H. Hardy rhapsodizes in [56]: ‘‘The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons.’’ Dence and Dence [32] wrote that ‘‘indirect proof [...] is sometimes the easiest way to show that a certain set is infinite’’ and speculated on why Euclid (allegedly) chose to use it in this case. Mukhopadhyay, Ghosh, and Sen [97] call it ‘‘one of the most elegant proofs in mathematics that uses the method of reductio ad absurdum.’’ Some authors suggest that Euclid’s result can be proved only by contradiction or by nonconstructive means. Dunham [39] writes, ‘‘Beyond that, all that is necessary to establish the infinitude of primes is an understanding of proof by contradiction.’’ He explains that idea and contrasts it with a direct existence proof of a trivial proposition, and then says, ‘‘Unfortunately, to prove the infinitude of primes we have no such direct option [...] we must resort to the indirect attack of proof by contradiction.’’ Ogilvy and Anderson [103] do not mention Euclid, and after explaining that there is no closed-form formula or recursion formula for a larger prime than those in a specified finite set, they state, ‘‘Such a constructive proof is not available to us at the present state of knowledge, for primes.’’ (emphasis theirs). This seems to suggest that the proof they give is nonconstructive. Would they have thought that if they had not given a proof by contradiction? Numerous others assert that Euclid’s original proof was by contradiction: Andrews [3], Apostol [5], Bach and Shallit [7], Barnett [8], Beiler [9], Bennett [10], Brassard and Bratley [14], Bromwich [16], Bruce and Giblin [18], Childs [21], Chinn and Davis [22], Crandall and Pomerance [23], Danesi [27], Devlin [34] [35], Dickson [36], Fisher [42], Gonza´les [49], G. H. Hardy [55], Herstein and Kaplansky [62], Hintikka [63], Hobbs and Perryman [64], Hoffman [65], Hull [67], Karatsuba [78], Kisaecanin [79], Klein [80], Kline [81], Kramer [84], Koshy [83], Lozansky and Rousseau [89], Maor [92], Meyerson [94], Moser [95], Nagel and Newman [98], Narkiewicz [99] [100], Patterson [106], Phillips [107], Popper [109] [110], Ratzan [112], Regis [114], L. Reid [116], Ribenboim [117] [118] [119], Sethuraman [125], Shahriari [127], Shklarsky, Chentzov, and Yaglom [128], Spencer [129], Tattersall [131], Urban [133], Vakil [134], Vanden Eynden [135], Velleman [136], Wallace [137], Ward and Everest [138], Wickless [140], and Zubair [147]. Humphreys and Prest [68] state clearly that Euclid’s proof is by contradiction three pages after they give what they say is his proof in a form that is not by contradiction. One of those, Devlin [34], says not only that Euclid’s proof is by contradiction, but that ‘‘Euclid claimed that this number N + 1 is a prime number’’, where N ¼ 2  3  5  7  11  . . .  P and P is a supposed ‘‘largest prime number’’. Devlin iterates that assertion: ‘‘Euclid claimed that this number N + 1 is a prime number’’, then, ‘‘So how did Euclid show that the number N + 1 is a prime number?’’,

and finally, ‘‘So N + 1 is indeed a prime number. { That’s Euclid’s proof.’’ In a later work, [35], Devlin corrects that assertion by qualifying the third of those three quoted sentences by saying the conclusion was based on the initial assumption, and then stating in a footnote, ‘‘It is not the case that numbers of the form N + 1 are always prime. The proof given is based on the (false) assumption that there is a largest prime.’’ But confusingly, he left the first two of the previously quoted sentences intact. That confusion would not have happened if the technique of proof by contradiction had not been used. Devlin, in an earlier account [33] of Euclid’s proof, does not assert it to be a proof by contradiction. De Smith [31] also says that Euclid proved that ‘‘the number q ¼ 2  3  5  7  . . .  p þ 1 formed by multiplying all the prime numbers less than or equal to a prime number p together, and then adding 1, is not divisible by any of these smaller primes, so must itself be prime.’’ In one of the mentioned references, Brassard and Brattley [14] first attribute the proof by contradiction to Euclid, then expand upon the benefits of rearranging the proof so that it is not by contradiction, the result of the rearrangement being the proof that Euclid actually gave. G. H. Hardy [56], asserting that the proof is by contradiction, goes on to say in a footnote, ‘‘The proof can be rearranged to avoid a reductio, and logicians of some schools would prefer that it should be.’’ Ribenboim [118], writes, ‘‘From this indirect proof one cannot deduce a method for generating prime numbers’’. The reader could easily get the impression that the proof is nonconstructive because it is indirect. We found the following others who were attributing the theorem to Euclid, and then were stating the theorem followed by the usual proof by contradiction, all phrased in such a way that reasonable readers would be likely to think they’re attributing the proof by contradiction to Euclid: Birkhoff and Mac Lane [12], Bressoud [15], Brown [17], Bruce and Giblin [18], Chahal [20], Daepp and Gorkin [26], Garrett [46], Gerstein [47], Grosswald [52], Jan [72], Lauritzen [85], Luca and Somer [90], Niven and Zuckerman [102], Pieprzyk, Seberry, and Hardjono [108], Rose [121], and Yan [146]. Goldrei [48] writes, ‘‘A classical argument is that found in Euclid [...] A modern version of this proof goes as follows’’ and gives the usual proof by contradiction. Biggs [11] writes about ‘‘that proof (attributed to Euclid)’’, and then says, ‘‘We repeat it here in slightly different form’’ and gives the usual proof by contradiction. We found eight sources asserting erroneously that Euclid used factorials in this proof. Boolos, Burgess, and Jeffrey [13] write, ‘‘the proof in Euclid shows that there is a prime y [ x with y B x! + 1’’. That short quote summarizes a proof obviously inspired by Euclid’s, and one that makes it clear that phrasing it as a proof by contradiction is unnecessary. But the attribution to Euclid of the explicit use of factorials cannot serve the history teacher’s purposes. Danzig [28] writes, ‘‘In this proof Euclid introduces for the first time in history what we call today factorial numbers’’. Grinstead and Snell [51] write, ‘‘One of the earliest uses of factorials occurred in Euclid’s proof that there are infinitely many prime numbers’’, and ‘‘Euclid argued that there must
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be a prime number between n and n! + 1’’. Epstein and Carnielli [41], Kac and Ulam [77], Muir [96], Tenenbaum and France [132], and Wiedijk [141] also say Euclid used factorials. Birkhoff and Mac Lane [12], Derbyshire [30], and Lovasz, Pelikan, and Vesztergombi [88] state that the theorem was known to Euclid and then give the proof involving n! + 1 without explicitly attributing that to Euclid, although they could easily be understood as meaning that that is what Euclid did. Three of the sources stating that Euclid used factorials assert that Euclid’s proof was by contradiction: Kac and Ulam [77], Muir [96], and Wiedijk [141]. Hogben [66] writes that Euclid used factorials ‘‘to prove some quite useless facts about prime numbers’’, without specifying what those facts were. That statement remains in the second edition and is omitted in the third and fourth. Derbyshire [30] attributes the theorem but not the proof to Euclid, and uses factorials. He does not explicitly say that he wants to present a proof by contradiction, but he does raise the question of whether there is any ‘‘biggest prime’’. Then he writes ‘‘Proof: Suppose N is a prime. Form this number: ð1  2  3  . . .N Þ þ 1.’’ He goes on to show that this number’s prime factors exceed N. His hypothesis that N is prime can be dispensed with: clearly if for every positive integer N there is a prime bigger than N, then infinitely many primes exist. Why would he include that hypothesis, which he never uses? It is as if the concept of a hypothetical ‘‘biggest prime’’, not mentioned in the proof itself but mentioned in the question the proof was to answer, acts like the hypothesis assumed at the beginning of a proof by contradiction. Its effect is only to complicate the proof needlessly. The earlier account given by Ogilvy and Anderson in [103] with no attribution to Euclid, is nearly identical in various details and notation to Derbyshire’s proof, including the assumption that N is prime, the use of the capital-N notation, and the explanation that 5 9 4 9 3 9 2 9 1 + 1 is divisible by 11. Menini and van Oystaeyen [93] write (note the parenthesized name after their theorem number),

THEOREM 23.4 (Euclid) Let R be a unique factorization domain. If R is not a field and if U(R) is finite, then R has infinitely many prime, pairwise not associated, elements. Then they give a proof by contradiction involving qk ¼ ðp1 . . .pn Þk þ 1 2 R . They stop short of explicitly attributing this proof to Euclid. Perhaps one could debate whether they attribute too much to Euclid. Whether their proof could be rephrased as a constructive proof by taking Euclid’s own words as a hint, we leave as an exercise. Narkiewicz [99] calls Euclid’s proof ‘‘fallacious’’, alleging that it shows only that any set of three primes does not contain all primes. This seems rather like saying that the expression ‘‘1 þ . . . þ n ¼ nðn þ 1Þ=2’’ does not include the case n B 2 because the dots imply an intermediate term. We think he construes the notation too literally. MacCarthy and Stidd [91] state in two separate footnotes, on pp. 71 and 72, that Euclid’s proof is by contradiction, but on p. 72 give what they say is Euclid’s proof in a way that clearly is not a proof by contradiction and is quite faithful to Euclid (although expressed in modern terminology). 48

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Williams [143] states that it is a proof by contradiction and that it is constructive. Apparently she meant that one lemma in the middle of the argument is by contradiction. The following state correctly that Euclid started with a finite set of prime numbers without initially assuming it to contain all prime numbers, but then say, erroneously, that Euclid took that set to be the smallest n primes, for some n: Crossley [25], Davenport [29], Driessen and Suarez [38], Hardy and Wright [57], Hersh and Davis [61], Ireland and Rosen [70], Koch [82], Weisstein [139], Wilson [145]. In particular, Davenport [29], Crossley [24], [25] and Weisstein [139] say that the new prime will always be larger than those in the initially considered finite set. Any finite set of odd primes is a counterexample, as are many other sets, such as {2, 3, 5, 31}, for which the new primes that Euclid’s technique yields are 7 and 19. In some cases, such as the set {5, 7}, the new primes are smaller than all of those in the set we started with. Ingham [69] presents Euclid’s proof very well, neither assuming at the outset that the set of all primes is finite, nor saying that the finite set initially considered had to be the smallest n primes for some n, but rather saying ‘‘any finite set of primes’’, and at the end says, ‘‘Hence there exists at least one prime distinct from those occurring in P ’’. He could have stopped there, but then goes on: ‘‘If there were only a finite number of primes altogether, we could take P to be the product of all primes, and a contradiction would result.’’ Similarly, Szabo´ [130] wrote of Euclid’s proof, ‘‘Proposition IX.20 is proved in a similar manner; given a finite set of primes, the existence of a prime number which lies outside it is established[...]’’. So far this is perfect. It does not say Euclid assumed only finitely many primes exist; it does not say the finite set contained only the smallest n primes for some n. Szabo´ goes on to say, ‘‘by showing that, if no such number exists, a contradiction follows.’’ Both Ingham and Szabo´ are right to say a contradiction would result, but the final words could leave unshaken the widespread impression that Euclid wrote a proof by contradiction. The following sources present Euclid’s proof faithfully (in some cases modulo translation into modern terminology and concepts). They do not state that it was, nor make it appear to be, a proof by contradiction, and they do not say that the finite set initially considered must contain just the smallest n primes for some n: Aigner and Ziegler [2], Aaboe [1], Bunt, Jones, and Bedient [19], du Sautoy [123], Dietzfelbinger [37], Dunham [40], Franze´n [45], Gerstein [47], Hasse and Zimmer [59], Jameson [71], Jensen [73], Johnston and Richman [75], Jones, Morris, and Pearson [76], Nathanson [101], Ore [104], Rademacher [111], Redmond [113], C. Reid [115], Roberts [120], Russo [122], Schumer [124], and Shanks [126]. Shanks [126] deserves an honorable mention for explicitly saying that the set of n primes that Euclid considered were ‘‘not necessarily consecutive’’. R. L. Wilder [142] preceded us in deploring the rearrangement of Euclid’s proof: It is somewhat amusing, I think, that some of us become so used to the ‘‘contra-positive’’ type of argument that we automatically take an argument that is ideally suited

to the constructive form of proof, and twist it into a reductio ad absurdum. For example, consider the classic theorem that there exists an infinity of prime numbers. As you know, Euclid showed how, given a certain finite set of prime numbers, one can construct a prime number not in the set, and it is no task to go a step further and make this number unique. Hence, it seems hardly necessary to argue that if it be assumed that the set of all prime numbers is finite, say k in number, then by Euclid’s method there can be shown to exist a (k + 1)th prime and hence a contradiction, inasmuch as Euclid’s method itself provides a valid procedure for constructing an infinite series of primes by recursion... Similarly, Franze´n, in a book on logic, [45], wrote: ‘‘Euclid’s proof that there are infinitely many primes is not an indirect proof[...] The argument is sometimes formulated as an indirect proof by replacing it with the assumption ‘Suppose q1,…,qn are all the primes’. However, since this assumption isn’t even used in the proof, the reformulation is pointless.’’

[9] Beiler, A. H., Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Courier Dover Publications, 1966, p. 212. [10] Bennett, D. J., Logic Made Easy: How to Know When Language Deceives You, W. W. Norton and Company, 2004, p. 33–34. [11] Biggs, N. I., Discrete Mathematics, Oxford University Press, 2002, p. 52. [12] Birkhoff, G. and Mac Lane, S., Algebra, Chelsea, 1999, p. 119. (The error does not appear in the four earlier editions published in the title A Survey of Modern Algebra, published in 1941, 1953, 1965, and 1977.) [13] Boolos, G., Burgess, J. P., and Jeffrey, R. C., Computability and Logic, Cambridge University Press, 2002, p. 79. [14] Brassard, G. and Bratley, P., Fundamentals of Algorithmics, Prentice Hall, 1996, p. 14. [15] Bressoud, David M., Factorization and Primality Testing, Springer, 1989, p. 3. [16] Bromwich, T. J. I. A., An Introduction to the Theory of Infinite Series, American Mathematical Society, 1991 (reprint of a book published by Macmillan in 1908 and 1926; the cited page is

Further Work for Historians When and how did the error become the prevailing doctrine? We have no answer.

identical to its couterpart in the first edition), p. 395. [17] Brown, J. R., Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures, Routledge, 1999, p. 1. [18] Bruce, J. W. and Giblin, P. J., Microcomputers and Mathemat-

ACKNOWLEDGMENTS

This paper owes a great debt to Jitse Niesen. He declined to be included as a coauthor after finding that we had already written a rough draft, but it was he who first called to our attention, in a discussion page on the Citizendium web site, the fact that Euclid’s proof was not by contradiction, by citing David Joyce’s translation on the web at hhttp:// aleph0.clarku.edu/%7Edjoyce/java/elements/bookIX/prop IX20.htmli. We are happy to thank John Baxter and Ezra Miller for useful discussions.

ics, Cambridge University Press, 1990, p. 146. [19] Bunt, L. N. H., Jones, P. S., and Bedient, J. D., The Historical Roots of Elementary Mathematics, Courier Dover Publications, 1988, p. 190. [20] Chahal, J. S., Topics in Number Theory, Springer, 1988, p. 4. [21] Childs, L., A Concrete Introduction to Higher Algebra, third edition, Springer, 2008, p. 63. [22] Chinn, W. G. and Davis, P. J., 3.1416 and All That, Birkhauser, 1985, p. 12. [23] Crandall, R. E. and Pomerance, C. B., Prime Numbers: A Computational Perspective, Springer, 2005, p. 6. [24] Crossley, J. N., The Emergence of Number, World Scientific,

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Review

Osmo Pekonen, Editor

The Archimedes Codex: How a Medieval Prayer Book is Revealing the True Genius of Antiquity’s Greatest Scientist by Reviel Netz and William Noel CAMBRIDGE, MA: DA CAPO PRESS, 2007, 313 + IX PP, US $27.50, ISBN 978-0-306-81580-5, ISBN 10:0-306-81580-X ¨ RAN FRIBERG REVIEWED BY JO

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected]

his is a fascinating account of the nearly unbelievable ups and downs of the ‘‘Archimedes Palimpsest,’’ a Byzantine codex containing, under the text of a prayer book, a small library of unique ancient texts, including exceptionally important Archimedes treatises. The account is written in alternating chapters by William Noel, Curator of Manuscripts at the Walters Art Museum, Baltimore, and Reviel Netz, Professor of Classics and Philosophy at Stanford University. The story of the three known codices with works of Archimedes is told in the chapters written by Noel. The story begins with Archimedes, a prominent citizen of Syracuse, a Greek city in Sicily, sending letters with a number of his mathematical works for further distribution to colleagues in Alexandria, such as Conon, Dositheus and Eratosthenes. Archimedes was killed in 212 BC, when Syracuse fell to the invading Romans, but copies of his works, written on papyrus scrolls, continued to circulate. Early in the sixth century A.D., Eutocius wrote

T

commentaries to some of Archimedes’s works: Sphere and Cylinder I-II, Measurement of the Circle and Balancing Planes. At about this time, the texts of some of Archimedes’s treatises that were still around were transferred from papyrus scrolls to a new and less fragile medium with greater capacity, codices made of stacks of folded parchment sheets stitched together. A new change of medium in the ninth century involved the transition from majuscule to minuscule, lowercase script. Manuscripts written in the older script were often destroyed. After centuries of neglect, a revival of interest in Archimedes’s texts in the Byzantine Renaissance in the ninth and tenth centuries may be one of the reasons for the survival of three codices with Archimedes treatises, now called Codices A, B and C. A and B both contain the treatises Balancing Planes and Quadrature of the Parabola, while B also contains On Floating Bodies, and A also contains Sphere and Cylinder, Measurement of the Circle, Spiral Lines, Conoids and Spheroids, and Sand-Reckoner. Today, both A and B are lost, but copies of A still survive, and so does a Latin translation of B, made in 1269 by William of Moerbeke, a Franciscan friar. After being lost for nearly a millennium (as is now known, after having spent several centuries being used as any other prayer book at the monastery of St. Sabas in the Judaean desert), Codex C sensationally resurfaced in 1899, when a few lines of a partially erased text visible under the text of a medieval prayer book were reproduced in a catalog of manuscripts in the library of the Metochion of the Holy Sepulcher in Constantinople. The Dane J. L. Heiberg, a prominent Greek scholar, recognized the lines as being taken from a work of Archimedes. Subsequently he went to Constantinople, where he could verify that the prayer book was a palimpsest, with the text of the prayer book written above

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and across the erased Greek text of several Archimedes treatises, of which two, Method and Stomachion, were previously unknown, while a third, On Floating Bodies, was previously known only in the Latin translation of Codex B. As much as could then be read of the text of the Method was published by Heiberg in 1907. Later, the prayer book with the Archimedes text, called the Archimedes Palimpsest, was lost again. Indeed, in 1938, when manuscripts from the Metochion in Constantinople were clandestinely transferred to the National Library of Greece in the middle of a conflict between Greeks and Turks, the Archimedes Palimpsest never made it to Athens with the others. A century after its first reappearance, the Archimedes Palimpsest resurfaced for the second time in 1998, when it was sold for two million dollars at an auction at Christie’s in New York to an anonymous buyer, Mr. B. The following year, William Noel at the Walters Art Museum was temporarily put in charge of the palimpsest and was appointed director of a salvation program of epic proportions, generously funded by Mr. B. The aim of the program was the restoration and conservation of the palimpsest, which had been very seriously damaged by careless handling before its reappearance in 1998, and the use of modern imaging techniques in a concerted effort to see more of the erased Archimedes text than Heiberg had been able to in 1907 using only a magnifying glass. Nigel Wilson and Reviel Netz were entrusted with the difficult task of reading the reconstituted Greek text and interpreting its mathematical content. W. Noel was going to try to find out more about the history of the palimpsest. There were many positive surprises and unexpected difficulties in this quest. For instance, Heiberg’s supposedly lost photos of many pages of the palimpsest, when it was still in relatively good condition, were found hidden for a century in the Danish Royal Library. One page of the palimpsest, apparently stolen from the Metochion in the 1840s by a well-known biblical scholar, was found in the Cambridge University Library. An account of the effort to find out when and by whom

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four pages of the palimpsest had been painted over with forged portraits of the Evangelists reads like an intricate detective story. It took the conservator Abigail Quandt four years of hard work to unbind the folios of the palimpsest, to clean and repair them, and to make them ready for imaging. Efforts to read the erased, damaged or concealed Archimedes text below the text of the prayer book and the forged portraits made use of successively more advanced techniques such as ‘‘matched spectral filtering,’’ ‘‘pseudocolor images,’’ ‘‘light emitting diodes,’’ ‘‘micro X-ray fluorescence imaging’’ and the use of synchrotron radiation with the Stanford Positron Accelerating Ring as a light source in order to produce ‘‘iron maps’’ of the ink in the hidden text. Even when the text of Archimedes’s treatises could be surely reconstituted, it was no simple matter to read the text. An example highlighting how difficult it could be is given by Netz on p. 113. There a statement which can be translated as, ‘‘Therefore as the KY to the YE, the YE to EG, and therefore as the on KD to the by KHD, the on AG to the by AEG,’’ meaning, in modern symbolic notations, that KY:YE:: YE:EG ? KD2: KY*YD:: AG2: AG*EG. However, a translation keeping the abbreviations actually used by Archimedes would take the following nearly unintelligible form: e v t’ KY S t’ YE, t’ YE S EG, K e v t’ on KD S t’ by KYD, t’ on AG S t’ by AEG. The first great breakthrough came in 2001 when R. Netz and Ken Saito were the first to read the entire text of Method, Prop. 14, in which Archimedes ingeniously proves that a certain fingernail-shaped cylindrical cut is to a circumscribed rectangular prism as a certain parabola is to a circumscribed rectangle. Surprisingly, Archimedes in his proof makes strict and explicit use of ‘‘actual infinities,’’ showing that triangular slices of the cylindrical cut are ‘‘equal in magnitude’’ in a one-to-one fashion to linear slices of the parabola. Elsewhere in Method, Archimedes uses a clever mechanical method based on his law of the balance and implicit or potential infinities. A second great breakthrough came in 2003 when the text on the severely damaged final folio of the palimpsest

was found to reveal that Archimedes in his Stomachion set out to compute in how many ways the 14 pieces of a square cut up in a certain way (as described in an Arabic manuscript from the seventeenth century) could be rearranged and still form a square. (In some ancient manuscripts the puzzle is called ‘‘Archimedes’s Box.’’) Surprisingly, this turns out to be a quite difficult problem in combinatorics. Except for some interesting initial steps, Archimedes’s solution of the problem is lost (although it is now possible to see that he understood that ‘‘there is not a small number of figures made of them’’), but a team effort of modern specialists in combinatorics found that there are 536 basic solutions, each one of which can be combined with 32 rotations to give the result 536 9 32 = 17,152 actual solutions. The reconstitution of the text of the palimpsest also allowed the reconstruction of the precise form of the diagrams in the Archimedes treatises (diagrams which Heiberg never bothered to copy). It also allowed the reading of the ‘‘colophon’’ on folio 1v, which states that the priest Ioannes Myronas finished his writing of the prayer book (on top of Archimedes’s treatises, a previously unknown speech of the Athenian orator Hyperides, and some other historically important texts) on April 14, 6737, Greek orthodox time (1229 AD). It is difficult to find anything negative to say about this captivating book, but it is strange that Netz, after an intriguing discussion of the logic of Greek diagrams (see pp. 99–105), chooses to replace Archimedes’s original diagram in Method, Prop. 14, by a less interesting modern diagram (see Fig. 8.7). Personally, I would also have liked to read more about the mathematical details of some of Archimedes’s proofs briefly discussed by R. Netz, instead of hand-waving comments such as (p. 147, after a brief discussion of Archimedes’s proof that the center of gravity of a triangle is located at the intersection of all three medians, one-third of the way along each median): ‘‘We turn away, in disbelief. But Archimedes is right.’’ It is similarly uninformative when a crucial step in the proof of Method, Prop. 1, is described as follows (pp. 194–195): ‘‘Archimedes, through

tremendous geometrical ingenuity, had succeeded in proving that in such an arrangement: The area of the triangle of the prism is to the triangle of the cylinder as the line of the rectangle is to the line of the parabola.’’ Actually, Archimedes himself begins his argument by writing, ‘‘And the (rectangle) by the MNL is equal to the (square) on NS. For this is obvious.’’ It is not too difficult to see how Archimedes reasoned here. With the notations in Figure 4, the area of the triangle cut off from the prism is to the area of the triangle cut off from the cylinder as NM2:NS2, since the two triangles are similar. The ‘‘line of the rectangle’’ is NM, and the ‘‘line of the parabola’’ is NL. It follows from the properties of a parabola and of a circle, respectively, that MN * MK = MZ2 and NS2 ¼ KS2
NK2 . Therefore,

NR2 ¼ KS2
NK2 ¼ MN2
MZ2 ¼ MN2
MN  MK ¼ MN  ðMN
MKÞ ¼ MN  MK

Archimedes, The Open Court Publ. Co., Chicago, 1909. [3] Netz, R., F. Acerbi, and N. Wilson, Towards a reconstruction of Archimedes’s

as claimed by Archimedes, and, consequently, NM2 :NS2 ¼ NM2 :MN  NK ¼ NM:NK Note, by the way, that Archimedes here, in his usual fashion, does not tell his readers how he came up with the idea.

Stomachion, Sciamvs 2 (2001), 67–99. [4] Netz, R., K. Saito, and N. Tchernetska, A new reading of Method proposition 14: Preliminary evidence from the Archimedes Palimpsest, Part 1, Sciamvs 2 (2001), 9–29. [5] Netz, R., K. Saito, and N. Tchernetska, A new reading of Method proposition 14: Preliminary evidence from the Archimedes Palimpsest, Part 2, Sciamvs 3 (2002), 109–125.

REFERENCES

[1] Dijksterhuis, E. J. Archimedes, Munksgard, Copenhagen, 1956. Revised ed., Princeton 1987. [2] Heiberg, J. L., Geometrical Solutions Derived from Mechanics: A Treatise of

Department of Mathematical Sciences Chalmers University of Technology Gothenburg, Sweden e-mail: [email protected]

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Mathematics and Culture in Europe by Mirella Manaresi (ed.) BERLIN, SPRINGER, 2007, 406 + X PP. + DVD, €49.95 ISBN 978-3-540-71963-2 REVIEWED BY MARCO ABATE

n 2002, the European Union, within the framework of the Socrates Program, funded a project called Diffusion and Improvement of Mathematical Knowledge in Europe (short title Mathematics in Europe), coordinated by Mirella Manaresi of the Department of Mathematics of the University of Bologna, and involving the universities of Bologna, Bochum (local coordinator Hubert Flenner), Paris VII (local coordinator Salomon Ofman), Cyprus (local coordinator Alekos Vidras), and Durham (local coordinator Brian Straughan). The aim of the project was twofold: On one side, a reconnaissance of the mathematical abilities of students just enrolled in the participating universities; and, on the other side, to explore, encourage, and promote contacts and exchanges between mathematics and other sciences, humanities and the arts, trying to increase at the same time the awareness of the importance of mathematics in the general public. To achieve these aims, a number of activities were organized in the participating universities; this book describes four of them, three based in Bologna and one transversal to the five universities. Let me start with the latter, related to the first aim mentioned above. In the first semester of the academic year 2003/2004, the project participants conducted a test of the mathematical knowledge of first-year students majoring in technical or scientific subjects in the five universities (and a few others); the first two papers in the book present the outcomes of the test (and of its repetition one year later). Unfortunately (as the authors themselves admit), the test was organized a bit naively, and as a result the data obtained are more tantalizing than interesting. The context

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information on the students was not collected in a uniform way, making a bit difficult the comparison between different countries. More importantly, the questions were not calibrated beforehand; in particular, a statistical analysis conducted later (and described in the second paper in this book) showed that the questionnaire was not suitable for detecting the knowledge of the weaker students. Predictably, the students in universities requiring an entrance examination (namely, Durham and Cyprus) fared better, and students enrolled in natural and life sciences fared worse. The data obtained seem, however, to suggest an essentially comparable basic mathematical preparation across the five countries, and, if confirmed by a more thorough and organized survey, this would be positive news for European students, meaning that mathematics would not hinder their studies in other countries’ universities (no more than in their own country, at least...). The description of the other three activities (a congress and two series of evening meetings, one devoted to Mathematics and Cinema and the second to Mathematics and Theatre) makes up the rest of the book—and the enclosed DVD. The first third of the volume contains the proceedings of the eponymous conference Mathematics and Culture in Europe held in Bologna on October 22–23, 2004 and organized by all partners in the project. Not all the talks given in the conference are represented in the book, but the DVD contains the audio/video recording of all of them. The audio quality is not exactly excellent, and the video editing is virtually nonexistent, but probably it could not be helped. On the other hand, the content of the DVD is elegantly presented and easy to navigate, and it works both for PCs and Macs, which is a big plus. The purpose of the conference was to try (I quote) ‘‘to convince the public of the importance of mathematical culture and make them aware of what being a mathematician means nowadays (...) showing a wide audience the central position occupied by mathematics, not only in the progress of technology and science, but also in many literary and artistic experiences.’’

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As a result, the articles cover a wide range of topics, from astronomy to tomography. Michele Emmer describes some aspects of contemporary architecture somewhat related to mathematics and topology (even though it seems that the use of the latter word by architects has more to do with its metaphorical resonances than its actual meaning), and Gian Marco Todesco presents mathematical techniques included in computer programs used to realize two-dimensional cartoons. In the eye-catchingly but deceptively titled paper ‘‘Painters, Murderers, Mathematicians,’’ Peter Deuflhard does not talk about the way mathematicians can kill painters, but about old and new ways of representing the interior of the human body. Another curiously named paper is ‘‘Two Examples of Unbounded and Compact Triply Extended Spaces’’ by Franco Ghione, where he uses Dante Alighieri, Bernhard Riemann and Giordano Bruno to promenade us through a solid torus and a threedimensional sphere (‘‘triply extended’’ in the title just means ‘‘three-dimensional’’). Carlton Baugh, Pete Edwards, and Carlos Frenk present a recent model of galaxy formation; James F. Blowey discusses teaching mathematics in the United Kingdom; and Apostolos Doxiadis suggests that the several manuals on movie scripting that have appeared in the last few years might be the beginning of a mathematical formalization of narrative. The first part of the book is completed by the transcript of a round-table on ‘‘Mathematics and Media: Tools, Expectations, Results’’ hosted by Simonetta Di Sieno during the conference (the DVD contains the complete recording of the round-table), and by an essay of Michele Emmer on Lucio Saffaro, an Italian artist particularly interested in the representation of polyhedra and whose work was exhibited in Bologna in 2004 (the DVD contains a whole section devoted to Saffaro and the geometry of polyhedra, including computer graphics shorts and the recording of three talks on this topic given by Michele Emmer, Franco Ghione, and Gian Marco Todesco). Another activity organized by the project’s participants in Bologna has been the screening of a series of movies about mathematicians and mathematics. What,

in my opinion, made this activity particularly interesting and fruitful is that every movie was followed by a public discussion, led by a mathematician, of the main mathematical topics presented (or, more often, suggested) in the movie, thus giving to the public (which, according to people attending the screenings, loved the idea) the possibility of seeing firsthand the mathematical realities behind the fiction. The second third of the book reports on this activity. Each movie shown in the series, (A Beautiful Mind by Ron Howard, The Bank by Robert Connolly, Moebius, by Gustavo R. Mosquera, Cube, by Vincenzo Natali, The Mirror Has Two Faces, by Barbra Streisand, Enigma, by Michael Apted, Fermat’s Last Theorem by Simon Singh and John Lynch, and No More Time by Ansano Giannarelli) is described, and the descriptions are followed by essays (from Marco Li Calzi, Riccardo Cesari, Massimo Ferri, Alberto Perelli, Michele Emmer, Mirella Manaresi, and Angelo Vistoli) on the mathematical aspects of the movie, providing short introductions to game theory, mathematical finance, topology, prime numbers, and the like. The public discussions were recorded and can be found on the DVD (but the movies, alas, cannot). Following the success of this activity, a similar one was organized around Mathematics and Theatre. The structure was the same: A staging of a play (or a

theatrical reading) followed by a public discussion led by a mathematician. Again, the public responded enthusiastically, and the resulting material is collected in the third part of the book (and in the DVD). The plays staged were Napoleon Magical Emperor, by Sergio Bustric; Proof, by David Auburn; Galois, by Luca Vigano`; Arcadia, by Tom Stoppard; and a theatrical reading of Uncle Petros and Goldbach’s Conjecture, by Apostolos Doxiadis. In the book, each description of a play is followed by one, two or sometimes even three essays (written by Sergio Bustric, Claude Viterbo, Angelo Savelli, Umberto Zannier, Andrea Paolucci, Franco La Polla, Angelo Vistoli, Luca Vigano`, Paolo Salmon, Leonardo Angelini, Francesco Giannini, and Laura Guidotti) discussing its mathematical, historical and/or theatrical aspects. The third part of the book is completed by two essays on mathematics and theater in general (by Piergiorgio Odifreddi and Michele Emmer), and by an interesting description of the genesis of the play Padre Saccheri by Maria Rosa Menzio on the mathematician Girolamo Saccheri. Unfortunately, it was not possible to include the play in the program (and so there is nothing about it on the DVD). Finally, the DVD contains material on yet another activity, a series of talks on Mathematics, Art, Science, and Technology addressed to the general

public, ranging from cryptography to Poincare´ ’s conjecture. To sum up: As often happens with this kind of collection, this book contains both very interesting essays and papers one can easily do without. However, having collected in a single volume descriptions, explanations, and references to supplementary readings on several recent and not-so-recent movies and plays about mathematics and mathematicians is definitely helpful, and would justify buying this book to anybody interested in the relationships between mathematics and art. A broader question is: Has the project reached its aim of increasing the awareness of the importance of mathematics in the general public? In Europe, and in Italy in particular, this has been only one of many activities realized with this aim in mind, and so it is difficult to single out the specific effect of any one of them. But one datum is clear: The number of students enrolling in mathematical studies every year in Italy has almost doubled since the beginning of the new millennium. So even if we cannot be sure that this particular project was completely successful, it definitely did not hurt.

Dipartimento di Matematica Universita` di Pisa Largo Pontecorvo 5, 56127 Pisa, Italy e-mail: [email protected]

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Linear Operators and Their Spectra by E. Brian Davies CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS, VOL. 106, 2007, XII + 451 PAGES, £ 42, ISBN 0-521-86629-4 REVIEWED BY N. P. LANDSMAN

rian Davies (King’s College, London) has had a long and distinguished career in the functional analysis/ mathematical physics interface, including the authorship of classics like Quantum Theory of Open Systems (1976), One-Parameter Semigroups (1980), Heat Kernels and Spectral Theory (1989) and Spectral Theory and Differential Operators (1995). He subsequently shocked at least his mathematical audience (and perhaps even his former self) by publishing Science in the Looking Glass: What Do Scientists Really Know? (2003), a critical survey of modern science which includes an assault on Platonism as a philosophy of mathematics, and to some extent even on mathematics itself, portrayed as a subject with shaky foundations and questionable relevance to science. I considered this popular book a gem on appearance and feared the worst for the mathematical output of its author, but fortunately an uninterrupted sequence of research papers showed that his critical attitude towards mathematics has not stopped at least Davies himself in pursuing the subject. Subsequently, the book under review, which is a continuation of the technical series of books just listed and does not even mention Science in the Looking Glass, has removed all possible doubts about Davies’s commitment to mathematics. The only nod towards his previous book (which makes a case for constructive mathematics in the style of Bishop) is a dry comment in the Preface to the effect that ‘‘the present book has a slight philosophical bias towards explicit bounds and away from abstract existence theorems’’ (which in any case is a rather common stance among so-called ‘‘hard analysts’’), followed by a statement of preference for constructive rather than general proofs (which, however, is quite unusual for functional analysts). But in fact, this preference is by no means just ‘‘philosophical,’’ as shown by the example of an ‘‘abstract existence theorem’’ Davies gives a few lines later, namely the nonemptiness of the spectrum of an operator. This is usually proved by reductio ad absurdum, which in particular means that the proof gives or suggests no procedure for actually finding explicit elements of the spectrum. Davies relates this to the phenomenon that the spectrum can be highly unstable under perturbation, a drawback that is not shared by the so-called pseudospectrum. It is no accident that this notion already appears in the Preface, as its incorporation is one of the distinguishing features of the book. So what is the pseudospectrum of an

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operator—or rather, to begin with, what is the spectrum of an operator and which kind of operators are considered in this book? We know from linear algebra that a linear map A : Cn ! Cn is injective if and only if it is surjective. Consequently, for any z 2 C, the map A - z is invertible (i.e., injective and surjective) if and only if z is not an eigenvalue of A, i.e., if and only if there exists no nonzero f 2 Cn such that Af = zf. Hence, the collection of eigenvalues of A may be characterized as the set of all z 2 C for which A - z is not invertible. It is the latter set that defines the spectrum Spec(A) also for bounded linear maps A : B ! B, where B is a Banach space (i.e. a complete normed vector space over C). As has been known since the beginning of the twentieth century, the possible definition of the spectrum as the collection of eigenvalues of A turns out to be too narrow for infinite-dimensional B, where injectivity of a linear map is no longer equivalent to surjectivity. Perhaps surprisingly, this concept of a spectrum turned out to be the key to operator theory, on which ‘‘everything else is based’’ (p. 14). One situation has been studied particularly well, from Hilbert to the present day, namely the case where B is a Hilbert space H and A : H ! H is self-adjoint (in that (Af, g) = (f, Ag) for all f ; g 2 H, in terms of the inner product (,) of H). More generally, the notion of self-adjointness makes sense for unbounded operators A : DðAÞ ! H, where the domain DðAÞ of A is a dense linear subspace of H, with essentially the same spectral theory as in the bounded case. Formalized by von Neumann (who brought to maturity ideas that had emerged from Hilbert and his school at Go¨ttingen), the spectral theory of unbounded self-adjoint operators has been developed partly in response to quantum mechanics, where fundamental observables like position, momentum, and the Hamiltonian are of this form. More generally, there is a fertile interplay between boundaryvalue problems for linear partial-differential equations and unbounded self-adjoint operators on Hilbert space—it remains a miracle of the history of science how much of the theory of the Schro¨dinger equation of quantum mechanics (first written down in 1926) was already clear from the context of classical mathematical physics, as contained in the legendary monograph Methoden der mathematischen Physik by Courant and Hilbert from 1924. Today, one can find dozens of excellent books (and probably hundreds of books altogether) on the spectral theory of self-adjoint operators, including one by Davies himself (1995). In contrast, the book under review is emphatically about the spectral theory of non-self-adjoint linear operators. What is meant by this is a priori not entirely clear, but the book pays attention to two different situations: 1. Non-self-adjoint operators on Hilbert space (where a comparison with the self-adjoint case is possible and often illuminating); 2. Operators on Banach space (where no such comparison can be made, so that the theory stands on its own). The thrust of the book lies in case 2 above, the spectral theory of general (linear) operators on Banach space. To set the stage, in the Preface the author laments that ‘‘Studying non-self-adjoint operators is like being a vet rather than a

THE MATHEMATICAL INTELLIGENCER Ó 2009 The Author(s). This article is published with open access at Springerlink.com

doctor: One has to acquire a much wider range of knowledge, and accept that one cannot expect to have as high a rate of success when confronted with particular cases.’’ Nonetheless, two general techniques appear to be widely applicable and could be said to dominate the book: One-parameter semigroups and pseudospectra. The principal innovative aspect of the book lies in its coverage of pseudospectra and related notions. According to the online Pseudospectra gateway by Mark Embree and Nick Trefethen (see http: == web:comlab:ox:ac:uk = pseudo spectra=), pseudospectra have been independently invented at least five times between 1974 and 1990, motivated by the following observation. The linear equation (A - k)x = b for x has a unique solution whenever k 62 SpecðAÞ, obviously given by x = (A - k)-1b. Now if b is perturbed to b0 with kb
b0 k  e (or, alternatively, if b is only known with finite precision e), then the corresponding solutions satisfy kx
x 0 k  ekðA
kÞ
1 k. For a self-adjoint operator A on a Hilbert space (more generally, for a normal operator), the number kðA
kÞ
1 k that evidently controls the stability of the solution turns out to be equal to the inverse of the distance between k and Spec(A), so that, roughly speaking, it is small if k is far from the spectrum and large if k is almost an eigenvalue of A. For general operators A, however, kðA
kÞ
1 k can be large even if k is far from Spec(A). This motivates the introduction of the e-pseudospectrum Spece ðAÞ  C of A, defined for each e [ 0 as Spece ðAÞ ¼ fz 2 C j kðA
zÞ
1 k  e
1 g (in fact, Davies also includes Spec(A) in this set). The pseudospectrum of A then consists of the parametrized family {Spece(A)}e[0. Keeping in mind that for self-adjoint A on Hilbert space one has k [ Spec(A) if and only if there exists a sequence (fn) with kfn k ¼ 1 for all n such that limn kðA
kÞfn k ¼ 0, it is interesting to know that k [ Spece(A) if and only if there exists a vector f with kf k ¼ 1 such that kðA
kÞf k  e. Pseudospectra will be new to most functional analysts and mathematical physicists, and Davies provides extensive coverage with a good mix of theory and examples. The conclusion is that phenomena that formerly had been regarded as pathological behaviour of the spectrum of nonself-adjoint operators, such as the existence of approximate eigenvalues far from the spectrum or the instability of the spectrum under small perturbations, disappear if one replaces the spectrum by the pseudospectrum. In addition, Davies discusses other generalizations of the spectrum, like

the hull and the numerical range of a bounded operator A on a Banach space B. One will look in vain for the notions of pseudospectrum, hull and numerical range in standard functional analysis texts, so Davies has done us a great service by explaining them through beautiful theorems and examples. More generally, his book is the first to offer a comprehensive survey of the spectral theory of non-self-adjoint operators, including both ‘‘classical’’ and ‘‘cutting edge’’ results, showing that this theory holds as much promise as the self-adjoint theory in both foundations and application. The scope of the book is truly enormous and is only partly reflected by listing the chapter titles: Elementary Operator Theory, Function Spaces, Fourier Transforms and Bases, Intermediate Operator Theory, Operators on Hilbert Space, One-Parameter Semigroups, Special Classes of Semigroup (sic), Resolvents and Generators, Quantitative Bounds on Operators, Quantitative Bounds on Semigroups, Perturbation Theory, Markov Chains and Graphs, Positive Semigroups, Non-SelfAdjoint Schro¨dinger Operators. My only criticisms would be that the organization of the book could have been better (for example, with a clearer chapterwise separation between Hilbert space and general Banach space results), and that historical or bibliographical notes are lacking. As the author states himself, the book is halfway between being a textbook and a monograph, which makes it difficult to say for whom the book is intended. I’d say that a firm background in Hilbert space theory and some feeling for basic Banach spaces such as Lp(X) and C(X) is necessary and sufficient to understand and appreciate this beautiful volume, which has no competitors.

OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Institute for Mathematics Astrophysics and Particle Physics Faculty of Science Radboud University Nijmegen Nijmegen, The Netherlands e-mail: [email protected]

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Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life: An Agony in Eight Fits by Robin Wilson NEW YORK, NY, LONDON, W. W. NORTON & COMPANY, INC., 2008, XII + 237 PP., US $24.95, ISBN 978-0-393-06027-0 REVIEWED BY JOHN J. WATKINS

t has been said that Queen Victoria was so charmed by Alice’s Adventures in Wonderland that she demanded to be sent the very next book written by Lewis Carroll. She was, however, not at all amused when the next book did in fact arrive: An Elementary Treatise on Determinants, with their Application to Simultaneous Linear Equations and Algebraical Geometry. The queen’s annoyance, and confusion, is entirely understandable. The author of both of these books was of course none other than Charles Lutwidge Dodgson—pronounced ‘‘Dod son’’—a lecturer in mathematics at Oxford University. It is indeed a shame that the monarch who gave her name to the ‘‘Victorian era’’ did not have available in her time a delightful new book to guide her, Lewis Carroll in Numberland, by Robin Wilson, for she then might not only have discovered that there is far more mathematics in Alice’s Adventures than she ever suspected, but a good bit more fun lurking in Dodgson’s treatise on determinants than she surely first guessed from its rather off-putting title. Charles Dodgson—under the pen name, Lewis Carroll, a name derived from his middle name Lutwidge and from Charles and which was first used during his early twenties as some of his poems, parodies, and stories began to appear in magazines—will forever

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be remembered as the author of the brilliant children’s fantasies Alice’s Adventures in Wonderland and Through the Looking Glass and What Alice Found There. Robin Wilson’s thoroughly enjoyable book, Lewis Carroll in Numberland, reveals Dodgson to us as a serious mathematician whose impressive body of work to be sure contained important academic books such as the previously mentioned one on determinants, and also as a serious mathematician who in his popular writing could draw on his endless supply of whimsy to help children develop their powers of mathematical and logical thinking. What makes Lewis Carroll in Numberland so enjoyable to read is the double pleasure of rediscovering on the one hand that Dodgson was in fact one of the funniest writers who ever put pen to paper (I’’m talking laughout-loud Monty Python funny), and on the other hand coming across really interesting mathematics that Dodgson either discovered or used in very creative ways. In addition, the book is beautifully illustrated with drawings from Alice’s Adventures and Through the Looking Glass, and also with many of Dodgson’s own photographs and other memorabilia such as an intricate maze he once constructed as a teen for a ‘‘magazine’’ produced for his family. In these family magazines, Dodgson showed early flashes of the sheer silliness that runs through his later work. One volume contained ‘‘Hints for Etiquette’’ such as ‘‘We do not recommend the practice of eating cheese with a knife and fork in one hand and a spoon and wine-glass in the other,’’ or ‘‘As a general rule, do not kick the shins of the opposite gentleman under the table, if personally unacquainted with him: Your pleasantry is liable to be misunderstood.’’ A decade later, this same sort of silliness is present during an encounter that Alice has with the White Knight, although this time there is also an underlying lesson for the reader in the logic of naming things: The White Knight is talking to Alice about a song of his named ‘‘The Aged Aged Man,’’ but as he patiently explains to Alice the name of his song is called ‘‘Haddock’s Eyes,’’ and yet the song is called ‘‘Ways and Means,’’ while the

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song itself is ‘‘A-Sitting on a Gate.’’ I defy anyone, especially a mathematician, to read this passage in Alice’s Adventures in Wonderland and not burst out laughing. In one of these family magazines, when he was 17, Dodgson raised a question he later called ‘‘Where does the day begin?’’ It is actually a rather serious question, but as usual Dodgson injects humor: ‘‘Supposing on Tuesday, it is morning at London; in another hour it would be Tuesday morning at the west of England; if the whole world were land we might go on tracing Tuesday Morning, Tuesday Morning all the way round, till in 24 hours we get to London again. But we know that at London 24 hours after Tuesday morning it is Wednesday morning. Where then, in its passage round the earth, does the day change its name? … some line would have to be fixed, where the inhabitant of one house would wake up and say ‘heigh ho! Tuesday morning!’ and the inhabitant of the next, (over the line) a few miles to the west would wake a few minutes afterwards and say ‘‘heigh ho! Wednesday morning!’’ What hopeless confusion the people who happen to live on the line would be in, it is not for me to say.’’ The International Dateline was established nearly fifty years later in 1884. Charles Dodgson, following in his father’s footsteps, went to Oxford University and studied at Christ Church where he excelled, receiving a First Class in Mathematics. He was soon appointed to a lectureship by the Dean of Christ Church, the Reverend Henry Liddell (which rhymes with ‘‘fiddle’’). It was six years later that Dodgson, while on a boating excursion with an Oxford friend, and the three Liddell daughters invented on the spot a story to enliven a lazy summer afternoon. At the day’s end, 10-year old Alice Liddell implored him, ‘‘Oh, Mr. Dodgson, I wish you would write out Alice’s adventures for me.’’ He stayed up nearly the whole night committing Alice’s story to paper and later added his own illustrations. Alice’s Adventures in Wonderland has never been out of print since.

Robin Wilson takes great care to show us the skill with which Dodgson weaves mathematical ideas into his stories. At times, Dodgson can seem merely whimsical though in fact there may be a darker underlying layer as when the White Queen asks Alice: ‘‘Can you do Addition? What’s one and one and one and one and one and one and one and one and one and one?’’ It is entirely understandable when Alice is forced to admit: ‘‘I don’t know. I lost count,’’ and certainly not remotely fair, albeit quite funny, when the Red Queen harshly concludes: ‘‘She can’t do Addition.’’ At other times, Dodgson could also be fairly demanding of the mathematical level of his audience. Here is Alice after falling down the rabbit-hole: ‘‘I’ll try if I know all the things I used to know. Let me see: Four times five is twelve, and four times six is thirteen and four times seven is—oh dear! I shall never get to twenty at that rate!’’ This seems like utter nonsense, but the mathematical ‘‘joke’’ here is that in base 18, we would write 4 9 5 as 12; then, in base 21, we would write 4 9 6 as 13. So, Alice seems headed for 20 after all since in base 24, we write 4 9 7 as 14; in base 27, we write 4 9 8 as 15; in base 30, we write 4 9 9 as 16; in base 33, we write 4 9 ‘‘ten’’ as 17; in base 36, we write 4 9 ‘‘eleven’’ as 18; in base 39, we write 4 9 ‘‘twelve’’ as 19. But, as Alice clearly saw coming well ahead of time, 4 9 ‘‘thirteen’’ is not 20 in base 42. Dodgson frequently used puzzles or bits of recreational mathematics in his teaching, and he also did this to amuse children, for example asking them to multiply the ‘‘magic’’ number 142857 by 2, 3, 4, 5, 6 and 7. The resulting numbers are: 285714, 428571, 571428, 714285, 857142, and 999999. Begin at the 1 in each of these numbers, except the last, and the same six digits appear in the same order. He also in his later years wrote extensively on logic under the name Lewis Carroll, making what might otherwise seem like serious study seem fun with terms such as ‘‘prim Misses,’’ ‘‘delusions’’ and ‘‘sillygisms.’’ Everyone is familiar with Aristotle’s famous syllogism that concludes ‘‘all Greeks are mortal,’’ but Dodgson is far more inventive with syllogisms such as these two:

No fossil can be No bald creature crossed in love needs a hairbrush An oyster may be No lizards have crossed in love hair for which the delusions are (I mean, the conclusions are) ‘‘oysters are not fossils’’ and ‘‘no lizard needs a hairbrush.’’ Bertrand Russell considered an exposition by Dodgson of Zeno’s paradox that appeared in the periodical Mind, a quarterly devoted to psychology and philosophy, and the following ‘‘barber paradox’’ to be Dodgson’s greatest contributions to logic. There are three barbers, Allen, Brown and Carr, who cannot all leave their barber’s shop at the same time. Suppose that: If Carr is out, then if Allen is out, Brown is in; and, if Allen is out, Brown is out. Can Carr go out? Lest you think there is an easy resolution to this paradox, Oxford’s Wykeham Professor of Logic, John Cook Wilson, finally admitted, seven years after Dodgson’s death, that he had incorrectly analyzed this problem and that Dodgson had gotten it right all along! Dodgson also wrote extensively on Euclid’s Elements, invented numerous codes, and discovered a method for evaluating determinants in which only 2 9 2 determinants need be calculated, called the ‘‘method of condensation,’’ a method that clearly deserves to be more widely known than it is. Since I happened to be teaching a course in number theory while I was reading this book, I was struck by the fact that Dodgson appears to have possessed the true spirit of a numbertheorist. At one point in 1890 he says he ‘‘chanced on a theorem (which seems true, though I cannot prove it), that if x2 + y2 be even, its half is the sum of two squares.’’ For example, 72 + 32 = 58 is even, so 29 should be the sum of two squares (which it is, 25 and 4). Fermat himself would have been delighted by such a ‘‘theorem.’’ Within a few days, Dodgson realized that ‘(x2 + y2) = [‘(x + y)]2 + [‘(x - y)]2. Similarly, Dodgson had said that a kindred theorem— that 2(x2 + y2) is always the sum of two squares—also seems unprovable; then, he found that 2(x2 + y2) = (x + y)2 + (x - y)2. Dodgson also included similar formulas as problems in a book published

in 1893 with the subtitle Pillow-Problems Thought out During Sleepless Nights. One of these asks the reader to prove that 3 times the sum of 3 squares is also the sum of 4 squares. Since Lagrange had already proved his celebrated theorem in 1770 that any number can be written as a sum of four squares, the point of this ‘‘pillow problem’’ is to do it as a mental exercise in your head, presumably in bed and with your eyes closed—a mathematician’s alternative to counting sheep, I suppose. I think I could actually manage to do this particular problem in my head since it is just a matter of producing the highly structured formula
3 x 2 þ y 2 þ z 2 ¼ ðx þ y þ z Þ2 þ ðx
yÞ2 þðy
z Þ2 þðz
x Þ2 : Another problem, however, asks the reader to prove that the sum of 2 different squares, multiplied by the sum of 2 different squares, gives the sum of 2 squares in 2 different ways. Thus, Dodgson—whose own powers of mental calculation were prodigious— expected readers to not only be able to verify the following remarkable formula, but also to discover it, in their heads:

a2 þ b2 c2 þ d 2 ¼ ðac  bd Þ2 þ ðad  bcÞ2 : This formula first appeared in 1225 in Fibonacci’s Liber quadratorum, but surely was known to Diophantus who wrote in Arithmetica: ‘‘It is the nature of 65 that it can be written in two different ways as a sum of two squares, as 16 + 49 and as 64 + 1; this happens because it is the product of 13 and 5, each of which is a sum of two squares.’’ Less than four weeks before his death at the age of 65, Dodgson wrote in his diary: ‘‘Sat up last night ‘till 4 a.m., over a tempting problem, sent me from New York, ‘to find three equal rational-sided right-angled triangles.’ I found two, whose sides are 20, 21, 29; 12, 35, 37: but could not find three.’’ Dodgson should have been able to find three triangles. Perhaps his difficulty was that he was sitting up, since for him this should have been a relatively easy ‘‘pillow-problem.’’ The key is that for any two positive integers s and t, if we define x, y and z by


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x = 2st, y = s2 - t2, z = s2 + t2, then x2 + y2 = z2, and we have a Pythagorean triangle whose area A is given by
A ¼ 1=2xy ¼ st s2
t 2 ¼ st ðs þ t Þðs
t Þ We know that Dodgson was aware of this idea because Euclid not only proved this fact in the Elements, but also proved that any primitive Pythagorean triangle must have this form where s and t are relatively prime and one of them is even and the other odd. So, all we have to do (in our heads!) is find three sets of values for s and t that

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yield the same product: s  t  (s + t)  (s - t). For example, for the values s = 6, t = 1 we get a product 6  1  7  5, and for the values s = 5, t = 2 we get a product 5  2  7  3, and we can easily see that these two products will be the same. These two sets of values, in fact, yield the two triangles Dodgson found, each having area 210. I’ll end by giving you a hint for solving this ‘‘pillow-problem’’ that kept Dodgson up until 4 a.m. along with a very strong recommendation that you also read Lewis Carroll in

Numberland. Here is the hint: For three triangles the area will be greater than 210, but all s and t values can be less than 10 for the smallest solution. Moreover, it turns out that there are infinitely many solutions, so this particular ‘‘pillow-problem’’ can be used night after night after night after night, just like sheep. Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903, USA e-mail: [email protected]

Symmetry by Marcus du Sautoy NEW YORK: HARPERCOLLINS, 2008, 384 PP., US $25.95. ISBN: 9780060789404; ISBN10: 0060789409 REVIEWED BY HAROLD R. PARKS

arcus du Sautoy is a serious mathematician; he is on the faculty at the University of Oxford, the institution from which he earned his doctorate. He is also the author of several expository books about mathematics aimed at a general audience. Additionally, du Sautoy writes for newspapers, radio, and television. So significant has been du Sautoy’s contribution to the popularization and exposition of mathematics that he has recently been selected to be the Simonyi Professor for the Public Understanding of Science, following the retirement of Richard Dawkins, the previous holder of that chair. Symmetry is du Sautoy’s latest book aimed at the general public. This book was published nearly simultaneously in Great Britain under the title Finding Moonshine. One presumes that the publisher felt the title Finding Moonshine might mislead the potential American reader. To an American, if ‘‘moonshine’’ does not refer to the light from the moon, then it probably means illegally distilled whiskey. Aside from the title, Symmetry does not appear to have been Americanized (maths is mathematics and football is soccer). The ‘‘moonshine’’ in the British title refers to the connection between the sporadic group called the Monster and modular functions. When first mooted, that connection between the Monster and modular functions was very conjectural. Indeed, it was a totally wild conjecture, possibly moonshine, i.e., a foolish idea. Nonetheless, some held out hope that such a moonshine connection could be established. Andrew Ogg even offered a bottle of Jack Daniel’s as a prize for the person who succeeded. Eventually, the reality of the moonshine connection was proved by Richard Borcherds, work for which Borcherds won a Fields Medal. John Horton Conway asserted that while his

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student, Borcherds, had proved the moonshine connection, Borcherds had not explained it. Consequently, Borcherds was not awarded the bottle of Jack Daniel’s. Superficially, Symmetry is organized around one year in the life of Marcus du Sautoy with each of the 12 chapters assigned to and named for a month. The starting point is du Sautoy’s fortieth birthday on August 26, 2005. He spent that day in the Sinai Desert on the edge of the Red Sea. The chapter opens at noon of that birthday and closes at midnight. Since the seventh month, August, is the first month in the Symmetry calendar, chapter n refers to month n + 6 mod 12. In fact, relatively little of the text of Symmetry is devoted to describing what actually happened to the author during 2005–2006. Instead, things du Sautoy did and places he went provide openings and segues for the author to discuss mathematics and its history and to regale the reader with descriptions of and anecdotes about contemporary mathematicians. Many of the descriptions and anecdotes are quite amusing. Sometimes a trip to one place leads to the author reminiscing about a trip earlier in his life. If one stops to think about it, the actual chronology is often confusing. Don’t worry too much about the precise dates; just relax and enjoy the flow of the story. As mentioned, the book begins on du Sautoy’s fortieth birthday. It is perhaps obligatory that the author must tell his readers that mathematicians who are 40 or older are presumed to be past their prime, and thus are no longer eligible to be awarded the Fields Medal. Those who are past 40 may not relish the recitation of this lore, but we have the consolation that du Sautoy, and many of the contemporary mathematicians that figure in the book, are with us in the over-the-hill club. Symmetry is a popularizing and expository book. Technical details are ruthlessly suppressed. Even though an important focus of the book is the connection between the Monster and modular functions, there is no formal definition of a group nor is there a definition of what constitutes a modular function. Given the nature of the book and the intended audience, the absence of formal definitions is no doubt appro-

priate: The general reader is not going to be enlightened by the actual definitions. The suppression of technical details is further exemplified by the fact that there are 13 displayed equations and formulas in the 353 pages of text (there are additional displays of numbers and initial segments of sequences). For mathematicians, that is an incredibly low number and density of displayed equations and formulas. For the general public, it may still be too many. The general, nontechnical reader is presumably walking around with some operational meaning of ‘‘symmetry’’ in his or her head. Group theory would play no part in that person’s operational definition. In the first chapter, du Sautoy introduces the nontechnical reader to the notion of a group via the symmetries of simple figures. Of course, this is done without giving an explicit definition of a group. Similarly, the crucial notion of a simple group is described only by analogy with a prime number. The second and third chapters continue to be quite concrete, focusing on Platonic solids and plane symmetries. These subjects are made particularly tangible in the context of the author’s visits to the British museum (in September) and to the Alhambra (in October). While most mathematicians know of the Alhambra and its symmetric tilings, one might not be familiar with the symmetrically carved Neolithic stone balls that have been found in the British isles. In November, the fourth month in the Symmetry calendar, du Sautoy went to a conference in Okinawa, Japan. In addition to being a chance to share mathematics, going to a conference can lead to meeting new and interesting people and to having other adventures. This trip appears to have been supplied with the full complement of such opportunities; it also gives the author the occasion to describe other conference experiences. The reader may be reminded of his or her similar conference experiences. Galois theory is the device du Sautoy uses to connect symmetry to algebra. This material occupies several chapters. Any nontechnical reader who is tackling a math book such as Symmetry would certainly have had some experience with solving polynomial

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equations. So this method of leading the reader along from symmetry to algebra is sound. By page count, most of the material is historical. The history of the search for solutions by radicals of cubic, quartic and quintic equations, involving Tartaglia, Cardano, Abel, and Galois, makes good, compelling reading. The April chapter, the ninth month in the Symmetry calendar, is devoted to symmetry in music. I am not particularly musical and prefer music that is tonal. Clearly, other readers will have a better appreciation than I did, but even for the nonmusical person, the chapter is interesting. The explanation of change ringing was remarkable for its clarity. Change ringing figures in the novel The Nine Tailors by Dorothy Sayers, but Sayers’s explanation was not nearly as lucid as du Sautoy’s. The May chapter should be extraordinarily interesting for many readers. The topic is the impact and usefulness of symmetry in pharmacology, virology, and coding. One might know that some particular organic molecules can occur in mirror image forms. Again, there is a mystery story by Dorothy Sayers, the solution of which hinges on this fact. It turns out that the physiological effects

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of the mirror image molecules can be similar or they can be very different. The impact of symmetry on virology is that symmetry can be used to compress the amount of information that is needed by a virus for replication. The final topic, Hamming codes, seems innocent enough, but the 24-bit code returns in the next chapter. The penultimate chapter starts with the story of Walter Feit and John Thompson proving Burnside’s oddprime conjecture (that every finite simple group of nonprime order must have even order). The proof, published in 1963, occupied a full issue of the Pacific Journal of Mathematics. At the time, the 255-page proof was considered enormous. The bulk of the story in this chapter is about the discovery of those sporadic groups that were not among the five found by E´mile Mathieu in the nineteenth century. It’s a great story, and, for the benefit of the general reader, there are no annoying details to detract from it. The final chapter, July, the twelfth month in the Symmetry calendar, discusses several topics. The saga of the sporadic groups is concluded by the construction of the last two such

groups: The Monster and the fourth Janko group. The completion of the classification of finite simple groups allowed the completion of The Atlas of Finite Groups by Conway, Curtis, Norton, Parker, and Wilson and its publication in 1985. The beginnings of the Atlas project in the mid 1970s are also described. There is more about moonshine and about Borcherds’s success in proving its existence. Finally, we learn that even if the author was too old to qualify for the Fields Medal, he did give an Invited Lecture at the Madrid ICM: In the Algebra Section, on August 26, 2006. Symmetry is enjoyable reading. John Horton Conway shows up often, and he is such an exuberant character that his presence adds much zest. There are no technical details, but du Sautoy makes his subject sound so compellingly interesting that even an old geometric analyst might get the urge to reacquaint himself with group theory.

Department of Mathematics Oregon State University Corvallis, OR 97331-4605, USA e-mail: [email protected]

Random Curves: Journeys of a Mathematician by Neal Koblitz BERLIN, HEIDELBERG: SPRINGER SCIENCE + BUSINESS MEDIA, 2008, 392 PP., US $34.95, ISBN 978-3-540-74077-3 REVIEWED BY BERNHELM BOOSS-BAVNBEK

All men of any condition who have done something of special worth or something that may truly resemble those things of special merit, should, if they are truthful and good people, write in their own hand the story of their lives, but they should not begin such a fine undertaking until they have passed the age of 40.

ith this verdict, the Renaissance goldsmith and sculptor Benvenuto Cellini (1500– 1571) opened his own autobiography [3, p. 5], probably composed between 1558 and 1567. Neal Koblitz, the author of the autobiographical memoirs Random Curves, is such a Renaissance personality: A renowned top mathematician, a prolific author of widely used text books in number theory and cryptography, a harsh, polemic writer (also for 30 years for this magazine) against ‘‘mathematics as propaganda,’’ on elementary school mathematics teaching, and on the mathematical and general cultural life in developing countries. No doubt, Koblitz has both ‘‘done something of special worth’’ in his scientific work and ‘‘something that may truly resemble those things of special merit,’’ though contrary to Cellini’s glorification of the bloody contemporary Florentine Medici dictatorship, Koblitz praises the rights and the virtues of the suppressed, minorities, women, children. Another title for his book could have been Another Look at Enlightened Self-Interest: Because that is what the book is about. For most of its pages, Random Curves delivers a long and fascinating array of very sharp, personal and uncompromising comments on the great

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political events of the second half of the 20th century: The Cold War, but also the quality of life in the former Soviet Union; the American aggression in Vietnam, but also the charisma of the solidarity and liberation movements; racism and civil rights movements; suppression in Africa and Latin America and counter forces; injustice, stupidity, brutality, arrogance of power—and the tactics and strength of the suppressed and how best to support them. Nothing passes without getting Koblitz’s wellargued ‘‘Another Look’’, be it a deserved rap for widely accepted conditions or a positive explanation of widely rejected circumstances. He is highly opinionated and displays a total absence of the politeness associated to fashionable social constructivism and relativism. Throughout the book, Koblitz is opinionated with good reason: He writes only of things he seems to have investigated thoroughly, and he is, more than most, aware of possible fatal consequences of careless disregard for logic and truth. There is no tolerance for superficial political thinking. Thoughtlessness is perceived by Koblitz as almost worse than bad will or selfishness, like the saying: ‘‘It is terrible to be knocked down by a car, but much more terrible to be trampled down by a hundred geese.’’ As in grading mathematical exercises, Koblitz vigorously discloses the slightest weakness in common arguments; and as in mathematical proofs, and contrary to common political arguing and military tactics, Koblitz always attacks his adversaries’ strongest positions. That makes reading Random Curves sometimes offending, often demanding, but always rewarding. Koblitz also displays a warm and human enlightened self-interest associated with his extensive work for solidarity and charity. He doesn’t underplay the contradiction between his almost hedonistic life as a highly gifted, tenured, respected and wellpaid university professor in mathematics and the miserable circumstances of the people he tries to help. Random Curves is not so much about his own sacrifices, renunciation, risks, punishments (though they are there), as about the emotional reward for his political activism and the wide range of possibilities an individual really has: I

did not tire of reading of his and his wife Ann Hibner’s happiness (she a math historian with a profound Kovalevskaia biography [7] and prolific anthropologist and gender researcher), about their travels, their organizational work, their endowments, their encounters with so many interesting, impressing and sympathetic people around the world. Koblitz is a good writer: People are described in a lively way, with a lot of humor, but always with high respect, whether he recalls the words of a schoolboy in rural Peru or of the Vietnamese Prime Minister. This human touch hopefully reconciles even a conservative reader who otherwise may feel repelled by the hard inexorable logic of Koblitz’s political arguing. There is not much about mathematics in the book, just a few rather sketchy comments about Koblitz’s personal path into number theory at Harvard, Princeton and Moscow; then just 32 pages on his seminal work on elliptic curve cryptography (ECC) and his continuing vendetta against claims by the proponents of the mainstream publickey cryptography algorithm RSA; and, finally, 21 pages on elementary math education and math teacher education. Contrary to model autobiographies like Norbert Wiener’s I Am a Mathematician [9] which enthusiastically discusses control theory, prediction, Fourier analysis and brain research, or Mark Kac’s Enigma of Chance [6] which explores the interface between different fields of mathematics and statistical mechanics and explains, for example, the combinatorial rules of phase transition, Koblitz seems convinced by Mark Kac’s proclamation [6, p. xiii]: ‘‘The autobiography of a mathematician must contain some mathematics. Yet a presentation in popular form of some of the problems and ideas with which I have been involved throughout my life is unfortunately an impossible task.’’ Nevertheless, Kac tried. That was good. Koblitz did not really try. That is a pity. Surely, Koblitz may have had good reasons for that restraint. He describes his life-long fight against overhyped ideas, often connected to improper numericity, when people dress up poor understanding or even fraudulent arguments by slick, pompous, false and misleading manipulation of logic


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and numbers. Understandably, but unfortunately, he restrains himself then from giving more detailed explanations of his own work. He must have been afraid that such explanations could give the impression of adequate description while in reality and by necessity remaining superficial. Perhaps it is not a bad choice Koblitz has made for this book, rather carefully describing the math environment instead of the mathematics itself. There, in the mathematician’s everyday life, he has observed much and has much to say: On the nice and generous personality of many mathematicians, how easy it was and is from the rather remote and protected position in a math department to mix with politics even on the outspoken left, and how to make use of ties with sympathetic professionals. Strangely enough, when writing about the mathematical life, he sees everything through rose-colored spectacles and seems ready to make any compromise, even to close his eyes and shut his ears, like so many otherwise very critical mathematicians, such as Laurent Schwartz [8]. For a possible explanation, I quote Elias Canetti [2]: ‘‘Don’t tell me who you are. I want to worship you.’’ Three Examples: I. With reference to C. P. Snow, Koblitz notices the gap between different cultures, and he praises, unreservedly, a math community with values solely associated to ‘‘intellectual achievements.’’ Snow was not so onesided, and even less so was his source Benjamin Disraeli [4], who coined the ‘‘Two Nations’’ concept when he was a social critical writer, before becoming the conservative British politician and Prime Minister. Where has Koblitz been the last 30 years? On what Mount Olympus? He doesn’t seem to be aware of the breakdown of the peer referee system in mathematical journals, now that all mathematicians are pressed by their deans to publish more and shorter articles; to make a small epsilon variation into a separate note at once rather than wait for full solution; to establish friendship circles of mutual citation for higher impact factors; not to waste time as a referee for uncredited and time-consuming reading, learning and checking, an activity previously 66

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considered highly rewarding. It seems that Koblitz doesn’t know about the desperate situation of the editors of formerly highly-respected journals (names can be provided) who are deserted by their referees and, consequently, must stick to short superficial checks of submitted papers by the opinion leaders in a field. (Often, it seems, the opinion leaders do not have the time to check more than whether and how they are quoted.) II. Even more mysterious is the author’s reluctance to address the role of mathematics in the various wars fought by his country, in particular in Vietnam three decades ago, then in Kosovo (Yugoslavia) and now in Iraq and Afghanistan. Koblitz brushes aside G. H. Hardy’s concerns regarding military applications of mathematics (long before the cryptography potential of number theory became evident), by one single claim regarding cryptography: ‘‘Earlier systems for scrambling messages worked well in military or diplomatic applications, where there was a fixed hierarchy of people who were authorized to know the secret keys. By the 1970s, with major sections of the economy rapidly becoming computerized, the limitations of classical cryptography were coming to the fore’’ (p. 297f). Koblitz elaborates that claim of the purely commercial relevance of mathematical work for publickey cryptography in a speculation about what, in Koblitz’s perspective, would really upset Hardy today, namely the war in Iraq—‘‘much more than the use of number theory in cryptography’’ (p. 320). I cannot judge. I’m not an expert on the high-speed cryptography now indispensable for real-time control of military operations, and I doubt that Koblitz is. More generally, it seems to me that the mathematics historian Jens Høyrup and I have put sufficiently rich material together [1] to expect a politically attentive mathematician to make the connection between mathematics and modern war when speaking of and against war. There is ample evidence that only the superstitious belief in the math-supported pin-point accuracy of modern weaponry could create the necessary public and political support for an aggression promising to be clean and gentle with one’s own

troops and civilians. Moreover, on April 9, 2003, one month into the Iraq war, the then president of Koblitz’s professional organization, the American Mathematical Society, took notice, not so much of the military role of mathematics in general, as of mathematics as a key component in the preservation of US military superiority, implicitly promising the loyal assistance of his members (and asking for adequate payment): ‘‘…for a military commander to have secure communications in the field depends on fundamental advances in number theory…. Future progress in this seemingly abstract area, by us or by hostile forces, could threaten the security of all these communications.’’ (David Eisenbud, statement to a US congressional subcommittee overseeing the funding of the US National Science Foundation [5]). III. Another strange aspect of Random Curves is what may appear as a naı¨ve perception of corporative business. Apparently, he sees only the blessings of setting the math supported turbo on modern capitalism. Once again, for Koblitz, clearly, capitalism has its bleak sides. He advocates socialist ideas and, to a surprisingly large extent, but rather convincingly, also socialist practice as he witnessed it. But he closes his eyes once again to the role of mathematics in the modern economy, for example, in the dawn of the present financial crisis. To be fair, the book was published early in 2008, a few months before the financial crisis became visible to a large population, and with it the fatal role of the math-supported belief in the security of hedge funds and investing in real estate. How come he closes his eyes to the role of mathematicians? Here we have a man, feeling responsible for his students, his product, and for the society, his customers. What customers? In recent years, financial business has most probably employed more than half of each year’s ‘‘products,’’ also from Koblitz’s department. A few mathematicians were concerned about the emerging contradiction between the math-based triumph of rational pricing of options and other derivatives and the evolving impenetrability of the financial markets. Apparently, Koblitz didn’t belong to them. Perhaps Koblitz is right: Perhaps we need not pay attention to the

mathematical aspects of military aggression and the capitalist economy. Perhaps it suffices to protest aggression and exploitation. It seems that Koblitz says, ‘‘Don’t mix!’’ I see his point, but I can’t agree. Who will be interested in this book? Any mathematician or historian with a desire to immerse herself/himself in the vanished world of the American civil-rights movement, in solidarity movements, in national liberation movements, and in the differences and parallels between intellectual and cultural life in different nations and different segments of society will find Random Curves absorbing. This is not a meticulous documentation of political moves and reactions like Noam Chomsky’s writing. Koblitz delivers mostly oral history, with its charm and its limitations. Happily, some stories sound really old and passe´. This is particularly true for Koblitz’s reported hardships in the US Civil-Rights Movement. Regarding other events, one is tempted to recall Zhou Enlai’s alleged quip to Henry Kissinger (not reported in Kissinger’s autobiography), ‘‘It is too early to say,’’ when asked for his assessment of the 1789 French Revolution. This may be partly true for Koblitz’s comments on the rise and fall of socialist ideas—in spite of the fact that he has passed the ominous age of 40 required by Cellini for a balanced view. Older liberal and left-wing mathematicians will recognize many of

Koblitz’s recollections and will be able to compare them with their own experiences. The book might also be attractive to young readers (possibly at the advanced high-school level but more probably college age) who like to read the intelligent and sensitive eyewitness and reflections ‘‘of a student and later a scientist caught up in the tumultuous events of his generation,’’ as the back cover reads. This is the kind of autobiography that I read avidly when I was a teenager, and although prior knowledge of mathematics and cryptography might be helpful, it is certainly not essential for the enjoyment of Koblitz’s moving stories.

[4] Benjamin Disraeli. Sybil: Or the Two Nations. Oxford World’s Classics, Oxford University Press, Oxford, 1998. [5] David Eisenbud. Statement to a US Congress

subcommittee

overseeing

the

funding of the US National Science Foundation, Notices of the AMS, June-July 2003: 704f; http://www.ams.org/notices/ 200306/inside.pdf. [6] Mark Kac. Enigmas of Chance: An Autobiography. Sloan Foundation Series, Harper and Row, New York, 1985. Published posthumously with a memoriam note by Gian-Carlo Rota. [7] Ann Hibner Koblitz. A Convergence of Lives: Sophia Kovalevskaia, Scientist, Writer, Revolutionary. Birkha¨user, Boston, 1983. [Reviewed in The Mathematical Intelligencer, 7/4:69–73, 1985.].

REFERENCES

[1] Bernhelm Booß-Bavnbek, Jens Høyrup, editors. Mathematics and War. Birkha¨user, Basel, 2003. All chapters can be downloaded for free at http://www.springer. com/birkhauser/historyofscience/book/978-37643-1634-1. [2] Elias Canetti. Nachtra¨ge aus Hampstead; Aus den Aufzeichnungen 1954–1971. Carl Hanser, Munich, 1994); Notes from Hamp-

[8] Laurent Schwartz. A Mathematician Grappling with His Century. Birkha¨user, Boston, 2001. A translation in English of Laurent Schwartz’s autobiography, Un mathe´maticien aux prises avec le sie`cle, originally published by editions Odile Jacob, Paris, 1997. [9] Norbert Wiener. I Am a Mathematician. Victor Gollancz Ltd., London, 1956.

stead: The Writer’s Notes: 1954–1971. Translated from German by John Hargraves. Farrar Straus Giroux, NY, 1998. [3] Benvenuto Cellini. La vita—My Life. Translated from Italian by Julia Conaway Bondanella

and

Peter

Bondanella,

Oxford

World’s Classics, Oxford University Press, Oxford, 2002.

Department of Science, Systems and Models / IMFUFA Roskilde University DK-4000 Roskilde Denmark e-mail: [email protected]


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Continuums by Robert Carr OAKVILLE, ONTARIO, CANADA: MOSAIC PRESS, 2008, 250 PP., US$22.00 ISBN: 0889628920

Goldman’s Theorem by R.J. Stern CARMANGAY, ALBERTA, CANADA. SAGA BOOKS, 2009, 220 PP., US$15.95 ISBN: 1897512228 REVIEWED BY ALEX KASMAN

ne criterion by which literature can be judged is the artistry of the author’s prose. By this standard, David Bajo’s recent novel The 351 Books of Irma Arcuri [1], about a mathematician looking for his lover who has voluntarily vanished from her own life, is arguably praiseworthy. However, like so many works of fiction involving mathematicians, that novel so grossly misrepresents mathematics and those who study it that it is difficult for a mathematically knowledgeable reader to fully appreciate it. This review will consider recently published first novels from two authors who may not be Bajo’s equal as a wordsmith, but have successfully captured the culture of mathematics and made worthwhile contributions to the growing catalog of ‘‘mathematical fiction’’. (See, for example, [2], where over 700 works of fiction with mathematical content are listed.) In Continuums by Robert Carr, mathematician Alexandra Jacobi-Semeu takes advantage of her international reputation to escape Ceaus¸escu’s Romania, but loses her daughter in the process. Although mathematics is at the heart of all three of the book’s story lines, it is the decisions people make and the consequences they have to live with that are really its main focus. When we first meet Alexandra in 1969, her life seems relatively pleasant. She is a very successful mathematics professor in Bucharest, married to an equally successful medical doctor, and the mother to a daughter, Ada, whom she adores. The only problem she faces is that her husband expects her to do all of the housework and resents the time

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that she spends away from the family doing research. Then, an unexpected announcement from her brother completely changes the dynamics of her life. Leonard Jacobi, who had trained to be a research mathematician but opted instead to do calculations for a government office, was more troubled than his sister by the oppressive politics of their home country and planned an escape. Once he was safely in Canada, the fact that she knew of his plans and did nothing to stop him was used against Alexandra by an omnipresent network of spies and party agents. Her fame prevents this from having much of an effect on her career, but her family life suffers terribly. As her marriage deteriorates, Alexandra becomes romantically involved with a Que´be´cois mathematician with whom she has written a seminal series of papers on analytic number theory. Finally, after several years of this longdistance affair, she secretly meets her lover at an airport when attending a conference abroad and escapes to Montre´al. However, despite achieving even more recognition for her research, she is in emotional turmoil over her decision to leave Ada in Romania with her father. This provides the main tension, and its resolution constitutes the climax of the novel. However, the story does not proceed in a simple ‘‘arc’’. Interestingly, Carr intertwines Alexandra’s story of decisions and consequences with two others, both also having mathematical components. One is the story of Asuero Aroso, Alexandra’s elderly teacher and mentor who is caring for his bedridden wife. He recalls his life story for Alexandra, beginning with his upbringing as a Jew in Istanbul, his youth as a math prodigy and his studies at Go¨ttingen under David Hilbert. The dramatic turning point in his life came when he rejected an offer from Richard Courant of a professorship at Go¨ttingen so that he could be with the woman he loved in Romania. He claims he was so confident of his mathematical genius that he was certain he would succeed no matter where he was. In retrospect, this does not seem to have worked out for the best. It is therefore significant that he advises Alexandra to leave for the sake of her research even at the risk of losing her family.

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Her brother’s story is also one of a mathematician living with the consequences of his dubious decisions. Leonard has reason to question his decision to defect because of the disastrous effect it had on his sister’s life. However, Leonard also seems to represent the dangers of not making decisions. At one point, he is accused of being unwilling to commit to anything: Not to his unfulfilled dream of being a researcher himself, not to the clerical jobs he has chosen instead, not to his family nor to his hobby of writing monographs on the lives of under-appreciated mathematicians. Because the novel describes Leonard’s research on these monographs, the reader has the opportunity to learn about the controversies surrounding some historical mathematicians such as Olinde Rodrigues and Hermann Grassmann. The book also contains many long discussions of mathematics itself, including advanced mathematical terminology. Sometimes, the mathematics is used metaphorically (e.g., comparing the independence of the continuum hypothesis to the freedom people have to make choices in their lives). At one point, ‘‘2@0 ¼ @1 ’’ (the continuum hypothesis stated concisely in Cantor’s notation) appears in dialogue between Alexandra and Leonard, as if it were possible to typeset spoken words in TeX. Unusually for mathematics in fiction, this is all handled perfectly. With one small exception (a metaphor about a couple who have nothing in common described as ‘‘two sets with no union’’ rather than ‘‘two sets with no intersection’’), the mathematical aspects of the text read as if they were written by a professional mathematician. The interaction between collaborators and the details of submitting papers for publication in mathematics journals are also entirely realistic. All of this leaves me wondering who the author, Robert Carr, really is and how he came to write this book. The impression of the author one gets from the brief biography on the back cover is intriguingly similar to the character of Leonard Jacobi. Like Leonard, Carr himself fled communist Romania, worked in a field that applies mathematics (aerospace engineering, rather than economics), and now lives in Toronto, Canada, where he writes

about mathematicians. At one point in the novel, Leonard tells his sister, ‘‘If I can’t be a mathematician, I’ll be ... a writer about mathematicians. It’s not the real thing, I’ll have only one foot in it, maybe only half a foot, but it’s better than no foot at all.’’ Perhaps this also explains how an engineer came to write a novel with so much mathematics in it. In many ways, Continuums is reminiscent of Uncle Petros and Goldbach’s Conjecture by Apostolos Doxiadis [3]. Both novels involve an aging, reclusive mathematician from the Near East recalling the successes and failures of his youth. Moreover, if my suspicion about Carr is correct, each book also represents the author’s own explanation of why he did not become a research mathematician despite his interest in the subject. Although it represents an entirely different genre of literature, R. J. Stern’s Goldman’s Theorem is similar in some ways to Continuums. Both of these novels involve the mathematics department at the Universite´ de Montre´al, both discuss Jewish culture, both contain a sexual relationship between two mathematicians, and both concern the proof of a famous theorem. Most importantly, both have a verisimilitude that is only possible when the author has an intimate and personal knowledge of the mathematics community. Perhaps this is less surprising in the case of Goldman’s Theorem, since Stern is a mathematics professor at Concordia University in Montre´al. However, Goldman’s Theorem is a broad farce, parodying the life of math professors by presenting outlandish characters and scenarios: The department chair who seems unable to utter a sentence free of obscenities, the dean who models herself after Elizabeth I, and the affectionate cougar who

voluntarily takes up residence at the protagonist’s home. The only thing in Continuums that comes close to such hyperbole is Aroso’s recollections of attending a dinner at David Hilbert’s house along with a young John von Neumann and being recognized by all as the smartest person in the room. (This is certainly a fantasy for an aspiring mathematician, although I am not certain whether we are expected to accept it uncritically or imagine that it involves some immodest exaggeration.) The title character of Stern’s novel is a ‘‘star’’ of math research, proudly stolen away from another university by the little known (and fictional) University of Northern Vermont. In this regard, he is very much like Alexandra JacobiSemeu at the nearby Universite´ de Montre´al, but that is where the similarity ends: In contrast to Alexandra’s research success, Goldman develops an infamous reputation for being entirely unproductive. In fact, the main character of the novel is not Goldman but his old friend Aitch Singleton. It was Singleton who helped bring his famously brilliant colleague to UNV, and it is he who is responsible for arranging the media circus that ensues when Goldman, under pressure to produce something spectacular, announces a proof of one of the famous Millennium problems. The wacky scenarios and quixotic characters are like those one might expect in a TV sit-com, but they are recognizable as caricatures of real people and situations to any reader who has spent time in a university math department. The mathematics which arises in this otherwise zany tale (including discussions of nonsmooth analysis and effective computability of the Travelling Salesman Problem) is undistorted and accurate.

The representations of mathematicians in many works of fiction are entirely unrealistic. It is refreshing for me to review two new books written by authors who really understand the culture of mathematics. They are different enough from each other that they may appeal to entirely different audiences. Goldman’s Theorem is an extremely entertaining burlesque of university life. Continuums does not sugarcoat or oversimplify either the moral dilemmas or the mathematics it describes, and so achieves an inspiring philosophical depth. In my opinion, Continuums is a more mature book than Goldman’s Theorem. This is not to say that Continuums has explicit sex and violence and so is suitable only for mature audiences. Quite the contrary, it is Goldman’s Theorem that has plenty of sex and violence. Carr’s book bravely ignores the marketing pressure to include much of these popular motifs and instead offers a thoroughly serious inspection of the human condition. However, both are recommended reading for anyone who would enjoy seeing a literary depiction of the world as it appears to mathematicians. REFERENCES

[1] Bajo, David The 351 Books of Irma Arcuri (2008) Viking Adult, New York [2] http://kasmana.people.cofc.edu/MATHFICT [3] Doxiadis, Apostolos Uncle Petros and Goldbach’s Conjecture (1992) Bloomsbury, New York & London

Department of Mathematics College of Charleston Charleston, SC 29424-0001 USA e-mail: [email protected]

Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009

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Osmo Pekonen, Editor

Group Theory in the Bedroom, and Other Mathematical Diversions by Brian Hayes NEW YORK: HILL AND WANG, 2008, 269 PP. US $25.00, ISBN-10: 0-8090-5219-9, ISBN-13: 978-0-8090-5219-6 REVIEWED BY JEANINE DAEMS AND IONICA SMEETS1

hen we first heard about Group Theory in the Bedroom, we became very enthusiastic. We were reminded of Mathematics and Sex by Clio Cresswell. We enjoyed that book a few years ago, but it contained too little mathematics to our taste (and surprisingly little about sex, for that matter). The title of Hayes’s book sounded very promising: More serious mathematics in the bedroom! However, this bedroom does not appear until the final chapter of the book, which deals with mattress flipping. And in that chapter, the author soon turns to group theory in the garage. We understand why the author chose this eye-catching title, even though it does not really cover the contents. In the early 1980s, Brian Hayes was working as an editor at Scientific American magazine. The magazine wanted to launch a new monthly column called Computer Recreations, and although he had no real knowledge of computers at the time, Hayes volunteered to write it. But he managed, and as he writes in the preface, ‘‘I discovered that the computer is not like the violin: It doesn’t take inborn genius or a lifetime of practice to get sweet music out of it.’’ Hayes wrote the column for just a few months, but his newly developed interest stayed, and later he wrote some pieces for Computer Language and The Sciences. Since 1993, he

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has written another column for American Scientist. The essays in Group Theory in the Bedroom are all reprints, with one appearing in The Sciences and the rest appearing in American Scientist. Some of the essays are over 10 years old and show their age. Hayes has added a section called Afterthoughts to each essay, in which he describes more recent developments and discusses some of the reactions he got from readers after first publication. The topics of the essays are quite diverse, although all of them have an algorithmic and/or mathematical flavor. Subjects include: The astronomical clock of Strasbourg Cathedral; randomness; statistics of wars; the history of gears and their relation to computing; the ternary system; and, obviously, group theory. Hayes is no more a mathematician than a computer scientist, but his fascination for the subject shows and he knows his audience well. ‘‘I’m not a mathematician, but I’ve been hanging around with some of them long enough to know how the game is played. Once you’ve solved a problem, the next step is to generalize it beyond all recognition.’’ Your two reviewers could not agree on one favorite chapter. Jeanine really enjoyed The Clock of Ages. This essay was written at the end of 1999, when people were preparing themselves for the upcoming New Year. ‘‘As the world spirals on toward 01-01-00, survivalists are hoarding cash, canned goods, and shotgun shells. It’s not the Rapture or the Revolution they await, but a technological apocalypse. Y2K!’’ Of course, the apocalypse did not happen on January 1, 2000. But how could the computer programmers of the 1960s and 1970s fail to look beyond 1999? Hayes gives them the benefit of the doubt. No programmer back then would think his programs would still be in use by the year 1999. The important questions of this chapter are whether there is any sense in building things to last, and to what extent? A very remarkable example of something that was definitely built to last for a very long time is the astronomical clock of Strasbourg Cathedral

(see ‘‘On Picturing the Past: Arithmetic and Geometry as Wings of the Mind’’ by Volker Remmert, this magazine, Vol. 31, No. 3). In its present form, it was started up in 1574. It is more an astronomical and calendar computer than a clock. It keeps track of a host of objects and events: The positions of 5,000 stars, the six inner planets, the current phase of the moon, sidereal time, local solar time, local lunar time, mean solar time, the present year, the day of the year (including February 29th in leap years), and the ‘‘movable feasts’’ of the ecclesiastical calendar among others. All this is done with gears, with the clock’s error being less than a second per century. Schwilgue´, the maker of the clock, was thinking long-term. He included parts in the leap-year mechanism that engage only every 400 years and that were first tested in the year 2000. The clock can represent the years until 9999, but Schwilgue´ suggested that after that there just should be a ‘‘1’’ painted on the left of the thousands digit. The existence of a clock like this raises the question: Is the building of multimillennial machines a good idea? Hayes is doubtful. Ionica found the story about the clocks rather dull, but she loved Inventing the Genetic Code. Hayes describes how physicists and mathematicians in the early 1950s tried to break the code of DNA right after biologists discovered that the language of the double helix consisted of just four letters: A, T, G and C. The big question was how these four letters coded 20 different amino acids. Many beautiful coding schemes were invented, very efficient codes with nice symmetries and error-correcting possibilities. With the discovery of the actual encoding scheme, Hayes admits that he was rather disappointed by nature’s real solution. He jokes that we might have been better off with the genetic code devised by one of the mathematicians: ‘‘Life would be a lot more reliable if Solomon Golomb were in charge.’’ We both liked the final chapter of the book, the one that gave the book its title. It presents a nice way to

1 Jeanine Daems and Ionica Smeets–often referred to as The Math Girls (wiskundemeisjes in Dutch)–are two female Dutch Ph.D students of mathematics who started, in March 2006, a hugely successful website devoted to cultural aspects of mathematics: http://www.wiskundemeisjes.nl. They have also appeared as celebrities on TV shows and in other media in The Netherlands.–O. P.

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THE MATHEMATICAL INTELLIGENCER Ó 2009 The Author(s). This article is published with open access at Springerlink.com

explain group theory. Group theory helps when the author can’t sleep at night: ‘‘Having run out of sheep the other night, I found myself counting the ways to flip a mattress.’’ In the long intervals between seasonal flips, he always forgets which way he flipped the mattress the last time, so he asks if there is a so-called golden rule for mattress flipping, i.e., a consistent rule telling you what to do that results in the mattress cycling through all its possible configurations. Using the transformations one could perform while flipping a mattress into another proper position and the structure of the mattress’s symmetry group, he answers the question. Then the author turns to group theory in the garage, asking for a similar golden rule for rotating the tires of a car. The structures of the Klein 4group and the cyclic 4-group are explained in the process. Hayes even discusses ways to flip a hypothetical cubical mattress, daring the readers to do the same with a 4- or even moredimensional cube. However, in the ‘‘Afterthoughts,’’ it becomes clear even this is not enough generalization for mathematicians. Readers have written him about polygon-shaped mattresses, circular mattresses and even a Mo¨bius mattress. It is not very clear what kind of audience this book aims at; we think a potential reader should be familiar with algorithmic thinking and some mathematics. Hayes is not wholly

consistent in the prior knowledge he assumes his readers have. Most explanations of the mathematics and algorithms are very clear, but some may be too brief for nonmathematicians. Luckily, his examples are very well chosen and most of them are appealing. For example, ‘‘To appreciate the value of randomness, imagine a world without it. What would replace the referee’s coin flip at the start of a football game?’’ Our main criticism is that 12 of these essays is too much of the same. Hayes likes computer simulations a bit more than we do, and after a few chapters we started groaning when he proudly presented another homemade diagram. At one point we were also slightly annoyed with his lack of understanding of the mathematics. In the (highly enjoyable) chapter The Easiest Hard Problem, Hayes counts the number of perfect partitions of a set of n integers. He nicely illustrates this problem with picking teams for a ball game where you want the teams to be as equal as possible. In mathematical terms: Given a set of integers, you want to divide it into two subsets that have the same sum of values. If the sum of all values in the original set is odd, then this is impossible. In this case, a perfect partition is given by two subsets whose sums differ by exactly one. Of course, Hayes could not resist doing a bunch of computer experiments, and he was surprised that ‘‘sets whose sum is an odd number have

about twice as many partitions, on average, as similar sets whose sum is an even number.’’ He emailed some top-range mathematicians about this ‘‘strange’’ phenomenon and they kindly helped him solve this ‘‘mystery.’’ This example is the exception, though. Hayes took a lot of effort to delve into the subjects, and his view of mathematics from the outside is refreshing. In many cases, it is charming to see his personal struggle with the material. We recommend this book to people who are already know a bit about mathematics and algorithms and who love to read about problems and questions in the real world and its generalizations. We think that, for those readers, the book is extremely suitable for reading in the bedroom. So, at least in this sense, the title fits! OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Mathematical Institute University of Leiden P.O. Box 9512, 2300 RA Leiden The Netherlands e-mail: [email protected] e-mail: [email protected]

Ó 2009 The Author(s). This article is published with open access at Springerlink.com, Volume 31, Number 4, 2009

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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2001 2001 saw the publication by Springer of my book Stamping Through Mathematics, depicting about 400 mathematical stamps with accompanying text. Since then, several new mathematical stamps have appeared, and the next few Stamp Corners will present some of these.

Binary Code is the means used for representing text or processing instructions for a computer. It is based on the binary number system of two digits 0 and 1, in which a decimal number such as 26 = (2 9 101) + (6 9 100) is represented by the binary number 11010 =

Binary Code (Bosnia)

(1 9 24) + (1 9 23) + (0 9 22) + (1 9 21) + (0 9 20).

Pierre Fermat (1601–1665) made substantial contributions to the development of analytic geometry but is mainly remembered for his contributions to number theory— notably, his ‘little theorem’ that ap - a is divisible by p for any natural number a and prime number p; his observation that any prime of the form 4n + 1 can be written as the sum of two squares; and his claim (the ‘last theorem’) that the equation xn + yn = zn has no nontrivial solutions in integers when n [ 2.

Fullerenes or buckyballs, named after the architect Buckminster Fuller, are chemical molecules that take the form of a polyhedron made up of pentagons and hexagons. All of these have just 12 pentagons, and the smallest, C60, corresponds to a truncated icosahedron.

Werner Heisenberg (1901–1976) took an algebraic approach to quantum theory, repre-

senting quantities such as position, momentum and energy by infinite matrices. Using the fact that matrix multiplication is noncommutative, Heisenberg deduced his famous Uncertainty Principle that it is theoretically impossible to determine the position and momentum of an electron at the same time.

Otto Yulievich Schmidt (1891–1956) was a distinguished algebraist who wrote a celebrated book on the theory of groups and contributed to the ‘Krull-Schmidt Theorem’ on groups with the ascending and descending chain conditions. A well-known polar explorer, he was rescued from a stranded ice-breaker during a polar scientific expedition.

Nasiraddin Tusi (1201-1274) constructed the first modern astronomical observatory and wrote influential treatises on logic, ethics and theology. He also investigated the sine rule for triangles and Euclid’s parallel postulate.

Pierre Fermat (France) Fullerene (Great Britain)

Werner Heisenberg (Germany)

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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Otto Schmidt (Belarus)

THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

Nasiraddin Tusi (Azerbaijan)