No title

Letter to the Editors

Bourbaki’s Structures and Structuralism The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

have read with interest Osmo Pekonen’s review of Amir Aczel’s book, The Artist and the Mathematician, in The Mathematical Intelligencer, Vol. 31 (2009), No. 3. I had already read the book and had been surprised again and again by Aczel’s complete freedom with historical facts (see, for example, his comparison of Andre´ Weil, born in 1906, with Alexander Grothendieck, born in 1928). But here I will concentrate on just one important point: The supposed relation of Bourbaki’s structures to structuralism. This is a pure intellectual fraud, propagated by many people from the social sciences and repeated by Aczel. Bourbaki’s structures and structuralism had independent births, even if we wave hands and refer to the Zeitgeist.

I

But let us be precise. The idea of structure appeared in mathematics before Bourbaki in the theory of abstract algebra of commutative fields (E. Steinitz, ‘‘Algebraische Theorie der Ko¨rper,’’ Jour. fu¨r die reine und angewandte Mathematik 137 (1910), 167– 309), in linear algebra, and also in the beginning of the theory of continuous groups with Elie Cartan. Bourbaki was directly inspired by them (Pierre Cartier, personal communication, April 2010). The word ‘‘structure’’ appeared independently in Claude Levi-Strauss’s book Anthropologie Structurale (1958). When structuralism became a fashion in the 1960s, referring to Bourbaki in structuralist essays was a way of giving some scientific credit and weight to works of variable quality. When I asked Claude Levi-Strauss about the origin of the word ‘‘structure’’ in his work, he answered (letter to the author, Nov. 16, 1990): ‘‘Ne croyez pas un instant que Bourbaki m’ait emprunte´ le terme ‘‘ structure’’ ou le contraire, il me vient de la linguistique et plus pre´cise´ment de l’Ecole de Prague.’’ (Do not believe for one minute that Bourbaki borrowed the word ‘‘structure’’ from me, or the contrary; it came to me from linguistics, more precisely, from the School of Prague.) This, I hope, puts an end to any discussion about the origin of ‘‘structures.’’

Jean-Michel Kantor Institut de Mathe´matiques de Jussieu 4, Place Jussieu, Case 24775005 Paris, France e-mail: [email protected]

Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

1

Note

Young Gauss Meets Dynamical Systems

Seventy years ago, A. L. O’Toole [11] recommended that teachers avoid the above derivation of the formula (2), considering it a mere trick that offers no insight. Instead, he called attention to the fundamental theorem of summation, a discrete variant of the Leibniz-Newton theorem: If there is a function f(x) such that ak = f(k + 1) - f(k) for k 2 f1; . . .; ng; then

CONSTANTIN P. NICULESCU

n X

M

ost people are convinced that doing mathematics is something like computing sums such as S ¼ 1 þ 2 þ 3 þ    þ 100:

But we know that one who does this by merely adding terms one after another is not seeing the forest for the trees. An anecdote about young Gauss tells us that he solved the above problem by noticing that pairwise addition of terms from opposite ends of the list yields identical intermediate sums. This famous story is well told by Hayes in [5], with references. A very convenient way to express Gauss’s idea is to write down the series twice, once in ascending and once in descending order, 1 þ 100 þ

2 99

þ 3 þ 98

þ þ

   þ 100  þ 1

and to sum columns before summing rows. Thus 2S ¼ ð1 þ 100Þ þ ð2 þ 99Þ þ    þ ð100 þ 1Þ ¼ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl 101 þ 101ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl þ    þ 101 ffl}

ak ¼ f ðn þ 1Þ  f ð1Þ ¼ f ðkÞjnþ1 1 :

k¼1

Indeed, this theorem provides a unifying approach for many interesting summation formulae (including those for arithmetic progressions and geometric progressions). However, determining the nature of the function f(x) is not always immediate. In the case of an arithmetic progression (1) we may choose f(x) as a second-degree polynomial, namely, r 3r f ðxÞ ¼ x 2 þ ða1  Þx þ C; 2 2 where C is an arbitrary constant. Though more limited, ‘‘Gauss’s trick’’ is much simpler, and besides, it provides a nice illustration of a key concept of contemporary mathematics, that of measurable dynamical system. Letting M ¼ f1; . . .; ng; we may consider the measurable space ðM; PðMÞ; lÞ; where P(M) is the power set of M and l is the counting measure on M, defined by the formula lðAÞ ¼ j Aj

100 times

for every A 2 PðMÞ:

Every real sequence a1 ; . . .; an of length n can be thought of as a function f : M ! R; given by f(k) = ak. Moreover, f is integrable with respect to l, and Z f ðkÞdl ¼ a1 þ    þ an :

¼ 10100; whence S ¼ 5050:

M

Of course, the same technique applies to any arithmetic progression a1 ; a2 ¼ a1 þ r; a3 ¼ a1 þ 2r; . . .; an ¼ a1 þ ðn  1Þr;

ð1Þ

T : M ! M;

and the result is the well-known summation formula nða1 þ an Þ : a1 þ a2 þ    þ an ¼ 2

ð2Þ

A similar idea can be used to sum up strings that are not necessarily arithmetic progressions. For example,       n n n a0 þ a1 þ    þ a ¼ 2n1 ða0 þ an Þ; 0 1 n n for every arithmetic progression a0 ; a1 ; . . .; an :

2

THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC

The main ingredient that makes possible an easy computation of the sum of an arithmetic progression is the existence of a nicely behaved map, namely, T ðkÞ ¼ n  k þ 1:

Indeed, the measure l is invariant under the map T in the sense that Z Z f ðkÞdl ¼ f ðT ðkÞÞdl ð3Þ M

M

regardless of the choice of f (for T is just a permutation of the summation indices). When f represents an arithmetic progression of length n, then there exists a positive constant C such that

f ðkÞ þ f ðT ðkÞÞ ¼ C;

for all k 2 M;

ð4Þ

and taking into account (3) we recover the summation formula (2) in the following equivalent form, Z 1 f ðkÞdl ¼ C j M j: 2 M The natural generalization of the reasoning above is to consider arbitrary triples (M, T, l), where M is an abstract space, l is a finite positive measure defined on a r-algebra R of subsets of M, and T : M ! M is a measurable map that is invariant under the action of l in the sense that (3) works for all f [ L1 (l). Such triples are usually called measurable dynamical systems. In this context, if f [ L1 (l) satisfies a formula like f ðT ðxÞÞ ¼ kf ðxÞ þ gðxÞ

ð5Þ

with k = 1, then the computation of $M f(x)dl, or rather of its expectation, Z 1 f ðtÞdlðtÞ; Eðf Þ ¼ lðMÞ M reduces to the computation of $M g(x)dl. For example, the integral of an odd function over an interval symmetric about the origin is zero; this corresponds to (5) for T(x) = -x, k = -1, and g = 0. Among the many practical implications of this remark, the following two seem especially important:

and Z

p=4

lnð1 þ tan xÞdx ¼

0

p ln 2 : 8

ð6Þ

In the first case, the measurable dynamical system under consideration is the triple consisting of the interval M = (0, ?), the map T(x) = 1/x, and the weighted Lebesgue dx measure 1þx 2 : The invariance of this measure with respect to T is assured by the change of variable formula, while the formula (5) becomes lnð1=xÞ ¼  ln x: In the second case, the measurable dynamical system is the triple ([0, p/4], p/4 - x, dx). For f ðxÞ ¼ lnð1 þ tan xÞ; the formula (5) becomes lnð1 þ tanðp=4  xÞÞ ¼  lnð1 þ tan xÞ þ ln 2 and thus Z Z p=4 lnð1 þ tan xÞdx ¼ 0

p=4

lnð1 þ tanðp=4  xÞÞdx

0

¼

Z

p=4

½ln 2  lnð1 þ tan xÞdx

0

¼

p ln 2  4

Z

p=4

lnð1 þ tan xÞdx; 0

whence (6). This formula admits a straightforward generalization: Z h lnð1 þ tan h tan xÞdx ¼ h lnðcos hÞ; 0

a) the Fourier series of any odd function is a series of sine functions; b) the barycenter of any body that admits an axis of symmetry lies on that axis. Two other instances of the formula (5) are Z 1 ln x dx ¼ 0 1 þ x2 0

for all h [ (-p/2, p/2). In the same manner we obtain the integral formulae Z Z p p p xf ðsin xÞdx ¼ f ðsin xÞdx; 2 0 0 Z

p

f ðsin xÞdx ¼ 2

0

Z

p=2

f ðsin xÞdx:

0

There is a relationship between the expectation of a function f and the values of the iterates of f under the action of T,

AUTHOR

......................................................................... CONSTANTIN P. NICULESCU received his

Ph.D. at the University of Bucharest; he has been teaching at the University of Craiova since 1976. He works on convex analysis (see his joint work with Lars-Erik Persson, Convex Functions and Their Applications), functional analysis, and dynamical systems. He also lectures on heuristic, the art and science of discovery and invention. His hobbies include reading, music, and gardening. Department of Mathematics University of Craiova Craiova, RO-200585 Romania e-mail: [email protected]

f;

f  T;

f  T 2 ; . . .;

expressed in the ergodic theorems. A sample is Weyl’s ergodic theorem; here M is the unit interval, l is the restriction of Lebesgue measure to the unit interval, and T : ½0; 1 ! ½0; 1 is the irrational translation defined by T ðxÞ ¼ fx þ ag; here fg denotes the fractional part and a [ 0 is some irrational number. The invariance of T is usually derived from the remark that the linear span of characteristic functions of subintervals of [0,1] is dense in L1 ð½0; 1Þ: Thus the verification of the invariance formula (3) reduces to the (trivial) case where f is such a characteristic function. The following result does not make use of the invariance property of T (but can be used to derive it).

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

3

T HEOREM 1. (Weyl’s Ergodic Theorem [10]). Suppose that a [ 0 is irrational. Then 1 1 NX f ðfx þ kagÞ ¼ N !1 N k¼0

Z

1

f ðtÞdt

lim

ð7Þ

0

for all Riemann integrable functions f : ½0; 1 ! R and all x 2 ½0; 1: P ROOF . It is easy to check that the above formula holds for each of the functions e2pint (n 2 ZÞ; and thus for linear combinations of them. By the Weierstrass approximation theorem (see [3]) it follows that the formula (7) actually holds for all continuous functions f : ½0; 1 ! C with f(0) = f(1). Now if I  ½0; 1 is a subinterval, then for each e [ 0 one can choose continuous real-valued functions g, h with g  vI  h such that Z 1 ðh  gÞdt\e: gð0Þ ¼ gð1Þ; hð0Þ ¼ hð1Þ and 0

continued fractions he considered the dynamical system consisting of the map  0 if x ¼ 0 ; ð8Þ G : ½0; 1Þ ! ½0; 1Þ; GðxÞ ¼ 1 if x 6¼ 0 x and the invariant measure dlðxÞ ¼

1 dx: ðlog 2Þð1 þ x Þ

In the variant of Lebesgue integrability, the convergence defined by the formula (7) still works, but only almost everywhere. This was noticed by A. Ya. Khinchin [6], but can be deduced also from another famous result, Birkhoff’s ergodic theorem, a large extension of Theorem 1. See [8] for details. It is Birkhoff’s result that reveals the true nature of the Gauss map (8) and a surprising property of continued fractions (first noticed by A. Ya. Khinchin [7]). A nice account of this story (and many others) may be found in the book of K. Dajani and C. Kraaikamp [2].

By the previous step we infer that 1 N 1 1 NX 1X vI ðfx þ kagÞ  lim vI ðfx þ kagÞ N !1 N N !1 N k¼0 k¼0 R1 R1 lies in ð 0 vI ðtÞdt  e; 0 vI ðtÞdt þ eÞ: As e [ 0 was arbitrarily fixed, this shows that the formula (7) works for vI (and thus for all step functions on [0,1]). The general case of a Riemann integrable function f can be settled in a similar way, by using Darboux integral sums.

lim

The convergence provided by Weyl’s ergodic theorem may be very slow. In fact, we already noticed that Z p=4 Z 1 lnðt þ 1Þ dt ¼ lnð1 þ tan xÞdx ¼ 0:27220. . .; t2 þ 1 0 0 whereas the approximating sequence in (7) offers this precision only for N [ 104. However, Weyl’s ergodic theorem has important arithmetic applications. A nice introduction is offered by the paper of P. Strzelecki [9]. Full details may be found in the monograph of R. Man˜e [8]. An inspection of the argument of Weyl’s ergodic theorem shows that the convergence (7) is uniform on [0,1] when f : ½0; 1 ! C is a continuous function with f(0) = f(1). It is worth mention that Gauss himself [4] was interested in the asymptotic behavior of dynamical systems involving the fractional part. In fact, in connection with the study of

4

THE MATHEMATICAL INTELLIGENCER

REFERENCES

[1] G. D. Birkhoff, Proof of the ergodic theorem, Proceedings of the National Academy of Sciences USA, 17 (1931), 656-660 [2] K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, The Mathematical Association of America, 2002. [3] K. R. Davidson and A. P. Donsig, Real Analysis with Real Applications, Prentice-Hall. Inc., Upper Saddle River, 2002. [4] C. F. Gauss, Mathematisches Tagebuch 1796-1814, Akademische Verlagsgesellschaft Geest & Portig K.G., Leipzig, 1976. [5] B. Hayes, Gauss’s Day of Reckoning. A famous story about the boy wonder of mathematics has taken on a life of its own, American Scientist 94 (2006), No. 3, pp. 200–205. Online: http:// www.americanscientist.org/template/AssetDetail/assetid/50686? &print=yes [6] A. Ya. Khinchin, Zur Birkhoffs Lo¨sung des Ergodenproblems, Math. Ann. 107 (1932), 485-488. [7] A. Ya. Khinchin, Metrische Kettenbruchprobleme, Compositio Math. 1 (1935), 361-382. [8] R. Man˜e, Ergodic Theory and Differentiable Dynamics, SpringerVerlag, 1987. [9] P. Strzelecki, On powers of 2, Newsletter European Mathematical Society No 52, June 2004, pp. 7-8. [10] H. Weyl, U¨ber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352. [11] A. L. O’Toole, Insights or Trick Methods?, National Mathematics Magazine, Vol. 15, No. 1 (Oct., 1940), pp. 35-38.

Viewpoint

One, Two, Many: Individuality and Collectivity in Mathematics MELVYN B. NATHANSON

The Viewpoint column offers readers of The Mathematical Intelligencer 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

was the way the story comported with the Romantic myth: Solitary genius, great accomplishment. This is a compelling narrative in science. We have the image of the young Newton, who watched a falling apple and discovered gravity as he sat, alone, in an orchard in Lincolnshire while Cambridge was closed because of an epidemic. We recall Galois, working desperately through the night to write down, before his duel the next morning, all of the mathematics he had discovered alone. There was Abel, isolated in Norway, his discovery of the unsolvability of the quintic ignored by the mathematical elite. And Einstein, exiled to a Swiss patent office, where he analyzed Brownian motion, explained the photoelectric effect, and discovered relativity. In a speech in 1933, Einstein said that being a lighthouse keeper would be a good occupation for a physicist. Stories such as these give Eric Temple Bell’s Men of Mathematics its hypnotic power, and inspire many young students to do research. Wiles did not follow the script perfectly. His initial manuscript contained a gap that was eventually filled by Wiles and his former student Richard Taylor. Within epsilon, however, Wiles solved Fermat in the best possible way. Intense solitary thought produces the best mathematics.

those of the author, and the publisher and editors-in-chief

Gel0 fand’s List

do not endorse them or accept responsibility for them.

Some of the greatest twentieth-century mathematicians, such as Andre´ Weil and Atle Selberg, had few joint papers. os Others, like Paul Erd} os and I. M. Gel0 fand, had many. Erd} was a master collaborator, with hundreds of co-authors. (Full disclosure: I am one of them.) Reviewing Erd} os’s number-theory papers, I find that in his early years, from his first published work in 1929 through 1945, most (60%) of his 112 papers were singly authored, and that most of his stunningly original papers in number theory were papers that he wrote by himself. In 1972–1973 I was in Moscow as a post-doc studying with Gel0 fand. In a conversation one day he told me there were only ten people in the world who really understood representation theory, and he proceeded to name them. It was an interesting list, with some unusual inclusions, and some striking exclusions. ‘‘Why is X not on the list?’’ I asked, mentioning the name of a really famous representation theorist. ‘‘He’s just an engineer,’’ was Gel0 fand’s disparaging reply. But the tenth name on the list was not a name, but a description: ‘‘Somewhere in China,’’ said Gel0 fand, ‘‘there is a young student, working alone, who understands representation theory.’’

Viewpoint should be submitted to one of the editors-inchief, Chandler Davis and Marjorie Senechal.

ermat’s last theorem’’ is famous because it is old and easily understood, but it is not particularly interesting. Many, perhaps most, mathematicians would agree with this statement, though they might add that it is nonetheless important because of the new mathematics created in the attempt to solve the problem. By solving Fermat, Andrew Wiles became one of the world’s best known mathematicians, along with John Nash, who achieved fame by being crazy, and Theodore Kaczynski, the Unabomber, by killing people. Wiles is known not only because of the problem he solved, but also because of how he solved it. He was not part of a corporate team. He did not work over coffee, by mail, or via the Internet with a group of collaborators. Instead, for many years, he worked alone in an attic study and did not talk to anyone about his ideas. This is the classical model of the artist, laboring in obscurity. (Not real obscurity, of course, since Wiles was, after all, a Princeton professor.) What made the solution of Fermat’s last theorem so powerful in the public and scientific imagination

F

‘‘

Bers Mafia A traditional form of mathematical collaboration is to join a school. Analogous to the political question, ‘‘Who’s your Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

5

rabbi?’’ (meaning ‘‘Who’s your boss? Who is the guy whom you support and who helps you in return?’’), there is the mathematical question, ‘‘Who’s your mafia?’’ The mafia is the group of scholars with whom you share research interests, with whom you socialize, whom you support, and who support you. In the New York area, for example, there is the self-described ‘‘Ahlfors-Bers mafia,’’ beautifully described in a series of articles about Lipman Bers that were published in a memorial issue of the Notices of the American Mathematical Society in 1995. Bers was an impressive and charismatic mathematician at New York University and Columbia University who created a community of graduate students, post-docs, and senior scientists who shared common research interests. Being a member of the Bers mafia was valuable both scientifically and professionally. As students of the master, members spoke a common language and pursued common research goals with similar mathematical tools. Members could easily read, understand, and appreciate each others’ papers, and their own work fed into and complemented the research of others. Notwithstanding sometimes intense internal group rivalries, members would write recommendations for each others’ job applications, review their papers and books, referee their grant proposals, and nominate and promote each other for prizes and invited lectures. Being part of a school made life easy. This is the strength and the weakness of the collective. Members of a mafia, protected and protecting, competing with other mafias, are better situated than those who work alone. Membership guarantees moderate success, but makes it hard to create an original style.

The Riemann Hypothesis The American Institute of Mathematics organized its first conference, ‘‘In Celebration of the Centenary of the Proof of the Prime Number Theorem: A Symposium on the Riemann Hypothesis,’’ at the University of Washington on August 12–15, 1996. According to its website, ‘‘the American Institute of Mathematics, a nonprofit organization, was founded in 1994 by Silicon Valley businessmen John Fry and Steve Sorenson, longtime supporters of mathematical research.’’ The story circulating at the meeting was that the businessmen funding AIM believed that the way to prove the Riemann hypothesis was the corporate model: To solve a problem, put together the right team of ‘‘experts’’ and they will quickly find a solution. At the AIM meeting, various experts (including Berry, Connes, Goldfeld, Heath-Brown, Iwaniec, Kurokawa, Montgomery, Odlyzko, Sarnak, and Selberg) described ideas for solving the Riemann hypothesis. I asked one of the organizers why the celebrated number theorist Z was not giving a lecture. The answer: Z had been invited, but declined to speak. Z had said that if he had an idea that he thought would solve the Riemann hypothesis, he certainly would not tell anyone because he wanted to solve it alone. This is a simple and basic human desire: Keep the glory for yourself. Thus, the AIM conference was really a series of lectures on ‘‘How not to solve the Riemann hypothesis.’’ It was a meeting of distinguished mathematicians describing methods that 6

THE MATHEMATICAL INTELLIGENCER

had failed, and the importance of the lectures was to learn what not to waste time on.

The Polymath Project The preceding examples are prologue to a discussion of a new, widely publicized Internet-based effort to achieve massive mathematical collaboration. Tim Gowers began this experiment on January 27, 2009, with the post ‘‘Is massively collaborative mathematics possible?’’ on his webblog http:// gowers.wordpress.com. He wrote, ‘‘Different people have different characteristics when it comes to research. Some like to throw out ideas, others to criticize them, others to work out details, others to re-explain ideas in a different language, others to formulate different but related problems, others to step back from a big muddle of ideas and fashion some more coherent picture out of them, and so on. A hugely collaborative project would make it possible for people to specialize. . .. In short, if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well.’’ This is the fundamental idea, which he restated explicitly as follows: ‘‘Suppose one had a forum . . . for the online discussion of a particular problem. . .. The ideal outcome would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed amongst bits of the brains of lots of interlinked people.’’ What makes Gowers’s polymath project noteworthy is its promise to produce extraordinary results—new theorems, methods, and ideas—that could not come from the ordinary collaboration of even a large number of first-rate scientists. Polymath succeeds if it produces a super-brain. Otherwise, it’s boring. In appropriately pseudo-scientific form, I would restate the ‘‘Gowers hypothesis’’ as follows: Let qual(w) denote the quality of the mathematical paper w, and let Qual(M) denote the quality of the mathematical papers written by the mathematician M. If w is a paper produced by the massive collaboration of a set M of mathematicians, then qualðwÞ [ supfQualðMÞ : M 2 Mg:

ð1Þ

Indeed, a reading of the many published articles and comments on massive collaboration suggests that its enthusiastic proponents believe the following much stronger statement: lim ðqualðwÞ  supfQualðMÞ : M 2 MgÞ ¼ 1:

jMj!1

ð2Þ

Superficially, at least, this might seem plausible, especially when suggested by one Fields Medalist (Gowers), and enthusiastically supported by another (Terry Tao). I assert that (1) and (2) are wrong, and that the opposite inequality is true: qualðwÞ\ supfQualðMÞ : M 2 Mg:

ð3Þ

First, some background. Massive mathematical collaboration is one of several recent experiments in scientific social networking. The ongoing projects to write computer code for GNU/Linux and to contribute articles on science and mathematics to Wikipedia are two successes. Another example is the DARPA Network Challenge. On December 5,

2009, the Defense Advanced Research Projects Agency (DARPA) tethered 10 red weather balloons at undisclosed but readily accessible locations across the United States, each balloon visible from a nearby highway, and offered a $40,000 prize to the first individual or team that could correctly give the latitude and longitude of each of the 10 balloons. In a press release, DARPA wrote that it had ‘‘announced the Network Challenge . . . to explore how broad-scope problems can be tackled using social networking tools. The Challenge explores basic research issues such as mobilization, collaboration, and trust in diverse social networking constructs and could serve to fuel innovation across a wide spectrum of applications.’’ In less than nine hours, the MIT Red Balloon Challenge Team won the prize. According to the DARPA final project report, ‘‘The geolocation of ten balloons in the United States by conventional intelligence methods is considered by many to be intractable; one senior analyst at the National Geospatial Intelligence Agency characterized the problem as impossible. A distributed human sensor approach built around social networks was recognized as a promising, nonconventional method of solving the problem, and the Network Challenge was designed to explore how quickly and effectively social networks could mobilize to solve the geo-location problem. The speed with which the Network Challenge was solved provides a quantitative measure for the effectiveness of emerging new forms of social media in mobilizing teams to solve an important problem.’’ The DARPA Challenge shows that, in certain situations, scientific networking can be extraordinarily effective, but there is a fundamental difference between the DARPA Network Challenge and massive mathematical collaboration. The difference is the difference between stupidity and creativity. The participants in the DARPA challenge had a stupid task to perform: Look for a big red balloon and, if you see one, report it. No intelligence required. Just do it. The widely dispersed members of the MIT team, like a colony of social ants, worked cooperatively and productively for the greater good, but didn’t create anything. Mathematics, however, requires intense thought. Individual mathematicians do have ‘‘to think all that hard.’’ Mathematicians create. In a recent magazine article (‘‘Massively collaborative mathematics,’’ Nature, October 15, 2009), Gowers and Michael Nielsen proclaimed, ‘‘The collaboration achieved far more than Gowers expected, and showcases what we think will be a powerful force in scientific discovery—the collaboration of many minds through the Internet.’’ They are wrong. Massive mathematical collaboration has so far failed to achieve its ambitious goal. Consider what massive mathematical collaboration has produced, and who produced it. Gowers proposed the problem of finding an elementary proof of the density version of the Hales–Jewett theorem, which is a fundamental result in combinatorial number theory and Ramsey theory. In a very short time, the blog team came up with a proof, chose a nom de plume (‘‘D. H. J. Polymath’’), wrote a paper, uploaded it to arXiv, and submitted it for publication. The paper is: D. H. J. Polymath, ‘‘A new proof of the density Hales–Jewett theorem,’’ arXiv: 0910.3926.

The abstract describes it clearly: ‘‘The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-coloring of the n-dimensional grid {1, . . . , k}n contains a combinatorial line. . .. The Hales–Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemere´di’s theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be.’’ A second, related paper by D. H. J. Polymath, ‘‘Density Hales–Jewett and Moser numbers,’’ arXiv: 1002.6374, has also been posted on arXiv. These papers are good, but obviously not of Fields Medal quality, so Nathanson’s inequality (3) is satisfied. A better experiment might be massive collaboration without the participation of mathematicians in the Fields Medal class. This would reduce the upper bound in Gowers’s inequality (1), and give it a better chance to hold. It is possible, however, that Internet collaboration can succeed only when controlled by a very small number of extremely smart people. Certainly, the leadership of Gowers and Tao is a strong inducement for a mathematician to play the massive participation game, because, inter alia, it allows one to claim joint authorship with Fields medalists. After writing the first paper, Gowers blogged, ‘‘Let me say that for me personally this has been one of the most exciting six weeks of my mathematical life. . .. There seemed to be such a lot of interest in the whole idea that I thought that there would be dozens of contributors, but instead the number settled down to a handful, all of whom I knew personally.’’ In other words, this became an ordinary, not a massive, collaboration. This is exactly how it was reported in Scientific American. On March 17, 2010, Davide Castelvecchi wrote, ‘‘In another way, however, the project was a bit of a disappointment. Just six people—all professional mathematicians and ‘usual suspects’ in the field—did most of the work. Among them was another Fields medalist and prolific blogger, Terence Tao of the University of California, Los Angeles.’’ Human beings are social animals. We enjoy working together, through conversation, letter writing, and e-mail. (More full disclosure: I’ve written many joint papers. One paper even has five authors. Collaboration can be fun.) But massive collaboration is supposed to achieve much more than ordinary collaboration. Its goal, as Gowers wrote, is the creation of a super-brain, and that won’t happen. Mathematicians, like other scientists, rejoice in unexpected new discoveries, and delight when new ideas produce new methods to solve old problems and create new ones. We usually don’t care how the breakthroughs are achieved. Still, I prefer one person working alone to two or three working collaboratively, and I find the notion of massive collaboration esthetically appalling. Better a discovery by an individual than the same discovery by a group. I would guess that even in the already interactive twentieth century, most of the new ideas in mathematics originated in papers written by a single author. A glance at MathSciNet Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

7

shows that only three of Tim Gowers’s papers have a coauthor. (But Terry Tao responded to this observation by noting that half of his many papers are collaborative.) In a contribution to a ‘‘New Ideas’’ issue of The New York Times Magazine on December 13, 2009, Jordan Ellenberg described massive mathematical collaboration with journalistic hyperbole: ‘‘By now we’re used to the idea that gigantic aggregates of human brains—especially when allowed to communicate nearly instantaneously via the Internet—can carry out fantastically difficult cognitive tasks, like writing an encyclopedia or mapping a social network. But some problems we still jealously guard as the province of individual beautiful minds: Writing a novel, choosing a spouse, creating a new mathematical theorem. The Polymath experiment suggests this prejudice may need to be rethought. In the near future, we might talk not only about the wisdom of crowds but also of their genius.’’ It is always good to rethink old prejudices, but sometimes the re-evaluation confirms the truth of the original

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prejudice. I predict that massive collaboration will produce useful results, but it will not meet the standard that Gowers set: No mathematical ‘‘super-brain’’ will evolve on the Internet and create new theories yielding brilliant solutions to important unsolved problems. Recalling Mark Kac’s famous division of mathematical geniuses into two classes, ordinary geniuses and magicians, one can imagine that massive collaboration will produce ordinary work and, possibly, in the future, even work of ordinary genius, but not magic. Work of ordinary genius is not a minor accomplishment, but magic is better.

Department of Mathematics Lehman College (CUNY) New York, NY 10468 USA e-mail: [email protected]

Viewpoint

Analyzing Massively Collaborative Mathematics Projects DINESH SARVATE, SUSANNE WETZEL, AND WAYNE PATTERSON The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editors-inchief endorses or accepts responsibility for them. Viewpoint should be submitted to one of the editorsin-chief, Chandler Davis or Marjorie Senechal.

T

he American late-night philosopher, David Letterman, sometimes had a segment on his television show entitled ‘‘Is This Anything?’’

In the increasingly rare segment, the stage curtain is raised to reveal an individual or team performing an unusual stunt, often accompanied by music from the CBS Orchestra … after about thirty seconds the curtain is lowered and Letterman discusses with [Paul] Shaffer whether the act was ‘something’ or ‘nothing’ … it was resurrected on the March 22, 2006, episode. A man balanced himself on a ladder and juggled: Paul voted a clear ‘nothing,’ and Dave was going to vote ‘something’ before he noticed a safety mat. Dave then concurred with Paul [1]. We have been developing techniques to analyze socalled ‘‘massively collaborative mathematics’’ or ‘‘polymath projects’’. These deserve attention if it is possible that new results can be obtained using them that could not be obtained by traditional methods. We will argue that there are other motivations for studying the new approach.

There has already been a great deal of discussion about the process, and some new results. But we may ask the question of the esteemed Mr. Letterman: ‘‘Is this anything?’’

The New Way is Opened There is hardly a scholarly pursuit that has not been affected in the last fifteen years by the aids to communication and inquiry afforded by the Web and other electronic innovations. It was in this context that Timothy Gowers, Cambridge mathematician and 1998 Fields Medal winner, put forth the challenge last year to the mathematics community to re-examine the way it conducts research [2]. He asked, ‘‘Is it possible to discover new theorems in mathematics, or improve on the proofs of existing theorems, by using a ‘polymath process,’ joining many researchers into a unit for a designated research objective? Can such a unit become more powerful than any one researcher?’’ This is quite unlike our usual behavior. Many mathematicians have been very successful in research working completely alone; others have profited from collaboration with one or two partners. But it is rare that more than three mathematicians work jointly, and in fact many in the field doubt that larger collaborations can be advantageous. Thus Gowers’s challenge ran counter to a centuriesold tradition, which might have made it unappealing. But it turned out that many mathematicians were ready and willing to join a massively collaborative mathematics project, given a good problem and clear rules. Unquestionably the invitation owed some of its attractiveness to the fact that it came from Gowers, a Fields Medalist, and to the eager support of another Fields Medalist, Terence Tao of UCLA. The experiment in massive collaboration does follow a trend that has reaped benefits in a number of other fields. ‘‘Crowd science’’ has had an impact on astronomy and is also being used in biology, oceanography, and environmental sciences [3]. However, the main motive for crowd science is that the phenomenon to be studied simply involves such massive datasets that the analysis is beyond the capacity of one person or a small group of persons. It is rare indeed that a problem in mathematics remotely resembles problems of that sort. But there is one remarkable difference and advantage in choosing massive collaboration as our topic of research: With the protocol established by the polymath leaders, all communication is maintained and made publicly available. This gives us data we can analyze by graph-theoretical

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models, in hopes of identifying crucial steps in progress toward the solution. Thus, the purpose of this article is to look for techniques giving insights into the functioning of the polymath approach to research, but the secondary motivation is the hope that the great visibility of the polymath process will give insights applicable also to mathematical research overall.

The Nature of Discovery in Mathematics The process of discovery in mathematics is not well understood. Certainly what is called the ‘‘scientific method’’ is not commonly used in mathematical discovery, nor is it clear that it should be. In the physical or life sciences, one may begin by formulating a hypothesis, and go on to gather and analyze data, getting experimental results in agreement or disagreement with the hypothesis. In mathematics, the amassing of data —instances of a conjecture, say— may not get one very far toward solving a problem. Another difference is that in the physical or life sciences, the role of any of the participants in a research project is normally well defined, whereas collaborative relations in mathematics may take very diverse forms. Consequently, one question that can be raised when trying to understand the nature of discovery in mathematics is the manner in which mathematicians communicate and work together. Perhaps the situation is much the same in related fields such as computer science and theoretical physics. To substantiate our statements about communication in these disciplines, we note that multiple authorship of scholarly articles is relatively very infrequent both in mathematics and in computer science (see Table 1). In the last ten volumes of the Annals of Mathematics [4], for example, 80% of the articles have one or two authors; in the last two volumes of the Journal of the Association for Computing Machinery [5], more than 86% have three or

fewer authors. A very different picture is seen in the most recent issue of the medical journal The Lancet [6]; the lead article has 16 authors, and the average is 7.5 authors per article throughout the issue. The average number of authors per article in the Annals analyzed is 1.9; in the JACM, 2.6. This is not a new phenomenon. In the Annals, beginning in 1884, the first 89 articles—running over the first 23 issues, and the first three volumes—were all single-author. (The first joint article was ‘‘Effect of Friction at ConnectingRod Bearings on the Forces Transmitted’’ by J. Burkitt Webb and D.S. Jacobus. Webb commented, ‘‘Professor Jacobus insists on my name appearing first in the article. I fully appreciate the courtesy, but it is hardly fair to himself, as he has done most of the work.’’)

A Promising Start Gowers has suggested a number of problems—the ‘‘Polymath Projects’’—that might be addressed by a large number of co-workers sharing an open blog. Someone not known to the other participants might perfectly well join such a team. Gowers and his co-author, Michael Nielsen, wrote in Nature [7, 8] about the results of the initial project, called Polymath1. This project resulted in an alternative proof of the density Hales-Jewett theorem (DHJ). The DHJ theorem [9] states (informally): for any positive integers n and c there is a number H such that if the cells of an H-dimensional n 9 n 9 n 9 … 9 n cube are colored with c colors, there must be one row, column, or diagonal of length n all of whose cells are the same color. That is, it says that the higher-dimensional, multi-player, n-{row, column, diagonal} generalization of the game of tic-tac-toe cannot end in a draw, independently of n and c and of who plays first, provided only that it is played in sufficiently high dimension H.

AUTHORS

......................................................................................................................................................... DINESH SARVATE was educated first in

SUSANNE WETZEL has a Ph.D. in Computer

India, and later completed a Ph.D. under Jennifer Seberry in Sydney, Australia. He has taught in Papua New Guinea, at various academic institutions in Thailand, at the University of Bombay, and held the Hugh Kelley Fellowship at Rhodes University, South Africa. He has been involved in numerous joint research projects aimed at bringing a new generation of students, including minorities, into research. One such project produced a new type of combinatorial designs, now known as Sarvate-Beam designs.

Science from Saarland University, a Diploma from Karlsruhe University, and also an honorary M.E. in Engineering from Stevens Institute of Technology. Before joining Stevens, she did industrial research in Germany, the United States, and Sweden. Her research interests center in cryptography and algorithmic number theory, and range from wireless security and privacy, to biometrics and lattice theory.

Department of Mathematics College of Charleston Charleston, SC 29424 USA e-mail: [email protected] 10

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Department of Computer Science Stevens Institute of Technology Hoboken, NJ 07030 USA email: [email protected]

Table 1. Comparison of authorship in leading mathematics and computer science journals Annals of Mathematics (10 recent volumes) Number of authors

Number of articles

Journal of the Association for Computing Machinery (2 recent volumes) % of articles

Number of articles

% of articles

1

99

36.00

9

2

121

44.00

22

33.33

3

40

14.55

26

39.39

4

14

5.09

7

10.61

5

1

0.36

2

3.03

Now the DHJ theorem was proved in 1991, so this is not a case of a massively collaborative discovery of a new fact. But specialists in the field were dissatisfied with the original proof of DHJ, so they regard an alternative proof as significant. Polymath1 has since been formally written up and will be published. This Polymath1 project began on February 1, 2009, and reached a conclusion, involving 27 participants making approximately 800 comments (in 170,000 words) over 37 days, with the satisfactory result noted previously. Another dozen or so polymath projects have since been proposed and/or implemented. Some have been fruitful, others have reached a dead end, and a number continue.

Analysis of Polymath Research We are interested in finding ways to analyze the process involved in a polymath project. One clear advantage in trying to analyze these projects is that the rules established by the initiators require that all relevant communications or posts be made through a Wiki or blog; thus after the fact,

......................................................................... received Bachelor’s through Doctoral Degrees in Mathematics from the University of Toronto and the University of Michigan; later he received a Master’s Degree in Computer Science from the University of New Brunswick. He has held teaching and administrative positions at four universities, beginning in his home town of Moncton, Canada, and leading to his present position as Professor and Assistant to the Dean at Howard. Among his many extra-academic labors, he was for 42 years Chair of the Board of Project SEED, an exemplary mathematics program for innercity children, and he has been seven times an invited speaker at the Baseball Hall of Fame in Cooperstown, New York.

WAYNE PATTERSON

Department of Computer Science Howard University Washington, DC 20059 USA e-mail: [email protected]

13.64

one may analyze the text to see how progress is being made, or not made. It is harder to study the genesis of ideas in mathematics, theoretical computer science, and theoretical physics than in other scientific disciplines. An article in these fields is the product of a single researcher or a small group, yet it may depend on many ideas, from a variety of contributors, which may shape the formulation of a theorem and clarify the intuition but which despite their cumulative importance are not considered publishable. The Gowers program bestows on the student of the research process situations in which the pattern of communication is uniquely visible.

Examples of Exceptional Forms of Collaboration in Mathematics Consider the eventual proof of Fermat’s Last Theorem, attributed to Andrew Wiles [10]. Although it has been written that the complete description of the proof would take about 1000 pages of text and by consequence would bring in the contribution of many others, yet none would consider Wiles’s result ‘‘massively collaborative’’. As is usual, many of the underlying components were established separately and published under the discoverer’s name or not at all. In contrast, we want to recall some examples from the past that do have some of the features of massive collaboration. Consider the number-theory result from 1992 (which could be considered an important step either in mathematics or computer science) of the complete factorization 9 of F9, the ninth Fermat number, 22 þ 1: This initiative, led by A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard [11], did involve approximately 700 collaborators. However, the roˆle of the many collaborators was to provide computational power rather than intellectual contribution. Each collaborator was sent by Lenstra a set of computations to perform with the relevant software and input data; after directing a significant amount of computational time on local machines, the collaborators returned their output to Lenstra et al. After they had assembled all of these data, and with a significant step involving inverting a 72,000-by-72,000 matrix, they were able to obtain the necessary results. Quite different is the influential collaboration in mathematics known as Bourbaki [12], in which a number of prominent mathematicians, mostly French, set out to write a series of books that would organize all of mathematics according to their philosophy. A third example is the paper ‘‘Maximal Ideals in an Algebra of Bounded Analytic Functions’’ by I.J. Schark [13].  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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A footnote in this paper indicated that I.J. Schark was ‘‘a pseudonym for a large group of mathematicians who discussed these problems during the 1957 Conference on Analytic Functions sponsored by the Institute of Advanced Study’’. Not in the paper, but only later in Mathematics Reviews [14], was it noted that the group consisted of Irving Kaplansky, John Wermer, Shizuo Kakutani, R. Creighton Buck, Halsey Royden, Andrew Gleason, Richard Arens, and Kenneth Hoffman. The three cases have some distant resemblance to ‘‘massive collaboration’’. In the F9 case, no individual intellectual contribution was solicited, and participation of collaborators was acknowledged in a group listing. With Bourbaki, the initiative was not designed to discover new results, but to reveal the logical structure of the subject; the identity of members was kept secret, and publication was only under the group pseudonym. In the I.J. Schark case, we can venture to hope that the few surviving members of the team will give an account of the process that led to the paper. In all these prior examples of success of an unusual collaboration in mathematics, we are left with the ultimate results, but without data for detailed analysis of what made them succeed.

A New Perspective? Gowers only presented his new perspective about a year ago, but the preliminary results are promising and a number of researchers have decided to participate. Here are some of the questions that we propose for consideration: 1. Can we describe and classify the types of problems that might benefit from a polymath approach? Something like Fermat’s Last Theorem (FLT) presumably could not, for only a handful of people would have the background knowledge necessary to play a meaningful part in discussion of the FLT. 2. Of the problems that might attract people to a polymath, which would likely engage an increasing number of them in one discussion, rather than its devolving into a dialogue between a few of them, or breaking up into several disjoint groups? 3. Is a blog the best mechanism for conducting a polymath? 4. How can we measure or keep track of the progress in a specific project? 5. How should a polymath decide on attribution in publishing? We will have something to say about all these questions.

The Graph Theory Model of a Polymath A major challenge inherent in our study is to determine how the sequence of ideas and suggestions flow and how they contribute to the ultimate solution—or not. On the one hand, the polymath process is infinitely richer in material to analyze than is a finished paper, or even a narrative describing a particular creative experience. In the DHJ polymath project, for example, we have a conversation that uses almost 300,000 words and occupies 931 pages of text before reaching an agreed-upon solution. 12

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Table 2. Comparison of the scope of two polymath projects Case

Identifier

Page length

File size

Words

I

IMOq6

194 pp

1883 MB

42,234 words

II

DHJ

931 pp

4075 MB

278,907 words

Under the current rules of procedure generally accepted by the various polymath projects, there are many potential approaches to an analysis that might reveal how one person influences a group, or one group influences another, or how one idea influences another (Table 2). The reduction of a thousand or so pages of text to a relatively simple graph-theoretic object clearly has one advantage: it is an object that can be digested. On the other hand, using a computer science term, this is undoubtedly ‘‘lossy compression’’. In considering this challenge, we began by modeling the process using a mapping to a graph, or more precisely to a sequence of graphs G(t) parametrized by time, t. In our definition, the vertices of each graph correspond to the participants, or posters, in the polymath project, and a (directed) edge (x,y) from vertex x to vertex y is defined if y, at a given stage, has acknowledged some contribution made by x—be the contribution a question raised, a suggestion, or a specific result enunciated. The t parameter could be considered an indexing of the posts, or the actual clock time of the post. We chose the former for simplicity, but the latter could be said to provide some additional information: it might be informative in later studies to record whether y’s recognition of x’s contribution came four hours later rather than one minute later. (A delayed recognition might mean that hard thought was needed before responding, for instance.) Nevertheless, we proceeded with mere sequential numbering, so that G(100), for example, represents the graph defined by the first 100 posts. Thus the graph-theoretic object of study is the collection {G(t) | t = 1, 2, …, tend } where tend is the sequential number of the last post. In building this model, it is necessary also to maintain other related data, for example, the identification of individual posts with the vertex associated to the poster. In our attempt not to introduce any bias into the analysis, we have labeled the vertices in each collection of graphs in a sequential fashion, from the time of that person’s first post. We have not identified the posters by name. Standard graph-theoretic techniques may answer some questions about how ideas flow or how they may spread or die. In other words, the graphs contain information about participation, about who credits whom with a contribution and when, and so on. Here are some of the analyses that can be performed easily, in fact can be automated relatively easily: Who is posting the most? The individual posts will have been collected, and the cardinality of the set of posts associated with a given poster will answer this question. Who influences others the most? This is determined by the out-degree at each vertex. Who uses others’ contributions the most? This is determined by the in-degree at each vertex.

Are there subgroups working independently? The connectivity of each graph G(t) answers this question. Does the frequency of contribution vary over time? By taking snapshots of the sequence of graphs at any point in time, one can determine the level of participation up to that time. Can we capture the spirit of the polymath by a graph reduction? It may be useful in analyzing the flow of information to consider subgraphs determined by the most frequent contributors according to some appropriate criterion. Going beyond simple analysis of graphs, we might burrow further into the text and isolate the point where the result is obtained, say at vertex x, and then perform further analysis of the subgraph of vertices and edges within a given distance of x. In our analysis, edges all have equal weight. It might be preferable to weight each communication according to its magnitude. We have not done so because we were trying to develop a technique that could partially be automated, and to assess the weight of an edge would depend on the level of expertise of the person assigning the weight. A further question to ask of the sequence of graphs is, Where are the critical steps? We may try to concentrate subsequent analysis on crucial edges and crucial values of t. Finally, if we are looking for qualitative differences in how collective efforts arrive at a goal, the tool we have defined is probably too blunt an instrument. Can we really tell, for example, if a participant has simply monitored all of the transactions without adding any ideas, and then swooped in to put the final piece into the puzzle? Can we really tell if a person who only rarely contributed a thought was the one with a breakthrough idea? We have constructed the graph sequences for two polymath projects called, respectively, IMOq6 and DHJ. We have analyzed each sequence of graphs, and have drawn some comparisons between the two projects analyzed.

Case Study I: International Mathematical Olympiad 2009 Question 6 (Mini-polymath1 or IMOq6) In late July 2009, Terence Tao [15] posted the following: ‘‘The International Mathematical Olympiad (IMO) consists of a set of six problems, to be solved in two sessions of four and a half hours each. Traditionally, the last problem (Problem 6) is significantly harder than the others. Problem 6 of the 2009 IMO, which was given out last Wednesday, reads as follows: ‘‘Problem 6. Let a1 ; a2 ; . . .; an be distinct positive integers and let M be a set of n  1 positive integers not containing s ¼ a1 þ a2 þ . . . þ an . A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1 ; a2 ; . . .; an in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.’’ Of the 565 participants in the Olympiad, only three managed to solve this problem completely. Tao felt that this problem ‘‘might make a nice ‘miniPolymath’ project to be solved collaboratively; it is significantly simpler than an unsolved research problem (in

particular, being an IMO problem, it is already known that there is a solution, which uses only elementary methods), and the problem is receptive to the incremental, one-trivialobservation-at-a-time polymath approach.’’ A number of ground rules were established, including rules to ensure that persons who had worked on or solved the problem didn’t participate; to ensure that polymath participants didn’t search the Web for solutions; that contributions should take place online; and to encourage collaboration rather than competition. We thought this exercise would be very helpful for developing our analytical approach. First, the problem is sufficiently challenging to engage a number of volunteers. Second, the problem is simply stated, so that potentially useful contributions can be made by persons without extraordinary specialized knowledge. This is confirmed by the large number of respondents jumping in within a very short window of time (70 persons with 278 posts in more than 35.43 hours). Finally, the collaborative process seems to have been similar to that seen in other polymath projects, so we can use this case to spot deficiencies in the mechanism for communication (including labeling issues for posts and inconsistencies in reproducing mathematical notation). In this example, we have labeled the vertices a1 through a70, ordered by their chronological appearance in the project. As described previously, when a researcher represented by vertex ai has a post that the researcher represented by vertex aj acknowledges (implicitly or explicitly) as valuable to the problem, then the edge (ai, aj) is created. There were 278 posts in this sample case, and each post represents a set of edges (possibly the empty set). Each edge established through the process can be given a time stamp (the time of the post) and an assessed value on some scale that will measure the level of contribution of the post. The number of vertices depends on the time, but by the end, the graph G(tend) had 70 vertices and 238 edges. There are various ways to obtain insight into the evolution of the polymath. We note two results from initial analysis. By dividing the interval [t0, tend] into five relatively even components according to polymath activity rather than time elapsed, we can see in a discrete fashion the level of activity in the polymath. In general, the in-degree of a vertex measures the level of posting activity by a participant, since an edge is only created when the poster has made a contribution another member recognizes. On the other hand, the out-degree of a vertex is a measure of the flow of information contributing to the solution of a problem, since it represents the incorporation of some other information coming from the origin of the edge. Thus ranking the vertices by in-degrees and out-degrees, and observing the changes over time, may indicate both the influences of ideas toward the solution and the activity of the participants (Table 3). Let us define tk as the time of the k th post; and Im = [t(50*(m-1)+1), t(50*m)] for m = 1, …, 4, and I5 = [t201,tend]. With these definitions, the ranking of the vertices by in-degrees and out-degrees follows: Comments: The in-degree rank of a1 began high and declined. This is inevitable, for the poster represented by a1 defined the problem and set the polymath into  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Table 3. Ranking of in-degrees and out-degrees of vertices in polymath project example Ranking

In-degrees

Cumulative ranking after interval

I1

I2

I3

I4

I5

Out-degrees I1

I2

I3

I4

I5

1

a1

a3

a3

a9

a9

a3

a3

a3

a3

a3

2

a4

a1

a49

a3

a3

a4

a1

a1

a1

a1

3

a3

a25

a1

a49

a49

a1

a4

a4

a49

a9

4

a17

a4

a9

a47

a36

a5

a14

a14

a4

a49

5

a19

a14

a25

a1

a47

a20

a2

a40

a2

a59

6

a25

a17

a32

a36

a23

a2

a5

a2

a14

a4

7

a20

a19

a4

a25

a1

a19

a6

a49

a40

a36

8

a8

a30

a14

a32

a32

a6

a20

a9

a9

a2

9

a13

a32

a17

a4

a53

a8

a23

a5

a36

a14

10

a15

a23

a23

a23

a25

a14

a25

a6

a5

a40

motion— but his initial comments decline as other ideas are contributed. On the other hand, a49’s contributions had little impact initially but rose in importance. With respect to the level of participation, a3 is a constant factor, serving as a sort of moderator, whereas a4’s involvement rises, and a14 grows and declines in his or her level of activity. It is interesting to note that there was consensus that the one who first got a complete proof was a59, whose posts didn’t begin until late in the process. Using Mathematica 7, we have developed an animation that shows how this sequence of graphs evolves over time. The single image shown in Figure 1 represents the graph G(236) using the previous notation, highlighting the vertex giving the proof. (Anyone wishing to view the full animation can do so at: www.howard.edu/csl.)

Figure 1. The graph G(236) describing the state of the IMOq6 Polymath project. 14

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Case Study II: The Density Hales-Jewett Theorem Polymath (Polymath1 or DHJ) The most celebrated example of a polymath project is DHJ Polymath. As mentioned previously, the effort resulted in a new proof of an important theorem in discrete mathematics, the density Hales-Jewett theorem. Using the same process as in the earlier case, we developed a sequence of graphs with 32 vertices and ultimately 555 edges. There were a total of 649 posts to the point where the main theorem seems to have been demonstrated. A noteworthy difference is that in DHJ, two posters accounted for 51.5% of the traffic; the most frequent poster, 28.4%. For the IMOq6 case, the comparable figures were 18.3% and 9.4%. Also, not unexpectedly, there were many more posts in DHJ, and the ‘‘bounce back’’ (multiple edges between certain pairs of posters) was much higher. The graph G(550) is shown in Figure 2, with the vertex enunciating the proof highlighted. Certainly there are many aspects of graph structure differentiating among the types of graphs depicted here. For example, more than twice as many persons chose to participate in the IMOq6 Polymath project as in the DHJ project. Presumably, this is a function of the difficulty of DHJ and the prior knowledge necessary to contribute. Also, a much smaller fraction of the posters for DHJ exchanged ideas among themselves. The time rate of posting is also an important characteristic. Here, the edges were created in the IMOq6 graph on average at 7.8 per hour, whereas in the DHJ graph the comparable figure is 0.9 per hour (Table 4).

Comparisons In the two cases that we have analyzed in considerable detail, the announcement of the solution and its subsequent validation by other participants, came about in considerably different ways.

Figure 2. The graph G(550) describing the state of the DHJ Polymath project.

Table 4. Comparison of the graphs describing IMOq6 and DHJ polymaths Criterion

IMOq6

DHJ

Number of participants (or posters)

70

32

Number of contributing posters

57

25

Number of posts

278

649

Time elapsed

35.43 hours

31 days, 4.27 hours

Number of edges

258

555

Frequency of posts by lead poster

9.4%

28.4%

Frequency of posts by two lead posters

18.3%

51.5%

Largest number of (undirected edges)

(a9, a49): 10

(b2, b5): 56

between any two posters

A casual inspection of the graph sequences indicates a clear difference in the manner of progress toward the eventual results. (This can be seen more vividly in the animation of G(t). In this article, the figures are displayed for t at the moment of clear success.) In the IMOq6 project, it is readily apparent that the contributors are much more dispersed. The subgraph of relatively frequent contributors is larger than the corresponding subgraph for DHJ. In DHJ, the first quartile of posters (by frequency of posting) are responsible for 87.1% of posts, and the first quartile of posters cited (vertex outdegrees) constitutes 74.5% of the references. Comparable figures for the IMOq6 are 67.6% and 68.3% (Table 5). One might locate key contributions by looking at subgraphs confined to the most frequent posters or to those who were referenced the most. Let us take advantage of the openness ensured by the polymath rules, to probe more deeply those posts that appear related to the announcement of the solution itself. First, in the case of IMOq6, the proof is first given (in the post numbered 216a) by poster a59, which is this person’s first post [16]. This post occurred 29 hours and 14 minutes after the problem had been posed. Discussion ensued, but within 5 hours four other posters had apparently reviewed the proof and were satisfied that it was correct. One poster (a53) had asked: ‘‘Well done… did any part of the thread help you?’’ Although a59 did not respond directly, he or she later indicated: ‘‘Thanks for the kind words. Actually I saw the key idea … in a faulty proof which only considered cases 3, 4 and which seemed to assume that you can ‘extend’ Mn to Mn+1. I wrote it up mainly to have a full documented proof … Acquaintance got faulty proof from mathlinks.ro discussion I believe. Have to admit I didn’t get all of that idea on my own.’’

Table 5. Concentrations of posts among posters divided into quartiles Quartile (cumulative frequencies)

DHJ Posts (%)

IMOq6 Out-degree (%)

Posts (%)

Out-degree (%)

1st

87.1

74.5

67.6

68.3

2nd

97.4

91.9

84.8

84.1

3rd

99.0

98.2

93.6

95.1

4th

100.0

100.0

100.0

100.0

One could express skepticism at this account. Is it possible that a59 assiduously monitored all the traffic up to post 216a, and then piped up with the trivial extension of some previous work—without specifically giving credit? Of course, analyzing text after the fact cannot allow us to see into the mind of the writer beyond what he or she has posted, nor are we told whether a59 had even read the earlier posts. But the available evidence from the quotes provided here would indicate that a59’s inspiration was from an outside source (which had been incorrect), and the approval of the others signified satisfaction with the result. In the case of the DHJ project, as the participants approached the solution, 90% of the references involved only four of the posters, b2, b5, b8, and b26.

Further Questions As promised previously, we will attempt to answer a number of questions that the polymath process inspires: 1. Identifying problems that might benefit from a polymath approach As the analysis of polymath proceeds, it will be interesting to consider a number of known theorems to analyze the prior knowledge necessary to contribute to a solution and the estimated size of the community that might contribute to a polymath solution. It should be noted that if the goal of a polymath is a new result, then it is too much to expect to be certain in advance of the level of background knowledge that will be required. In order to develop a technique for analysis, we will first consider existing polymaths. The current examples have involved combinatorics (Ramsey Theory), number theory, complexity theory, and functional analysis (Banach spaces). By studying other known proofs, we can estimate the potential community of recruits for a polymath. Let us illustrate with a specific hard problem. Example: Prove the correctness of the algorithm for the Rivest-Shamir-Adleman Public Key Cryptosystem [17]. The proof can be divided into a number of components. In a meta-description of the proof, we annotate each step with an estimate of the prerequisites for participating in it (Table 6). Thus, given the assumption for RSA that it is possible for cryptographic purposes to find prime numbers of the appropriate size, the pool for potential polymath participants would include persons with a knowledge base including ALG [ ENT [ ICS [ AA—thus including the set of all persons, say, with the equivalent background of an undergraduate mathematics (and possibly computer science) major. If the assumption that finding prime numbers is a given is not made, then the pool of participants would likely be reduced to ALG [ GNT [ ICS [ AA. In order to estimate the size of the pool of participants in a polymath, a useful point of reference would be a comparison of the number of persons with degrees in mathematics or computer science at various levels. This is not to deny that  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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analysis of the importance of various contributors to the given project. For example, an inconsistent labeling methodology for the blog hampered the communication in a number of cases. In addition, the inability of some respondents to post mathematical notation also rendered the analysis more difficult. One of the goals of this research will be to identify and/or develop tools that will reduce these barriers to communication in the virtual organization.

Table 6. Analysis of the steps of the RSA correctness proof Step

Underlying information Requisite knowledge for step

Selection of prime numbers

Graduate number

p, q

theory (GNT)

Computation of n = p 9 q Computation of /(n)

College algebra (ALG) Knowledge that /(n) = (p-1)(q-1)

Selection of e such that

Elementary number theory (ENT) Introductory computer

GCD(e, /(n)) = 1

science course (ICS)

Selection of d such that

4. How can we measure or classify the progress in a specific project?

Introductory computer

e 9 d : 1 (mod n)

science course

Verification that the cipher

Knowledge of the

c : me (mod n) satisfies

Little Fermat

m : cd (mod n)

Theorem

Abstract algebra (AA)

persons without formal degrees in these disciplines might have mathematical or computational insight, but some challenges will be difficult to understand and therefore to follow without some basic knowledge, as in the previously described RSA example. So it would seem that numbers of graduates could provide an indication of the size of potential pools for polymath projects. The following data are from the most recent National Science Foundation reports on degrees awarded (2007) [18] (Table 7). In the RSA example, the pool for solving the problem, not assuming methods to find primes, would be at best about a third the size of the problem without this assumption. This analysis will be performed on a wide variety of problems to develop criteria for the potential of a polymath project to launch and continue successfully. 2. How does the level of participation in a polymath grow or diminish over time? One assessment of the distribution of the level of participation will begin with the data collection as described in our case studies. Since the data structure G(t) changes over time (where time is a discrete parameter), we can analyze the characteristics of G as a function of t. We can also answer questions globally by analyzing G(tend) to be able to answer questions such as, what percentage of the comments come from the highest 10% of respondents in terms of numbers of posts? For example, in the two case studies IMOq6 and DHJ, the most frequently posting are shown in Table 8. The IMOq6 respondents are identified as ai and the DHJ respondents as bj 3. Is a blog the best mechanism for conducting the polymath? Analysis of the case studies shows that the blog is lacking in many components that would permit deeper

As opposed to a static analysis after a polymath has been closed for a specific result, it is worth considering methods for real-time analysis using the data structure models we have described here. The objective will be to incorporate into the virtual organization model automatic updates of the data structure parameters in real time. For example, we have pointed out that in Case Study I above, the person who posted the ultimate accepted solution had made few posts beforehand, so our method left us unable to analyze the flow of information leading to it. 5. How can a polymath decide on attribution for the research contribution in publishing? This will have to be resolved if the polymath method is to be accepted. Most scholars want their contributions to be recognized, both for their own sense of the value of their research and also for practical considerations such as jobs and promotions. In the case of the DHJ polymath, the new proof will get normal publication, but under a joint pseudonym. This may have satisfied the Bourbaki, or other near-massive collaborators recalled above, but will it satisfy future polymath participants? And will the novel excitement of the massive collaboration be as satisfying as the joy of mastering a problem oneself or sharing the experience with a co-worker one sees? It remains to be seen if (for example) assistant professors facing tenure decisions will want to participate in a polymath if their level of contribution will not be explicitly described. On the other hand, in the mathematical sciences it would not only fly in the face of tradition to publish papers with 70 co-authors (as in the Mini-polymath1 case study), but it would leave the community wondering which co-authors did the essential work. Remember the data we cited earlier regarding the Annals of Mathematics and the Journal of the Association for Computing Machinery. Terence Tao has given in his blog some preliminary conclusions on the experience [19]: ‘‘There is no shortage of potential interest in polymath projects. I was impressed by how the project could

Table 7. Degrees awarded in the United States in mathematics and computer science (2007) Discipline

Bachelor’s graduates

Master’s graduates

Ratio M.S./B.S.

Ph.D. graduates

Ratio Ph.D./M.S.

Ratio Ph.D./B.S.

Computer science

42,596

16,314

0.38

1597

0.10

0.04

Mathematics

15,551

5,035

0.32

1356

0.27

0.09

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THE MATHEMATICAL INTELLIGENCER

Table 8. Frequency of posts to the polymaths IMOq6 and DHJ IMOq6 respondent

Number of posts

DHJ respondent

Number of posts

a36

26

b2

184

a3

25

b5

150

a9

20

b8

61

a53

18

b3

54

a21

17

b20

33

a1

12

b4

30

a49

12

b10

Total

130

23 535

with guidance from professionals, can reinstate mathematical research as a pastime. In the cases before us, as a poster is not necessarily identified, it is not known whether the participants are new to research on the question or are the ‘‘usual suspects’’. One can often see in the text that some of the participants seem to know each other well, but also that this is not always the case. At least one of the contributors to DHJ is a high-school teacher [8]. Taking everything into account, our overall conclusion — in the terms we used at the beginning — is that Polymath is something.

round up a dozen interested and qualified participants in a matter of hours; this is one particular strength of the polymath paradigm … There is an increasing temptation to work offline as the project develops. In the rules of the polymath projects to date, the idea is for participants to avoid working ‘‘offline’’ for too long, instead reporting all partial progress and thoughts on the blog and/or the wiki as it occurs. This ideal seems to be adhered to well in the first phases of the project, when the ‘‘easy’’ but essential observations are being made … Without leadership or organisation, the big picture can be obscured by chaos. … Polymath projects tend to generate multiple solutions to a problem, rather than a single solution … Polymath progress is both very fast and very slow. I’ve noticed something paradoxical about these projects. On the one hand, progress can be very fast in the sense that ideas get tossed out there at a rapid rate; also, with all the proofreaders, errors in arguments get picked up much quicker than when only one mathematician is involved.’’

We may not know exactly what it is, or how to ensure that a given project will have beneficial results to the community, but it seems clear that with the successes already claimed and the interest aroused in the community, there is reason enough to encourage more polymath projects. We hope they do prosper, because we are convinced they are a rich lode of information about the interactions and communication that sustain all mathematical research.

REFERENCES

[1] List of David Letterman Sketches, ‘‘Is this anything?’’ Wikipedia, http://en.wikipedia.org/wiki/List_of_David_Letterman_sketches. [2] Timothy Gowers, Gowers’ Weblog, ‘‘Is massively collaborative mathematics possible?’’ http://gowers.wordpress.com/2009/01/ 27/is-massively-collaborative-mathematics-possible/. [3] Jeffrey R. Young, ‘‘Crowd science reaches new heights,’’ Chronicle of Higher Education, May 28, 2010. [4] Annals of Mathematics, Princeton, New Jersey, http://annals. math.princeton.edu. [5] Journal of the Association for Computing Machinery, New York, New York, jacm.acm.org/. [6] The Lancet, London, England, www.thelancet.com/.

Lessons Learned and Conclusions So does the polymath model give us ‘‘anything’’ or ‘‘nothing’’? For an inquiry such as ours, it does allow a new visibility of the assembling of component ideas into a conclusion. If one tries to make the same inquiry concerning a conventional paper, one only rarely can get a ‘‘rough draft’’ describing some of the thought processes leading to the result. In a polymath, the established protocol provides for communication of even casual thoughts, so that one can get the feel of the intermediate stages—if the polymath succeeds—of the steps toward the conclusion. Which communities benefit from a polymath? Since participation is open to all, they give new scholars a rare opportunity to participate in research. It is clear even from a few projects that the attractiveness of joining varies with the topic and perhaps with its complexity. We have mentioned that IMOq6 got 70 participants, and DHJ got 32. Polymath2 had only 8 participants whereas Polymath4 had 40. It seems quite possible that polymaths may attract a wider public to mathematics. In the past, mathematics was enjoyed by American presidents, and for that matter by Napoleon Bonaparte. Perhaps the polymath experience,

[7] Timothy Gowers and Michael Nielsen, ‘‘Massively collaborative mathematics,’’ Nature 461, 879-881 (15 October 2009), doi: 10.1038/461879a; Published online 14 October 2009. [8] Jordan Ellenberg, ‘‘Massively collaborative mathematics,’’ New York Times, New York, 13 December 2009. [9] Hillel Furstenberg and Yitzhak Katznelson, ‘‘A density version of the Hales–Jewett theorem,’’ J. d’Analyse Math. 57, 64–119 (1991). [10] Andrew Wiles, ‘‘Modular elliptic curves and Fermat’s Last Theorem’’ (PDF), Annals of Mathematics 141(3): 443–551 (1995). doi: 10.2307/2118559. ISSN0003486X. OCLC37032255. http:// math.stanford.edu/*lekheng/flt/wiles.pdf. [11] A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard, ‘‘The factorization of the ninth Fermat number,’’ Math. Comp. 61, 319–349 (1993). Addendum, Math. Comp. 64, 1357 (1995). [12] J. Dieudonne´, ‘‘The work of Nicolas Bourbaki,’’ Amer. Math. Monthly 77, 134–145 (1970). [13] I.J. Schark, ‘‘Maximal ideals in an algebra of bounded analytic functions,’’ J. Math. Mech. 10, 735–746 (1961). [14] W. Rudin, Math. Reviews, MR0125442 (23 #A2744). [15] Terence Tao, ‘‘IMO 2009 Q6 as a mini-polymath project,’’ http://terrytao.wordpress.com/2009/07/20/imo-2009-q6-as-a-minipolymath-project/.

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[16] Terence Tao, ‘‘IMO 2009 Q6 mini-polymath project cont.,’’

[18] National Science Foundation, SRS Publications and Data,

http://terrytao.wordpress.com/2009/07/21/imo-2009-q6mini-polymath-project-cont/.

http://nsf.gov/statistics/degrees/. [19] Terence Tao, ‘‘IMO 2009 Q6 mini-polymath project: impressions,

[17] R. Rivest, A. Shamir, and L. Adleman, ‘‘A method for obtaining

18

reflections, analysis,’’

Terence

Tao’s

Blog,

http://terrytao.

digital signatures and public-key cryptosystems,’’ Communica-

wordpress.com/2009/07/22/imo-2009-q6-mini-polymath-

tions of the ACM 21(2), 120–126 (1978).

project-impressions-reflections-analysis/.

THE MATHEMATICAL INTELLIGENCER

Viewpoint

Numbers as Moments of Multisets: A New-Old Formulation of Arithmetic I. GRATTAN-GUINNESS

The Viewpoint column offers readers of The Mathematical Intelligencer 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 the publisher and editors-in-chief do not endorse them or accept responsibility for them. Viewpoint should be submitted to one of the editors-in-chief, Chandler Davis and Marjorie Senechal.

umbers and arithmetic have played a central role in the development of mathematics seemingly since antiquity and in all cultures. Nevertheless, numbers remain elusive objects, if indeed they are objects at all. Here I propose a formulation of integers and arithmetic that seems to be new, although all its components are known. The account involves distinguishing multisets from sets, and moments of wholes from their parts, and a fresh analysis of counting.

N

Motivation Taking Numbers for Granted It is not possible even to summarise the long and rich history of numbers here, but a few points can be made.1 Among the various traditions, the construal of the positive integers as multiples of a given unit by Euclid and other ancient Greeks has been very influential, seemingly more so than a kind of converse position in which numbers were multiples that could be divided into factors. Later, Immanuel Kant proposed an alternative view in which all mathematics was regarded as synthetic a priori (i.e., contentual but independent of experience), a line that still has its sympathisers. Very many texts over the centuries have presented or taught numbers and arithmetic, but the authors’ concern with the nature of numbers usually appears to have been very slight. In a typical example of its time written by an eminent textbook writer [Lamande´ 2004], S. F. Lacroix [1830] devoted less than the first three of his 154 pages to numbers as such, and then only said that arithmetic is the science of discrete magnitudes starting with ‘un, deux, trois, […] neuf’. After that, the book was taken up with kinds of real number, number words and systems, notations, ways of executing arithmetical operations including some shortcuts, and applications to money and to weights and measures. Lacroix’s former student at the Ecole Polytechnique, A.-L. Cauchy, was a little more elaborate. In his own teaching there he split real numbers of all kinds into ‘positive’ and ‘negative quantities’, and carefully laid out their formal laws of combination; he also followed the multiples tradition in regarding a number itself ‘as arising from the absolute measure of magnitudes’ in comparison ‘with another magnitude of the same kind taken for unit’ [1821, 1-2 and Note 1]. Common to these and many other authors is the zero status of zero! Even Cauchy left it out, although in his early pages he also affirmed that an infinitesimal ‘has zero as limit’ [p. 4]! We must not do nothing about zero.

1

The historical literature on numbers and arithmetic up to the middle of the 19th century is quite large but often very limited on numbers as such — like the original literature, it seems! For example, in the history of British teaching of arithmetic, more than 400 years in Britain [Yeldham 1936] does not raise the issue at all. De Morgan [1970] is a remarkable bibliography of arithmetic books, but little more; Number [2001] describes far fewer works but provides more commentary. The more useful general histories of arithmetic include Klein [1968], Scriba [1968], Menninger [1969], Gericke [1970] and Crossley [1980].

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The Critical Phase Partly because of Cauchy’s emphasis on rigour in mathematical analysis, in the late 19th century set theory and mathematical logic came to the fore, especially for foundational purposes. Arithmetic was a principal interest, due partly to concern with rigour and partly to the recognition that the traditional link of numbers with magnitudes was no longer adequate for the ever-widening scope of mathematics. Major outcomes included the Peano axioms (which turned out to capture only progressions); Richard Dedekind’s theory of chains; Georg Cantor’s definitions of cardinals and ordinals by abstractions from general sets; the logicist grounding of arithmetic in the predicate calculus of mathematical logic by Gottlob Frege and then in the new century by A. N. Whitehead and Bertrand Russell [1910-1913] in Principia mathematica (hereafter, ‘PM’); and the definitions of non-negative integers in axiomatic set theory by Ernst Zermelo (‘ZF’), with a different approach taken later by Johann von Neumann (‘NBG’). Parallel with these developments was the advocacy of axiomatisation of mathematical theories in general, especially by David Hilbert; arithmetic was the main target theory, but no definition of integers was adopted. During the later 20th century some variant and alternative theories of numbers were proposed, such as construing them to be quantifiers [Bostock 1974, ch. 5]. Some of these theories of positive or non-negative cardinals were extended to include ordinals, negatives, rationals and irrationals, and transfinite numbers, each kind with its own laws of combination.2 We shall do the same.

Aims The guiding considerations behind the formulation are these: 1) reject the assumption that numbers and arithmetic are a priori knowledge, and instead 2) give a central place to the character of our physical universe, especially that in many (but not all) circumstances it happens to be an environment where objects can endure and be distinguishable from each other; 3) specify the type of generality that arithmetic should display, and note also limitations; 4) handle collections with multiset theory, which admits multiple membership for its members, instead of set theory; 5) deploy the distinction between parts and moments of a multiset, to 6) propose that non-negative cardinal integers are moments of multisets (and ordinals of well-ordered multisets), so as to

2

7) analyse afresh the process of counting, in order to clarify the nature of counting while not giving priority to ordinals; and also to 8) propose a similar distinction between theories and notions in mathematical theories, in order both to 9) outline a formulation of the arithmetic of real numbers (including the negatives), largely by adapting known techniques; and to 10) account for the applicability of arithmetic as well as its ‘pure’ characteristics. Ancillary assumptions include these: 11) while demanding due rigour and exactitude, not to make any special claim for the certainty of mathematical (and logical) knowledge; and 12) admit temporal as well as classical logic in certain circumstances if desired. The focus lies exclusively upon numbers; I do not consider numerals or number words as such (important topics, of course),3 or numerical experiences such as catching the bus on route 73, which does not involve the number 73. Apart from the occasional footnote, I also pass over the interesting issue of the (un)congeniality of the formulation with the various ‘isms’ that philosophers affirm or deny — realism, empiricism, Platonism, positivism, predicativism, nominalism, and the (un-)like — and the relationships between mathematical equality and philosophical identity.4 Issues for mathematics education are also usually elided; some are addressed in Grattan-Guinness [1998].

Preparation Experience is Unavoidable The essential initial steps are to recognise as contingencies that a) the universe that we humans inhabit is composed of (usually) enduring and distinguishable objects; b) we humans are capable of conceiving and acting upon this distinguishability, and thereby comprehending objects both individually and collectively, and of handling them in various ways. The first contingency persuades us to give arithmetic primacy over all other branches of mathematics; by contrast, a sentient culture living in a fiery environment would use arithmetic for special tasks such as counting distant stars

The historical and philosophical literature on these developments is substantial. There are competent editions and/or biographies for several of the principal figures. For historical source books, see especially van Heijenoort [1967]. Texts that cover quite a lot of the history and contain many further references include Hallett [1984], Ferreiro´s [1999], Grattan-Guinness [2000], and Gabbay and Woods [2004]. An excellent survey of axiomatic set theories and related mathematical logic, including their foundations for the positive cardinals and ordinals and the attendant versions of arithmetic, is provided by Quine [1969]. Some of both the history and the philosophy of arithmetic is handled in Grattan-Guinness [1994, esp. part 5]. 3 Cajori [1928, pts. 2-3] contains a fine catalogue of symbols used in arithmetic. Chrisomalis [2010] is an impressive multicultural history. 4 Various older philosophies of arithmetic are criticized by Frege [1884]. Several classical philosophies of mathematics are represented in Benacerraf and Putnam [1983]; certain recent ones pay some attention to mathematical practice, including education [passim in Tymoczko 1985, Mancosu 2008, Lo¨we and Mu¨ller 2010].

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THE MATHEMATICAL INTELLIGENCER

and would give primacy to, say, topology and dynamics in their mathematics. In stressing these features as contingencies I follow anthropologists and educational psychologists much more than philosophers, who place them among the principles of a priori knowledge, or at least formulate them without recourse to empirical factors.5 Finding this appellation very misleading, I demur especially from linking mathematics and logics with knowledge that is true in all possible worlds, for this designation is useless for all purposes unless one lays down minimal conditions for an environment to be taken as a world in the first place. This category should be reduced to propositions that are analytical in Kant’s sense, such as ‘a red bird is a bird’. The Omnipresence of Integers and of Arithmetic Mathematical theories come in two distinct kinds of generality [Grattan-Guinness 2011b]. A theory T is omnipresent over some domain of reference D when it obtains in almost all circumstances available in D. This is the most common sense of which the word ‘general’ is used; relative to mathematics as a whole set theory and arithmetic are two important examples. By contrast, T is multipresent over D when it finds there a wide range of applications and circumstances in which it is usable; however, each application is quite localised. Abstract algebras are important examples. The domain of reference of a multipresent theory may be even wider than that of an omnipresent one, but it is more scattered. The distinction has many consequences for foundational aspects of mathematics, especially when it is applied to the growth of mathematical theories over time and not just to one ‘static’ state, which is a severe limitation evident in most other philosophies of mathematics. It applies also to logics, which play a role in our formulation: logical deduction will be taken as dealing with the processing of information rather than with the transmission of inference [Saguillo 2009].

The Strangely Distinct Histories of Set Theory and Multiset Theory Set theory. As mentioned, a prominent example of an omnipresent theory is set theory. The chief founder was Cantor [Dauben 1979]: starting out in the early 1870s from a technical problem in mathematical analysis, by the mid1880s he had developed both an elaborate point set topology on the line and arithmetics of transfinitely large cardinal and ordinal numbers; the two parts were linked together by notions such as derived sets and the (unproven) continuum hypothesis. Then he saw a wider vision for ‘general set theory’, where sets could have members of any kind. In particular, he proposed the grounding of arithmetic in this theory in which he took any well-ordered set, abstracted away the natures of its members to leave bare the corresponding ordinal number, and then abstracted away the ordinality to reveal the cardinal number. Frege [1892] recognised the philosophical fragility of relying upon

5

abstraction as a ground for knowledge that is supposedly as certain as arithmetic [Dauben 1979, 220-226]. Until the mid-1890s Cantor’s set theory received some support, some opposition, and much indifference; then reception improved quite rapidly, and not only in Germanspeaking countries. Some supporters were dogmatic: for example, ‘Set theory is the foundation of all mathematics’ [Hausdorff 1914, 1]. However, it suffered from two limitations. Firstly and well recognised, it admitted paradoxes, which had to be dealt with. Secondly and entirely overlooked, its omnipresence was compromised because it allowed for only single membership of its members. This oversight is very strange, since it is very easy to find examples in mathematics when multiple membership (hereafter, ‘multiplicity’) of members to a multiset is required; for example, repeated roots of a polynomial equation, and repeated prime factors of an integer in number theory. Such cases refute the logicist theses as Frege and Whitehead/ Russell had conceived them; for they both adopted a comprehension principle that asserted that to every paradox-free predicate there corresponds a set, whereas in these cases a multiset is needed. Multiset theory. The strangeness is accentuated by the fact that also in the mid-1880s the English mathematician A. B. Kempe had published with the Royal Society a very long paper [1886] on multisets, prefaced by an extensive summary [1885]. Using the name ‘heaps’, he was fully aware of their generality, noting in his opening pages ‘units […] which come under consideration in a variety of garbs — as material objects, intervals or periods of time, processes of thought, points, lines, statements, relationships, arrangements, algebraical expressions, operators, operations, &c., &c., occupy various positions, and are otherwise variously circumstanced’. He did not cite Cantor’s concurrent work and seems not to have known about it; conversely, Cantor never cited him. Indeed, despite the importance of his paper, its reception was minute; even John Venn, excellent logician and bibliographer, passed it over, although he entertained the possibility of multiple membership of classes [1894, 39-50]. The reviewing journal Jahrbuch u¨ber die Fortschritte der Mathematik failed to cover the main paper, and gave one bibliographical sentence to the summary version. Kempe’s only serious reader was the American polymath C. S. Peirce, who had used multisets intuitively in his algebra of logic [Walsh 1997]; yet he did not take up that link but instead, in his main reaction, reinterpreted some of Kempe’s cases to create the first version of his topology of syntax, which he later called ‘existential graphs’ [GrattanGuinness 2002]! Kempe included multisets in a characterisation of mathematics in his Presidential address [1894] to the London Mathematical Society, without impact; presumably he did not draw it to the attention of younger set-theorists such as Whitehead and Russell. Thus multiset theory has been recreated in modern times without influence from Kempe, not even on Hailperin [1976], although at first he sometimes used the word ‘heaps’ and applied his theory to aspects of the work of George Boole and

Moser [1987] is a nice collection of modern writings on a priori and kindred kinds of knowledge, with a good preface.

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Peirce.6 It now plays roles in several branches of mathematics, logics, and computer science, and here and there elsewhere [Brink 1987, Blizard 1989] — but far less widely than it deserves. Rado [1975] contains a careful presentation of several basic properties, placed within the framework of ZF in order to avoid paradoxes; the axiomatisation is elaborated in Blizard [1985]. Lake [1976] proposes to use NBG instead, and to render the theory in functional terms. The avoidance of paradoxes is serious, for the theory admits ones of its own. Russell’s paradox, that the set of all sets that do not belong to themselves does and does not belong to itself, extends to a schema about the multiset of all multisets that do not belong to themselves with any given multiplicity belongs and does not belong to itself with that multiplicity. It is easy to find manifestations of multisets in ordinary life; for one example among many, words like ‘letters’ that contain repeated letters, phrases or sentences such as ‘the cat sat on the mat’ that use repeated words, repeated digits in a numeral, and repeated notes in music (note the selfreference of ‘repeated’ here!). So I shall use multiset theory in my formulation of arithmetic, allowing for order if desired. A set is the particular kind of multiset in which only single membership is to be found.

The Supremacy of Classical Logic? The logic that usually attended those foundational studies was the traditional two-valued kind; propositions were true or false, no other options. But one of the consequences was that logical pluralism began slowly to emerge from the late 1900s onwards, especially with L. E. J. Brouwer, Hugh MacColl, Jan Łukasiewicz, and C. I. Lewis [Grattan-Guinness 2011a]; traditional logic gradually became multipresent rather than omnipresent in the expanding domain of reference of logics [Beall and Restall 2005]. Yet one opportunity for logical pluralism was long missed. We often make statements about taking sequences of actions and decisions in some specific order in time, with ‘and’ meaning ‘and then’; for example, ‘I walked down the street and took a left turn’ is not equivalent to ‘I took a left turn and walked down the street’. Here we are using a temporal logic, which differs from classical logic in not being commutative with respect to conjunction (among other differences).7 For this reason alone I regard as unconvincing the claim that classical logic is omnipresent and admit temporal logic if desired; it plays optional roles twice below.

The Supremacy of Axiomatic Set Theories? Set theory played a large role in invigorating the study of the foundations of mathematics around 1900; some principal examples were mentioned previously. Certainty and omnipresence were prime hopes, and a prioricity was sought. A striking example of the latter desire is provided by Russell’s logicist foundation of arithmetic in mathematical logic, which included set theory. In order to avoid the paradoxes, he stratified sets in a certain way using a theory of types, and his positivist orientation forced him to assume that the individuals that were located in its base type were empirical objects — and for him this was a ‘defect in logical purity’ [Grattan-Guinness 2000, 399-401]. Interestingly, much later his logicist colleague Whitehead explicitly appealed to experience in a sketch of a revised version of their logicism when he took as primitive the physical act of beholding an object; his motive was that it ‘expresses that recognition of individuality which is involved in counting’ [1934, 282]. Along with these foundational studies has come a significant consensus that axiomatic set theory and mathematical logic are the sole means of grounding arithmetic. An influential example is Paul Benacerraf’s paper [1965] on ‘what numbers could not be’: after finding that the two axiomatisations of set theory, ZF and NBG, were equally competent in fulfilling the desired aims for numbers and arithmetic, he concluded that numbers were nothing at all: ‘Any object can play the role of 3’ in a progression [p. 70]. But this reaction is surely excessive, just because axiomatic set theories do not deliver arithmetic as expected; other possibilities are available.

Parts and Moments We turn now to Edmund Husserl, student of Weierstrass, follower of the psychologist Franz Brentano, and junior colleague of Cantor. Husserl was a major early practitioner of ‘phenomenology’, a kind of philosophy in which special attention is paid to different manners of perceiving parts and wholes. One of his main interests from the mid 1880s onwards was to examine the foundations of arithmetic; numbers are primitive, but they can be analysed from experiential and psychological points of view. Husserl distinguished between parts and moments of a whole, which he called a ‘manifold’.8 For example, consider a manifold composed of a plastic bag containing some oranges and some apples. Among its parts are each apple, the bag, and the smallest orange, while moments of the manifold include the weight of the bagful, and the price that I paid for it at the market stall. Parts and moments may themselves have parts and moments; for example, my surprise at the price that I had to pay for the fruit, the skin of that smallest orange, the colour of its skin, and the weight of the bag. Some parts and moments may be empty; the pears in the bag, or the nonexistent blueness of the colour of the skin of the oranges. The difference between parts and moments rests upon separability and inseparability: I can remove a part from the whole, but not any moment. (Husserl sometimes spoke of ‘dependent’ and ‘independent’ parts of the whole.) For example, I can take an apple out of the bag in a way that I cannot take away the weight of the bagful from the bag.

6

Personal communications from Hailperin; similarly for Rado later in this paragraph. This use of temporal logic is to be distinguished from the expression within classical logic of the temporal order of events in terms of ordered sets or values [Russell 1936], which is a useful technical procedure but not a philosophical reduction. 8 Husserl took some initial steps [1891]; I do not record this history, or the prehistory, which traces back to Aristotle. The literature on Husserl and phenomenology is quite substantial, although the theory of moments is not prominent; see, for example, Hartimo [2010, esp. ch. 1] and Centrone [2010], both valuable in general. Smith [1982] is an excellent compendium of modern commentaries and some original texts, not by Husserl, on parts and moments. 7

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Among other contexts, in logic propositions P and Q form parts of the compound proposition (P & Q), but the conjunction connective is a moment of it; so are the other logical connectives, and also the quantifiers in predicate logic. In language, in a proposition each word is a part except for the verb, which is a moment (John drops/ buys/purchases the plastic bag of fruit). In mathematics, in an arithmetical proposition such as 2 + 3 = 5 the equality is a moment of the proposition in providing the verb; further, 2 and 3 are parts of the sum, whereas addition itself is a moment. In a union U of multisets each multiset is a part of U but each union is a moment of it. The theory of parts and moments of wholes should be far better known among mathematicians, logicians, and philosophers, for it impinges quite widely on their concerns. We now apply it to our formulation of arithmetic, in which wholes are taken to be multisets.

occupy: not only physical ones such as stones and fingers, but also transient ones such as waterfalls, experiential ones such as psychological states and musical keys, historical events of every kind, parts of scientific theories, characters in plays and novels, the gaps between words in written texts and the short pauses in well-executed speech acts, and so on and on. In particular, I include everyday numbers and arithmetic, which supplies us with many goods and services; for example, tallying, the numbers of the pages upon which this article is printed, special uses of numbers in mathematical theories such as ‘x-23’ and ‘log5x’, and the increasingly multipresent digit strings that identify ‘numbers’ for security, credit cards, telephones, websites, and many other contexts. I write the numerals of these numbers in the form ‘ 37 ’. A diverting kind of case was mentioned previously, when a numeral is treated as a multiset composed of its digits; thus ‘2010’ contains 1 1 , 1 2 and 2 0 s.

Formulation

An example. With this background consider, as an example, the multiset M* composed (in any order) of the fingers of my left hand (but not the thumb), a certain dinner jacket considered 3 times, the fear of running over tortoises when driving along the road, an imagined traffic sign outside my house, a pair of transistors, and the music written for clarinets in Benjamin Britten’s Cello Symphony ( 1965 ) regarded as a totality. M* is the union of its 6 constituent sub-multisets, where each union is a moment of the compound (sub-)multiset thereby created. The cardinality of M* is 4 + 3 + 1 + 1 + 2 + 1, which itself is a multiset of cardinals because of the 3 1s. The total, 12, is the sum of the cardinalities of the 6 constituent sub-multisets; each summation is a moment of the addition thereby created. Among parts of the cardinality of M*, the cardinal 0 is associated with absence, such as the lack of peacocks in M*; the cardinal 1 links to, say, any 1 instance of the dinner jacket, the non-existent traffic sign,9 or my middle finger; and the fear, the music, and one transistor, make up a 3. M* is deliberately polyglot in its (multiple) members, but multisets defined by intensions such as properties or relations are equally admissible. Further, each multiplicity in M* is known; this may not always be the case, but the principle of associating integers with multisets is not affected.

The theories of numbers advocated so far treat them as ‘objects’ of some sort, whether linked to quantities, or to multiples of a unit, or to counting, or to set theory and/or logic. By contrast, in the formulation that now follows, numbers are moments of multisets, and thereby they are not objects in those senses (compare Bostock [1974, ch. 1] on quantifiers). I choose to emulate the logicist construction of arithmetic in several respects, which are considered at suitable places; some of the points made pertain also to (axiomatic) set theory. All the main steps are described, but not all the details; an axiomatic version would make more precise both the required assumptions and the definability of some notions in terms of other notions. We start with cardinals.

Non-negative Finite Cardinals Principles. Let M be a multiset, considered either in isolation or as part of a larger multiset; it contains distinguishable objects as members, each one in some multiplicity, and no order is imposed on them. To avoid paradoxes, it is embedded in some modified version of an axiom system of set theory, as was mentioned earlier. Then the (unique) cardinality of M is the moment of it specified by the sum of the multiplicities of all its members. 0 is the cardinality of any empty multiset. 1 is the cardinality of any unit set, that is, of any multiset containing only one distinguished object once; it is a moment of that unit set, not of its sole member. Each finite cardinal c from 1 onwards is the cardinality of any multiset isomorphic with the set C := {0, 1,…, c - 1}. Should two multisets possess the same record of multiplicities of their respective members, then they are isomultiplous; sets are an important example. Integers could be defined for each kind of multiset if desired. To obtain the required omnipresence for these cardinals, the invariable and distinguishable (multi-)members of M must be objects of all kinds in the environment that we happen to

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Handling the multisets. The logicists grounded sets in the mathematical logic of propositional functions and relations, with a comprehension principle linking the two. The same strategy is followed here, modified to allow for multiple membership. The determination of membership of M is a matter of perception, as phenomenologists rightly stress. To reconsider M*, if I switch attention to my left hand from its fingers to the knuckles on each finger, then the cardinality of the new multiset jumps to 20. The distinction between the (multi-)setas-one and (multi-)set-as-many relates to this point; for example, if I take each note in the clarinet music as an object, then the cardinality of the corresponding multiset would be considerably larger (and how do we count tied notes?).

9

Here is an example of the philosophical differences noted earlier. I grant imaginary objects some kind of existence, and so give the traffic sign the multiplicity 1; empiricists might banish them from all discourse, and so deny them membership of a multiset.

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The association of cardinals with multisets does not require Cantor-like abstractions to be made from their members. Instead, we emulate the converse method usually followed in axiomatics of taking schematic letters a, b, c, … to denote uninterpreted objects that satisfy certain laws (such as a•(b•c) = (a•b)•c in group theory), and ignoring any other properties. That is, here we disregard all properties of the members of a multiset (including any numberings or listings of them that may have been imposed previously); we note only invariability, distinguishability and multiplicity, and structural features such as ordered pairhood,10 to form a sort of Urmultiset, and associate cardinals with it.11 The Statuses of Zero, Counting, and One Zero is not nothing. The integers from 1 onwards are often called ‘natural numbers’, which implies that 0 is not to be found in nature; sadly, 0 has long been identified with nothing.12 Major authors on arithmetic in antiquity — Iamblichos, Euclid, Boethius — ignored it, though Euclid’s geometry needs 0 as the angle between a curve and a tangent at the point of contact. Again, when the Christians numbered the years in the 6th century, B. C. 1 was followed by A. D. 1. The situation did not improve: we recall that Cauchy slipped in 0 without explanation. Even Cantor had great difficulty with 0 because of his reliance upon abstraction mentioned above; he always started his number sequences with 1.13 Dedekind also took 1 as ‘base-number’; he came to understand the empty set and unit sets rather late, did not relate them to 0, and never published what he had found [Sinaceur 1971]. Peano began his sequences sometimes with 0 and sometimes with 1, seemingly without reflection. Hilbert is no better than Cauchy over 0. Among other cases, Jean Piaget, strongly influenced by Russell, nevertheless ignored 0 completely in his study [1941] of numbers in children! Much ado about 0, indeed. Frege and Whitehead/Russell were to help sort it out by explicitly recognising the tri-distinction between 0, the empty set Ø, and literally nothing. One consequence was that they defined 0 as {Ø}, and recognised any positive cardinal c as a set of sets equinumerous with the above set C = {0, 1,…, c – 1}.14 Our formulation follows suit with the association of 0 with any empty (Ur)multiset, and a similar role for C. A variant formulation of arithmetic would carry emulation further by defining a cardinal as the multiset of the appropriate equinumerous multisets; however, such integers are objects, and the crucial property of inseparability is lost. Ordinals, and the practice of counting. So far no order has been imposed upon the members of M, so that the 10

associated integers are cardinals. However, if order matters, then we impose well-order on the members, including on their multiplicities, drawing upon a temporal logic if felt appropriate. Then we can specify ‘ordinalities’ as moments of these well-ordered (Ur)multisets, including 0 for the empty ones.15 We can also take in Russell’s much neglected generalisation, the ‘relation-arithmetic’ of ordinally similar relations [PM, pt. 4]. Cardinal numbers take epistemological precedence over ordinals here because they do not presume any order. However, enthusiasts for the converse priority, such as NBG, point to the importance in arithmetic of counting, which has not been stressed here. One reply is that counting belongs more to tallying and everyday arithmetic than to foundations; for example, to find out how many apples are in that plastic bag one would of course count them rather than try to perceive some mysterious moment about them. But how does counting work? This question needs attention. Zero counts in arithmetic! Lacroix gave the normal explanation of counting, whether of cardinals or ordinals; we count from 1, 2, … up to completion at, say, 13 apples in the plastic bag. (The accuracy of the count is not at issue here.) Now this procedure cannot ground 13 either as a cardinal or as an ordinal because it involves 13; indeed, we count the apples by assuming the sequence Q of positive finite cardinals (or ordinals) and complete the count by establishing a (well-ordered) isomorphism between the apples and the corresponding initial sequence of Q (assuming that our culture has developed Q sufficiently far); counting up the multiplicities to find the cardinality of M* above is a typical example. This sequence is usually not interpreted, and resembles an Urmultiset. Russell made these points to show that counting ‘is quite irrelevant to the theory of Arithmetic’ [1903, 133]; yet, ironically his theory clarifies the process of counting itself, as we now see. Whether or not we are concerned with foundations for counting, we must recognise that counting always has a pre-counting stage (as I shall call it), when we decide to count the apples in the bag rather than the oranges, say, or the chairs in the room where the bag is placed [Crowson 1970, 2]. During this stage, which may last only a very short time but still occurs, we have counted 0 apples; then we count from 0; in the case of the apples, to 1 and then on to 13. Further, sometimes we may count from 0 to 0; for example, if we count the number of pears in the bag.16 In many contexts counting explicitly starts from 0. Games and sports are a rich source. For example, in a football match the pre-counting stage ends when the referee blows his

We might invoke the definition of the ordered pair (a, b) as {{a}, {a, b}} (compare [PM, *55]); it can be iterated to n-tuples. Model-theorists replace the rather informal talk of uninterpreted objects by variables quantifiable over suitably wide ranges; this procedure could be imitated here. 12 See Rotman [1987] on this point. Kaplan [1999], a history of 0 both as number and numeral, may be used with some caution. 13 Nevertheless, when Cantor proposed his definitive notation for the smallest transfinite cardinal in 1895, he changed it on proof from ‘@1’ to ‘@0’, without explanation to his editor Felix Klein [Grattan-Guinness 2000, 113]. 14 Frege spoke of integers as ‘extensions of concepts’; his definitions were nominal [1884, ch. 4]. The definitions in PM were cast in terms of sets of sets [*100-*103], contextually reducible to propositional functions [*2001]. These are interesting instances of the difference between Platonism and positivism [Grattan-Guinness 2000, 349-351]; Robinson missed them in his critique of definitions in PM [Robinson 1950, ch. 7]. 15 PM is not very helpful here, in that the conception of series prevents 1 from being an ordinal [*15301]; 0 and 2 are safe [*5603, *5602]. 16 I distinguish counting from numbering, where conventions often prevail; for example, whether I choose to number the chapters of my next book from 0 or from 1 (or from 16 or …). 11

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opening whistle, starting the 1st of its 2 halves; but the scoreline is still 0-0, and 0 penalty cards have been issued so far. Measures of time provide many examples: a 9-month old baby has completed 9 months but 0 years; again, at 10.00 a.m. we agree to meet 2 hours later at midday. But counting days or nights is a context where inconsistency over 0 or 1 does obtain across societies. On this Monday I look forward 2 days to Wednesday, which means that I count today as the 0th day; but the ancient Romans, who did not have a numeral for 0, said the equivalent of looking ahead 3 days to Wednesday. Now both forms of counting days can be found in the former Roman Empire; for ‘fortnight’ (itself short for ‘fourteen nights’) is rendered in French as ‘quinze jours’, with the corresponding words also in (modern and maybe also ancient) Greek and Latin, and in Italian, Spanish, Portuguese, and Welsh. The numbering of years regretted above is another case of counting from 1. Presumably the unfortunate tradition of regarding 0 as nothing has led to this misunderstanding of counting. While rejecting counting as a ground, Russell’s (and Frege’s) characterisation of a cardinal in terms of the set C of all its predecessors from 0 captured it precisely; and our formulation follows suit. The other foundations of arithmetic mentioned earlier also handle it; for example, in ZF the definitions 0 := {Ø} and n := {n}iterations of the operation of forming the unit set. The ubiquity of 1. Table 1 lists a remarkable multitude of words and phrases linked to oneness in at least one sense or use: cardinal, ordinal, relational, technical, or cultural.17 There are far more words than are linked to zero, the other small integers, or infinity. Maybe someone can write a cultural history of oneness one day; it would be a notable contribution. A source will have been the very influential tradition of arithmetic that established 1 as a basic unit, as in Euclid’s arithmetic and Whitehead’s beholding, and construed all higher finite cardinals or ordinals as multiples of it, starting with 2. I acknowledge this tradition without feeling beholden to it, but it will now play a role in our treatment of rational numbers.

Other Numbers, and Arithmetic Operations on non-negative numbers, especially rational numbers. Let I(M) be the cardinal that is associated with M. Equinumerousness and inequality are determined by identifying (or not) isomorphisms between the members of (well-ordered) multisets. As for the arithmetical operations, subtraction I(M) - I(N)), M ) N, requires M to be partitioned into the multiset N and its complement. Addition (I(M) + I(P)) comes from the cardinality of the union multiset of M and P, allowing for multiple memberships when required; further additions require 2 or more

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Table 1. 1s various a(n)

monocracy

singleton

any

monocrotic

singular

alone

monocycle

sole

bachelor

monodrama

solitary

being

monody

solo

definite

monogamy

soloist

ecumenism

monolith

somebody/one

ego

monologue

somewhere

entire

monomania

that

first(ly)

monophony

the

her/himself

monopole

this

hegemony

monopoly

unaccompanied

I

monorail

unary

identity

monotheism

undividable

indissoluble

monotone

unicameral

individual

myself

unicorn

indivisible

next

unicity

inimitable

on your own

unicycle

initial

on your tod

uniform

inseparable

once

unify

insular

one

unilateral

integral

oneself

unipole

invariant

oneness

unique

isolated

one-off

unisex

isometry

one-one correspondence

unison

isomorphism

oneself

unit

kithless

one-sided

unitarianism

linear

one-to-one

unity

lone

one-way street

universal

lonely

only

univocal

loner

premier(e)

unmarried

lonesome

previous

Urtext

monad

primacy

whole

monadic

primary

yourself

monarch

prime

mono

same

monochord

selfhood

monochrome

single

associations. Multiplication comes as iterated addition, or as the cardinality of the multiset composed of I(M) instances of I(P) or as I(P) instances of I(M); and division follows from associating (I(M)/I(N)) with those multisets Q for which I(M) = I(N)I(Q)). The rational numbers are usually held to be a trivial extension of division: just as 3 9 8 = 24 grounds 24/8 = 3, so 25 9 40 = 125 9 8 grounds 25/8 = 125/40.18 However, the latter equation surely has only the status of a formal law stating the equality of two rational numbers; the concept of

17

Frege dwelt upon 1 at length [1884, pt. 3]; so did Husserl [1891, ch. 8]. ‘Once’ belongs to a special vocabulary that pertains to ‘guzinter’ integers, as they are affectionately known. They occur in, for example, ‘5 goes into 17 thrice with remainder 2’. Some students of arithmetic regard them as distinct from cardinals and ordinals. They are prominent in Euclid’s algorithm, which links to that strangely fugitive part of mathematics, continued fractions [Fowler 1999, esp. ch. 9]. 18 See, for example, Quine [1969, 119] or Russell [1919, 64]. PM captures the rational order-type (compact, denumerable, no first or last term) before homing in on a theory of rational numbers r/s based upon sets of the rth and the sth powers of pairs of relations that generate series [*273, *303]. My approach has some kinship with Russell’s [1903, 149-150]. On the rich history of rational numbers see Benoit and others [1992].

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rational number itself has not been formed since I(M)/I(N) is defined only if I(N) is a factor of I(M), and the equation as such is meaningless. The multiples tradition of counting helps us here. Let us associate the rational number I(M)/1 with the ordered pair (M, U) of multisets, where U is a unit set. Then the converse pair (U, M) furnishes the inverse fraction 1/I(M) (favoured by the ancient Egyptians, by the way), and any multiple of it is the rational number I(V)/I(M), where V is the multiset containing just the sole member of U I(V) times.19 Thus, for example, 3/5 is 3 9 (1/5), that is, literally ‘3 5ths’.20 This treatment of rational numbers relies on the property of closure. A multiset of distinguishable objects a, b, … of any kind is closed with respect to an operation ‘•’ when for all a and b, a•b is an object of the same kind. The restricted version above of division as a rewrite of multiplication is closed because I(N) is a factor of I(M), so that I(M)/I(N) is always a positive integer; by contrast, the extended version of division as rational numbers was not closed. However, the conversion of each integer I(M) into the rational number I(M)/1 leads on to the multiset of the rational numbers, which is closed with respect to division. Irrational and transfinite numbers are defined by appropriate versions of well-known procedures: for example, Dedekind cuts and Cantor’s principles of generating transfinite ordinals. They include assuming axioms when necessary, such as infinity and choice. Quine [1969] is a dependable guidebook to the requirements for the theories mentioned above, and also for two of his own systems; the handling of (ir)rational exponentiation [ch. 6] is especially careful. Bostock [1979], on (ir)rational numbers, may also be useful.

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Negative and complex numbers. The long-running distrust of negative numbers among (some) mathematicians was not always prejudicial or unthinking: in particular, it is not easy to explain the product rule for negatives other than as a formal law (Cauchy’s procedure, for example). But the hostility surely exposed the paucity in those times of vectorial thinking, which provides all sorts of contexts where negative numbers seem to be quite legitimate: not only spatial directions as used in coordinate geometry and dynamics but also temporal ones of days past and days to come (including 0 for today, as we saw) and conceptual 19

directions such as economic credit and debit.21 Kant [1763] made a worthy effort to justify the negatives in terms of a dialectical opposition to the positives. Some might even regard the precedence of the positives over the negatives as only a convention, like that of normally placing north instead of south at the top of a map of the world.22 In this formulation we associate the negative ordinal integers with inversely well-ordered multisets, again assuming a temporal logic if felt helpful;23 0 links to empty well-ordered (Ur)multisets, counting goes backwards, and subtraction of integers now becomes a closed operation over all the integers. Then we may emulate Whitehead and Russell, who obtained the negative real numbers by deploying the converse of the relations used to generate the positives [PM, *300, *307]. Alternatively but perhaps fainter-hearted, we adopt a formal law that assumes that for any positive real number p there exists a negative n such that p + n = 0. Another extension is to complex numbers a + ib, where each real number is given by b = 0. The mathematical reward is that, unlike the reals, these numbers are closed with respect not only to the arithmetical operations but also to the taking of roots of any finite order. Omnipresence is not universality. One way or another, every number, cardinal or ordinal, is either the unique moment of some kind of multiset or an ordered pair of them, or relies upon a formal law linking it to numbers already specified. But if the members of the multisets are not invariable and distinguishable, then the association of a number is impossible or becomes somewhat arbitrary, whatever approach to arithmetic is adopted. Contexts involving continua provide many examples, such as counting the colours of the rainbow. Counting fails for other reasons: when the target objects are too numerous, say, or move too rapidly, or where the music is too irregular in rhythm to let us count the beats. These are subjective contexts; objective ones include trying to study the individual components of a cloud. In all these situations arithmetic is not suitable mathematics.24 Aspects of Arithmetic Notions and theories. The elaboration of arithmetic, and of attendant theories such as number theory and numerology,

This version of rational numbers can be built up using that primitive and often overlooked notion, the ratio of two quantities (not necessarily numbers), which occurs quite often in ordinary and in mathematical life and plays a major role in Euclid’s Elements. Ratios are not numbers; in particular, they differ from rational numbers as they do not add or subtract, although they have laws that are structurally similar to multiplication and division. For example, the ratio (5:7) compounded with (7:11) is the same as the ratio (5:11), but 4 ± (5:7) are both meaningless. 20 Note the use of ‘5th’ to denote here an inverse fraction, in contrast to its quite different use as an ordinal integer; both can occur, as in ‘the 5th 5th’ of my income. Special uses of words arise in the cases of 2 and 4, with ‘the 2nd half’ of the football game, and the ‘4th quarter’ of the year. 21 For a collection of texts on negative numbers see Arcavi and Bruckheimer [1983]. 22 The alternative kind of map is indeed available in Australia! Maps and charts are also a striking non-example of negatives; for latitudes are presented as positive angles to north and south of the equator, and longitudes to east and west of Greenwich. 23 Hailperin [1976] allows for negative membership of multisets in his theory of ‘signed heaps’, such as when a poker player borrows some chips of a given colour from the bank: but if inversely infinite multiplicity were permitted, there would arise some unwelcome results akin to the paradox of grounded sets [Quine 1969, 37]. Hailperin created his theory to back a claim that Boole distinguished between ‘interpretable’ and ‘uninterpretable’ formulae in his logic by the absence or possible presence in them of multi-classes [1976, ch. 3]. 24 One substitute in some circumstances is fuzzy set theory, which handles vague predicates such as ‘is a large crowd of people’ [Dubois and Prade 1980]. The usual version maps fuzzy membership onto real numbers within [0, 1]; but in Grattan-Guinness [1976] I was one of the pioneers to advocate mapping onto sub-intervals of [0, 1], and this alternative has gained some support. Lake [1976] carries over to this theory his advocacy of functional versions of theories of collections. Another option is the theory of quasi-sets, which are collections of elements that have cardinality but not a definitive number of elements. It is used in quantum physics to provide sortal (that is, count-upable) predicates of entities that are both separable and indistinguishable [Krause and French 2007].

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proceeds unaffected; and the distinction between parts and moments can play another role. In a recent paper on methods of theory-building in mathematics I imitated it when making a distinction between mathematical (sub-) theories as parts of a mathematical theory and mathematical notions resembling inseparable moments of it, and claimed that both categories are important [Grattan-Guinness 2008]. Examples of ‘ubiquitous’ notions that pertain to arithmetic and its kin include linearity, duality, the proof method by infinite descent, convexity (in the geometry of numbers), and partitioning.

BIBLIOGRAPHY

Metatheory. The distinction D between a mathematical theory (of any kind) and its metamathematics is omnipresent. Thus it applies to this formulation of arithmetic in suitable adaptations of the established metatheories, whether formalised Hilbert-style to establish (in)consistency, (in)completeness and the like, or otherwise. In particular, Go¨del’s theorem [1931] about the incompletability of firstorder arithmetic, itself a source of D, is unchanged; he even starts with 0! Thus, for example, if I describe the revised version of counting as 1) addressing the counting exercise and 2) executing it from 0, I have not contradicted the theory itself by starting from 1; for I am in its metatheory, which begins with its own pre-metacounting stage from 0. Maybe the conflation of counting with metacounting has helped us to misunderstand it. Model theory seems also to be basically unaffected.

Benoit, P. and others (Eds.). 1992. Histoire des fractions, fractions d’histoire, Basel: Birkha¨user.

Arcavi, A. and Bruckheimer, M. 1983. The negative numbers: a historical source-work collection for pre- and in-service mathematics teachers courses, Weizmann Institute of Science, Israel: Department of Science Teaching. Beall, J. L. and Restall, G. 2005. Logical pluralism, Oxford: Oxford University Press. Benacerraf, P. 1965. ‘What numbers could not be’, The philosophical review, 74, 47-73. [Repr. in Putnam and Benacerraf [1983], 272294.]. Benacerraf, P. and Putnam, H. (Eds.). 1983. Philosophy of mathematics, 2nd ed., Cambridge: Cambridge University Press. [1st ed., 1964.].

Strategies for teaching and imparting. The distinction between parts and moments is evident in many fields of life, of which arithmetic, language, and logics are only three; and those between sets and multisets, and between omnipresence and multipresence, should also be brought out in presentations of mathematics. All three distinctions ought to be much more widely known, and not only in the contexts discussed here. Regarding teaching at school level, the omnipresence of the invariability and distinguishability of objects should be emphasised already in the teaching of young children (preferably using shorter words than these!); this policy was advocated strongly by Karl Bu¨hler [1930, ch. 3]. The continuing nonsenses of overlooking zero, or teaching that it is nothing, should stop; and counting should be explained more precisely. The property of closure could be emphasised in suitable contexts. But it may be prudent to delay the full formulation of arithmetic until the pupils are in their midto late teens, and to restrict it to the non-negative cardinal and ordinal integers; rely upon formal laws to get further, like Cauchy. At all events, it should be more enlightening to portray integers as inseparable associates of multisets instead of as quantities somehow linked to magnitudes, or as elusive objects coming before us from the a priori, or the results of counting away, un, deux, trois,… .

Blizard, W. D. 1985. ‘Generalizations of the concept of set: a formal theory of multisets’, Oxford University, doctoral dissertation. Blizard, W. D. 1989. ‘Multiset theory’, Notre Dame journal of formal logic, 30, 36-66. Bostock, D. 1974, 1979. Logic and arithmetic, 2 vols., Oxford: Clarendon Press. Brink, C. 1987. ‘Some background on multisets’, Australian National University, Canberra: Research School of Social Sciences, Report TR-ARP-2/87. Bu¨hler, K. 1930. The mental development of the child. A summary of modern psychological theory (trans. O. Oeser), London: Kegan Paul, Trench, Trubner; New York: Harcourt, Brace. [German original in eds. from 1919.]. Cajori, F. 1928. A history of mathematical notations, vol. 1, La Salle: Open Court. Cauchy, A.-L. 1821. Cours d’analyse a` l’Ecole Royale Polytechnique, Paris: De Bure. [Repr. Bologna: CLUEB, 1992, with intro. by U. Bottazzini.]. Chrisomalis, S. 2010. Numerical notation: a comparative history, Cambridge: Cambridge University Press. Crossley, J. N. 1980. The emergence of number, Steel’s Creek, Australia: Upside Down A Book Company. Crowson, R. A. 1970. Classification and biology, London: Heinemann Educational Books. Dauben, J. W. 1979. Georg Cantor, Cambridge, Mass. and London: Harvard University Press. [Repr. Princeton: Princeton University Press, 1990.]. De Morgan, A. 1970. Rara arithmetica: a catalogue of the arithmetics written before the year MDCI …, 4th ed., New York: Chelsea. [Includes ‘Addenda’ by D. E. Smith.]. Dubois, D. and Prade, H. 1980. Fuzzy sets and systems: theory and applications, New York: Academic Press. Ferreiro´s, J. 1999. Labyrinth of thought: a history of set theory and its role in modern mathematics, Basel: Birkha¨user. Fowler, D. 1999. The mathematics of Plato’s Academy, 2nd ed., Oxford: Clarendon Press. Frege, F. L. G. 1884. Die Grundlagen der Arithmetik. Eine logischmathematische Untersuchung u¨ber den Begriff der Zahl, Breslau: Ko¨bner. [Various reprs. and transs.]. Frege, F. L. G. 1892. Review of a pamphlet by Cantor, Zeitschrift fu¨r

ACKNOWLEDGMENTS

Philosophie und philosophische Kritik, 100, 269-272. [Repr. in

For discussions and comments on draft material I am indebted to Joseph Brenner, Mirja Hartimo, and Paola Cantu.

Kleine Schriften (ed. I. Angelelli), Hildesheim: Olms, 1967, 163-166.].

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Gabbay, D. and Woods, J. (Eds.). 2004. The rise of modern logic from

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Leibniz to Frege, Amsterdam: Elsevier. Gericke, H. 1970. Geschichte des Zahlbegriffs, Mannheim: Biblio-

Kaplan, R. 1999. The nothing that is: a natural history of zero, London:

graphisches Institut. Go¨del, K. 1931. ‘U¨ber formal unentscheidbare Sa¨tze der Principia

Kempe, A. B. 1885. ‘A memoir introductory to a general theory of

Allen Lane The Penguin Press.

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mathematical form’, Proceedings of the Royal Society of London,

ematik und Physik, 38, 173–198. [Many reprs. and transs.]

38, 393-401.

Grattan-Guinness, I. 1976. ‘Fuzzy membership mapped onto intervals

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and many-valued quantities’, Zeitschrift fu¨r mathematische Logik

Philosophical transactions of the Royal Society of London, 177, 1-70.

und Grundlagen der Mathematik, 22, 149-160. Grattan-Guinness, I. (Ed.). 1994. Companion encyclopaedia of the history

Kempe, A. B. 1894. ‘Mathematics’, Proceedings of the London Mathematical Society, (1)26, 5-15.

and philosophy of the mathematical sciences, 2 vols., London:

Klein, J. 1968. Greek mathematical thought and the origin of algebra,

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Cambridge, Mass: The MIT Press.

Grattan-Guinness, I. 1998. ‘Some neglected niches in the understand-

Krause, D. and French, S. 2007. ‘Quantum sortal predicates’,

ing and teaching of numbers and number systems’, Zentralblatt fu¨r

Synthese, 154, 417-430. Lacroix, S. F. 1830. Traite´ e´le´mentaire d’arithme´tique, a` l’usage de l’Ecole Centrale des Quatre-Nations, 18th ed., Paris: Bachelier.

Didaktik der Mathematik, 30, 12-19. [Repr. as Routes of learning: highways, pathways and byways in the history of mathematics, Baltimore: Johns Hopkins University Press, 2009, ch. 12.]. Grattan-Guinness, I. 2000. The search for mathematical roots, 18701940: logics, set theories and the foundations of mathematics from Cantor through Russell to Go¨del, Princeton: Princeton University Press. Y

Grattan-Guinness, I. 2002. ‘Re-interpreting ‘‘ ’’: Kempe on multisets and Peirce on graphs, 1886-1905’, Transactions of the C. S. Peirce Society, 38, 327-350.

Lake, J. 1976. ‘Sets, fuzzy sets, multisets and functions’, Journal of the London Mathematical Society, (2)12, 323-326. Lamande´, P. 2004. ‘La conception des nombres en France autour de 1800 : L’oeuvre didactique de Sylvestre Franc¸ois Lacroix’, Revue d’histoire des mathe´matiques, 10, 45-106. Lo¨we, B. and Mu¨ller, T. (Eds.). 2010. PhiMSAMP. Philosophy of mathematics: sociological aspects and mathematical practice, London: College Publications.

Grattan-Guinness, I. 2008. ‘Solving Wigner’s mystery: the reasonable (though perhaps limited) effectiveness of mathematics in the

Mancosu, P. (Ed.). 2008. The philosophy of mathematical practice, Oxford: Oxford University Press.

natural sciences’, The mathematical intelligencer, 30, no. 3, 7-17.

Menninger, K. 1969. Number words and number symbols, Cam-

[Repr. as Corroborations and criticisms: forays with the philosophy of Karl Popper, London: College Publications, 2010, ch. 12.]. Grattan-Guinness, I. 2011a. ‘Was Hugh MacColl a logical pluralist or a logical monist? A case study in the slow emergence of metatheorising’, Fundamenta scientiae, to appear. Grattan-Guinness, I. 2011b. ‘Omnipresence, multipresence and ubiquity: types of generality in and around mathematics and logics’, in preparation. Hailperin, T. 1976. Boole’s logic and probability, 1st ed., Amsterdam: North-Holland. [2nd ed. 1986.]. Hallett, M. 1984. Cantorian set theory and limitation of size, Oxford: Clarendon Press. Hausdorff, F. 1914. Grundzu¨ge der Mengenlehre, Leipzig: de Gruyter. [Repr. New York: Chelsea, 1949.]. Husserl, E. 1891. Philosophie der Arithmetik. Logische und psycho-

bridge, Mass.: The MIT Press. Moser, P. K. (Ed.). 1997. A priori knowledge, Oxford: Oxford University Press. Number 2001. Maß, Zahl und Gewicht. Mathematik als Schlu¨ssel zu Weltversta¨ndnis und Weltbeherrschung (ed. K. Reich, M. Folkerts und E. Knobloch), 2nd ed., Wiesbaden: Harrasowitz. Piaget, J. 1941. La gene`se du nombre chez l’enfant, Neuchaˆtel: Delachaux et Niestle´. [English trans.: The child’s conception of number, London: Routledge and Kegan, Paul, 1952.]. Quine, W. V. O. 1969. Set theory and its logic, 2nd ed., Cambridge, Mass.: Harvard University Press. Rado, R. 1975. ‘The cardinal module and some theorems on families of sets’, Annali di matematica pura ed applicata, (4)102, 135-154. Robinson, R. G. F. 1950. Definition, Oxford: Clarendon Press. Rotman, B. 1987. Signifying nothing: the semiotics of zero, London: MacMillan.

logische Untersuchungen, Halle: Pfeffer. [English trans. in [2003].]

Russell, B. A. W. 1903. The principles of mathematics, Cambridge:

Husserl, E. 1901. Logische Untersuchungen, vol. 2, Ist ed., Halle:

Cambridge University Press. [Repr. London: Allen and Unwin,

Niemeyer. [English translation of 2nd ed.: Logical investigations (translator J. N. Findlay), vol. 2, London: Routledge and Kegan Paul, 1970.] Husserl, E. 1994. Early writings on the philosophy of logic and mathematics (translated and edited by D. Willard), Dordrecht: Kluwer.

1937 with new intro.]. Russell, B. A. W. 1919. Introduction to mathematical philosophy, London: Allen and Unwin. Russell, B. A. W. 1936. ‘On order in time’, Proceedings of the Cambridge Philosophical Society, 32, 216-228. [Repr. in Logic and knowledge, London: Allen and Unwin, 1956, 345-363; and in

Husserl, E. 2003. Philosophy of arithmetic. Psychological and logical

Collected papers, vol. 10, London: Routledge, 1996, 122-137.].

investigations, with supplementary texts from 1887–1901 (trans-

Saguillo, J. M. 2009. ‘Methodological practice and complementary

lated and edited by D. Willard), Dordrecht: Kluwer. Kant, I. 1763. Versuch den Begriff der negativen Gro¨ßen in die Weltweisheit einzufu¨hren, Ko¨nigsberg: Kanter. [Various reprs.

concepts of logical consequence: Tarski’s model-theoretic con-

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sequence and Corcoran’s information-theoretic consequence’, History and philosophy of logic, 30, 49-68.

Scriba, C. J. 1968. The concept of number, Mannheim: Bibliograph-

Whitehead, A. N. 1934. ‘Indication, classes, number, validation’, Mind,

isches Institut. Sinaceur, M.-A. 1971. ‘Appartenance et inclusion. Un ine´dit de

new ser., 43, 281-297, 543 [corrigenda. Repr. in Essays in science and philosophy [ed. D. Runes], New York: Philosophical

Richard Dedekind’, Revue d’histoire des sciences, 24, 247-254. Smith, B. (Ed.) 1982. Parts and Moments. Studies in logic and formal ontology, Munich: Philosophia. Tymoczko, T. 1985. New directions in the philosophy of mathematics, Basel: Birkha¨user. van Heijenoort, J. (Ed.). 1967. From Frege to Go¨del. A source book in mathematical logic, Cambridge, Mass.: Harvard University Press. Venn, J. 1894. Symbolic logic, 2nd ed., London: Macmillan. [Repr. New York: Chelsea, 1970.]. Walsh, A. 1997. ‘Differentiation and infinitesimal relatives in Peirce’s 1870 paper on logic: a new interpretation’, History and philosophy of logic, 18, 61-78.

Library, 1948, 227-240.]. Whitehead, A. N. and Russell, B. A. W. PM. Principia mathematica, 3 vols., 1st ed., Cambridge: Cambridge University Press, 19101913. [2nd ed. 1925-1927.]. Yeldham, F. 1936. The teaching of arithmetic through four hundred years, London: Harrap. Middlesex University Business School, The Burroughs, Hendon, London NW4 4BT, England, UK e-mail: [email protected]

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Mathematically Bent

Colin Adams, Editor

Hardy and Ramanujan COLIN ADAMS The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of 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.

â Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected] 1

C. P. Snow, Variety of Men. London: Penguin Books, Ltd, 1969.

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THE MATHEMATICAL INTELLIGENCER Ó 2010 Springer Science+Business Media, LLC

Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘‘I thought the number of my taxicab was 1729. It seemed to me a rather dull number.’’ To which Ramanujan replied: ‘‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’’ —C. P. Snow1 amanujan was in poor health from the day of his arrival in England. The climate was one for which he was ill prepared. He never complained, but his nasal drip was a constant reminder of how imprudent it might have been for me to have brought him from India. Eventually, he ended up in the care of a clinic, where I would regularly go to visit him. One day, I arrived to find him in bed adding large numbers without the aid of paper and pencil. He used a convoluted algorithm involving his fingers and toes as well as several nurses who had to position themselves at various points around the room under his instructions. Upon my arrival, he thanked the nurses politely and then waved them out. I sat down, but, as usual, found myself unable to begin a casual conversation. This personality defect originated with my nanny, Mrs. Hanscomb, who used to shock me with a large car battery whenever I commented on the weather. It was her firmly held belief that trivial banter should be strongly discouraged. Unfortunately, Ramanujan had also never learned the fundamentals of social engagement, so after 10 minutes of silence and exchanges of expectant glances, I hazarded an opening gambit. ‘‘Ramanujan, I see you had two nurses taking care of you today.’’ A glazed look passed over his face. ‘‘Hardy,’’ he said, ‘‘2 is the smallest divisor of 2,136,575,432.’’ ‘‘Yes, that is true, ‘‘ I replied, trying not to let on how trivial I found his observation to be. Although brilliant, his utter lack of mathematical training meant that he often did not realize whether or not a given assertion was significant. ‘‘Did you come by taxi today?’’ he asked. ‘‘Yes, I always come by taxi.’’

R

‘‘And what was its number?’’ ‘‘It was 1556’’. ‘‘Oh, this is a great disappointment, Hardy. I have been so looking forward to your taxi number all morning, and this number is not an interesting number.’’ ‘‘Listen, Ramanujan,’’ I replied. ‘‘I spent 30 minutes at the taxi stand waiting to get a good number. But Littlewood grabbed 1361, and there wasn’t a prime left in the bunch. I cannot spend my entire day trying to find a cab with a number that will amuse you.’’ ‘‘Oh, Hardy. I apologize. 1556 isn’t such a bad number. At the very least, it is the first number that is 4 times a prime whose digits add to 20.’’ ‘‘Yes, it is that,’’ I replied, feigning awareness of this fact. ‘‘Hardy, I must tell you. I am very hungry. Is there not any edible food in this entire miserable country?’’ ‘‘But Ramanujan,’’ I replied, ‘‘They have left you a kidney pie on your tray. They already cut it into three pieces.’’ ‘‘That makes each a third of the whole, Hardy,’’ responded Ramanujan. ‘‘In fact, that does not follow, Ramanujan. I did not say that the pieces were equally sized. It could, for instance, be the case that one piece is half of the pie and the other two are each a quarter.’’ ‘‘But, Hardy, look at the pie. They are three equally sized pieces, each a third.’’ This was the essence of the problem with Ramanujan. He often arrived at conclusions based on evidence he observed in the real world, rather than relying entirely on abstract mathematics. He reminded me of the great English batsman Braddock, who was brilliant when he was facing the right direction and laughable the rest of the time. We sat in silence for the next 10 minutes, both doing our best to avoid eye contact. Finally, Ramanujan spoke. ‘‘Tell me Hardy, what is your favorite number?’’ ‘‘You asked me that, yesterday,’’ I replied. ‘‘I ask you it every day,’’ he retorted. ‘‘You know what I mean. What is your favorite number today?’’ I knew that I could only embarrass myself with an answer. Either Ramanujan would use his prodigious mathematical talents to instantaneously uncover the amazing properties of my number or he would tease me if it had none. But given his circumstances, I felt obliged to respond. I decided to narrow my choices to numbers no larger than 100,000. Otherwise, it would take me too long to reach a conclusion. Ramanujan waited with that expectant look I had come to dread. I eliminated the even numbers, which are simply a product of some other number and 2, so why not just consider the other number. I repeated this process with numbers divisible by 3, 4, and 5, until I realized at this rate, I might have none left. So I then eliminated the primes, being much too obvious. Then I discarded numbers divisible by higher powers of primes, and numbers divisible by primes that were themselves the sum of the prime divisors of the number. I subsequently eliminated those numbers whose digits, when reversed, yielded a number that was the sum of n other numbers, all of which themselves were palindromes. Continuing in this manner, I eventually whittled the options down until I found myself with only one number remaining. So I said, ‘‘Well of course, Ramanujan, my favorite number is 67,789.’’

‘‘Of course,’’ replied Ramanujan, beaming. ‘‘The digits of which yield the famous riddle. Why was 6 afraid of 7? Because 7 ate 9.’’ ‘‘Exactly,’’ I replied, trying not to let on that I had never heard this riddle before. People tended not to tell me riddles because of the choking noises I made when I laughed. Ramanujan suddenly surprised me by reaching out and grabbing my hand. For such a sick man, he had a unexpectedly strong grip. I held my hand as limply as possible. ‘‘Hardy,’’ he said. ‘‘I have known you now for quite some time. Do you think I could call you by your first name?’’ I found this request quite awkward. For I was not one to promote familiarity. Who knows where it might lead? But given the situation, I had little choice. ‘‘I suppose so. If that is your desire.’’ ‘‘What is it?’’ ‘‘What is what?’’ ‘‘Your first name.’’ ‘‘Well, it is G. H.’’ ‘‘No, Hardy, those are your initials. What is your actual first name?’’ I began to sweat. ‘‘I prefer not to divulge it.’’ ‘‘I am imploring you, from my sickbed, tell me your name.’’ ‘‘Very well, then, if you must know, it is Godfrey.’’ ‘‘Godfrey? Godfrey? Hardy, that is a wonderful name. I do not understand why you dislike it so.’’ I was not about to explain to Ramanujan the many ways that Mrs. Hanscomb had tormented me over my name, and always with that infernal battery. But, at any rate, I decided I had had enough for one day, and it was time to get back to my mathematics. ‘‘Good Lord, Ramanujan, look at the time. I must go at once.’’ ‘‘I am sorry, Hardy. Please stay. I will not call you Godfrey. Tell me again how useless the mathematics is that you do. That so entertains me. How nothing you have done will ever prove relevant to cryptography or quantum physics or any of the other applied scientific endeavors. And how you revel in that fact.’’ ‘‘I am sorry, Ramanujan, but I absolutely must run. You know I must prove three theorems before tea, and they are serving crumpets today, and the afternoon cricket match begins right after that. The cricketers would be very disappointed if I were not observing from my customary viewpoint. But I shall return tomorrow.’’ ‘‘Very well, Hardy, but please do try to find a cab with an interesting number. You know what that means to me.’’ ‘‘You know, Ramanujan, I believe I shall bicycle tomorrow.’’ He looked as if he might cry. But I fought down the urge to give in and waved goodbye as I slipped out the door. That evening, as I watched the cricket match, I was struck with an overwhelming case of remorse. Ramanujan had done nothing wrong to deserve his fate and the least I could do was to support him in his time of crisis. The next day, I found myself at the taxi stand an hour early. I picked out a beauty and then paid the driver to sit with Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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me in the cab until it was time to go. Upon arrival at the clinic, I was so pleased with the number that I leapt from the cab, rushed into the building and burst into Ramanujan’s room. But I was surprised to find there was no Ramanujan. The bed was empty. The nurse who was changing the sheets looked up and just shook her head sadly. And so, brokenhearted, I leaned against the doorframe. Ramanujan was gone. We had lost one of the greatest mathematical geniuses of all time. Memories of all of the fun we had had together flooded into my head. Laughing uncontrollably over the inappropriately named perfect numbers. Making fun of Boyer’s attempted proof of the Goldbach Conjecture. Giggling behind his back when Hall believed us that 92,650,699 was a prime. Those were special days.

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And most difficult of all, I would have no one with whom to share the lovely number of my cab. I regretted terribly my thoughtless behavior from the previous day. I should have encouraged Ramanujan to call me Godfrey and perhaps, even asked him for his first name. I owed him an apology, an apology from one mathematician to another. But the sad truth was that it was an apology I would never be able to deliver. Now it was too late. The nurse cleared her throat. I turned and she smiled at me kindly, preparing to deliver some consoling words, perhaps with the hope of initiating a personal relationship. I immediately scurried out the door, hopped into the cab and returned to the safety of my rooms at Cambridge.

Mathematical Entertainments Michael Kleber and Ravi Vakil, Editors

Mathematical Vanity Plates

drove away and I was unable to follow. I’ll never know if he was aware that the state had assigned one of the most important numbers in mathematics to his car. How I envied him! Indeed, 65536 is not just an ordinary power of 2, and not simply the order of the multiplicative group in the Galois field of the largest known Fermat prime. It is

DONALD E. KNUTH

22

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome. Extracted from the author’s book Selected Papers on Fun and Games (2010) CSLI Publications, Stanford. his story began in the spring of 1967 when I made a visit to Madison, Wisconsin, in order to give a lecture at the university. I was driving near the campus on a fine, sunny day. There was moderate traffic, and when a red light stopped me I happened to glance at the license plate of the car ahead. My jaw dropped—wow! It said ‘‘H65536’’. I can still picture that plate in my mind, as plain as day, although more than forty years have passed; in fact, with modern software, I now can recreate its exact appearance:

T

I started to jump out of my car, thinking that I might approach the driver to ask him if he knew what a fantastic license number he owned. But the light turned green; he

â

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]

22

22

¼ 22

20

;

a number of huge importance in computer algorithms. Already in 1967 I had become a big fan of this number, and my appreciation has continued to grow as I continue to learn more.

The Old Rules In those days each state of the USA required its drivers to purchase new license plates with new license numbers every few years, assigning the values in sequence. Thus your chance of getting any particular number n was at most 1/n. People began to covet the small numbers—not because small numbers are mathematically interesting (which they are), but because small numbers are easy to remember. Small numbers also connote rank: The governor was number 1. The New York Times reported in 1959 [39] that Low-numbered ‘‘vanity’’ plates have had a long and distinguished history in the automobile business. They have traditionally been obtained by politicians for their friends and campaign contributors. They have long been doggedly sought after by celebrities, egomaniacs and men who have everything. Last year about 69,000 New Yorkers paid $5 extra for special low-numbered plates. About 5 million automobiles were registered in New York state at that time; New York plates contained one or two letters to encode a county, together with a serial number. Clever people didn’t actually have to settle for a random number on their licenses, however. It was perfectly legal in many states, including New York and California, to get your plates in a remote county, far from where you lived. During a vacation to a sparsely populated area, you could drop in to the local office and obtain decent numbers without much competition. Furthermore, your initials might well be available in some rural county. My father-in-law, who lived at 525 Summit Street in the medium-sized city of Fostoria, Ohio, was able to drive Ohio plates reading 525 S (his address) ever since the 1950s, because he was a friend of the local people who were in charge of auto registration. Beginning in 1967, Wisconsin motorists could obtain plates of the forms A9, A99, A999, A9999, or AA999 by paying a surcharge, where A denotes any letter and 9

 2010 The Author(s). This article is published with open access at Springerlink.com, Volume 33, Number 1, 2011

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denotes any digit. Thus a person could pay to have G256, say, but not H65536. I like to imagine that the man who acquired the plate I saw in Madison had obtained it on a day when he knew that the 65,500s were being distributed. By carefully trading places with people behind him in line, he could then have seized the opportunity to acquire the license number of his dreams. Wisconsonites are friendly folks who would have been quite willing to accommodate such a whim—unless of course two mathematicians had been in the same line. Every state in the USA has its own esoteric rules about license tags, except in one respect: All plates made since 1956 or so have the same size, 600 9 1200 , so that automobile manufacturers know how much space to allocate for plate display on their cars. When California adopted the standard size, it had to rethink the format of license numbers because the existing plates were somewhat larger. The old ‘digit-letter-digitdigit-digit-digit-digit’ style was capable of distinguishing 26,000,000 vehicles; for example, a license such as

would be typical. But a maximum of six characters was desirable on the new plates, and California had more than 5.5 million registered cars. So the authorities introduced a new tagging scheme in which all passenger cars were identified

by three black letters followed by three black digits, on a yellow base, potentially allowing 2303 = 12,167,000 combinations. (The letters I, O, and Q were excluded.) These plates were distributed in batches of 1000 to district offices; thus you could easily identify where any given license had been obtained, by knowing local letter codes. By 1963 it was time for new plates, and all Californians got new numbers again. The new scheme featured three yellow letters followed by three yellow digits, on a black base; my wife and I received

(meaning ‘‘Whiz by at 65 miles per hour’’—the legal speed limit.) I didn’t realize until later that, with this plate, we already owned three of the six characters in H65536. Notice that if I had been just 66 places earlier in the queue, my car would have been repeatedly asking, ‘‘WHY 999’’? Late in 1962, a determined young man had driven from Los Angeles to the small town of Alturas in the rugged northeastern corner of California, where he stood in line for 42 hours, braving both cold and rain to win the ultimate prize: California license AAA 000, the first of the new series [1]. I remember seeing him interviewed on television when he substituted this trophy for his former plate (which was WWW 333).

Three-Letter Words

AUTHOR

......................................................................... DONALD E. KNUTH is Professer Emeritus

of The Art of Computer Programming at Stanford University, where he supervised the Ph.D. dissertations of 28 students since becoming a professor in 1968. He is the author of numerous books, including 4 + e volumes (so far) of The Art of Computer Programming, five volumes of Computers & Typesetting, 1/3 volume of Concrete Mathematics, eight volumes of selected papers, and a collection of nontechnical essays entitled 3:16 Bible Texts Illuminated. His software systems TEX and METAFONT are extensively used for book publishing throughout the world. Computer Science Department Gates Building 4B Stanford University Stanford CA 94305 USA http://www.cs-faculty.stanford.edu/~knuth

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The 1963 licenses began to allow the letter Q as well as the vowels I and O, in the first and second positions, thus forming many new three-letter combinations with familiar connotations. Back in those days LOL didn’t make anybody laugh; but the license ONE 234 was quickly spotted on a black Corvette. Staunch Democrats refused to accept GOP, but HIS and HER plates became popular with two-car families. More serious, however, were three-letter combinations that were patently offensive, like (censored), (censored), (censored), and (censored). California’s license czars had foreseen this problem already in 1955, when they asked Prof. Emeneau of the linguistics department at UC Berkeley to find all of the unsuitable words in the set fA; B; C; D; E; F; G; H; J; K; L; M; N; P; R; S; T; U; V; W; X; Y; Zg3 . He and his helpers ‘‘came up with 152 no-nos,’’ including the word YES [27]. California’s Department of Motor Vehicles (DMV) has always disallowed CHP (California Highway Patrol) and CIA (Central Intelligence Agency) on license plates; but DMV is OK. Perhaps they don’t know that DMV stands for Deutsche Mathematiker-Vereinigung.

Four-Letter Words American license plates rarely contained more than two letters during the first half of the 20th century. The reason may have been that license numbers were regarded as, well, numbers. Or perhaps officials realized that the

possibilities for mischief grow exponentially as the number of letters increases. At any rate, motorists continued to want more and more varieties of personalized license tags, until finally the concept of vanity plates has broadened: Instead of simply having small numbers and/or our own initials, we’re now free to express ourselves in any tasteful way. Seeds of this revolution were planted in 1937, when Connecticut motorists with a good driving record were allowed to get plates containing just their three initials. Four-letter combinations appeared in Connecticut during the 1940s [2]. New Hampshire issued vanity plates of up to five letters in 1957; and Vermont followed soon after, allowing letters and numbers to be mixed. Several years went by before the idea of decorating automobiles with personal names began to evolve into the display of personal statements. For example, when vanity plates of up to five letters became available in the District of Columbia in 1964, people’s first choices were the one-letter plates from A to Z; numbers 1, 2, 3 were pre-reserved for the mayor, deputy mayor, and city council chair. (I don’t think 0 was permitted.) Five years later, Colman McCarthy [32] surveyed the 6000 choices that DC drivers had reserved during their first years of plate-naming freedom. He found that most motorists chose their initials or their first names, in order to make their cars look swanky. A second group, reacting against ‘‘faceless numbers sent by a faceless computer,’’ wanted to be identified in a more personal way using nicknames; the B-names in this category were BABS, BABY, BALDY, BEBE, BIRDY, BOBO, BOOFY, BUBI, BUNNY, BUTCH, and BUZZ. But the new folk art of devising witty licenses was also beginning to develop in DC: A man named Carl Levin got C-11 for himself and S-11 for his wife Sonia. A man named Ware had two cars, respectively AWARE and BWARE. A retired Navy captain’s Buick sported the license AWOL, received as a gift when he stepped down as head of the Citizen’s Association of Georgetown. Uplifting messages like FAITH, HOPE, LOVE, JOY, MERCY, PEACE (and PAX) were also in evidence. Although YES was forbidden in California, the District of Columbia had both YES and NO as well as OUI; also OH NO, SORRY, and OOPS. Somebody in the nation’s capital had MONEY, and I imagine somebody else had POWER—but McCarthy didn’t speak to that.

California Vanities California embraced self-designed license plates in 1970, but with a new twist: Motorists paid $25 for this privilege, and $10 per year afterwards, all earmarked for a special fund to fight pollution; thus the plates became officially known as environmental plates. Up to six letters and/or numbers were permissible, plus an optional space. Among the first requests were PAID 4, OREGON, TURTLE, WHY WAR, GRANNY, and 32 FORD. Initial applications were collected for 30 days so that the most popular choices could be awarded by lot; but afterwards the rule was to be strictly first-come, first-served. The most wanted words, among about 10,000 applications initially received, were PEACE, SMILE, JAGUAR, GEORGE, TBIRD, BOB, LAWYER, MARGIE, MORGAN, LARRY, SNOOPY, LOVE, in decreasing order of popularity.

By December, the number-one Christmas gift in southern California was a new license plate with a message like HOHOHO or XMAS 70. Californians purchased more than 65 thousand personalized plates during the first 1.5 years, thereby raising more than $1.5 million for the environmental fund; about 1500 new requests were being processed each week. By 1977 the fund had received $18 million, and a total of 423,213 plates had been issued before April 1979, although 55,523 of them had not been renewed. Narrower letters and numbers were adopted in 1979 so that the licenses could contain up to seven characters; thus many more choices became possible, and about 600 new applications began to be filed every day. The millionth environmental plate was produced in 1982, at which time about 750,000 were actually in circulation.

A Complete List One of my cherished possessions is the official list [8] of all California environmental plates that were current on 21 July 1981, obtained from a friendly administrator in 1982 when I explained that I was a computer science professor interested in database research. Altogether 665,571 entries appear in this list, and I estimate via random sampling that about 40% of them have length 7. What a wealth of ingenuity exudes from almost every page of this list! For example, I wondered if anybody had wanted to put BONFIRE on his vanity plate; sure enough, there it was—together with BONFYR and BONFYRE, as well as FALO, FALO1, and FALO2. To get an inkling of California’s 836-page collection, vintage 1981, let’s consider page 316, which contains 798 entries from HOWZEIT to HRNDEZ, including HOWZIT, HOW2WIN, HOXIE, HOY, HOYLES, HOZWIFE, HP, HPBMW, HPBOOKS, HPBOSS, HPBRDAY, HPENNY, HPNOSIS, HPOWER, HPTYHOP, HPYTRLS, HRAKA, HRBLOCK, HRBONUS, HRCULES, HRDCORE, HRDROCK, HRDRSR, HRDTOP, HRDWARE, HRD2GIT, HRD2PAS, HRD2PLZ, HRD24GT, HRGOD, HRH, HRIZON, HRLEQIN, HRMIONE, HRMMPH, HRMNIZE, HRNBLWR. (Spaces are suppressed here because they’re ignored in the DMV’s test for equality of names.) One thing that’s immediately clear is that people with the same idea have been forced to spell it in different ways. We find plates like PLEYBOY, JIGOLO, FORSAIL, STOLIN, JILLOPY, LAEMON, NOWHEY, and POETIQ; of course this is just poetic license. All 24 letters of the Greek alphabet are present in [8]— except for ZETA and XI, which are represented by ZETA7 and XICHI. And they often occur with ‘‘subscripts’’: ALPHA0, ALPHA1,…, ALPHA8, as well as ALPHA01 and ALPHA99 (which I guess were chosen by statisticians or social scientists). Latin scholars have contributed ERGOSUM, ERGOIAM, and ERGOIGO. We find both PUBLISH and PERISH, as well as HANSEL and GRETEL, PEARL and OYSTER, GOLD and SILVER, DONNER and BLITZEN, etc. Yet there also are surprising gaps: The English numerals ONE, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, ELEVEN are present and accounted for, but

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TWO and TWELVE are lacking. Nobody in 1981 had even thought of VEHICLE. There was neither YIN nor YANG. Music lovers had named their plates after HAYDN, MOZART, B8HOVEN, BRAHMS, and MAHLER; also BACH, BACHJS, JSBACH, JSBACH1, and PDQBACH. But not (yet) BERLIOZ, DVORAK, or MILHAUD. A Beethoven fan had selected OPUS 132. ADAGIO, ALLEGRO, ANDANTE, LARGO, and PRESTO? Yes. LENTO and VIVACE? No.

Other States Vanity-plate fever has continued to spread until all 50 states of the USA have long since flowed with the stream; in fact, the last state to inaugurate ‘‘own-choice’’ licenses was West Virginia in 1982. But even today, Massachusetts requires all numbers to be preceded by letters; in that state you can say MITPHD or MIT69 but not MY69MG, and you must use at least two letters. Illinois is similar, but it insists on a space separating letters from numbers. Statistics from all 50 states and the District of Columbia, based on the total number of registered motor vehicles in 2005 as well as the total number that were ‘‘vanitized’’ as of 2006 or 2007, show that the current fraction of vanity plates in the USA was 9292843/242991747&3.8% at that time, with greatest penetration in Virginia (1065217/6578773&16.2%) and least in Texas (97315/17347615&0.6%). California ranked 22nd, with a ratio of 1136772/32592000&3.5%. Thus vanity plates have clearly become thoroughly integrated into American pop culture. For a nice survey of vintage 1986 plates, see Eckler [13]. A complete list of current South Dakota vanity plates, containing 15,483 entries from 001 to ZZZZOOM, was posted on the Internet on 25 October 2009 [30], and it’s interesting to compare it to the California list of 1981. The interval from page 316 of [8], discussed above, intersects the South Dakota data in only three cases: HPYDAYS, HPYTRLS, HRDROCK. There are 22 other South Dakota plates in the corresponding interval, including HPBD2ME, HPYGIRL, HRDLUCK, HRDLY, HRHOTRD, and HRMSWAY.

The Challenge So how do mathematicians fit into this fast-growing trend? Dentists can advertise their specialty with ‘‘dental plates’’ such as DENTIST, 2TH DR, DRILLER, SMILE, MOLARS, CUSPID, GUMSAVR, NO DK, FLOSSEM, CROWNS. Physicists now sport licenses like PHYSICS, NUCLEON, DELTA S, GLAST, XRAY BMR, CY N TIST, E PLUS, and E MINUS [11]. And of course E COLI [8] belongs to a biologist. If you, like me, don’t already own a vanity plate, what would be your first choice? What’s the briefest and best way to represent yourself to the world? If there’s something that’s uniquely you, there’s a greater chance that it won’t already be reserved. Ideally your choice should bring a smile to people who see it. They’ll think, ‘‘Aha! A person who loves mathematics!’’ Alternatively, you might choose something that will please and amaze specialists, although the masses won’t get the point; that can still lead to teachable moments. Many of the things we deal with as mathematicians involve words that are much too long: DIFFERENTIAL, MATHEMATICS, DETERMINANT, CONJUGATE, INTEGRATION, 36

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COMPUTATION, LOGARITHM, PERMUTATION, ARITHMETIC, ALGORITHM, COUNTEREXAMPLE, etc., etc. There are some shorter terms like MATHS, ALGEBRA, SINE, COSINE, TANGENT, MATRIX, GROUP; but they’re in [8], already claimed ages ago. What’s left? I remember discussing this question with my son, long ago when he was in elementary school and vanity plates were fairly new. I told him that, ha ha, our car should say VANITY. (I naı¨vely believed that this was an original thought.) But he had a much better idea: How about letting the plate be entirely blank? At that very moment I realized that he was a budding mathematician, wise beyond his years. For indeed, what could be more vain, yet more rich in mathematical properties, than the empty set? This idea trumps even the governor’s number 1. Unfortunately, though, the empty string is too short a word for state bureaucrats to understand. And even if they did issue a blank plate when you left the form blank, what do you think highway patrolmen would do when you drove by? Not everybody understands the empty set, alas. On the other hand—surprise—I happened to spot 7 SPACES, on 8 August 2004. Aha, I learned, the empty plate does exist! (Moreover, 7 BLANKS is still available in California, as of 5 November 2009.)

The Character Set Before we investigate the possibilities further, we need to know the ground rules, which differ from state to state. The most important limitation is the total number of characters. Wyoming residents are expected to be most creative: They must express themselves in four symbols or less, in order to leave room for a county code and the famous bronco-buster logo. (Unfortunately, this state and some others have recently switched to ‘‘digital plate technology,’’ produced by laser printing instead of embossing, with an atrocious font. It looks cheap, perhaps because it is. If I lived there I’m afraid I would choose the word UGLY until they reverted to the beautiful style of 2002.) The maximum message length in Alaska, Connecticut, Hawaii, Kentucky, Maine, Massachusetts, Missouri, Oregon, Rhode Island, and Texas is 6. All other states have a limit of 7, except for New York, North Carolina, and West Virginia, where you can (gasp) go up to 8. Only in the latter three states can you claim GEOMETRY or TOPOLOGY. What can those 4 or 6 or 7 or 8 characters be? All 26 uppercase letters from A to Z are legal, and New Mexico ˜. The digits 2 through 9 are obviously all OK allows also N too. But 1 is equivalent to I in Louisiana and Minnesota; and 0 presents a really sticky problem. Consider the following nineteen entries from [8]: OO, O0; OOO, O0O, O00; OOOO, O000; OOOOO, O000O, 00O00, 0OOO0; OOOOOO, OO00OO; OOOOOOO, OOOO00O, OOO0OOO, OOO0O0O, O00000O, O000000. There are 240 or 241 combinations of O and 0 that don’t conflict with plates in the ordinary sequential series; by July of 1981, Californians had reserved those 19. (I’m not sure whether 0OOO000 would have been legal.) Such plates must have caused nightmares for police officers because they’re so hard to distinguish. Therefore O and 0 have now become identical

in California; also in Colorado, Louisiana, Minnesota, Nevada, North Carolina, Vermont, and Wisconsin. I don’t know when this change was made, or if cars with visually ambiguous licenses are still on the road in California. South Dakota still considers O and 0 to be distinct; I know this because [30] lists both O0O0O0O and OOOOOOO but no other combinations of length 7. Massachusetts does too, because AUTO is taken but AUT0 is not. The most interesting variation between states, vanitywise, is the set of allowable delimiters or punctuation marks that can appear. My eyes popped in 1985 when I first saw New Hampshire plates containing + signs as well as - signs, opening up a whole new world of mathematical vanities. Complex analysts in the Granite State can adorn their cars with X+IY. (California now has + signs too; see below.) New Hampshire had previously used heavier, red-cross-like symbols on licenses specifically for ambulances; in 1974 they reserved AMB for ambulance plates, and eventually they made vanity pluses available for use by anybody [18]. Furthermore, New Hampshire drivers are able to use ampersands. Hence they can construct the wonderful formula -X&X, which yields the least significant 1-bit, if any, in the binary representation of X (see [26, Eq. 7.1.3–(37)]). I checked both X+IY and -X&X on New Hampshire’s website in November 2009, and both of them were available. But X&-X was illegal, because consecutive punctuation marks are forbidden. Most states allow only letters and digits. But a minus sign (or hyphen) is permissible in Alabama, Colorado, Delaware, Florida, Kentucky, Louisiana, Minnesota, Missouri, New Mexico, North Carolina, Oklahoma, Oregon, Pennsylvania, Virginia, and Washington, as well as New Hampshire. An ampersand is legal in Delaware, New Hampshire, North Carolina, North Dakota, South Carolina, and Virginia. You can use a dot in Colorado, Connecticut, Louisiana, and North Carolina. Apostrophes are OK in Missouri, New Mexico, and (again) North Carolina. North Carolina, in fact, is Vanity Plate Heaven. We’ve seen that tarheels are able to enjoy up to 8 characters that include pluses (+), minuses (-), dots (.), and apostrophes (’); and in fact they can also have number signs (#), question marks (?), dollar signs ($), asterisks (*), slashes (/), equals signs (=), at signs (@), colons (:), double quotes ("), commas (,), and exclamation points (!)! Unfortunately for mathematicians, the list stops there; parentheses, and the relational signs\and[, aren’t permitted. Unfortunately for TEX users, backslashes and curly braces are lacking too. But hey, $720=6!$ in this state. Calculus and physics teachers can have $dx/dt$, etc. (All special characters are ignored when comparing two plates; thus 7206 and dxdt would be indistinguishable from those punctuated examples, and so would #7+2/0-6 and dx/dt?, both of which happen to be presently available—subject to approval by the authorities.) I wonder if a fallacious proposition such as $120=6!$ would also be acceptable to the North Carolina censors. Probably it would, thereby setting back education in the Southeast. Even worse, I fear that somebody will ask for BAD@MATH and be proud to display it. But let’s not be

pessimistic; North Carolina deserves applause for leaping way ahead of everybody else. Besides ordinary punctuation, a few special characters are also available. New Mexico has an enchanting Zia Sun symbol, which their website illustrates with the example VAN ITY. New York offers a blob in the shape of its empire, which I won’t illustrate here. Since 1994, California has allowed a single delimiter to appear on vanity plates, taking the place of a letter or digit. There are four choices: Drivers can use either a plus sign or . The senior co-editorone of the unique symbols in-chief of this journal could be C LER D if he moved to California; Steve Smale could be LE BODY. Both of these are currently up for grabs, as are X+IY, I MATHS, and L SPACE. The rules for spaces (I mean blank spaces, not L*spaces) are too complicated to explain here. Suffice it to say that most states allow you to insert them in order to improve readability.

Examples on the Road Let’s look now at how some mathematicians and/or their friends have risen to the occasion by meeting these constraints. Cathy Seeley, who was president of the National Council of Teachers of Mathematics (NCTM) from 2004 to 2006, likes her license plate so much that she included it in an illustrated lecture that I found on the Internet:

Her plate sits in a holder that was in fact produced by the NCTM, saying ‘‘Do math and you can do anything!’’ (I’d amend that to ‘‘Do math and learn to write, and you can do anything’’—but my version wouldn’t fit the frame.) She told me about two other nice examples: Ed Rathmell, a math professor at the University of Northern Iowa who does a lot of work with education, is MATH ED. And Gail Englert, a middle-school math teacher in Norfolk, Virginia, has IEDUK8M, with a nice play on ‘K–8’ in education. David Eisenbud, who was president of the American Mathematical Society (AMS) during 2003 and 2004, received an appropriate plate from his wife shortly after he became director of the Mathematical Sciences Research Institute in 1997:

Incidentally, when I spotted the California plate I AM PAMS on 2 January 2001, I realized that it did not refer to

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Proceedings of the American Mathematical Society—because Clifford J. Earle, Jr. was at Cornell. Vanity plates that name the basic words of our discipline are more difficult to come by, because more people try to reserve them. Victor Miller has been doubly successful in this regard, because he not only owned New York ALGEBRA in 1978, he purchased New Jersey ALGEBRA in 1993! He still drives it:

Dave Bayer is another lucky mathematician who obtained the plate of his dreams, perfectly suited to his work on geometric invariants [4]:

He is evidently not alone, because the California list from 1981 [8] already contained SYZYGY, SYZYGY1, SYZYGY2, SYZYGY3, SYZYGYS, and SYZYGYX, as well as SYZ WIZ. The key words NUMBER, THEORY, and THEOREM were also unavailable to Californians who sought new vanity plates in 1981. But eventually two of them came back into play: Fan Chung and her husband Ron Graham (the AMS president in 1993 and 1994) now have personal plates, acquired in 1999, that are amazingly appropriate when parked side by side:

The existence of Ron’s plate implies that THEOREM is once again unavailable; so I used the DMV website to try for LEMMA. It, too, is currently taken, and so is DILEMMA. So are MATRIX and VECTOR; but not TENSOR. (According to [30], NUMBER, THEORY, THEOREM, LEMMA, DILEMMA, MATRIX, VECTOR, and TENSOR are all presently available in South Dakota, as are DO MATH, MATH SCI, ALGEBRA, and SYZYGY. This observation doesn’t prove that South Dakotans are anti-mathematics; indeed, [30] does list MATH ROX, NUMBERS, and NUMB3RS.) Fan and Ron also share a recreational vehicle with a tag that commemorates their joint work on quasi-random graphs and other quasi-random structures [9]: 38

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(This license plate incidentally illustrates the unfortunate fact that California has never figured out how to make a decent-looking letter Q. From a typographic standpoint, Wisconsin and other states would actually be better places to advertise QRANDOM research.) Ron tells me that he often sees 2+2 R 4 in the parking lot of the math building at UCSD; another one is S CUBED. At a higher level he also encounters RC FLOW, representing Ricci flow—the key idea with which Grisha Perelman resolved Poincare´’s conjecture [36]. Andy Magid has said [31] that he’s the only licensed Galois theorist in Oklahoma, because he drives GALOIS. ‘‘Most people who read or have to record my car tag, such as tow truck operators or highway patrol officers, do so without comment or even accurate pronunciation. But on occasion it does provoke welcome conversation.’’ What about the names of other famous mathematicians? I checked a few, to see what California drivers have selected, and got a shock: name

taken in 1981?

taken in 2009?

FERMAT

no

yes

EULER

yes

no

LAPLACE

yes

yes

FOURIER

no

yes

GAUSS

no

yes

GALOIS

no

no

HILBERT

no

yes

Alas—my personal hero, Leonhard Euler, has waned in popularity out west. (But David Robbins did have New Jersey EULER for many years.) The license plate BIG OH has been spotted in New York, and it’s also present in California. I love that notation [24]. At present New Yorkers, like me, are less enthusiastic about LITTLE OH. My colleague Ingram Olkin in Stanford’s Department of Statistics received STAT PRO as a gift from his children about ten years ago. His daughter Julia now has the California plate SOLV4X; she really wanted SOLVE4X, but it was already taken in 1981 [8]. California’s 1981 list had OPTIMUM,MINIMUM, and MAXIMUM; also OPTIMAL and MINIMAL but not yet MAXIMAL. It included both MINMAX and MINIMAX, as well as MAXIMIN and—my favorite—MAXIMOM. There was a Z AXIS on the road, but no X AXIS or Y AXIS. Somebody had DY DX; another had DU DS but without the space. Fifty-two of the California environmental plates in 1981 began with MATH. Some of them, like MATHER, MATHEW, and MATHIS, were surnames that aren’t relevant to our discussion; MATHIEU may, however, have belonged to a group theorist. Noteworthy are MATHBIZ, MATHMAN, MATHPRO, MATHS,

Figure 2. Shades of Leibniz and Boole.

Figure 1. NCTM prizewinner Carol Bohlin.

MATHWIZ, and MATH4U. I fear that MATHANX was supposed to suggest anxiety rather than thank-you-dear-ma; and MATHOS was perhaps a feminine form of PATHOS. A light-hearted competition for the best vanity plate with a mathematical theme was organized by the NCTM at the beginning of 2005, and prizes were awarded at their meeting in April of that year. First place went to a ‘‘pair o’ docs’’ named Carol and Roy Bohlin, who teach at Fresno State University and always carry a trunkful of material for the math classes and workshops that they teach (see Figure 1).

Small Integers Of course license plates have traditionally featured numbers rather than letters, and nothing can be more mathematical than numbers. Therefore mathematical vanity plates often involve carefully selected numbers. Let’s start at the beginning with 0. We’ve already noted that California’s database from 1981 [8] had many combinations of 0 and O. It also included DOUBLE0, DOUBLEO, DBL00; in fact, somebody even claimed 0X0 EQ 1! There was a vote for 00NUKES. My favorite from [8] in this category, however, is 00MPH—a plate that is not only pronounceable, it is semantically equivalent to ST0PPED, if you think about it. Moving up, the use of 1 as a cardinal number was quite common, as in 1 EGO FIX, 1 FOW VEY, 1 HONKER, 1 LITER, 1 MOMENT. The ordinal 1 appeared too, in 1ST AID, 1ST ALTO, 1ST HALF, 1ST LAP, 1ST N 10, 1ST VIOL. The plate 1F100 may have denoted a hypergeometric function; but I don’t know the significance of 1DELTA1, 1OMEGA1, 1SIGMA1 (all found in [8]). Laurence C. Brevard tells the following story (see Figure 2): I got 1 OR 0 in Texas in 1982 but the picture shows the 1984 plate. Back then you got new plates every year instead of the stickers they use now. . . . After I moved to Oregon in 2001 I got the same plate ‘‘number’’ there. People consistently thought the OR stood for Oregon. Sigh. . . Someone else has this plate in California! I also have had the domain 1or0.com since 1998. May I suggest ORBITS? (Sorry.) Bill Ragsdale sent me a picture of the license

which he spotted

during a trip to China in 1984. He theorizes that ‘‘China is moving to binary license plates due to the difficulty of their character set.’’ The website of Utah’s Division of Motor Vehicles [40] provides helpful examples of personalized plates, including 2XX3XY, which certainly looks mathematical. Upon further inspection, however, this one actually turns out to be genomical: The driver has two girls and three boys. Indeed, I’ve rarely seen license plates that celebrate the beautiful mathematical properties of small integers. Why haven’t people chosen messages such as 0 IS NONE, 0 IS LOG1, 1 IS UNIT, 2 IS EVEN, 3D WORLD, 4 COLORS, 5 IS F5, PERFECT 6, 7 FRIEZES, 8 IS CUBE, or PAPPUS 9? A large territory remains to be explored. (In [8] I do find 1 IS ALL and 10ISBUF; but the latter surely was chosen by a tennis buff.) Franc¸ois Le Lionnais wrote a classic book [28] that is filled with good reasons to like particular numbers, and David Wells has written a similar but more elementary sequel [41]. (See also De Koninck [12].) There’s a well-known proof by induction that all nonnegative integers are interesting; for if this statement were false, the smallest noninteresting number would certainly be quite interesting. QED. Following this reasoning we can conclude that the number 62 is interesting, because it’s the smallest integer that appears neither in [28] nor in [41]. The more recent book [12] lists only prosaic facts about 62. But John Conway has discovered that 62 also has a far more interesting property, namely that it’s the least n such that no number is exactly n times the sum of its digits. (See [38], sequence A003635.) We could make a vanity plate from that fact: WHATS 62. On the other hand, Gordon Garb told me a cautionary tale. After having been excited and inspired by a reference to a paper by Li and Yorke entitled ‘‘Period three implies chaos’’ [29], he once decided to acquire the California vanity plate PERIOD 3. Unfortunately, his choice didn’t turn out to be as cool as he had hoped: In the years that I had it on my vehicle, nobody ever got the Chaos reference. I explained it many times when friends asked, but what fun is an inside joke if you always have to explain it? My future wife told me years later that she just assumed I was a hockey fan. . . . I replaced it with a vanity plate that simply has my typical login name.

Bond; James Bond What is the most-wanted three-digit number on a vanity plate? The winner, hands down, is 007. Uncountably many motorists have apparently dreamed of masquerading as Ian Fleming’s immortal character James Bond. In the 1981 list [8], for example, one can find AGNT007, BONDOO7, BOND

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007, DBLOH7, DBL07, DOUBLO7, DUBLE07, D007, JAMES07, JBOND, JB007, JMSBOND, JMSB007, J007, OHOHSVN, OHOH7, OO7, OO7BOND, OO7LIVZ, OOSEVEN, OOVII, O07, 007LIVS, 00SVN, and probably more that I’ve missed. (The choice 007 was not allowed; nobody had yet taken 0O7.) In 1966, Sol Golomb created Bond’s illustrious binary cousin, Secret Agent 00111, when he introduced the concept of run-length encoding for sequences of bits [16]. Several of the exciting and bizarre exploits of this intrepid information trafficker were revealed later, in an introductory textbook [17]. Agent 00111 either remained undercover in 1981 or failed to own a California vehicle at that time; but he or she is now driving somewhere in the state, and also in Florida.

Real Numbers Mathematics doesn’t limit itself to whole numbers, of course; many others are out there, including SURREAL ones [8, 22]. What do you think is the first noninteger number that most people think of? You’re right: It’s p. To verify this hypothesis, I looked up the license plates that are currently available in Florida, Michigan, Nevada, and North Carolina, which are among the few states that allow motorists to choose plates that contain seven digits and no letters. In each case the combinations 3141592 and 3141593 have already been taken, but 3141591 and 3141594 have not. (Recall that p&3.1415926535897932.) This cannot be just a coincidence. In North Carolina, which allows up to eight digits, I checked for 31415926 and 31415927 as well, but those plates are still available. The p-fanciers of that state have most likely taken advantage of their typographic freedom by selecting plates with the decimal point included: 3.141592 and 3.141593. Arizona motorists have currently claimed 314, 3141, 31415, 314159, and 3141592, but not 3141593. Have they perhaps been basing their choices on the successive version numbers of TEX [23]? Although California doesn’t permit plates that are entirely numeric, the 1981 list [8] does include QT314, QT31415, and QT31416, as well as PI R SQ and PI R2. The people have spoken: p wins! Figure 3 is a historic photo from 1987 that shows four people who have made significant contributions to the high-precision evaluation of p: Gene Salamin, Yasumasa Kanada, David Bailey, and Bill Gosper. They’re gathered around Bailey’s appropriately numbered car. At that time Kanada held the world record, having recently computed p to 134,214,700 decimal places. He had helped with the first calculation that exceeded 2 million places, in 1981, but that record didn’t last long; Gosper had topped 17 million places in 1985, using some ideas of Salamin, and Bailey had surpassed 29 million in 1986, before Kanada got back in the lead [3]. More than two further decades of continued progress have led to the astonishing present record of nearly 2.7 trillion decimal places, announced on 31 December 2009 [5]. This major feat was achieved by Fabrice Bellard on a personal desktop computer after 131 days of calculation. 40

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Figure 3. Four mathematicians with a nonrandom license plate.

David Bailey’s current license shows p in a less familiar guise:

Hexadecimal notation, which makes p equal to ð3:243F6A8885A3. . .Þ16 , nicely meets California’s stipulation that all license plates must contain letters. And it’s also especially appropriate for Bailey, who helped to discover the remarkable formula   1 X 1 4 2 1 1    ; p¼ 16k 8k þ 1 8k þ 4 8k þ 5 8k þ 6 k¼0 by which the nth hexadecimal digit of p can be efficiently computed without evaluating the previous n - 1 [3]. Martin Davis currently drives around Berkeley with the vanity plate E I PYE. (He is also [email protected].) That’s a very nice formula; but I think my own preference would be SQRT 2PI, which happens to be currently available if I decide to go for it. While writing this essay I tried to find other familiar pffiffiffi constants ( 2; e; /; c) by querying the appropriate websites in Arizona, Florida, Michigan, Nevada, and North Carolina. But I encountered only a few hits: Arizona drivers have reserved 271828, 1618PHI, and 1618033; North Carolina drivers have reserved 1.414214, 2.718282, and 1.618034; otherwise nothing. From this limited sample it appears that fans of Euler’s constant have not yet arisen to promote their cause, and that rounding is preferred in the East but not the West. A rich vein of important numbers remains to be claimed, vanitywise.

Sometimes people obtain mathematically significant license plates purely by accident, without making a personal selection. A striking example of this phenomenon is the case of Michel Goemans, who received the following innocuous-looking plate from the Massachusetts Registry of Motor Vehicles when he and his wife purchased a Subaru at the beginning of September 1993:

Two weeks later, Michel got together with his former student David Williamson, and they suddenly realized how to solve a problem that they had been working on for some years: to get good approximations for maximum cut and satisfiability problems by exploiting semidefinite programming. Lo and behold, their new method—which led to a famous, award-winning paper [15]—yielded the approximation factor .878! There it was, right on the license, with C, S, and W standing respectively for cut, satisfiability, and Williamson.

Large Numbers Let’s return now to the scenario we began with, a license plate that bore the desirable number 65536. Mathematicians have traditionally befriended numbers that are much smaller than this, because smaller numbers tend to have more interesting properties. (Or perhaps because smaller numbers have properties that are easier to discover without computer assistance.) Le Lionnais considered this situation in his postlude to [28], saying ‘‘Tous les nombres sont remarquables, mais peu ont e´te´ remarque´s.’’ His book discusses 219 integers between 20 and 220, having a total of 574 ‘‘properties,’’ with the distribution of k-bit numbers that is shown in Figure 4 for 1 B k B 20. In this illustration the black bars stand for numbers and the gray bars stand for properties; for example, when k = 1 the sole number is 1 and he mentions 14 of its properties. The sole number listed for k = 20 is 604800, the number of elements in the Hall–Janko group (the fifth sporadic group); this number also has the property that some of its divisors yield an interesting ‘‘congruence cover.’’ A congruence cover is a set of integer pairs (a1, d1), . . ., (as, ds) with d1 \ . . . \ ds such that every integer is congruent to ak

Figure 4. Remarkable numbers (black) and remarkable properties (gray) in [28].

(modulo dk) for some k. For example, the simplest congruence cover [14] is fð0; 2Þ; ð0; 3Þ; ð1; 4Þ; ð1; 6Þ; ð11; 12Þg: Robert Churchhouse [10] found a congruence cover for which d1 = 9, ds = d124 = 2700, and lcm(d1, . . ., ds) = 604800; when Le Lionnais wrote [28], Churchhouse’s example had the largest known value of d1. (Erdo¨s had conjectured that d1 could be arbitrarily large. His conjecture remains open, and carries a $1000 reward for the solver. A cover with d1 = 40 and s&1050 has recently been found [33].) What other properties does 604800 have, besides the two that were featured by Le Lionnais? For this question mathematicians can now turn to Neil Sloane’s wonderful On-Line Encyclopedia of Integer Sequences [38], which tells us for example (in sequence A053401) that there are 604800 seconds in a week. Sequence A001715 of the OEIS reminds us that 604800 = 10!/3!; from this fact we can conclude, with a hint from A091478, that exactly 604800 simple graphs on 5 labeled vertices have 7 labeled edges. Furthermore we learn from sequences A055981, A058295, A060593, and A080497 that 604800 is the number of ways to write an 11-cycle as the product of two 11-cycles on the same elements [6], and that 604800 can not only be expressed as 5! 7! and as 12!/d(12!) but also as ð1  2  3  4  5  6  7  8  9  10  11Þ =ð1 þ 2 þ 3 þ 4 þ 5 þ 6 þ 7 þ 8 þ 9 þ 10 þ 11Þ and—via prime numbers—as ð17  2Þð17  3Þð17  5Þð17  7Þð17  11Þð17  13Þ: Altogether the number 604800 appears explicitly in 78 sequences of the current OEIS list, so it possesses 78 ‘‘OEIS properties.’’ I can well imagine that Marshall Hall, who was my Ph.D. advisor long ago, would have been delighted to drive an automobile whose license plates bore the number 604800. These considerations beg us to ask, ‘‘What numbers greater than, say, 10000, have the most OEIS properties?’’ I posed this question to Sloane in 2001, and he told me how to answer it by downloading a stripped version of the database. The current champion numbers, by this criterion, are shown in Table 1. Several conclusions can readily be drawn from this table. First, we notice that the magic number 65536 of my Madison experience is right up there, nearly tied for the lead. Second, almost all of these property-rich numbers are round in G. H. Hardy’s sense: They are ‘‘the product of a considerable number of comparatively small factors’’ [20, page 48]. The only exceptions are 10001 and 11111, which are oriented to radix-10 notation. Indeed, all of the champions other than 10001 and 11111 are powers of 2, 3, 5, 6, 7, 10, or 11, except for 8!, 9!, and 10080 (which is twice 7!). Table 1 ranks a number n by counting only the sequences in which the OEIS database lists n explicitly; it doesn’t count all the sequences to which n actually belongs. For example, A005843 is the sequence of even numbers, which explicitly lists only 0, 2, 4, 6, . . ., 120; a number like

 2010 The Author(s). This article is published with open access at Springerlink.com,, Volume 33, Number 1, 2011

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Table 1. Numbers [10000 with the most OEIS properties value

props

value

props

16384

646

46656

321

65536

638

1000000

316

32768

621

362880

305

262144

577

10080

301

40320

508

100000

291

15625

415

531441

288

1048576

387

14641

288

19683

365

10001

282

16777216

360

524288

278

2097152

348

4194304

258

59049

337

11111

255

131072

328

16807

254

604800 is even but doesn’t have the OEIS property of evenness. Being near the beginning of a sequence seems to make membership stronger, because the fact that 2 is even is noteworthy for 2 but the fact that 120 is even is basically ‘‘duh’’ for 120. Therefore I tried another experiment in which the successive values of each OEIS sequence are assigned weights 1, 1/2, 1/4, 1/8, . . .. Table 2 shows the integers that currently are most important by this criterion, and again 65536 ranks near the top. Here the tendency to prefer ‘‘roundness’’ is even more pronounced than before: Every number in Table 2, with the exception of 10001, is a power of 2, 3, 5, 6, 7, or 10. I’m willing to admit that such numbers are important. Yet somehow the criteria used to generate Tables 1 and 2 leave me dissatisfied from the standpoint of license-plate desirability. One reason is that many of the OEIS sequences are not really very interesting at all, propertywise. For instance, the Hall– Janko number 604800 occurs near the beginning of A002677, but only because it’s the denominator of an obscure constant called M 011 3 [37]. Frankly, I couldn’t care less. Numbers are often in fact especially interesting when they’re at the end of a sequence, not the beginning. For example, 65537 is interesting because it’s the largest known Fermat prime (A092506, A019434); 43112609 is interesting because it’s the binary length of the largest prime number ever discovered, (11. . .1)2 [42]. (Fans of 43112609 can’t use

it on a license plate, however, except in North Carolina, because it is eight digits long.) I imagine that Richard Brent’s favorite number is 1568705, which is currently the last element of A064411, because he was surprised to discover it in the continued fraction for ec [7]; remarkably, this 7-digit value arises rather early on, as the 4294th partial quotient of a number that probably isn’t rational. In my own case, if I had had a chance to choose my favorite 5-digit number to put on a license plate in 1969, shortly after I had seen the plate in Madison, my choice would not have been 65536; I would definitely have chosen 12509 instead. Why? Because I had just completed extensive calculations leading to the conclusion that 12509 is the smallest n such that l(n) \ l*(n), where l(n) is the length of the shortest addition chain for n and l*(n) is the length of the shortest ‘‘star chain’’ (see [21], §4.6.3). Before I had done these computations, such integers were known to exist because of a theorem due to W. Hansen [19], but the smallest example that could be based on his theorem was the gigantic value n = 26103 + (21016 + 1)(22032 + 1). At once 12509 became my favorite 5-digit number. I bet Neill Clift’s favorite 8-digit number is 30958077, because he discovered in 2007 that it’s the least n such that l(n) = l(2n) = l(4n). Incidentally, the largest integer in the California list [8] was 9GOOGOL, namely 9 9 10100. But it was trumped by ALEPH0 and ALEPH1.

Computer Science In my day job I profess to be a computer scientist, not a mathematician, although there is definitely a soft spot for mathematics in my heart. Thus my closest colleagues have a computer-oriented rather than math-oriented perspective in their predilections for personalized plates. For example, Gio Wiederhold drives D8ABASE. Vaughan Pratt chose DUELITY, because his work makes considerable use of DUALITY (which was already taken when he tried to get it) and because his duality also applies to games. The most famous license plate from Stanford’s Computer Science Department is undoubtedly the one by which the late Gene Golub aptly described himself:

Table 2. Numbers [ 10000 with the heaviest OEIS properties value

weight

value

262144

27.9

390625

weight

16777216

27.6

68719476736

13.5

65536

22.5

4194304

13.2

531441

21.5

78125

13.2

59049

20.8

16807

13.1

19683

20.4

823543

12.9

15625

19.8

177147

12.6

1048576

18.6

117649

12.1

16384

18.5

10001

11.8

46656

18.2

9765625

11.6

13.5

387420489

15.1

100000

11.6

32768

15.1

2097152

11.3

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THE MATHEMATICAL INTELLIGENCER

This much-photographed plate, now his epitaph, highlights the leading roles that he played with respect to the computation, application, and popularization of the singularvalue decomposition of rectangular matrices. Matrix fans also probably know about Jack Dongarra’s Tennessee plate, LAPACK, commemorating the popular package of linear algebra routines to which he has made many contributions. (In fact, he previously owned LAPACK in Illinois, and LINPACK in New Mexico before that.) The most celebrated problem of theoretical computer science—‘‘Is P equal to NP?’’—should probably show up on the

highway too. But at present nobody has voted either for P EQ NP or for P NEQ NP, at least not in California or Massachusetts or New York or in any other state that I checked. (With two cars, you could hedge your bet and acquire both plates.) The California list of 1981 [8] included some lowbrow computer-related examples such as A HACK, ALGOL 60, CODER, HACKERS, HACKIT, I CMPUTE, I COBOL, PC SALES; also TEXWEB, which startled me when I spotted it during a visit to Marin County in 1984. By 1986, a car bearing PC WIDOW was already on the road in the East [13]. Armando Stettner, an engineer at Digital Equipment Corporation (DEC) who was working on the UNIXr operating system, decided in 1983 to get New Hampshire license plates that said UNIX. People in those days were clamoring for software licenses that would allow them to use this proprietary system legitimately, so he convinced DEC to make mock ‘‘UNIX licenses’’ to be handed out at conventions. These souvenir plates became wildly popular [35]. As I was writing the present essay I happened to see the California license UPSTART, which microcomputer pioneer Lee Felsenstein has owned since he created a ‘‘startup’’ in 1986. I also came across TURING1, proudly driven by the Computer History Museum curator Dag Spicer. Dag told me that TURING itself was unavailable when he made his choice. I’ve also heard about two clever vanity-plate ideas based on programming languages, both of which happen to be presently available in California. The first one, DO 4 TRAN, will be readily understood by any FORTRAN programmer. But the second, 4TH IF H, needs a bit of explanation: It means ‘‘Honk (H) if you love Forth,’’ in perfectly decent Forth-language syntax (when followed by THEN). I must confess that, when I was following a car several years ago whose rear license read ENOFILE, it took me a minute to realize that the driver wasn’t necessarily a programmer. Somebody in California is now driving a car whose license reads CDR CAR. Maybe it’s a man named Charles Dudley Robinson. But I hope it’s actually a LISP programmer, ideally one who knows also that the left and right halves of a machine word, when LISP was first implemented on the IBM 704 computer, were obtained by the respective instructions CDR (contents of the decrement field of a register number) and CAR (contents of the address field of a register number). Other plates refer to computer graphics, or to the Internet, or to user interfaces, artificial intelligence, robotics, networking, texting, etc. But that’s a topic for another essay, to be published perhaps in the Information Technological Intelligencer.

Other Countries Canada began to catch the US-style vanity plate craze in the 1980s, beginning in Ontario and Prince Edward Island, where motorists had already been allowed to choose their own standard-format letters and numbers since 1973. By 2007, about 3% of all Canadian plates were vanitized—not counting the provinces of Quebec and Newfoundland/ Labrador, which have so far held out against such freedom of choice. (Que´be´cois can, however, display anything they

want on the fronts of their cars, because the official plates appear only in back.) Ontario’s vanity plates allow up to 8 characters, but they are subject to special restrictions in order to enhance readability by law enforcement officials [34]: The letters A, S, G are respectively equivalent to the digits 4, 5, 6; and O is equivalent to both 0 and Q. Thus the license 666SAGAS would preclude 255 others such as GG65A645, and 00GOOGOL would preclude 971 lookalikes. Furthermore, you can’t have more than four equivalent characters in a row, as in GRRRRR or XXXXXXX. Nothing like the US or Canadian freedom to vanitize is possible in England, where the number plates are subject to severe syntactical restrictions. Britishers do, however, sometimes try; for example, a popular singer named Jess Conrad reportedly once threatened a duel in Regents Park in order to acquire the plate JC21. When Britain introduced auto registration in 1903, Bertrand Russell’s older brother Frank famously waited in line all night so that his car could be identified as A1. (He later became Under-Secretary of State for Transport, responsible for abolishing British speed limits.) More recently, the comedian Jimmy Tarbuck was known for driving COM 1C; hairstylist Nicky Clarke owns H41 RD0; James Bond could be 13 OND; and a yellow Mercedes convertible supposedly says ORG 45M. Somebody claims to have reserved L1 NUX in 1993, and L7 NUX was being auctioned in 2001. Similar remarks apply to licenses in Germany. Therefore I was astonished and thrilled when the plate

was presented to me as a surprise gift several years ago, after I’d given some lectures about the MMIX computer [25] at the University of Applied Sciences in Munich. Indeed, the German prefix MM is available only in a small nearby village, where one of my hosts happened to have the connections necessary to acquire this miraculously perfect combination on my behalf. Notice also the elegant typography. Vanity plates, American style, have however spread to the northern shores of Europe, beginning in Scandinavia. I think Sweden was first, in 1993 or so, followed soon after by Finland, Denmark, and Iceland. Norway will begin to issue personlige bilskilt in 2010. Meanwhile Latvia, Poland, Luxembourg, Slovenia, and even Austria have jumped on the bandwagon. Rumor has it that the Netherlands will be vanitized next. Will their plates have room for WISKUNDE? At the opposite end of the world, ‘‘true’’ vanity plates do exist nowadays in Australia, New Zealand, and Hong Kong; they have also occasionally been issued in the Philippines for special events. Japanese motorists live with a rather bizarre system in which they have freedom only to choose a serial number from 1 to 9999, leading to license plates of the forms . . .a; : :ab; :abc, or abcd, where a, b, c, and d are digits with a = 0. Furthermore, cd is never 42 or 49, because those numbers connote ‘‘death’’ or ‘‘bitter death’’ in Japanese. (In Japan the fact that u(49) = 42 is bad news.) Beginning in 1999, monthly lotteries were held for

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the 26 most popular numbers . . .s; :sss; s000; ssss, 12-34 and 56-78, where s 2{1,2,3,5,7,8}; this list was later decreased to only 13 entries. If you want one of the other values, you can pay extra to get your favorite. On the continent of Africa one can reportedly obtain vanity plates in Liberia and Uganda. But South America is still holding out. Indeed, US-style vanity plates seem to be presently unavailable anywhere in Latin America except in Puerto Rico, where they are called tablillas especiales personalizadas.

Conclusion

[5] Fabrice Bellard, ‘‘Pi computation record,’’ http://bellard.org/pi/pi 2700e9/announce.html [accessed January 2010]. [6] G. Boccara, ‘‘Nombre de representations d’une permutation comme produit de deux cycles de longueurs donnees,’’ Discrete Mathematics 29 (1980), 105–134. [7] Richard P. Brent, ‘‘Computation of the regular continued fraction for Euler’s constant,’’ Mathematics of Computation 31 (1977), 771–777. [8] State of California, Department of Motor Vehicles, Environmental License Plate Numbers (21 July 1981). [9] F. R. K. Chung and R. L. Graham, ‘‘Quasi-random set systems,’’ Journal of the Amer. Math. Soc. 4 (1991), 151–196. [10] R. F. Churchhouse, ‘‘Covering sets and systems of congru-

We’ve seen that vanity-plate fever is sweeping through many parts of the world, and that this phenomenon presents remarkable challenges to mathematicians. One of the main unresolved problems is to determine the integers of 5 to 8 digits that are most ‘‘interesting,’’ in some reasonably mathematical sense. As I did this research I learned about six or seven available plates that would suit me well and make me happy. But unfortunately I have only one car, and I can’t decide which of the plates to live without. So I guess I’ll just continue to fantasize about the possibilities.

5, 3 (August 2008), 22–27. [12] Jean-Marie De Koninck, Those Fascinating Numbers (American

ACKNOWLEDGMENTS

[15] Michel X. Goemans and David P. Williamson, ‘‘Improved

ences,’’ in Computers in Mathematical Research, edited by R. F. Churchhouse and J.-C. Herz (Amsterdam: North-Holland, 1968), 20–36. [11] Matt Cunningham, ‘‘A bumper crop of physics plates,’’ Symmetry

Mathematical Society, 2009). [13] Faith W. Eckler, ‘‘Vanity of vanities,’’ Word Ways 19 (1986), 195–198; All is vanity,’’ Word Ways 20 (1987), 141–143. [14] Paul Erdo¨s, ‘‘On integers of the form 2k +p and some related problems,’’ Summa Brasiliensis mathematicæ 2 (1950), 113–123.

The networking skills of Eugene Miya were especially helpful in bringing many choice examples to my attention, and I’ve also been helped by dozens of other people in casual conversations about the subject. Andrew Turnbull provided important historical information. David Bailey, Dave Bayer, Carol Bohlin, Laurence Brevard, David Eisenbud, Michel Goemans, Ron Graham, Victor Miller, Bill Ragsdale, and Cathy Seeley contributed photos. Gay Dillin of NCTM sent details of the 2005 contest. The four-author p photograph was taken by Raul Mendez. I found the PROF SVD image on Wikimedia Commons, where it had been posted by Da Troll.

approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,’’ Journal of the ACM 42 (1995), 1115–1145. [16] Solomon W. Golomb, ‘‘Run-length encodings,’’ IEEE Transactions on Information Theory IT-12 (1966), 399–401. [17] Solomon W. Golomb, Robert E. Peile, and Robert A. Scholtz, Basic Concepts in Information Theory and Coding: The Adventures of Secret Agent 00111 (New York: Plenum, 1994). [18] Gerry Griffin, ‘‘New Hampshire license plate museum,’’ panel 3, http://nhlpm.com/3.html [accessed November 2009] [19] Walter Hansen, ‘‘Zum Scholz–Brauerschen Problem,’’ Journal fu¨r die reine und angewandte Mathematik 202 (1959), 129–136. [20] G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Sug-

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.

gested by His Life and Work (Cambridge, England: Cambridge Univ. Press, 1940). [21] Donald E. Knuth, Seminumerical Algorithms, Volume 2 of The Art of Computer Programming (Reading, Mass.: Addison–Wesley, 1969). [22] Donald E. Knuth, Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness (Reading, Mass.: Addison–Wesley, 1974).

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[1] Anonymous, ‘‘First new California plates appear,’’ Los Angeles Times (6 December 1962), A1. [2] Anonymous, ‘‘Playful plates: Connecticut tags identify owners,’’ Life 31, 23 (3 December 1951), 133. [3] D. H. Bailey, J. M. Borwein, P. B. Borwein, and S. Plouffe, ‘‘The quest for pi,’’ The Mathematical Intelligencer 19, 1 (Winter 1997), 50–57.

[23] Donald E. Knuth, TEX: The Program, Volume B of Computers & Typesetting (Reading, Mass.: Addison–Wesley, 1986), 2. [Versions 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926 were released respectively in 1991, 1992, 1993, 1995, 2002, 2008.] [24] Donald E. Knuth, ‘‘Teach calculus with Big O,’’ Notices of the Amer. Math. Soc. 45 (1998), 687–688. [25] Donald E. Knuth, MMIX: A RISC Computer for the New Millen-

[4] David Bayer and Michael Stillman, ‘‘On the complexity of com-

nium, Volume 1, Fascicle 1 of new material for The Art of

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Computer Programming (Upper Saddle River, New Jersey:

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THE MATHEMATICAL INTELLIGENCER

[26] Donald E. Knuth, Combinatorial Algorithms, part 1: Volume 4A of The Art of Computer Programming (Upper Saddle River, New Jersey: Addison–Wesley, 2010). [27] David Larsen, ‘‘Words you can’t drive by,’’ Los Angeles Times (20 January 1970), B1, B8.

[35] The Open Group, ‘‘The history of the UNIXr license plate,’’ http://www.unix.org/license-plate.html 2009].

[accessed

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[36] Grisha Perelman, ‘‘The entropy formula for the Ricci flow and its geometric applications,’’ http://arxiv.org/abs/math/0211159

[28] Franc¸ois Le Lionnais, Les Nombres Remarquables (Paris: Hermann, 1983).

[37] Herbert E. Salzer, ‘‘Tables of coefficients for obtaining central differences from the derivatives,’’ Journal of Mathematics and

[29] Tien-Yien Li and James A. Yorke, ‘‘Period three implies chaos,’’ Amer. Math. Monthly 82 (1975), 985–992.

Physics 42 (1963), insert facing page 163. [38] Neil J. A. Sloane, The On-Line Encyclopedia of Integer Sequen-

[30] Megan Luther, ‘‘A license to make a statement,’’ Argus Leader (25 October 2009), appendix, http://www.argusleader.com/assets/ pdf/DF1451091023.PDF [accessed November 2009].

ces, http://oeis.org/ [accessed November 2009]. [39] Gay Talese, ‘‘State unit urges car law changes,’’ The New York Times (16 January 1959), 12.

[31] Andy Magid, ‘‘Mathematics and the public,’’ Notices of the Amer. Math. Soc. 51 (2004), 1181.

[40] UTAH.GOV services, ‘‘Personalized license plates,’’ http:// dmv.utah.gov/licensepersonalized.html

[32] Colman McCarthy, ‘‘A PLATE 2 CALL YOUR OWN,’’ The Washington Post Times Herald (2 March 1969), Potomac magazine, 25–27.

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2009]. [41] David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Harmondsworth, Middlesex, England: Penguin Books,

[33] Pace P. Nielsen, ‘‘A covering system whose smallest modulus is 40,’’ Journal of Number Theory 129 (2009), 640–666.

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Interview with Professor Ngoˆ Ba’o Chaˆu NEAL KOBLITZ n January 1989 I interviewed Hoa`ng Tu: y, who was then Vietnam’s most prominent living mathematician; that interview was published in The Mathematical Intelligencer the following year. At the time, a particular point of pride for friends of Vietnam — a point that I asked Hoa`ng Tu: y to comment on — was that the previous year at the International Mathematical Olympiad (IMO) in Australia the Vietnamese team had had its best performance since Vietnam started competing in the IMO in 1974; Vietnam came in 5th, ahead of the 6th-place American team. What I did not know was that this result was in large part due to the perfect score achieved by a 16-year-old by the name of Ngoˆ Ba’o Chaˆu. Fast-forward 22 years, and on August 19, 2010, at the International Congress of Mathematicians in Hyderabad, the President of India formally bestowed the Fields Medal on Professor Ngoˆ Ba’o Chaˆu. The Medal was given in recognition of his proof of the ‘‘Fundamental Lemma’’ about automorphic forms that had been a central unsolved problem in mathematics since it was conjectured by Robert Langlands and Diana Shelstad in the early 1980s. Ngoˆ Ba’o Chaˆu reached adulthood during a transition period from a time when most top math students in Vietnam went to the Soviet Union or Eastern Europe for advanced study to a time when they are more likely to go to Western Europe, Australia, or North America. Although he received his undergraduate and graduate education in France, Ngoˆ Ba’o Chaˆu has maintained close ties to Vietnam, and he has a longstanding association with the Hanoi Mathematical Institute. In 2005 at the age of 33 he became the youngest person ever given the title of Full Professor in Vietnam. In recent years he has been a professor at Universite´ Paris-Sud and at the Institute for Advanced Study in Princeton. In September 2010 he took a position at the University of Chicago.

I

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THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC

NK: Please tell us about your early life — what schools you attended and who had important influences on you. At what age did you decide to become a mathematician? NBC: I went to an experimental elementary school that had been created by a revolutionary pedagogue named H Ngo: c Ða: i. While one can debate whether the rudiments of set theory should be introduced before elementary arithmetic and whether multiplication should be defined using the cartesian product, our experience at this elementary school was particularly refreshing. The teacher-student relationship was not based on authority as it is in traditional Vietnamese schools, and we were encouraged to express ourselves very freely.

Figure 1. Smt. Pratibha Devisingh Patil, President of India, presents the Fields Medal to Prof. Ngoˆ Ba’o Chaˆu at the ICM meeting in Hyderabad.

Figure 2. Problem 6 of the 1988 International Mathematical Olympiad.

At the end of my elementary school years, my father came back from the Soviet Union after getting his doctorate in applied mathematics. He was not enthusiastic about H Ngo: c Ða: i’s approach to education, and he decided to pull me out of the experimental school. I then started a more traditional curriculum in the ‘‘chuyeˆn toa´n,’’ the special classes for gifted students in mathematics. At first this was at the Tr ng V ng Middle School next to our house, and later it was at a selective high-school attached to Vietnam National University. It was not easy for me to adapt to the concrete and challenging problems in the chuyeˆn toa´n, since I was more familiar with the pretty abstract mathematics taught in the experimental school. NK: I understand that both your parents are scientists. Did they influence you to want to become a mathematician? NBC: My parents’ influence on me was quite indirect. My father’s young colleagues enjoyed coming to our house and having animated conversations with my parents. During the breaks, they taught me mathematics. It was quite a revelation. I realized that I liked doing mathematics. NK: Tell us about the International Mathematical Olympiad in Australia in 1988. Were you surprised that you got a perfect score and a Gold Medal? NBC: I still remember that I was stunned when I succeeded in carrying out an elaborate Fermat descent in the difficult 6th problem. Ten years later, I tried again to solve

AUTHOR

......................................................................... NEAL KOBLITZ received his Ph.D. from

Princeton in 1974, and since 1979 he has been at the University of Washington in Seattle. He works in number theory and cryptography, and has also written extensively on educational issues. He is the author of six books, of which the last one, Random Curves: Journeys of a Mathematician (Springer, 2007), is autobiographical. Neal and his wife Ann have been visiting Vietnam regularly since 1978, working mainly with the Hanoi Mathematical Institute and the Vietnam Women’s Union. Two chapters of Random Curves are devoted to Vietnam, as are several opinion pieces on Neal’s webpage. http://www.math. washington.edu/*koblitz. Department of Mathematics University of Washington Seattle, WA 98195 USA e-mail: [email protected]

it, but I couldn’t. Otherwise, I was not really surprised with the Gold Medal because at that time I was good at solving Olympiad problems. NK: In Vietnam there is a great deal of popular interest in the team’s performance at the IMO. Did you become a celebrity after the 1988 IMO, and again after you received another Gold Medal at the 1989 IMO? I’ve heard that in Vietnam the top math competitors are as famous as movie stars in the West. Is this true? NBC: In those years in Vietnam there were not as many movie stars as there are now, and consequently there was more room for high-school math competitors to become personally famous. I think that your comparison is exaggerated, but it is true that some of our math team members were featured in the media to such an extent that their names remained in the public memory even if they never became professional mathematicians. NK: Why did you decide to go to a university in France? NBC: I was prepared to go to study in Hungary because I liked combinatorics very much. But the agreement between Hungary and our country went kaput after the fall of the Berlin Wall. It also happened that a French professor visited the Institute of Mechanics, where my father worked. My father’s colleagues talked to him about my Gold Medals in the Olympiads, and he decided to fight to get me a scholarship to study in France. NK: Did you already know French when you left for the university? Did you have any difficulties adjusting to student life in France? NBC: I knew only some of the rudiments of French from my grandfather. The first year in France was rough because I was not prepared for the gap between an isolated country that had been through thirty years of war and a brilliant country like France. NK: How did French students compare with the students you had known in Vietnam? NBC: In the first year I was in a regular class at Paris VI. The math and physics taught to first-year students at Paris VI were not difficult, fortunately, and so it left some time for me to improve my French. The students were obviously not as good as the ones in my selective high-school in Hanoi; but the following year I was sent to the Ecole Normale Supe´rieure, and that was a completely different story. NK: When you were a student, what were your interests outside of mathematics? NBC: I read a lot of books. Reading has always been my favorite leisure activity. NK: At what point did you become interested in the Langlands program? NBC: At that time a lot of students at Ecole Normale were attracted to arithmetic geometry and the Langlands program. This was probably a side effect of Wiles’s proof of the Shimura-Taniyama-Weil conjecture. NK: Can you describe the general idea of the Fundamental Lemma in words that a nonspecialist can understand? NBC: The Fundamental Lemma is basically an identity of two numbers, each of them defined by a complicated integral. This identity is deeply rooted in the arithmetic structure of the Arthur–Selberg trace formula. At the beginning it was thought to be a technical problem that Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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needed to be solved by a great deal of hard but direct computation. This turned out not to be the case. In the meantime, the Fundamental Lemma has also acquired more and more importance, since many of the advances in the Langlands program rely on its validity. NK: In 2004 you were given the prestigious Clay Research Award (along with Ge´rard Laumon) for your success in proving the Fundamental Lemma in the unitary case. At that time did you think that you would be able to go further? At what point did you start to believe that you would be able to prove the Fundamental Lemma in its entirety? NBC: In my Ph.D. thesis I solved a problem quite similar to the Fundamental Lemma and came to understand that the key to a solution may be a geometric model for the trace formula. In 2003 I realized that the geometric model for the trace formula for the Lie algebra is actually the Hitchin fibration. At that time I was convinced that I was on the right track for a proof of the Fundamental Lemma. In some sense, I absorbed the ideology, but the realization remained very difficult. The most difficult part of the proof was a certain technical statement about perverse sheaves. Laumon and I were able to prove it in the unitary case in 2004. After that the general case still had to wait a long time. NK: Did the main idea for your final proof in 2008 come to you all at once, or gradually over a period of time? NBC: I kept trying to generalize the proof in the unitary case, until the end of 2006, when I became convinced that it was impossible. At that time a conversation with Mark Goresky of IAS provided me with the missing piece of my puzzle. It took me one more year to come up with the complete proof. NK: Some people have noted the comparison between you and S. S. Chern, and have even started to think of you as the ‘‘S. S. Chern of Vietnam.’’ There are some interesting similarities — for example, Chern also became a professor at the University of Chicago at age 38. Would you like to play a role in Vietnam that is similar to what Chern did in China? NBC: The comparison with Chern is very flattering. He is certainly a model for me to follow. NK: Deputy Prime Minister Nguy n Thi n Nhaˆn has said that he hopes that you will become the head of the new Advanced Mathematics Institute in Hanoi that is being planned. Will you agree to take on this responsibility? NBC: There will be a Board of Directors, and it seems likely that I will serve on it. NK: How will you divide your time between the University of Chicago and Hanoi? NBC: I plan to spend summers in Hanoi. During the year I will fly to Hanoi for short visits one or two times. I can help the Board of Directors to select members for the new institute without being physically in Hanoi. Other colleagues will help to run the Institute on a day-to-day basis. NK: What are the objectives of the new institute, and how will it be different from the Hanoi Mathematical Institute? NBC: We will try to attract Vietnamese mathematicians from abroad as well as mathematicians of other 48

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nationalities to our institute to work on projects, with preference for joint projects with Vietnamese mathematicians. The visits can be for 3 months, 6 months, or a year. There will be no permanent members except for the Board of Directors. NK: What are the main steps that must be taken in order to improve mathematical research in Vietnam? NBC: The first step will be to construct an alternative for young Vietnamese mathematicians. It should be possible, at least on a temporary basis, for them to earn their living by doing research in mathematics in their own country. We hope that enough scientific and personal ties can be built during their stay at the new institute that some of them will take a job in a Vietnamese university. At the same time we will have to convince the universities to make reasonable offers to young scientists so that accepting a job is not scientific suicide. NK: Traditionally, Vietnam has been much stronger in the theoretical areas of mathematics than in the applied areas. What should be done to improve research in applied areas? How much of the focus of the new institute will be on applied research? NBC: We will very much welcome joint projects between mathematicians and researchers in related areas such as computer science, theoretical physics, biology, economics, and so on. NK: What would you do to improve undergraduate education at the leading Vietnamese universities, such as Vietnam National University (VNU)? Although VNU has some good mathematicians on its faculty, there seem to be many problems. For example, because salaries are low, many professors need to work at a second job in order to supplement their income. Also, they do not have their own offices. As a result, math professors almost never meet with students outside of formal lectures. What can be done to improve conditions at VNU and other universities? NBC: This is a topic that can be debated at length, but in my opinion the main reason for the problems in undergraduate education here in Vietnam is that there are not enough good professors. It should be made clear to the Vietnamese university presidents that their top priority should be the recruitment of young and talented people. And at a higher level, the science and education ministries should encourage the universities to do this using a strong system of grants. NK: Often the most competitive international applicants to U.S. Ph.D. programs are students who have already obtained a Masters degree in their own country. But Vietnam does not support strong Masters level programs in mathematics. For example, Masters students cannot easily find financial support for their studies. Are there any plans to expand Masters programs in mathematics at Vietnamese universities (with financial support for students), and will the new Advanced Mathematics Institute give Masters degrees? NBC: I agree that it is very important to develop good Masters level programs in Vietnam. We already have an International Masters program run jointly by the Institute of Mathematics and the Hanoi Pedagogical University (HPU). We recruit around 20 students each year. They spend the first year in Hanoi and the second year in Europe, and are

supported by fellowships from the Ministry of Education. Upon graduation they are granted Masters degrees from the European university where they spend their second year. We would love also to be able to grant them our own Masters degrees, but this is impossible under the current administrative rules. The new Advanced Mathematics Institute will not have its own Masters program, but obviously we will encourage its members to give lectures in the existing Masters programs as well as participate at a modest level in the undergraduate programs at VNU and HPU. NK: In the Soviet Union and in France, most of the leading mathematicians worked in institutes that had little or no role in undergraduate education; in contrast, in the U.S. most mathematicians work in universities and teach at the elementary as well as advanced level. For obvious reasons, Vietnam’s system is closer to the Soviet and French systems than to the American one; that is, many of Vietnam’s leading mathematicians do little or no undergraduate teaching. Do you see this changing? How would you propose to better integrate teaching and research in Vietnam? NBC: It is true that in Vietnam research and teaching are still regarded as separate activities supervised by two different ministries. This is not an ideal situation for mathematics and the basic sciences, since research and teaching are sources of inspiration for each other, and in fact it is very difficult to separate teaching at advanced levels from doing research. But we’d rather spend our time and energy on concrete projects that involve both teaching and research than fight against an administrative rule. NK: The mathematics examination for entrance to Vietnamese universities is extremely difficult. To an American it seems absolutely astonishing that a large number of graduating high-school students in Vietnam are able to get high marks on such an exam. On the positive side, this shows the high mathematical ability of young people in Vietnam. On the negative side, most of them see mathematics as a hurdle that must be surmounted in order to be admitted to a good university, and they have no interest in continuing their study of mathematics after that. So there is a big drop in mathematical activity after high-school. Are you in favor of changes in the entrance examination? NBC: My impression is that the pressures of the entrance examination have been easing in recent years. The Ministry of Education should be given credit for this evolution. There are more and more children from poor backgrounds who succeed at the entrance exam. I think that currently the entrance exam is not the crucial problem. The problem is the overall state of the house, not the size of the door. NK: Vietnamese schools teach formal mathematics at a very high level, but in a way that makes the subject seem far removed from practical life. As a result, on the one hand young people are extremely competitive in theoretical mathematics. For example, Vietnam has often done well at the IMOs—and you personally played a big role in this success in 1988 and 1989. But on the other hand, Vietnam does not participate in the Mathematical Contest in Modeling (in contrast to China, which has had many successful teams in the MCM). What can be done to help young people better understand mathematics as an area with important applications in the real world?

NBC: It is true that a lot more has to be done so that mathematics is not perceived by the public as a selection tool, but as a way to understand the world. We will have to learn more about what has been done in other countries before implementing concrete projects in Vietnam. We will think about how this can be worked out in our national plan for mathematics. Separately, we are also setting up a ‘‘Foundation for the Joy of Learning,’’ which may also become involved in such projects. NK: At present there are relatively few women mathematicians in Vietnam. For example, over 30 women scientists have received Vietnam’s Kovalevskaia Prize over the years, but none of them have been research mathematicians. NBC: I don’t think that the developed countries do any better on this. NK: What you say is undoubtedly true of some of the developed countries, but it is certainly not true of France or the U.S. In the United States, where women earn about 30% of the Ph.D.s in math, experience has shown that positive steps to encourage girls and women can have a major impact. That is why the percentage of women has climbed from roughly 5% to 30% over the last 40 or 50 years. In Vietnam’s case, you mentioned that progress has been made in increasing the number of children from poor backgrounds who obtain a higher education. So it is somewhat surprising that similar progress has not been made in increasing female participation in advanced mathematical studies. Would you be in favor of special efforts to attract more women to mathematics and support and encourage them at various stages of their careers? NBC: It happens too often that a Vietnamese woman has to make a painful choice between engaging in a scientific career and having a family life. Something should be done so that they do not need to face such a choice. I have to admit that I have never thought about this problem seriously enough to offer you a sensible answer. NK: For many years Vietnam has had more mathematical ties with Western countries such as France, Germany, and the U.S., than it has had with other countries of Asia. What ideas do you have for increasing Vietnam’s mathematical collaboration with other Asian countries, such as India and China?

Figure 3. Ngoˆ Ba’o Chaˆu with his mother, Tr n L u Vaˆn Hi n, and his father, Ngoˆ Huy C n. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Figure 6. Ngoˆ Ba’o Chaˆu is third from the left in this photo of Vietnam’s 1988 IMO team.

Figure 4.

Figure 5.

NBC: This year the second of the annual Pan Asian Number Theory conferences is being organized in Kyoto. There will be more and more regional cooperation in

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the coming years because everybody agrees on its importance. NK: In an interview with the Vietnamese newspaper Tu i Tr , you said that one of your top priorities will be giving opportunities to young people. However, at present there are very few activities in Vietnam that put senior Vietnamese scientists in direct informal contact with young people. When in school and even at the university, Vietnamese students have very little idea of what it’s like to be a researcher. They do not have role models, unless their own parents are scientists. The danger is that the youth will be influenced entirely by imported youth culture that comes from the mass media, and will not carry on the scholarly, scientific, and mathematical traditions of Vietnam’s older generations. What ideas do you have for transmitting these traditions to the younger generation, and stimulating their desire to lead the life of a mathematician or scientist? NBC: I agree with you that we need to do more to popularize science to the younger generation, either by direct contact or through the mass media. I have been agreeably surprised to find that the Vietnamese mass media are quite receptive to our message.

Mathematical Communities

Martin Gardner (1914–2010) MARJORIE SENECHAL

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of ‘‘mathematical community’’ is the broadest. We include ‘‘schools’’ of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

â Please send all submissions to Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA e-mail: [email protected] 1

Martin Gardner insisted he wasn’t a mathematician, but we all know better.

2

‘‘Car Talk,’’ National Public Radio (USA), June 5, 2010.

A

dieu, Puzzler King: Martin Gardner must be the 1 only mathematician whose passing has been mourned on a popular national radio show:

This week on Car Talk, Tom and Ray [Magliozzi] pay tribute to late puzzler mentor Martin Gardner, and they do it in true Magliozzi fashion—by recalling an embarrassing on-air moment, and then promptly swiping one of Martin’s classic puzzles.2 Car Talk, now in its 33rd year, is broadcast on more than 370 stations to some two million weekly listeners. Though our reach is somewhat smaller, we too wish to record our affection, admiration, and respect. Through ‘‘Mathematical Games’’ (his long-running column in Scientific American) and through his many books and his vast correspondence, Martin Gardner broadened and nurtured the mathematical community. Always looking for interesting mathematics for his essays and books, he searched widely, corresponded widely, and circulated information widely. When Robert Ammann, an amateur mathematician working in a Boston post office, sent Gardner his remarkable sketches of several sets of aperiodic tiles, Gardner showed them to Roger Penrose and others, asking for comments; this is how tiling theorists became aware of the ‘‘mysterious Mr. Ammann’’ and his remarkable discoveries. When I decided to write about Ammann for this column, I wrote to Martin Gardner. He generously sent me his entire Ammann file, with permission to use it as I liked. Thus it is fitting that The Mathematical Intelligencer pay tribute to Martin Gardner in a Mathematical Communities column. I have invited our past and present editors who had contact with him, or were influenced by him, to contribute to this bouquet. Like the other contributors, I regret that I never met Martin Gardner in person. But some people did, and one of them, Istva´n Hargittai, interviewed him for The Mathematical Intelligencer in 1996. You can read ‘‘A Great Communicator of Mathematics and Other Games: A Conversation with Martin Gardner’’ online, if you don’t have Volume 19, No. 4, on hand (Figure 1). Martin Gardner was a generous friend whom we never met in person. We have always tried to imagine how much bigger his impact on us would have been if we had met, but then realized that his influence was entirely born out

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Figure 1. Martin Gardner, 1996. Photograph by Istva´n Hargittai.

of his devotion to rational discourse that found its manifestation in the creative interpretation of things mathematical, including shapes, puzzles, and magic. Doing what he liked, thinking about what interested him, and communicating his insights, he has influenced mathematics more than most mathematicians by inspiring young people to enter the profession. The number and devotion of his ‘‘descendants’’ is legend. To us, as publishers, he was more than an author; he was a generous, critical advisor and an encouraging supporter through suggestions and endorsements. We still miss not having met him. Alice and Klaus Peters Martin Gardner is one of three people who had a huge and positive influence on my mathematical career. The second was one of my school mathematics teachers, Gordon Radford, who gave up almost all of his free time to teach me and four friends about mathematics that was outside the syllabus. The third was Christopher Zeeman, founding professor at Warwick University, where I worked for 40 years, and still maintain an office now that I have ‘retired’. I never met Martin, but we occasionally corresponded, and I read, and still own, dozens of his books. As a teenager, I was an avid reader of his Mathematical Games column in Scientific American. I was desperate to locate anything about mathematics that I could find, outside the school syllabus, 52

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and Martin’s column provided a regular, reliable source of wonderful ideas. It taught me that mathematics can be fun, but subliminally it also taught me a deeper message: that new, interesting mathematics is constantly being created. And his clear, organised style of writing also taught me that it is possible to explain difficult mathematics to nonspecialists. Not easy, but possible. The main things to remember are: you have to understand what you’re talking about, and you have to be aware that your audience doesn’t. When I was a PhD student, a bunch of us started a small mathematics magazine, Manifold. We duplicated it late at night on Department equipment (no computers or inkjet printers then—even a photocopier was exotic and expensive). It achieved a circulation of 600, and continued for 20 issues over about 12 years. Martin was one of our early subscribers, and once or twice he gave the magazine a mention in his column. Later, after I had become the fourth person to write what by then had become Manifold’s Mathematical Recreations column, and started publishing collections as books, Martin was kind enough to write a preface to Another Fine Math You’ve Got Me Into. The biggest problem I encountered when writing the column—and this is intended as admiration, not criticism—is that he had beaten me to almost all of the really interesting material. I still use his writings as a source of inspiration for two puzzle columns that I write regularly, and several of my books have also benefited from his work. One of the most recent, Professor Stewart’s Cabinet of Mathematical Curiosities, goes back to a series of notebooks that I collected when I was in my teens. Looking back, I find a lot of material came from Martin’s column, as is only to be expected, because his taste and mine were very similar. Probably because I got a lot of it from him. There were many other dimensions to Martin Gardner: writer, philosopher, magician, and scourge of pseudoscientists everywhere. He was unique, and he is irreplaceable. One measure of his success is that all over the world, professional mathematicians, as well as the reading public, will mourn his passing. He wasn’t trained as a mathematician, he didn’t hold a university position, and he didn’t publish research papers. But he was one of us, and we will all miss him. Ian Stewart It may well happen that more copies of Martin Gardner’s books were printed in the USSR than everywhere else together. At first this sounds dubious: The Annotated Alice (English edition) is said to have sold more than a million copies. On the other hand, the 5th edition of the Russian translation of Mathematics Magic and Mystery (which I have on my shelf) was printed in a run of 700,000 copies, and many of his other books were very popular, too (Figure 2). In high-school we played ‘‘Hex’’ during the classes and programmed ‘‘Life’’ after them – all this because of MG. Most teachers of mathematics make it painful, not fun (as it was intended to be); MG is a rare exception. He won a permanent place in the hearts of several generations of Russian-speaking mathematicians, both amateurs and professionals. From Russia, with gratitude, Alexander Shen

Gardner was also a well-regarded authority on the writings of Charles Dodgson (Lewis Carroll). His Annotatated Alice (on Alice’s Adventures in Wonderland and Through the Looking-Glass) was probably his best-known book, selling over one million copies, and he also wrote an Annotated Snark (on Carroll’s lengthy poem, ‘‘The Hunting of the Snark’’) and many articles. For mathematicians, probably his most interesting publication in this field was The Universe in a Handkerchief (Copernicus, Springer, 1996), which described and analysed Carroll’s mathematical puzzles, his use of mathematics in the Alice books, and his number-play and word-play, and the publication has recently been reissued in paperback form. This last work was particularly useful to me when I wrote my recent book Lewis Carroll in Numberland. Robin Wilson

Figure 2. Mathematics Magic and Mystery (5th Russian edition).

I am one of a multitude of mathematicians (and people working in the mathematical sciences in a broad sense), whom Martin Gardner initiated into a lifelong infatuation with the beauty and elegance of mathematics. His influence and legacy cannot be overstated. I am reminded of a joke: what father-son combination has contributed the most to mathematics? Answer: Gauss and his father. Not Gauss and one of his sons: the potential tie is broken by the fact that if Gauss’ father had not existed, Gauss would not have existed. Mathematicians and other pedants may quibble about details, but the principle is sound: one’s contribution to mathematics must be judged by what mathematics would not have happened had one never existed. By this metric, through his inspiration of young people, Martin Gardner stands as one of the giants of twentieth-century mathematics, despite never holding a degree in the subject. Ravi Vakil Although he is mainly known to mathematicians for his Scientific American columns and his mathematical books, which did so much to popularise and communicate mathematical ideas to the interested general public, Martin

Everyone who takes delight in mathematical puzzles and patterns is mourning the passing of Martin Gardner, whose column in Scientific American was such a garden of delights. I remember my fascination as an undergraduate with Conway’s Game of Life as Martin Gardner unveiled it in a column in 1970. Back then we played ‘‘Life’’ on a Go board and in ‘‘real’’ time, just like any other board game. I never had the pleasure of meeting Martin Gardner, so I can only offer some reflections from afar. Clearly, he was a writer with a cause, one that appealed to a wide swath of humanity, not just those who shared his passion for mathematics. Culturally and intellectually he represented the kind of clear-headed, good-humored nature that one associates with another native Oklahoman, a fellow named Will Rogers. My father, who taught counseling psychology at Oklahoma University, was always telling me about Martin Gardner, one of the fellow residents of Norman whom he most admired. As an avid reader of The Skeptical Inquirer, my dad also loved to tell us about the exploits of the magician, James Randi (‘‘The Amazing Randi’’), who turned his talents to exposing the fraudulent claims of selfprofessed psychics, in particular the mind-bending feats of Yuri Geller. Martin Gardner, a founding editor of The Skeptical Inquirer, stood at the forefront of that movement. Little wonder that he chose to dedicate his final volume of essays to James Randi, as ‘‘the world’s foremost debunker of bogus science and charlatans who claim paranormal powers.’’ Little surprise that my dad sent me a copy of that book, complete with Martin Gardner’s signature, as a Christmas present last year. I did not realize the range of his intellectual interests until I started reading this fascinating collection of twenty-four essays written over the course of his singular career. Many deal with his favorite themes in science and mathematics, but he also touches on sensitive political and religious issues, delivering his messages with wise words that reflect his tolerance and folksy sense of humor. In his opening essay, ‘‘Ann Coulter takes on Darwin,’’ he even manages to strike a civil tone when writing about the poisonous rhetoric of one of America’s foremost hate-mongers. Perhaps he wanted to confront right-wing hypocrisy up front, because he only returns to political themes toward the end of the volume.

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Much that falls between the book’s two ends has a distinctly serene air about it. Indeed, that’s the mood evoked by the book’s Darwinian title, When You Were a Tadpole and I Was a Fish, which is also the title of the eighteenth essay. That essay is dedicated to a forgotten journalist named Langdon Smith, who wrote a truly memorable poem, ‘‘Evolution.’’ Its opening stanza recalls a happy time: When you were a tadpole and I was a fish, In the Paleozoic time, And side by side on the ebbing tide We sprawled through the ooze and slime, Or skittered with many a caudal flip Through the depths of the Cambrian fen, My heart was rife with the joy of life For I loved you even then. This whimsical yet worshipful verse clearly resonated deeply with Gardner’s own sense of the human experience. The final essay is an exchange of letters on socialism published in The Norman Transcript immediately before and after Barack Obama’s election. Annoyed by the McCain campaign’s effort to label the Democrat a socialist, Gardner

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showed that every plank of the platform set forth by the Socialist Party (in the 1920s, ‘30s, and ‘40s, when Norman Thomas ran for president) had since been adopted by practically all leading members of the Republican Party, including McCain. Gardner was, of course, ruthlessly attacked for expressing such an outrageous opinion. As Norman Thomas said, ‘‘Most Americans don’t know the difference between socialism, communism, and rheumatism.’’ The Martin Gardner in these essays is someone who saw life as a whole, embraced what he found good about it, and accepted the rest with equanimity. David Rowe

Alice and Klaus Peters were the Founding Editors of The Mathematical Intelligencer. Ian Stewart was the European Editor of The Mathematical Intelligencer from 1985 to 1991 and the editor of ‘‘The Mathematical Tourist’’ column from 1992 to 1997. Alexander Shen edited ‘‘Mathematical Entertainments’’column from 1997 to 2001. Today, Ravi Vakil coedits ‘‘Mathematical Entertainments’’ with Michael Kleber, Robin Wilson edits ‘‘The Stamp Corner,’’ and David Rowe edits ‘‘Years Ago.’’

Years Ago

David E. Rowe, Editor

Puzzles and Paradoxes and Their (Sometimes) Profounder Implications DAVID E. ROWE Years Ago features essays by historians and mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or ‘‘schools’’ (as in schools of fish), the idea is always the same to shed new light on the mathematics of the past. Submissions are welcome.

â

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

In Remembrance of Martin Gardner

M

artin Gardner had a special affinity for the enchantment of games, puzzles, and simple ideas that lead to unforeseen possibilities. He was also clearly drawn to like-minded spirits of the present and past who shared his delight in such phenomena. By imbibing their lore and showcasing their findings in Scientific American, he celebrated creativity while making a signal contribution to our collective culture. For what better way could anyone convey the unlikely idea that doing mathematics can be fun? An avid reader, Gardner often stumbled upon many lost gems of the remote and not so distant past. In this column I endeavor to follow just a few of his leads. Not long ago, I wrote about the largely forgotten German mathematician, Victor Schlegel, a leading disciple of a once obscure Gymnasium teacher named Hermann Grassmann (Rowe 2010). Schlegel’s name, in fact, pops up in one of Gardner’s delightful essays on the Fibonacci numbers (Gardner 2006), in which he shows how these are tied to an amusing series of dissection problems for plane figures. The standard paradox arises when one divides an 8-by-8 square into four pieces and then on reassembling these discovers that they form a 5-by-13 rectangle (Fig. 1). As Gardner’s Dr. Matrix reveals, one can produce any number of such paradoxes by making use of a property of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, . . ., namely that the square of any Fibonacci number differs by one from the product of the preceding and succeeding terms. The case illustrated previously is just 82 + 1 = 5 9 13. Now according to Gardner, Schlegel was the first ‘‘to generalize the square-rectangle paradox’’ by making use of this insight, a finding he published in 1879. Martin Gardner also mentions others who dabbled in this funny business, including one of his personal favorites, Lewis Carroll. But he does not indicate where and when this paradox seems to have first popped up. Schlegel published his version in the widely read Zeitschrift fu¨r Mathematik und Physik, edited by Otto Schlo¨milch. Thumbing back about ten years in that journal, we encounter a little note by Schlo¨milch himself (see Fig. 2), which shows that the original paradox was known at least ten years earlier. So in all likelihood Schlegel knew about this oddity through Schlo¨milch’s earlier note, which would also explain why he chose to publish his observations regarding the general square-rectangle paradox in the latter’s journal. But moving forward in time, it is natural to wonder how this special dissection problem came to be known in wider circles, beginning with the German-speaking world. Indeed, I would like to show that there is much more to this tale than might first meet the eye. By the 1890s some sharp-minded mathematicians had begun to re-examine the foundations of Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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plane geometry. In doing so, they came to realize that the classical approach to the dissection of plane rectilinear figures invoked a general axiom in order to get around absurdities of the type we are discussing. This led them to ask: might there not be a simpler way?

Implications for Foundations of Geometry

Figure 1. Drawing, made by a famous physicist around 1912–1913, showing the original dissection paradox (CPAE 3 1993, 584).

Figure 2. Possibly the first published version of the SquareRectangle Paradox, Otto Schlo¨milch, Zeitschrift fu¨r Mathematik und Physik, 1869. 56

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According to Euclid, when comparing magnitudes of the same type—those which have a ratio to one another—it is always the case that ‘‘the whole is greater than the part.’’ Yet since Euclid’s theory of plane areas is based on dissection into congruent figures—rather than a theory of measure for areas—a mathematician keen on rigor would clearly like to rely on purely geometrical principles, avoiding the general axioms for magnitudes needed for a general theory of ratios and proportions (Euclid’s Book V). Let us now consider dissection paradoxes from this foundational point of view. Twenty years after Schlegel’s article appeared, an algebraist by the name of Hilbert offered a lecture course on the foundations of Euclidean geometry in Go¨ttingen. Hilbert’s first doctoral student, Otto Blumenthal, recalled how he and others reacted with much surprise at this announcement; they had never heard Hilbert talk about anything other than algebraic number fields (Blumenthal 1935, 402). In 1898 projective and non-Euclidean geometry were still very much in vogue, of course, but Euclidean geometry? What, his students wondered, could he possibly have up his sleeve? What they heard during the following winter semester was Hilbert’s warm-up act for a Festschrift essay that would eventually lead to the many editions of his Grundlagen der Geometrie. We remember this classic today because with it Hilbert helped launch a movement to modernize the axiomatic method, an approach he extended to a wide range of disciplines that came to be called exact sciences. But if we look more closely at his immediate goals and the material he presented in his original lecture course, some surprising elements come into view (Hilbert 2004). A central concern for Hilbert was to establish new foundations for a theory of area for plane rectilinear figures by recasting some key results found in the first two books of Euclid’s Elements. This endeavor had much in common with efforts of several leading geometers who investigated the status of the continuum during the 1890s, though Hilbert hoped to finesse the deeper problems by showing that much of classical geometry can be done without appealing to continuity arguments. Thus his overall strategy aimed to recover standard results in what he called elementary geometry without any reliance on continuity assumptions or other axioms taken from a theory of magnitudes. Euclid had, in fact, made explicit use of the latter in the form of common notions, which included the axiom that the whole is always greater than the part. A special case of this had also been assumed by Wilhelm Killing in his book on foundations of geometry, a work Hilbert studied carefully and recommended to students at the beginning of the lecture course (Toepell 1986, 148). Its importance in the present connection should not be overlooked, even though Hilbert did just

that in the pages of his Grundlagen der Geometrie (Hilbert 1899). In chapter 4 he introduced a new theory of content for plane rectilinear figures. This was essentially modeled on Euclid’s theory with the important exception that Hilbert based his on a purely geometrical theory of proportion. He developed the latter by introducing a segment arithmetic that gave him a number field whose properties he derived from Pappus’s Theorem (or Pascal’s Theorem for two lines), a result he proved without recourse to continuity axioms. Hilbert called this purely geometrical theory of content ‘‘one of the most remarkable applications of Pascal’s theorem in elementary geometry.’’ What he did not say was that this theory enabled him to circumvent Killing’s postulate, which states that if a rectangle is decomposed into n rectilinear figures, then it is impossible to fill the rectangle with only n – 1 of these (Toepell 1986, 138). Surely to complete a jigsaw puzzle we need all of its pieces, and just as surely for the pieces of a rectangle. But this, in effect, is the very situation that arises in the original rectangle-square paradox, for if we divide it into 65 squares, then allegedly only 64 are required to fill out the square formed from the four pieces of the original rectangle! So did Hilbert know anything about this curious anomaly? Absolutely, he even referred to it explicitly in notes he wrote just prior to the time he offered his lecture course on foundations of Euclidean geometry (Toepell 1986, 140). What is more, he cited the book in which he found this figure: Mathematische Mußestunden (Mathematical Pastimes, 1898) by Hermann Schubert, perhaps best known today as the inventor of the Schubert calculus in enumerative geometry. In his own time, though, Schubert’s books on recreational mathematics were widely read, and some were also translated into English (Schubert 1903). Within the German-speaking world, Schubert’s writings helped to popularize various games and puzzles, including the rectangle-square paradox.

Doodlings of a Physicist A testimonial of sorts can be found in the pages of a notebook kept by a certain A. Einstein, better known for his physical thought experiments. By 1911 he had already speculated that light rays in the vicinity of the sun would be deflected slightly by the solar gravitational field. His notebook contains sketches from the period 1912–1913, and sure enough we find drawings and calculations directly related to this phenomenon (see Fig. 3, bottom). It seems he even thought about the possibility of gravitational lensing back in those days (Renn, Sauer, Stachel 1997). Mixed together with these daring new speculations about the bending of light rays were some of Einstein’s doodles on an array of mathematical curiosities, including those found in Figure 1. On the pages of this notebook we find (CPAE 3 1993, 584–585), next to the dissection of a square ‘‘proving’’ that 64 = 65, allusions to several other samples from Schubert’s garden of puzzles and problems: 1. The Ko¨nigsberg bridge problem, solved by Euler, which arose from the local challenge of trying to cross each of

Figure 3. Calculations of light deflection along with a curious absurdity in elementary geometry (CPAE 3 1993, 585).

the seven bridges in the town of Ko¨nigsberg just once. Euler treated this problem as a plane graph and demonstrated that it contains no cycle connecting all the vertices; hence the problem is unsolvable. 2. The same graph problem posed for the 20 vertices of a pentagon dodecahedron. Einstein presumably read about this in chapter 4 of Schubert’s book, which deals with ‘‘Hamiltonian roundtrips,’’ beginning with these remarks: ‘‘In the year 1859 two concentration games appeared in London, presented by the famous mathematician Hamilton, inventor of the quaternions. The first was called The Traveler on the Dodecahedron or a Trip around the World, the other, The Icosahedron Game’’ (Schubert 1898, 68). 3. A standard algebraic ‘‘proof’’ that a = 2a, followed by another similar one leading to the conclusion that for any numbers a, b one always can show that a = b. At the top of Figure 3 we find a dissected triangle above which Einstein has written: ‘‘Alle Dreiecke sind gleichschenklig’’ (all triangles are isosceles). The ‘‘proof’’ begins by bisecting the angle at the apex, erecting the perpendicular bisector of the base, and finding their point P of intersection. From P we drop perpendiculars to the

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remaining two sides and draw lines to the end points of the base. The triangle has now been dissected into six smaller ones, and these ‘‘clearly’’ come in three pairs of congruent triangles, revealing that the two sides are of equal length! This little geometrical jest was very well known years ago. In fact, Hilbert presented it at the very outset of his lecture course of 1898–1899 on the foundations of Euclidean geometry. He wanted to make the point that rigorous geometric arguments should never draw conclusions merely by appeal to suggestive figures, however carefully drawn. Schubert drew his triangle very nicely indeed, making evident to the eye that the critically important point P can never fall inside the triangle! Among Einstein’s most famous ideas were his thought experiments with moving trains and elevators suspended in empty space, images he developed to explain key facets of relativity theory (Einstein 1917). He also introduced the thought experiments on time dilation that came to be widely known as the space-time paradoxes involving clocks and the twins who age at different rates. Given his predilection for whimsical humor and his keen eye for intellectual puzzles, it should perhaps come as no surprise that he took special delight in mathematical paradoxes, even those of a more trivial variety.

On Machian Thought Experiments One of Einstein’s most controversial ideas stemmed from his reading of Ernst Mach, who famously criticized Newton’s notion of absolute space as meaningless metaphysics. Newton was, indeed, a deeply metaphysical thinker, yet he also appealed to experimental evidence to support his belief in absolute space. His famous thought experiment with a rotating water bucket was meant to show that when a spinning object experiences inertial forces, these stem from its motion with respect to space itself. Einstein, like Mach before him, thought otherwise, coining what he called the relativity of inertia, which he later reformulated as Mach’s Principle. According to this, the metric properties of space, including its special inertial frames, are induced by global mass and energy. Einstein was deeply attracted to this idea during the very years when he was groping for a generalized theory of relativity that could unite gravity and inertia. By 1916, after he had found what he was looking for, he tried to find a global solution for his gravitational field equations, one that would yield a static cosmology, but to no avail. (In these efforts he was supported by Jakob Grommer, one of Hilbert’s many students.) Then Einstein hit upon the possibility of modifying his field equations by adding what became known as the ‘‘cosmological constant.’’ By February 1917 he unveiled his cylindrical universe, a cosmological model whose underlying topology was S3 9 R. Riemann, in his celebrated Habilitation lecture of 1854, had hinted that space might in fact be finite and closed, possessing the topology of a 3sphere. Einstein now showed not only how, but why this was the case. He was much enamored by his new creation, in no little measure because it served to implement Mach’s principle so nicely. Indeed, there was a simple correlation between the total mass M in the universe and the global curvature constant. 58

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Yet not everyone found this pretty model so convincing, and some found the Machian arguments for it highly objectionable. Moreover, Einstein soon learned from the Dutch astronomer Willem de Sitter that his new field equations admitted another solution that led to another cosmological model, a 4-manifold of constant curvature. The problem for Einstein was that this de Sitter universe contained no matter at all! In fact, de Sitter invented it in the course of an ongoing debate with Einstein over Mach’s principle, an idea that de Sitter took to be just as metaphysical as Newton’s arguments for an absolute space. How, he asked, could an astronomer ever measure the influence of so-called distant matter? Einstein was no astronomer, so he perhaps did not require an answer. Nor did this issue concern him much as the years went on. For by 1930, in the wake of Edwin Hubble’s measurements of extragalactic redshift effects, Einstein and de Sitter could easily reach a truce. Dropping the cosmological constant—which he famously called the ‘‘greatest mistake I ever made’’—Einstein left his static model behind. New cosmological models for an expanding universe were already on the drawing boards, while fresh data rendered the older ones (nearly) obsolete. Gone, too, was the initial motivation for Mach’s principle, an idea that began to look more and more dated. By 1949, when he wrote an autobiographical sketch, Einstein noted that . . . for a long time I considered [Mach’s conception] as, in principle, the correct one. It presupposes implicitly, however, that the basic theory should be of the general type of Newton’s mechanics . . . The attempt at such a solution does not fit into a consistent field theory, as will be immediately recognized. (Einstein 1949, 29) Nevertheless, Mach’s Principle enjoyed a kind of renaissance after Einstein’s death in 1955. During the latter half of the century one of its foremost champions was the British cosmologist Dennis Sciama, who as a fellow at the Institute for Advanced Study once had the opportunity to speak with Einstein. Many years later, he recalled their conversation: . . . at the end of my year at the Princeton Institute, April 1955, I wanted to see him of course, and I plucked up courage only at the end of the year to go and see him. It was literally a week before he died, and I was with him for over an hour and a half. That was a great experience for me. Originally, of course, the very phrase Mach’s Principle was Einstein’s own phrase for that idea, and he’d used the principle as the guiding light for constructing general relativity. But he later came to feel that the principle wasn’t so important, and in the autobiographical notes . . . he had said that he came to disown Mach’s Principle. I started out a bit nervous of course. I’d read that he had a hearty laugh and a simple sense of humor, . . . So knowing that, I went to see him and I said, ‘Professor Einstein, I’ve come to talk about Mach’s Principle and I’ve come to defend your former self.’ And it worked: he said, ‘‘Ho, ho, ho, that is gut, Ja!’’ Like that, really laughed. So that put me a bit at my ease. So then I talked about my way of doing Mach’s Principle and he talked

Figure 4. Invisible Hands of Distant Galaxies, as conceived by Martin Gardner and drawn by Anthony Ravielli (Gardner 1976, 124–125).

about his work and his doubts about quantum theory and so on. (Sciama 1978) Dennis Sciama took the view that general relativity failed to attain what Einstein originally set out to achieve, namely to provide a physical theory of inertial effects. He sketched his ideas for such a theory in his 1953 paper, ‘‘On the Origin of Inertia,’’ but unfortunately we don’t know what Einstein thought about this approach, if anything. What we do know is that Martin Gardner paid close attention to Sciama’s speculations, which he described in his best-seller, The Relativity Explosion (Gardner 1976), the eighth chapter of which is entitled ‘‘Mach’s Principle.’’ According to Sciama’s calculations, effects that arise from rotation and acceleration with respect to the celestial compass of inertia are induced by distant matter fields, which are almost entirely extragalactic. Mach and his contemporaries spoke about motion relative to the fixed stars, whereas Sciama estimated that all the stars in our galaxy contribute roughly one ten-millionth to the inertial effects measured in the vicinity of our planet. This gave Martin Gardner the inspiration for a wonderful thought experiment of his own, which he described as follows (Fig. 4): I once owned a small glass-topped puzzle, shaped like a square and containing four steel balls. Each ball rested on a groove that ran from the square’s center to one of its corners. The problem was to get all four balls into the corners at the same time. The only way to solve it was by placing the puzzle flat on a table and spinning it. Centrifugal force did the trick. If Sciama is right, this puzzle could not be solved in this way if it were not for the existence of billions of galaxies at enormous distances from our own.’’ (Gardner 1976, 125–126)

REFERENCES

Blumenthal, Otto. 1935. ‘‘Lebensgeschichte,’’ David Hilbert Gesammelte Abhandlungen, vol. 3, Berlin: Springer-Verlag, 388–429. CPAE 3. 1993. The Collected Papers of Albert Einstein, vol. 3, M. Klein, A. Kox, J. Renn, R. Schulmann, eds., Princeton: Princeton University Press. Einstein, Albert. 1917. U¨ber die spezielle und die allgemeine Relativita¨tstheorie. (Gemeinversta¨ndlich), Braunschweig: Vieweg. Einstein, Albert. 1949. ‘‘Autobiographical Notes,’’ in (Schilpp 1949, 1–95). Gardner, Martin. 1976. The Relativity Explosion, New York: Vintage Books. Gardner, Martin. 2006. ‘‘The Fibonacci Sequence,’’ Journal of Recreational Mathematics, 34: 183–190; reprinted in (Gardner 2009, 106–113). Gardner, Martin. 2009. When You Were a Tadpole and I Was a Fish, New York: Hill and Wang. Hilbert, David. 1899. Grundlagen der Geometrie, in (Hilbert 2004). Hilbert, David. 1932–1935. David Hilbert Gesammelte Abhandlungen, 3 vols., Berlin: Springer-Verlag. Hilbert, David. 2004. David Hilbert’s Lectures on the Foundations of Geometry, 1891–1902, Michael Hallett and Ulrich Majer, eds., Berlin, Heidelberg, New York: Springer. Renn, Ju¨rgen, Sauer, Tilman, and Stachel, John. 1997. ‘‘The Origin of Gravitational Lensing: A Postscript to Einstein’s 1936 Science Paper,’’ Science 275: 184–186. Rowe, David. 2010. ‘‘Debating Grassmann’s Mathematics: Schlegel vs. Klein,’’ Mathematical Intelligencer 32(1): 41–48. Schilpp, Paul Arthur, ed. 1949. Albert Einstein: Philosopher-Scientist, vol. 1., New York: Harper Torchbooks. Schubert, Hermann. 1898. Mathematische Mußestunden, Leipzig: Teubner.

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Schubert, Hermann. 1903. Mathematical Essays and Recreations,

Toepell, M.-M. 1986. U¨ber die Entstehung von David Hilberts

Chicago: Open Court. Sciama, Dennis. 1978. ‘‘Interview with Dennis Sciama,’’ with Spencer

Grundlagen der Geometrie. Studien zur Wissenschafts-, Sozialund Bildungsgeschichte der Mathematik, Bd. 2, Go¨ttingen:

Weart, Middletown, Conn., 14 April 1978. Archive for History of Quantum Physics, http://www.aip.org/history/ohilist/4871.html.

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Vandenhoeck & Ruprecht.

Cooking the Classics IAN STEWART

n Chapter 17 of Mathematical Carnival [3] Martin Gardner tells us that ‘‘When a mathematical puzzle is found to contain a major flaw—when the answer is wrong, when there is no answer, or when, contrary to claims, there is more than one answer or a better answer— the puzzle is said to be ‘cooked’.’’ Gardner gives several examples, the simplest being a puzzle he himself set in a children’s book: in the array of numbers

I

9 5 3 1

9 5 3 1

9 5 3 1

circle six digits to make the total of circled numbers equal 21. This is impossible on grounds of parity. Gardner’s answer, in effect cooking his own puzzle, is to turn the page upside down and circle the three 6’s and the three 1’s that then appear. But a reader, Howard Wilkerson, circled each of the 3’s, one of the 1’s, and then drew a big circle round the other two 1’s (giving 11). This is better, since upside down 3’s and 5’s don’t look like digits. Gardner calls this kind of cook a quibble-cook. It exploits an imprecise definition of the question to obtain an unexpected answer. Mathematics is well up to speed on precise definitions these days. Even so, the mathematics that we teach to our students, indeed tell to each other, is also sometimes open to cookery—especially some of our classic theorems, which occasionally have become cliche´s. Over the years, I’ve complied a mental list of cooked classics—a few contentious, all open to debate, all a matter of taste. I am now taking the dangerous step of committing them to print. At the very least, they might be offered to students as exercises, or for class discussion, to avoid giving the impression that the classic proof is holy writ. Many of them have a dynamical systems flavour, even when the topic is number theory. A few don’t. I’ll start with a couple of warm-up examples, which will be familiar to most of you.

deducing that both p and q must be even. The argument runs like this: p2 ¼ 2q 2 so p must be even, say p = 2k. But then 2k 2 ¼ q 2 so q must be even, which is a contradiction. One problem with this proof is that ‘lowest terms’ involves existence and uniqueness of prime factorization, but that can be got round by defining ‘lowest terms’ to mean ‘minimise q’. If you’re prepared to accept existence and uniqueness of prime factorization, then it’s more informative to observe that p2 ¼ 2q 2 has an even power of 2 on the lefthand side, but an odd power on the right. Better still, prove:

T HEOREM 0.1 A rational number a is a perfect square if and only if every prime occurs to an even power in the factorization of a. This does involve extending prime factorization to rationals, allowing negative powers, but that’s easy and the proof is trivial. I think this theorem puts the topic into an appropriate context, and makes the whole idea much clearer p than ffiffiffi a rather artificial argument tailored specifically to 2. Sometimes generalities are better than examples. However, if you don’t want to go that route, prime factorization can be eliminated completely by using what is in effect the original Greek proof, thereby ‘classic-ing the cook’: pffiffiffi Suppose that 2 ¼ p=q where q is as small as possible. Then p [ q and 2q [ p. Since pffiffiffi 2  2 pffiffiffi pffiffiffi ¼ 2 21 we have pffiffiffi 2  p=q 2q  p 2¼ ¼ p=q  1 pq

Root Two is Irrational

pffiffiffi The classic proof of the irrationality of 2 proceeds by pffiffiffi contradiction, assuming that 2 ¼ p=q in lowest terms, and

which, since p - q \ q, is a contradiction.

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The GCD Is a Z-Linear Combination A basic property of the integers is:

T HEOREM 0.2 If g = gcd(a, b) then there exist p, q such that g = pa + qb One favoured route to this result is to set up the Division Algorithm and the Euclidean Algorithm, and proceed inductively. This gets quite complicated. An alternative is to use some ring theory, note that Z is a principal ideal domain, and consider the ideal generated by {a, b}. But this involves a fair amount of machinery. However, a bare hands version of the PID proof is quick and simple, and avoids both algorithms: Let k be the smallest positive integer of the form pa + qb. Clearly g divides k. I claim that k divides a. To see why, choose the largest m such that mk  a. (If you don’t like this step, choose the smallest m such that (m + 1)k [ a.) If mk = a then 0 \ s = a - mk \ k (or else we can replace m by m + 1). Now k  s ¼ pa þ qb  s ¼ pa þ qb þ km  a ¼ pa þ qb  a þ mðpa þ qbÞ ¼ p0 a þ q 0 b for suitable p0 , q0 , contrary to the definition of k. This contradiction proves that k divides a. Similarly, k divides b. In fact, we can define the GCD this way, and prove existence alongside the ‘linear combination’ property.

The GCD as a Dynamical System Let’s get more ambitious. I’ve always found the Euclidean algorithm slightly complicated—not to understand or perform, but to argue about theoretically. Expressing the GCD as an integer linear combination involves a complicated

AUTHOR

......................................................................... IAN STEWART, Emeritus Professor of

Mathematics at Warwick University, is the author of many research papers and books for broad audiences. Currently he works on pattern formation and network dynamics. A Fellow of the Royal Society, his awards include the Royal Society’s Faraday Medal, the Gold Medal of the Institute for Mathematics and Its Applications, the Public Understanding of Science Award of AAAS, and the Zeeman Medal. His nonmathematical interests include science fiction, Egyptology, and geology. Mathematics Institute University of Warwick Coventry, CV4 7AL UK e-mail: [email protected]

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induction, working backward through the algorithm, and somehow the point gets lost. Now, the place where recursion comes into its own in mathematics is dynamical systems. And it is straightforward to turn the Euclidean algorithm into a dynamical system, by doing something slightly more simple-minded. As a bonus, we don’t need the division algorithm. Define a map / : N  N ! N  N by /ðx; yÞ ¼ ðmaxðx; yÞ  minðx; yÞ; minðx; yÞÞ I will prove that if (a, b) = (0, 0), then after finitely many steps the iterates of (a, b) reach a fixed point (d, 0) where d is the GCD of a and b. First, I establish several simple facts: 1) /(x, y) = (0, 0) if and only if (x, y) = (0, 0). 2) Define ||(x, y)|| = x + y. Then jj/ðx; yÞjj  jjðx; yÞjj with equality if and only if x = 0 or y = 0. 3) The fixed points of / are precisely the points (z, 0). 4) If (x, y) = (0, 0) then / preserves the GCD. That is, gcd(/(x, y)) = gcd(x, y). The proofs are trivial: I give the fourth, which is typical. I claim that z divides both x and y if and only if it divides the two components of /(x, y). If z|x and z|y then z|max(x, y) and z|min(x, y). So z divides the two components of /(x, y). If z| (max (x, y) - min (x, y)) and z|min (x, y) then z|(max (x, y) - min (x, y) + min (x, y)) = max (x, y). Therefore z divides both x and y.

T HEOREM 0.3 Let (a, b) = (0, 0). Then there exists t  1 such that /t(a, b) = (d, 0), and then d = gcd(a, b). P ROOF . The norm ||(x, y)|| is a Liapunov function for /: that is, it decreases when / is applied—strictly unless x = 0 or y = 0. Since x; y 2 N there must exist some t such that jj/tþ1 ða; bÞjj ¼ jj/t ða; bÞjj. Then /t(a, b) = (0, d) or (d, 0) for some d. Since /(0, d) = (d, 0), we have /t+1(a, b) = (d, 0). Since GCD is a conserved quantity for the dynamics, gcd(a, b) = gcd(d, 0) = d.

T HEOREM 0.4 If d = gcd(a, b) then there exist p, q such that d = pa + qb. P ROOF . Let X  N  N consist of all pairs ðp1 a þ q1 b; p2 a þ q2 bÞ for p1 ; p2 ; q1 ; q2 2 Z. It is trivial to prove that X is invariant under /: that is, /ðX Þ  X . Since ða; bÞ 2 X , so is /t(a, b) for all t  0. So ðd; 0Þ 2 X , implying that d = pa + qb for some p, q. This is all rather cute, and it gets cuter. The map / also has a scaling property: /ðka; kbÞ ¼ k/ða; bÞ for k 2 N. So X is the disjoint union of subsets X k , where X 0 ¼ fð0; 0Þg X k ¼ fða; bÞ : gcdða; bÞ ¼ kgðk [ 0Þ

Moreover, each X k is / - invariant, and (aside from k = 0) the dynamics of / on X k is conjugate to that of / on X 1 via the map ða; bÞ 7! ðka; kbÞ. So we can understand the dynamics of / by restricting attention to X 1 , the set of all pairs of coprime natural numbers. Each such pair has a uniquely defined height, which is the smallest t for which /t(a, b) = (1, 0). And we can classify pairs by increasing height, using:

L EMMA 0.5 /(a, b) = (c, d) if and only if (a, b) = (c + d, d) or (a, b) = (d, c + d). That is, /-1(c, d) = {(c + d, d), (d, c + d)}. We then find: Height 0 : (1,0) Height 1 : (0,1) Height 2 : (1,1) Height 3 : (2,1), (1,2) Height 4 : (3,1), (1,3), (3,2), (2,3) Height 5 : (4,1), (1,4), (4,3), (3,4), (5,2), (2,5), (5,3), (3,5) So there are 2n-2 pairs of height n when n  2. The matrix of heights is curious: 1 2 3 4 5 6 7 8 9

2  3  4  5  6

3 3  4 4  5 5 

4  4  5  5  6

5 6 4  4  5   6 6  5 7 5  6 

7 5 5 5 5 7  8 6

8  5  5  8  9

9 6  6 6  6 9 

Here the entry in row a, column b is the height of (a, b), and we have written  whenever a, b are not coprime. The reason for the  symbols is that the entire infinite matrix has a recursive structure: the part marked with  symbols repeats the same entries, but the row and column positions are scaled. (This is the decomposition into the X k mentioned earlier.) The form of this matrix is not obvious, and it could be worth investigating. The map / is a formal version of the procedure known to the ancient Greeks as anthyphaeresis (Fowler [2]), in which squares are trimmed off a rectangle until the remaining part is too small, at which point smaller squares are trimmed, and so on. It is of course well known that this procedure is equivalent to the Euclidean algorithm, which in turn is equivalent p toffiffiffi the continued fraction expansion. The Greek proof that 2 is irrational occupies similar territory. The map / makes sense on Rþ  Rþ , and is also related to the continued fraction of x/y or y/x. The norm remains a Liapunov function, so there are no periodic points. However, there are periodic points if we also rescale (x, y), and these occur when x/y is a quadratic irrational.

Euler’s Formula The famous formula e ip ¼ 1

is often presented as something of a mystery. We all know how to justify it, but the usual approaches lack motivation and present it as some sort of accident. There is at least one way to make it natural and inevitable, using differential equations. Quite a bit of machinery needs to be set up to do this, but it’s all good stuff in its own right. Consider the ODE dz ¼ iz dt in the complex plane, with initial conditions z(0) = 1. The solution is z(t) = eit. (You can define the exponential this way.) Now consider the geometrical meaning of the equation. Since iz is orthogonal to z, the tangent vector to a solution at a point z(t) is at right angles to the radius vector from 0 to z(t). It follows (and can be checked by a simple calculation—convert to polars) that • The unit circle centre 0 through z(t) is invariant under the flow (dr/dt = 0) • In polar coordinates, dh/dt = 1. Therefore the unit circle is the trajectory of the solution when z(0) = 1, and t is arc length in radians, so the point z(t) moves at uniform speed along the circle. The Greek definition of p tells us that the circumference of the circle is 2p, so halfway round occurs when t = p. But halfway round is the point z = -1. Hence eip = -1. All the ingredients of this proof are well known and form part of various standard approaches to the trigonometric and exponential functions. But the overall package seems not to get much prominence. Its big advantage is to explain why circles (the definition of p) have anything to do with the exponential.

Infinitude of Primes Euclid’s proof that the number of primes exceeds any finite bound is wonderfully clever, but I’ve always felt that considering p1 . . . pk þ 1 is something of a rabbit out of a hat. What follows pretty much explains how the rabbit got into the hat. As a warm-up, suppose that the only primes were 2, 3, and 5. Then a systematic list of all products of powers of these would yield all possible numbers. The list is most easily generated in non-numerical order, something like 1; 2; 3; 5; 2:2; 2:3; 2:5; 3:3; 3:5; 5:5;   . We want to prove that something is missing, and a good way to do that is to count how many numbers this process yields, up to some limit N. The number 1 occurs once. Multiples of 2 occur bN2 c times. Multiples of 3 occur bN3 c times. Multiples of 5 occur bN5 c times. However, we are overcounting since (for instance) multiples of 6 are multiples of 2 and multiples of 3. So we

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N must subtract multiples of 2.3, which occur b2:3 c times. But then... Well, you can see what’s coming. By the inclusionexclusion principle the number of numbers from 1 to N (which of course is N) has to be

N N N N N N N c 1 þ b c þ b cb c  b c  b c  b c þ b 2 3 5 2:3 2:5 3:5 2:3:5 exactly. Oh, but those floor functions are a pain. So let’s make N be exactly divisible by all the denominators; that is, N is a multiple of 2.3.5. Better still, why not set N = 2.3.5 = 30? Then the expression becomes

which isn’t 30. The generalization is now obvious. Suppose that p1 ; . . .; pk is the entire (finite) list of primes. Let N ¼ p1 . . .pk . The same argument using the inclusion-exclusion principle now implies that the number of numbers between 1 and N, namely N, satisfies X N XN X N  þ  ... N ¼1þ pi pi pj i;j;k pi pj pk i i;j where the subscripts in all sums are unequal. Therefore ! X 1 X1 X 1 þ  þ ... 1¼N 1 pi pi pj i;j;k pi pj pk i i;j     1 1 ... 1  ¼N 1 p1 p    k  1 1 . . .pk 1  ¼ p1 1  p1 pk ¼ ðp1  1Þ. . .ðpk  1Þ which is absurd unless k = 1 and p1 = 2. But now 3 is missing from the list of primes. Having now realized that some numbers are missing, we quickly notice that an obvious missing number is N  1 ¼ p1 . . . pk  1. Having seen that, we reconstruct Euclid’s proof when we decide that a remainder of 1 is easier to explain than a remainder of -1. There are, of course, innumerable proofs of the infinitude of primes. A very simple one, related to the above calculation, is to compute the Euler /-function of N ¼ p1 . . . pk . Supposing that every number between 2 and N - 1 is a multiple of some pi, clearly /(N) = 1. On the other hand, /ðN Þ ¼ /ðp1 Þ. . ./ðpk Þ ¼ ðp1  1Þ. . .ðpk  1Þ So all pi = 2, and k = 1, as before.

Solution of Polynomials by Radicals It might seem unlikely that such a tried and tested area as the solution of the quadratic, cubic, and quartic could have anything new to offer. However, some minor variations on the traditional themes are possible. Quadratics As a warm-up, I solve the general quadratic

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x 2 þ px þ q ¼ ðx þ aÞ2  b2 This leads directly to 2a ¼ p a2  b2 ¼ q Therefore a = p/2, so b2 ¼ p2 =4  q, so b ¼  Now (0.1) becomes

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 =4  q .

0 ¼ ðx þ aÞ2  b2 ¼ ðx þ a þ bÞðx þ a  bÞ

1 þ 15 þ 10 þ 6  2  3  5 þ 1 ¼ 23

x 2 þ px þ q ¼ 0

by an unorthodox method. The idea is that when it comes to the crunch, the only quadratic we can factorise is x 2  k2 . So we seek a, b such that

ð0:1Þ

so x = -a ± b. That is, p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼   p2 =4  q 2 which is the traditional formula. (What else?)

Quartics If a trick works, use it again. Thus emboldened, I attempt the general quartic x 4 þ px 3 þ qx 2 þ rx þ s ¼ 0

ð0:2Þ

by the same method. We seek a, b, c, d such that x 4 þ px 3 þ qx 2 þ rx þ s ¼ ðx 2 þ ax þ bÞ2  ðcx þ dÞ2 which leads to 2a ¼ p 2

a þ 2b  c2 ¼ q 2ab  2cd ¼ r b2  d 2 ¼ s Clearly we must set a = p/2. Set b ¼ ðq þ c2  p2 =4Þ=2 to solve the second equation for b in terms of c, and r þ 2ab r þ pðq þ c2  p2 =4Þ=2 ¼ 2c 2c 2r þ pðq þ c2  p2 =4Þ ¼ 4c

d¼

to solve the third equation for d in terms of c. Finally, substitute all of this into the fourth equation:   2r þ pðq þ c2  p2 =4Þ 2 ððq þ c2  p2 =4Þ=2Þ2  s 4c ¼0 and (with hindsight) multiply through by c2 to remove denominators. This yields  4    1 q 3p2 4 3p p2 q 2 pr  c þ  þ þ  s c2 0 ¼ c6 þ 4 2 16 4 64 4q 4   p6 p4 q p2 p3 r pqr r 2  þ  þ þ   4 256 32 16q 2 16 4 which (miraculously) is a cubic in c2. (It is of course a variant of Ferrari’s resolvent cubic.)

Therefore we can solve quartics provided we can solve cubics. Cubics The same trick seems not to work, partly because 3 is odd. A variant succeeds in reducing the cubic to... the same cubic. The classic trick is to reduce the equation to x 3 þ mx þ n ¼ 0 by translating x and then making a clever substitution. As a variation, don’t bother: consider x 3 þ px 2 þ qx þ r ¼ 0 and substitute x ¼ az þ b þ cz

1

(For motivation, consider the traditional x = t + 1/t for palindromic polynomials.) Then x 3 þ px 2 þ qx þ r ¼ Az 3 þ Bz 2 þ Cz þ D þ Ez 1 þ Fz 2 þ Gz 3

Trisection of Angles The usual proof that angles cannot be trisected (see for example [6]) relies on a cubic equation for cosð2p=9Þ and the multiplicativity of the degree of a field extension. Here’s an alternative using less machinery and a more natural setting. Identify Euclid’s plane with the complex plane C. Define z 2 C to be constructible if it can be constructed from Q  R  C by ruler and compass. (Note that we don’t consider the real and imaginary parts separately.) The usual coordinate calculations prove:

L E M MA 0.6 A point z is constructible if and only if there is a finite sequence of complex numbers a1 ; . . .; ak such that a21 2 Q, a2j 2 Qða1 ; . . .; aj1 Þ for j ¼ 2; . . .; k, and z 2 Qða1 ; . . .; ak Þ. Here, as usual, Qð. . .Þ denoted the subfield of C generated by the contents of the parentheses. Observe that if K is a subfield of C and a2 [ K, then

where

K ðaÞ ¼ fx þ ay : x; y 2 K g A ¼ a3

I now prove:

B ¼ a2 ð3b þ pÞ C ¼ að3b2 þ 3ac þ 2bp þ qÞ D ¼ b3 þ 6abc þ b2 p þ 2acp þ bq þ r

T HEOREM 0.7 The primitive 9th root of unity f ¼ e 2pi=9 is not constructible.

E ¼ cð3b2 þ 3ac þ 2bp þ qÞ F ¼ c2 ð3b þ pÞ G ¼ c3 A fortuitous coincidence now smacks us between the eyes. Making two expressions 3b + p and 3b2 + 3ac + 2bp + q vanish causes the four coefficients of z 2 ; z; z 1 ; z 2 to vanish. What luck! This happens when p2  3q a¼ 9c

p b¼ 3

and c is a free parameter. (All we need is 9ac = p2 - 3q.) Now the cubic becomes Az 3 þ D þ Gz 3 ¼ 0 so Az 6 þ Dz 3 þ G ¼ 0 which is quadratic in z3 so can be solved by radicals. Then substitute to get x. Quintics Thanks to Abel, Galois, and their predecessors, we know there isn’t a formula. I’ve concocted a stripped-down proof using very little technical machinery, but it’s about ten pages long. It’s not so much a cook as an attempt to reverseengineer what the algebraists from Legendre to Kronecker already knew, and reassemble the bits that are needed. It proves that x5 - 80x + 30 = 0 can’t be solved by radicals. I may publish it elsewhere, once it’s been polished up.

Since f trisects x = e2pi/3 it follows that the angle 2p/3 cannot be trisected using ruler and compass. It remains to prove the theorem without using multiplicativity of the degree of a field extension.

P ROOF . Assume for a contradiction that f is constructible. Define a tower of subfields Q ¼ K0 QðxÞ ¼ K1 K2    Ks such that f 2 Ks and Kj ¼ Kj1 ðaj Þ where a2j 2 Kj1 , for j ¼ 2;p . .ffiffi.;ffi s. Note that the p same goes for j = 1 since x ¼ ffiffiffi ð1 þ i 3Þ=2 so QðxÞ ¼ Qð 3Þ. Such a tower exists if and only if f is constructible. Choose one for which s is minimal. Then pffiffiffi f¼aþb b where a, b, b [ Ks-1. Minimality implies that b = 0, whence also a = 0. But f3 ¼ x, so pffiffiffi pffiffiffi x ¼ ða þ b bÞ3 ¼ ða3 þ 3ab2 bÞ þ ð3a2 b þ b3 bÞ b If 3a2 b þ b3 b 6¼ 0 then pffiffiffi x  a3  3ab2 b b¼ 3a2 b þ b3 b which lies in Ks-1, contrary to minimality. Therefore 3a2 b þ b3 b ¼ 0, so x ¼ a3 þ 3ab2 b

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But now pffiffiffi pffiffiffi ða  b bÞ3 ¼ ða3 þ 3ab2 bÞ  ð3a2 b þ b3 bÞ b ¼ ða3 þ 3ab2 bÞ ¼ x The cube roots of x are f; xf; x2 f. Therefore pffiffiffi a  b b ¼ xc f where c = 1 or c = 2. (We can’t have c = 0 since b = 0.) But we already know that pffiffiffi aþb b¼f Adding, we get f ¼ bð1 þ xc Þ=2a which is in Ks-1, a contradiction.

Two Squares Theorem Fermat’s Two Squares Therorem, proved by Euler, states that any prime of the form 4k + 1 is a sum of two squares. There’s more, but this is the hard part. The traditional approaches either use quadratic residues or prove that some multiple of the prime is a sum of two squares and use descent. The following proof must be well known to numbertheorists, but I’ve not seen it in the texts. It is short, conceptual, and straightforward. Recall that the Gaussian integers Z½i comprise all complex numbers a + bi, where a; b 2 Z. There is a norm N ða þ biÞ ¼ a2 þ b2 and this is multiplicative: N ðxyÞ ¼ N ðxÞN ðyÞ The Gaussian integers form a unique factorisation domain: in fact the norm provides a Euclidean algorithm and this can be proved quickly by elementary means. Let p 2 Z be prime (in the usual sense). We claim that if p : 1 (mod 4) then p is not prime in Z½i . For a contradiction, suppose that p = 4k + 1 is a Gaussian prime. Then Z½i =p is a field. It has a subfield Z=p, which does not contain i, since if it did, i would be real, indeed in Z. The multiplicative group of this subfield is cyclic of order 4k so has an element a of order 4. Now the quartic polynomial t4 - 1 has at least five distinct zeros: 1; a; a2 ; a3 , and i. This is a contradiction. Now N(p) = p2, so multiplicativity of the norm implies that p ¼ q1 q2 where q1 ; q2 are prime in Z½i . Since p is real, q2 ¼ q1 . Let q = a + bi. Then

There is a nice analogue for Z½x where x is a cube root of unity, and now we prove that primes 6k + 1 are of the form a2 þ b2  ab, or equivalently a2 þ 3b2 . Of course quadratic reciprocity gives far more—but that doesn’t count as ‘elementary’.

‘Give Me a Place to Stand, and I Will Move the Earth’ So, famously, said Archimedes. I claim he already had a place to stand. This is a quibble-cook, I think. Archimedes didn’t get anything wrong. Just missed the point. Archimedes was dramatizing the law of the lever, and what he had in mind was basically Figure 1. I don’t think he was interested in the position of the Earth in space, but he wanted the pivot point to be fixed, and in order to apply the law of the lever he needed uniform gravity, contrary to astronomical fact. He also needed a perfectly rigid lever of zero mass. No matter. I don’t want to get into discussions about inertia or other quibbles. Let’s grant him all those things. My question is: when the Earth moves, how far does it move? Assume Archimedes can exert a force sufficient to lift his own weight, say 100kg. The mass of the Earth is about 6.1024kg. If the pivot is 1 metre from the Earth then the Law of the Lever tells us that distance from the pivot to Archimedes is 6 9 1022 metres, and his lever is 1 + 6 9 1022 metres long. If Archimedes moves his end of the lever one metre, similar triangles tell us that the Earth moves 1.6 9 10-23 metres. A proton has diameter 10-15 metres... OK, but it still moves, dammit! True. But suppose that instead of this huge and improbable apparatus, Archimedes stands on the surface of the Earth and jumps. For every metre he clears, the Earth moves 1.6 9 10-23 metres the other way (action/reaction). Basically, this has exactly the same effect as a lever 1 + 6 9 1022 metres long—about 1.6 million light years, or about two thirds of the way to the Andromeda Galaxy.

The Reals are Uncountable I like the ‘diagonal’ proof, but it does need some intricate maneuvers with infinite decimals. Suppose R is countable with R ¼ faj : j ¼ 1; 2; . . .g. Define two functions R ! R: f ðxÞ ¼ 1 8x 2 R  1 if jx  aj j  2j for some j gðxÞ ¼ 0 otherwise

p ¼ ða þ biÞða  biÞ ¼ a2 þ b2 Strictly speaking, we get this up to a unit, but the units are ±1, ±i. Since p and a2 þ b2 are real and positive, the equation follows. A tactical variant is to observe that Z½i =p is Z=p½t =ht 2 þ 1i, and t2 + 1 is reducible (with a zero a) so the quotient cannot be a field. This is marginally more elegant but slightly less direct. 66

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Figure 1. Archimedes’s lever.

Then f(x) = g(x) for all x 2 R. But Z 1 Z 1 1 X f ¼1 g 2j ¼ 1 1

1

j¼1

Of course we need the Lebesgue integral to make this work, and for that we need Lebesgue measure on R. Now the cook gets cooked, because all we have to do is prove that a countable subset of R has measure zero. So here’s a topological proof using less machinery. With the same assumptions, choose a sequence of non-empty closed intervals Aj  ½0; 1 such that aj 62 Aj and Ajþ1  Aj . (This is easy.) Then A1 \ A2 \ A3 \    is non-empty (by compactness of [0,1]) but contains none of the aj.

Squares and Rectangles The next item for cooking is a problem discussed by Terence Tao in his book on problem-solving [7]. My treatment is not suitable for his intended audience. Still, here goes. The problem is about four rectangles, of equal area, that fit together to form a big rectangle as in Figure 2, leaving a rectangular hole (shaded). The problem is to prove that if the outer rectangle is actually square, then so is the shaded hole. Tao assumes that the outer square has side 1, and homes in on a key fact that cracks the problem wide open: the sides x, y of each smaller rectangle must sum to 1. But how to prove this? He studies how the equal-area condition propagates information from each rectangle to the next, points out that four such steps must lead back to the original rectangle, does some calculations, and out pops x + y = 1. I’m a dynamical systems person, and yet again this looks suspiciously like a dynamical systems problem to me. The function that maps each rectangle to the next has a period-4 point, and we have to deduce that this is really a fixed-point. (All four rectangles have to be congruent if the construction is to work with a surrounding square; this is not immediately apparent from Tao’s treatment. The condition x + y = 1 turns out to be irrelevant, though true.) Anyway, when I pursued this line of attack, it led to a continued fraction and some simple one-dimensional dynamics, like so: Suppose that the surrounding square has unit side, and let the sides of the first rectangle be x0 ; y0 , which lie in (0,1) to

xy 1−y R2 R1

1−y

R3 R0 x Figure 2. Configuration of four squares.

y

avoid trivial cases. Let the common area of the four rectangles be A. Then 4A \ 1 if the picture is to be believed, otherwise the rectangles would overlap or the shaded region would have zero area. So A\ 14. Following Tao, we observe that   A ðx1 ; y1 Þ ¼ 1  y0 ; 1  y0 where A ¼ x0 y0 , and in general  ðxiþ1 ; yiþ1 Þ ¼ 1  yi ;

A 1  yi



for all i because area is preserved. Already we see a discrete dynamical system on R2 . Moreover, ðx0 ; y0 Þ is a period-4 point, otherwise the rectangles would not fit together correctly. Although this is a 2-dimensional system, it reduces to one dimension because xi yi ¼ A. So yi ¼ A=xi , and what matters is the function / defined by /ðxÞ ¼ 1 

A x

which determines the trajectory of the x-coordinate. The 4-periodicity tells us that /4 ðx0 Þ ¼ x0 . It is therefore natural to work out what /4(x) is. Being lazy, and prone to computational errors, I did this using Mathematica, and the result was what always happens to me when I use Mathematica: /ðxÞ ¼ 1 

A x

/2 ðxÞ ¼ 1 

A 1  Ax

/3 ðxÞ ¼ 1 

A A 1  1 A x

A /4 ðxÞ ¼ 1  1  1A A

1A x

I always forget that you have to tell Mathematica to simplify and expand expressions the way a human mathematician would. Here it was just substituting them unchanged, and getting needlessly complicated. I wanted to set x = /4(x) and solve, but I also wanted to keep track of what was going on, so I tried to get the machine to simplify the expression— which never works the way I expect it to, somehow. The result was uninformative in any case. I then realised that Mathematica was telling me something, which I wouldn’t have noticed if the software had helpfully simplified the formula as it went along. You don’t have to be a genius to see that we are developing a continued fraction, whose terms visibly have period one, not four. It is obvious what all iterates of / look like. Moreover, if the limit of this continued fraction is z, then clearly z ¼ 1  Az, so z is actually a fixed point of /. In a little more detail:

L EMMA 0.8 If the sequence of all iterates of a periodic point converges, then it must be a fixed-point.  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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P ROOF . Let x0 be periodic with period p. We know that /n(x0) converges, and its limit n satifies /(n) = n. But the subsequence /np(x0) is constant, with every term equal to x0. So x0 = n and is a fixed-point. But now we are done aside from some routine checking. If x is a period-4 point of /, then z = x, so x is a fixedpoint. Therefore all four rectangles are congruent, the entire figure has rotational symmetry through p/2, and the shaded part is square. Rigour demands a little care with convergence. The continued fraction is not ‘regular’, with 1 on top in place of A, so maybe it doesn’t converge. Actually, it does. This follows easily from a few simple features of the dynamics of /, illustrated in Figure 3. The fixed-points of / are the solutions of the quadratic x2 - x + A = 0, namely pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  1  4A k¼ p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1  4A l¼ 2 Bearing in mind that A\ 14, these solutions are real. It is routine to establish the inequalities pffiffiffiffi 0\A\k\ A\l\1 The graph of / on [0,1] looks like Figure 3. There is a geometric symmetry in the problem: the two sides x0 ; y0 of R0 are interchangeable. The symmetry is nonlinear, since it maps x to A/x. We use it in a p trivial ffiffiffiffi way: without loss of generality we can choose x0 [ A, that is, make it the longer side pffiffiffiof ffi R0. The map / preserves this property. Moreover, if A\x\l, then x \ /(x) \ l, and if l \ x \ 1, then l \ /(x) \ x. (These facts are clear from the figure, and can be checked by routine calculations.) By monotonicity, the sequence /n(x) converges for all pffiffiffiffi x 2 ð A; 1Þ. Since the limit is a fixed-point of /, it must be l. Now we can appeal to Lemma 0.8 and we are done. Note that x + y = 1 plays no role in this approach. We gain something by this method, too. we start with pffiffiffiSuppose ffi any initial rectangle, taking x0 [ A, and go round and round the square forming new rectangles. Although in

Figure 3. Graph of / showing fixed-points. 68

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general this sequence does not have period 4, it converges toward a rectangle of period 4, indeed period 1.

Courant-Robbins Train This is metacookery, perhaps meta-metacookery. The issues are complicated and what gets cooked is not the obvious statement. It can be uncooked, but this seems like luck more than judgement, and doesn’t happen in very similar questions. I mention this one because it opens up an important general issue. Whenever I try to explain what’s wrong, someone always writes in and complains that with a bit of extra work the conclusion remains correct. That’s true (and this remark goes right back to Poston [4] when he pointed out the difficulty in 1976) but the whole point is that you need the extra work, contrary to what Courant and Robbins say. Indeed, if I wanted to be pedantic I could point out that they don’t state the precise hypotheses needed for their argument, so it’s not clear how generally they think it applies. Courant and Robbins [1] state the problem like this: Suppose a train travels from station A to station B along a straight section of track. The journey need not be of uniform speed or acceleration... But the exact motion of the train is supposed to be known in advance. On the floor... a rod is pivoted so that it may move without friction either forward or backward until it touches the floor. If it touches the floor, we assume that it remains on the floor henceforth; this will be the case if the rod does not bounce. Is it possible to place the rod in such a position that, if it is released at the instant when the train starts and allowed to move solely under the influence of gravity and the motion of the train, it will not fall to the floor during the entire journey from A to B ? The answer they give is ‘yes’, and the reasoning is continuity: No detailed knowledge of the laws of dynamics is needed; only the following assumption of a physical nature need be granted: The motion of the rod depends continuously on its initial position. Figure 4 is basically theirs, with added variables and wheels removed. To spell out the proof: starting at angle 0, the rod stays there; the same goes for angle p. So [0, p] maps continuously to [0, p], so the image contains all possible angles between 0 and p. Actually, the important lesson in this example is that boundary conditions can destroy ‘intuitive’ continuity properties. As it happens, they don’t—in the simplest model for this problem. But they do in slightly more complicated (and more realistic) models, and in very similar problems. The offhand reference to an ‘assumption of a physical nature’ could easily lead readers to think that the assumption is harmless, and would apply in all similar problems. That’s not so. How justified, then, is the continuity assumption? What do they mean when they state that ‘the motion of the rod’ is a continuous function of the initial position? There are at least three meanings: the entire trajectory (continuity in some function space), its location after some fixed time

(X,Y ) mg x

θ

π θ

α 0

Α t Figure 4. The Courant-Robbins train.

before imposing the absorbing boundary conditions, its location after some fixed time after imposing the absorbing boundary conditions. These could have different continuity properties. If the rod is free to rotate to any angle in the circle, there’s no great difficulty—and the property of continuity holds for any ODE with well-behaved solutions for all time. Singularities can make the question ill-posed, but they don’t occur here. However, as Poston [4] pointed out, the ‘absorbing boundary conditions’ at angles 0, p are more problematic. In fact, for a wide class of ODEs, the boundary conditions destroy continuity, as we’ll shortly see. So the train problem is ‘cooked’– not because Courant and Robbins gave the wrong answer; not even because they gave the wrong reason for it. It is cooked because they made no effort at all to justify the reason, citing physical intuition. Never forget that ODEs and PDEs have boundary conditions. And those may have a dramatic effect on continuity properties of solutions. Suppose, for instance, that Courant and Robbins had modified the problem to allow the wind to blow, with a prespecified velocity, depending smoothly on time. Or (see below) allowed the train’s floor not to be flat—which, incidentally, they don’t specify, although their picture points that way. Most readers would probably have accepted the same ‘physical assumption’, but this time it would be plain wrong. Poston remarks that the only way he can see to salvage the argument is to impose some stringent conditions: perfectly level track, no springs in the wheels of the train... Then you still have to explain why those conditions do the trick. This is nontrivial, and can’t just be dismissed as a simple property based on physical considerations. The problem is worth analysing in detail. Consider a smooth vector field on a circle, depending smoothly on time t 2 R. Then the associated flow w determines diffeomorphisms wt of the circle for all t. So the timet flow, for any given t, is smooth. For any given position function F(t) for the train, the time taken to go from A to B is determined. So the map from initial state to final state, for the position h of the rod, is a smooth diffeomorphism. Suppose, however, that the flow looks like Figure 5, which is entirely reasonable for a general smooth ODE. If we now impose the absorbing boundary conditions, we find that all initial conditions lead, after finite time, to states h = 0, p. That is, the rod hits the floor. As I said, I’m less interested in developing conditions on the mechanics that ensure such things cannot happen, than in observing that in general ODEs they do happen, robustly. That alone makes the continuity assumption far from

Figure 5. Why absorbing boundary conditions can destroy continuity.

obvious. Let’s see why, which will also explain the conditions under which Courant and Robbins are correct about continuity. Figure 4 shows about the simplest model of the train that I can invent. The train itself is reduced to a point A that moves along the horizontal line; its location is x relative to the origin. The rod is inclined at angle h, and we assume its mass m is concentrated at the end. To remove various constants, choose units to make the length of the rod equal to 1, the mass m = 1, and the acceleration due to gravity g = 1. Assume that the position of the train at time t is a prespecified function F(t), which we take to be of class C2 to avoid analytic issues. Take coordinates in a moving frame attached to point A. This introduces a ‘fictitious force’ F€ðtÞ, and aside from this we now have a simple pendulum. The angular position h satisifes the ODE € h ¼ F€ðtÞ sin h  cos h and we take initial conditions h ¼ h0 ; h_ ¼ 0 at time t = 0. (I use h_ ¼ 0 because Courant and Robbins say ‘released’. It’s not essential.) The absorbing boundary conditions can destroy continuity if there exists h0 for which the trajectory is tangent to the boundary h = 0, while h  0 locally along the trajectory, as in Figure 6(b). Then we would expect to be able to arrange F(t) to make nearby h0 hit the boundary transversely or miss it altogether as in Figures 6 (a, c). Similar problems arise at the other boundary h = p if h  p locally. The tangency condition on the boundary at 0 implies that h = 0 and h_ ¼ 0. But when h = 0, the equation of motion implies that € h ¼ 1, so locally h  0. Similarly, at the p boundary, € h ¼ 1. So the problematic kind of ‘grazing’ trajectory cannot occur. Even now, it is a nontrivial exercise to prove that the final position is a continuous function of the initial one when the absorbing boundary conditions hold. There might be other sources of discontinuity, for all we know. So technically Courant and Robbins are absolutely right, because they make continuity an explicit assumption. But continuity fails if we make apparently harmless modifications to the question. One of the simplest such modifications is to place the boundaries at p/4, 3p/4. Most readers would still be happy to accept the continuity assumption. However, if we take F(t) = t4/12 so that F€ðtÞ ¼ t 2 , which is not exactly rocket science, then numerical experiments find a grazing trajectory with initial conditions close to h0 = 1.0664. So the continuity  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Figure 6. Four trajectories. Initial conditions respectively are h(0) = 0.2, 1.0664, 1.1, 1.3.

Figure 7. Four trajectories. Initial conditions respectively are h(0) = 0.9, 0.943974 , 1, 1.1.

argument might break down. Further experiments indicate that it does, and there is no intermediate position that stays upright (Figure 6). For the next experiment, put the boundaries at p/50, 49p/50. Again we find numerical evidence for a grazing trajectory, and there is no intermediate position that stays upright beyond about 6 seconds (Figure 7). These results accord with physical intuition, of course. With a perfectly horizontal track, no applied acceleration can lift the rod off the floor if it is instantaneously at rest at h = 0, and the same goes for h = p. But if the rod is slightly above the horizontal position, a suitable acceleration could lift it. This is why pretty much the only boundaries that 70

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produce continuity are the ones employed by Courant and Robbins. Simple estimates prove:

T HEOREM 0.9 Suppose 0 \ a \ p and the boundaries are at a, p - a. Suppose F€ðtÞ ¼ t 2 . Then the rod hits the floor after a finite time T(a). _ P ROOF . Let vðtÞ ¼ hðtÞ be the angular velocity. Then the ODE reduces to the system _ ¼ vðtÞ hðtÞ _ ¼ t 2 sin hðtÞ  cos hðtÞ vðtÞ

If t [ 2 then 2

_  ð2  1Þ sin a ¼ 3 sin a vðtÞ Therefore vð2 þ tÞ  1 provided V þ 3t sin a  1 where V ¼ minfvð2Þ : hð0Þ 2 ½a; p  a ; vð0Þ ¼ 0g Therefore we must take t

1V 3 sin a

_  1 then hðt þ p  2aÞ  p  a, so the rod has Now, if hðtÞ hit the boundary. Thus the rod hits the boundary after at most time T ðaÞ ¼ p  2a þ 2 þ

1V 3 sin a

It remains to prove that V is finite. (It might be -?.) But _  cos a vðtÞ

problem still arises if distinct fluid particles are allowed to occupy the same location. The usual assumption of fluid dynamics, that particles don’t do that, implies that the flow of a body of fluid preserves its topological type. We can cook this elegant but slightly artificial example by using the invariant p0 instead—the space of connected components. Now there is a more commonplace, but equally dramatic scenario: the same appeal to continuity implies that you can’t empty a normal-shaped wine bottle into two or more glasses. Initially the wine forms one connected component, and a continuous map can’t increase the number of components. Once again, though, we must ask: is continuity of the flow justified? It surely depends on the boundary conditions, not just the PDE. Tyres and bottles have boundaries, and physically sensible boundary conditions could–perhaps should— allow the fluid to ‘break’ when its surface meets the boundary in the right way, perhaps tangentially. Without this boundary condition, the flow would be continuous, but with it, the fluid can split discontinuously. I don’t know enough about the Navier-Stokes equation to analyse this possibility, which is a question for experts; I imagine that the answer depends on the precise physical assumptions encoded in the PDE and its boundary conditions.

so vðtÞ  t cos a and vð2Þ  2 cos a Therefore V  2 cos a. Thus we may take T ðaÞ ¼ p  2a þ 2 þ

1 þ 2 cos a 3 sin a

Since we are taking F(t) = t4/12, the stations can be placed at A = 0, B = T4(a)/12. Note that T ðaÞ ! 1 as a ! 0. Poston [4] says: ‘Given the usual laws, the only physical assumptions I can find which guarantee a nonfalling history are that the pivot is perfect with the movement of the train perfectly, totally horizontal. (Not just level track: the train must have no springs.)’ And he adds: ‘Courant and Robbins did not make a silly mistake, but Dynamical Systems has progressed... it would be silly now.’

Pouring from a Toroidal Bottle I wonder if something similar is going on in a provocative article by Sarkaria [5] published in the Intelligencer in 2001. This discusses the hypothesis of continuity in the motion of matter, tracing it back to Anaxagoras, and demonstrates that ‘matter and motion cannot both be assumed continuous’. The example cited is the impossibility of emptying ‘a tyretube filled with water into a bucket in any finite length of time’. The proof is that a homotopically nontrival loop in the tyre would flow to define a homotopically trivial loop in the bucket. That is, the fundamental group p1 is an obstacle to emptying the bottle. Sarkaria observes that a similar

Afterword As I said at the start, this is a personal compilation and I’m not making any claims of originality or superiority. Variations on proofs tend to be rediscovered over and over again, and I’m sure that historians will shortly be telling me that the approach to cubics was known to some Portuguese algebraist in 1573, whereas experts in number theory will quietly draw me aside to explain that Serre knew everything in this article in the 1950s. Still, I think it’s a useful exercise to find alternatives to the classic proofs, some of which have become cliche´s and some of which do make a bit of a mouthful of things that can be done more clearly and more simply. I’d be interested to see other examples where the classics can be cooked. Maybe it would be worth setting up a website on mathematical cookery. REFERENCES

[1] H. Courant and H. Robbins. What is Mathematics?, Oxford: Oxford University Press, 1941. [2] D.H. Fowler. the Mathematics of Plato’s Academy, Oxford: Oxford University Press, 1987. [3] M. Gardner. Mathematical Carnival, New York: Knopf, 1975. [4] T. Poston. Au Courant with differential equations, Manifold 18 (1976) 6–9. [5] K.S. Sarkaria. A topological paradox of motion, Mathematical Intelligencer 23 vol. 4 (2001) 66–68. [6] I. Stewart. Galois Theory, Boca Raton: Chapman and Hall/CRC, 2004. [7] T. Tao. Solving Mathematical Problems — a Personal Perspective, Oxford: Oxford University Press, 2006.

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The Problem of Malfatti: Two Centuries of Debate MARCO ANDREATTA, ANDRA´S BEZDEK

ianfrancesco Malfatti (Figure 1) was a brilliant Italian mathematician born in 1731 in a small village in the Italian Alps, Ala, near Trento. He first studied at a Jesuit school in Verona, then at the University of Bologna. Malfatti was one of the founders of the Department of Mathematics of the University of Ferrara. He died in Ferrara in 1807. As a very active intellectual in the Age of Enlightenment, he devoted himself to the promotion of many new ideas and wrote many papers in different fields of mathematics including algebra, calculus, geometry, and probability theory. He played an important role in the creation of the Nuova Enciclopedia Italiana (1779), in the spirit of the French Encyclope´die edited by Diderot and d’Alembert. His mathematical papers were collected by the Italian Mathematical Society in the volume [7]. His historical figure has been discussed in a series of papers in [1]. This paper was inspired by a conference in 2007 commemorating the 200th anniversary of Malfatti’s death, organized by the municipality of Ala and the mathematics departments of Ferrara and Trento. Malfatti appears in the mathematical literature of the last two centuries mostly in connection with a problem he raised and discussed in a paper in 1803 (Figure 2): how to pack three non-overlapping circles of maximum total area in a given triangle? Malfatti assumed that the solution consisted of three mutually tangent circles, each also tangent to two edges of the triangle (now called Malfatti’s arrangement) and in his paper he constructed such arrangements (for a historical overview see [3]). In 1994 Zalgaller and Los [11] disproved Malfatti’s original assumption and showed that the greedy arrangement is always the best one. The detailed story of this 200-year-old

AND

JAN P. BORON´SKI

G

Figure 1. Gianfrancesco Malfatti (1731–1807).

problem is worth telling because it has many paradigms typical of research in mathematics, including the way one formulates a problem, how one interprets it or solves it, and what one should consider trivial and what one should not. In the following section we give the history of the problem. The section after that contains a new non-analytic solution for the problem of maximizing the total area of two disjoint circles contained in a given triangle. In the last section we generalize the two-circle problem for certain regions other than triangles. Our non-analytic approach shows that in various situations the greedy arrangements are the best ones.

Malfatti’s Marble Problem and Its History The term stereotomy in the title of Malfatti’s paper (from the Greek stereo = rseqeo, which means solid, rigid and

M. Andreatta was supported by a grant from Italian Miur-Prin, A. Bezdek was supported by OTKA Grant 68398.

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Figure 3. Malfatti’s triplet.

Figure 2. Title page of the paper ‘‘On a stereotomy problem’’.

tomy = solia, which means cut, section) refers to the art of cutting solids into certain figures or sections, as arches, and the like; it refers especially to the art of stonecutting. The first lines of the paper are more specific about the problem: ‘‘… given a triangular right prism of whatsoever material, say marble, take out from it three cylinders with the same heights of the prism but of maximum total volume, that is to say with the minimum scrap of material with respect to the volume …’’. We summarize the rest of the page with some observations. 1. Malfatti noted that his problem can be reduced, via a stereotomy, to a problem in plane geometry. Though not explicitly stated in the paper, the reduced problem is: Given a triangle find three non-overlapping circles inside it of total maximum area. The literature refers to this problem as Malfatti’s marble problem. 2. Then, without any justification, Malfatti ‘‘…observed that the problem reduces to the inscription of three circles in a triangle in such a way that each circle touches the other

two and at the same time two sides of the triangle…’’. Today we know that Malfatti’s intuition was wrong: this geometric configuration (Figure 3), ‘‘Malfatti’s configuration,’’ does not solve his marble problem. Yet the remaining part of Malfatti’s work was correct. 3. Malfatti constructed the unique three-circle arrangement that today bears his name. In his words ‘‘…Undertaken therefore the solution of this second problem, I found myself plunged into prolix calculations and harsh formulae…’’. The progress made on Malfatti’s marble problem and Malfatti’s construction problem should be separated from each other. Malfatti’s construction problem: It is believed that Jacob Bernoulli considered this question for isosceles triangles a century before Malfatti. The problem can also be found in Japanese temple geometry, where it is attributed to Chokuyen Naonobu Ajima (1732–1798). Malfatti’s approach was algebraic. He computed the coordinates of the centers of the circles involved, and noticed that the values of the expressions can be constructed using ruler and compass (the reader can find the explicit solution of Malfatti in his paper [6] and also in Section 5 of the more recent book [8]). In 1826 Steiner published an elegant solution of Malfatti’s construction problem. He also considered several variations,

AUTHORS

......................................................................... ............................................................................... MARCO ANDREATTA is Professor of

´ S BEZDEK is Professor of the MathANDRA

Geometry at the University of Trento, Italy. He received his ‘‘Laurea’’ in 1981. After post docs in Italy and the United States, he taught at the University of Milan before coming to Trento in 1991. His main field of research is algebraic geometry. He lives in the Italian Alps, very near to Malfatti’s birth place; besides mathematics he likes walking, cycling, and skiing in the mountains.

ematics at Auburn University, Alabama, and Senior Research Fellow at the Alfre´d Re´nyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary. He received his PhD degree from Ohio State University in Columbus in 1986 and his Doctor of Academy title (DSc) from the Hungarian Academy of Sciences in 2006 for his work in discrete geometry. He enjoys swimming and playing tennis.

Dipartimento di Matematica Universita´ di Trento via Sommarive, 38123 Trento Italy e-mail: [email protected]

MTA Re´nyi Institute 13-15 Re´altanoda u., Budapest Hungary e-mail: [email protected]  2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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including analogous problems where the sides of the triangle are replaced by circular arcs, or when these arcs are placed on a sphere. In 1811 Gergonne asked about the existence of a similar extremal arrangement in three-dimensional space, using a tetrahedron and four spheres instead of a triangle and three circles. The extremal arrangement of spheres was constructed by Sansone in 1968. In the nineteenth century many mathematicians, including Cayley, Schellbach, and Clebsch, worked on various generalizations. Malfatti’s marble problem: In 1930 Lob and Richmond [10] observed that in an equilateral triangle the triangle’s inscribed circle together with two smaller circles, each inscribed in one of the three components left uncovered by the first circle, produces greater total area than Malfatti’s arrangement. Eves [2] pointed out that in a very tall triangle placing three circles on top of each other also produces greater total area. We say that n circles in a given region form a greedy arrangement, if they are the result of the n-step process, where at each step one chooses the largest circle which does not overlap the previously selected circles and is contained by the given region. Goldberg [5], see also [4], outlined a numerical argument that the greedy arrangement is always better than Malfatti’s. Mellisen conjectured, in [9].

C ONJECTURE 1 The greedy arrangement has the largest total area among arrangements of n non-overlapping circles in a triangle. Malfatti’s marble problem is the case n = 3; it was settled by Zalgaller and Los [11]. Following [11] we say that a system of n non-overlapping circles in a triangle is a rigid arrangement if it is not possible to continuously deform one of the circles in order to increase its radius, without moving the others and keeping all circles non-overlapping.

......................................................................... ´ SKI JAN P. BORON

graduated from Silesian University at Katowice, Poland, majoring in Mathematics in 2005, and he received his PhD from Auburn University, Alabama, in 2010, under the guidance of Professor Krystyna Kuperberg. At Auburn he was a recipient of College of Sciences and Mathematics Dean’s Award for Outstanding PhD Graduate Student Research. His research interests include topological fixed-point theory, dynamical systems, and continuum theory. Among his hobbies are mountain biking, skiing, hiking, music, and movies. Mathematics & Statistics Auburn University Auburn, AL 36849-5310 USA e-mail: [email protected] 74

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It is evident that the solution of Malfatti’s marble problem is in the class of rigid arrangements. Note also that every circle of a rigid arrangement has at least three points of contact either with the other circles or with the sides of the triangle; moreover, these points of contact do not lie on one closed semicircle of the boundary of the circle under consideration. Zalgaller and Los showed, by an elaborate case analysis, that if n = 3, then with the exception of the greedy triplet, all rigid configurations allow local area improvements.

Two Circles in a Triangle We now consider the Malfatti marble problem for n = 2, arranging two non-overlapping circles of maximum total area in a given triangle.

T HEOREM 1 The greedy arrangement has the largest total area among pairs of non-overlapping circles in a triangle. This problem is not difficult; an analytic solution is explained in [9] and a similar solution is also included implicitly in the work of Los and Zalgaller [11]. We present a new non-analytic solution, which will lead to several generalizations. Let us note first that the greedy arrangement consists of the inscribed circle and the one touching the two longer sides and the inscribed circle.

P ROOF OF T HEOREM 1 Let ABC be the given triangle. Assume that two tangent circles are arranged in the triangle ABC so that the first circle touches the sides AB and AC and the second circle touches the sides AC and BC (Figure 4). This is the rigid arrangement of two circles. Let r be the radius of the first and let R be the radius of the second circle. If both circles are held fixed by their contact points, then R is uniquely determined by r. Denote by R(r) the function that describes the relation between the radii r and R. We will prove that the total area function (r2 + R2(r))p is convex. Therefore the area function attains its maximum at the end points of the admissible interval of r, which implies that the greedy arrangement is the best. A real-valued function f (x) is midpoint-convex on an interval for any two numbers x, x0 from its domain,
iff ðxÞþf ðx 0 Þ xþx 0 . It is known that any continuous, f 2  2

Figure 4. Two circles’ rigid arrangement.

2 In other words, the circle centered at M of radius r1 þr 2 and 2Þ must overlap. the one centered at N of radius Rðr1 ÞþRðr 2 Consequently, r þ r  Rðr Þ þ Rðr Þ 1 2 1 2 : \ R 2 2

Two Circles in Other Regions and Other Generalizations Figure 5. Comparison of two circles’ rigid arrangements.

midpoint-convex function is convex. It is also known that if both f (x) and g(x) are convex functions, then i) f (x) + g(x) is also convex, furthermore ii) if in addition to being convex, f (x) is also increasing, then f (g(x)) is convex. Thus all we need to show is:

L EMMA 1 R(r) is a midpoint-convex function of r. Let r1, R(r1) and r2, R(r2) be the radii of two pairs of circles satisfying the conditions of Lemma 1. Denote by O1, O10 and similarly by O2, O20 the centers of these circles (see Figure 5). Clearly the following equalities hold: jO1 O01 j ¼ r1 þ Rðr1 Þ and

jO2 O02 j ¼ r2 þ Rðr2 Þ

Let us recall the following elementary geometric exercise,

E XERCISE 1 Show that in any quadrilateral the sum of the lengths of two opposite sides is at least twice the distance between the midpoints of the remaining two sides. Solution of Exercise 1: Let ABCD be any quadrilateral (Figure 6); it can be convex, concave, or self-intersecting and can have collinear or even coinciding vertices. Let M and N be the midpoints of side AB and CD. Reflect B through the midpoint N to get B0 . Obviously 2|MN| = |AB0 | B |AD| + |DB0 | = |AD| + |BC|, which is what we wanted to show. Applying this exercise to the quadrilateral O1O2O20 O10 , with O1O0 1, and O2O20 being the opposite sides and M and N being the midpoints of the two remaining sides, we get jMN j\

It is natural to ask whether the greedy arrangement still gives the largest total area when the circles are placed in regions other than a triangle. Mellisen [9] showed a pentagon (Figure 7a) where the greedy arrangement clearly does not win. In this section we call a region a concave triangle if it is bounded by three concave curves (Figure 7b). We will prove

T HEOREM 2 The greedy arrangement has the largest total area among pairs of non-overlapping circles in any concave triangle (Figure 7b). When we proved Theorem 1 we looked at a pair of (rigid) circles which were different from the greedy arrangement and which could not be improved by changing only one of them (see Figure 4 again). We considered a continuous change of the two circles. It turned out (Lemma 1) that as we changed the radius r of one of the circles, the radius R(r) of the other circle, as a function, changed in a convex manner. This essentially meant that the total area could be improved locally. The bottom line is that any generalization of Lemma 1 could lead to a new theorem. First of all, the proof of Lemma 1 remains true word by word if the two circles are not restricted to the triangle (Figure 8a). The exact same proof remains valid if the two circles are allowed to increase or decrease maintaining contact not with the sides of the triangle but with the sides of two angular sectors (Figure 8b). Formally we have

jO1 O01 j þ jO2 O02 j r1 þ r2 Rðr1 Þ þ Rðr2 Þ ¼ : þ 2 2 2

(a)

(b)

Figure 7. Convex and concave containers.

(a) Figure 6. Exercise 1.

(b)

(c)

Figure 8. Proof of Theorem 2.  2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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L EMMA 2 Let AA0 and CC 0 be two non-intersecting segments in a triangle ABC. Assume two tangent circles of radii r, R are given so that the first circle touches the sides AB and the interior of the segment AA0 , and the second circle touches the sides BC and the interior of the segment CC 0 . If both circles are held fixed by the contact points, then R is determined by r and the function which describes this relation is convex in r.

P ROOF OF T HEOREM 2, by contradiction. Assume that the maximum total area is achieved by a pair of circles different from the greedy arrangement. Then the circles touch each other and they also touch exactly two of the concave curves (Figure 8c). The tangent lines of the circles at these contact points (if they are not unique, choose any of them) together with the circles satisfy the conditions of Lemma 2 and thus allow local area improvement. Assume one needs to arrange two non-overlapping spheres of greatest total volume in a given tetrahedron. The centers of spheres which are touching the same three faces of the tetrahedron are on a line and lines corresponding to different triples of the faces meet at the incenter of the tetrahedron. Thus the two-sphere marble problem leads to Lemma 1 again, and we have

R EMARK 1 The greedy arrangement has the largest total area among pairs of non-overlapping spheres in a tetrahedron. Assume one needs to arrange two non-overlapping circles of greatest total area in a triangle of the hyperbolic plane. Among steps of the Euclidean proof of Theorems 1 and 2 only the elementary geometric fact ‘‘for any quadrilateral (which can be convex, concave or self intersecting, or degenerate) the sum of the lengths of two opposite sides is at least twice the distance between the midpoints of the remaining two sides’’ needs to be questioned. Since that inequality holds in the hyperbolic plane too, we have

Figure 9. Two circles in a spherical line.

standard proof fails now is the above-mentioned elementary geometric inequality, which on the sphere can be proved only for self-intersecting quadrilaterals. This, in view of the application, means (details are omitted here) that

R EMARK 3 Theorem 1 remains true on the sphere if the diameter of the spherical triangle is less than p4 :

REFERENCES

[1] Atti del Convegno, Gianfranco Malfatti nella cultura del suo tempo. Univ. di Ferrara. 1982. [2] H. Eves, A Survey of Geometry, Vol. 2, Allyn and Bacon, Boston, 1965, 245–247. [3] A. Fiocca, Il problema di Malfatti nella letteratura matematica dell’800, Ann. Univ. Ferrara-sez VII. vol. XXVI, 1980. [4] H. Gabai and E. Liban, On Goldberg’s inequality associated with the Malfatti problem, Math. Mag., 41 (1968), 251–252. [5] M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241–247. [6] G. Malfatti, Memoria sopra un problema sterotomico, Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, 10 (1803), 235–244. [7] G. Malfatti, Opere I e II, a cura dell’Unione Matematica Italiana, ed. Cremonese, 1981.

R EMARK 2 Theorem 1 and 2 remain true in the hyper-

[8] G. Martin, Geometric constructions, UTM Springer, 1998.

bolic plane.

[9] H. Mellisen, Packing and covering with Circles, thesis, Univ. of Utrecht, 1997.

Consider the analogous problem of placing two circles in a spherical triangle. A straightforward computation shows that in a lune with sufficiently small angle the symmetrical arrangement has larger total area than that of the the greedy arrangement (Figure 9). The reason that the

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[10] H. Lob and H.W. Richmond, The solution of Malfatti’s problem for a triangle, Proc. London Math. Soc., 30 (1930), 287–304. [11] V. A. Zalgaller and G. A. Los, The solution of Malfatti’s problem, Journal of Mathematical Sciences, 72, No. 4, (1994), 3163– 3177.

Spinoza and the Icosahedron Eugene A. Katz

This monument to Baruch Spinoza by the sculptor Nicolas Dings was unveiled in Amsterdam in 2008. The Platonic icosahedron represents the classical ‘‘element’’ water and

symbolizes the Universe polished by human thinking. It also alludes to Spinoza’s job as a lens polisher. Photo by E.A. Katz.

Dept. of Solar Energy and Environmental Physics Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boker Campus, 84990, Israel e-mail: [email protected] Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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The Mathematical Tourist

Dirk Huylebrouck, Editor

The Geometry of Christopher Wren and Robert Hooke: A Walking Tour in London MARIA ZACK Does your hometown have any mathematical tourist 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.

â

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

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ondon contains a wealth of architectural styles and buildings from many periods; for those interested in architecture it is a wonderful place to explore on foot. After the Great Fire of 1666 destroyed most of the old walled city of London, Sir Christopher Wren and Robert Hooke were appointed to the group charged with rebuilding London. Wren and Hooke, both founding members of the Royal Society, began a lifelong friendship and pattern of collaboration during their student days at Oxford. Until the Great Fire, they were best known for their work in mathematics, physics, and astronomy. Wren was at one time the Savilian Professor of Astronomy at Oxford. As longtime Curator of Experiments for the Royal Society, Hooke had ‘‘to furnish them every day, on which they met, with three or four considerable experiments.’’ However, although the massive effort to rebuild London was led by both men, Wren’s enduring reputation is as an architect and Hooke’s is as a scientist. In the mid-seventeenth century, Inigo Jones brought the notion of ‘‘classical architecture’’ with all of its columns, domes, and Palladian windows to England. Many of these design elements had been rejected in the past as too ‘‘Catholic’’ for Henry VIII’s ‘‘protestant’’ England. But after the Great Fire, Wren and Hooke popularized this very geometric design style, using it in royal, public, and private buildings, as well as in places of worship. Wren has been credited with designing St. Paul’s Cathedral and more than fifty parish churches that were rebuilt in the City of London. Current scholarship indicates that the parish churches were most likely designed by Wren, Hooke, and others who worked under them. It is a difficult matter to determine attributions for most of the churches. Some hints can be found in parish vestry minutes and others in Hooke’s diaries (he was an avid diarist). It is clear from Hooke’s diaries that for decades he and Wren met nearly daily to discuss both architectural and scientific matters, which suggests that the design process was collaborative, with details executed by their apprentices. We know, from diaries, writings, and library lists, that Wren and Hooke had access to the canon of classical architectural texts of their day. These books by Alberti, Vitruvius, and Palladio all advocate geometry, proportion, and symmetry as the basis of architectural design. When seeking connections between Wren’s mathematics and architecture, many authors point to the strong use of geometry in his buildings. Derek Whiteside says of Wren, ‘‘Perhaps his greatest mathematical gift was his visual sensitivity and feeling for form which are obvious in his architectural designs and scientific sketches.’’ Wren himself states in Tract I: Geometrical Figures are naturally more beautiful than other irregular; in this all consent as to a Law of Nature. Of

L

geometrical Figures, the Square and the Circle are most beautiful; next, the Parallelogram and the Oval. Straight Lines are more beautiful than curve. Let’s take a walking tour of two of the most mathematical of Wren and Hooke’s remaining structures in London: St. Paul’s Cathedral (predominantly a Wren design) and the Monument to the Great Fire (fundamentally a Hooke design). Along the way we will take a brief look at a few of the other buildings designed by Hooke and Wren.

The Tour To help visualize the tour, I’ve appended a map of London (see Figure 10 at the end of this article). Begin your tour at St. Paul’s Cathedral (number 1 on the map). This church is justifiably known as Wren’s masterpiece. Harold Dorn and Robert Mark have analyzed the structural aspects of St. Paul’s Cathedral with a particular focus on Wren’s triple dome design. Wren probably saw double domes, based on a 400-year tradition, during his 1665 1666 visit to Paris. A careful analysis of the dome, piers, and buttresses indicates that the dome was overengineered and that the buttresses (and hence the second level fac¸ade that conceals them) were not necessary. Certainly Wren was aware of Leonardo da Vinci’s analysis of the cracks in domes in Codex Arundel, in particular the cracks in the dome of St. Peter’s in Rome that came from what we call hoop stress today. Wren’s clever triple dome design, with the central cone carrying the weight of the lantern for St. Paul’s, was an elegant solution to the problems found in St. Peter’s (Figure 1 shows the triple domes as they were built). However, there is very little documentary evidence that any of the engineering was based on more than the

AUTHOR

......................................................................... MARIA ZACK received her undergraduate

Figure 1. The triple dome (drawing by Lisa Gilbertson).

rules of thumb and intuition used by craftsmen for centuries. Dorn and Mark point out an intriguing bit of information: the shape of the conic dome in St. Paul’s is approximately a catenary. We know that Wren and Hooke were aware of the properties of the catenary. In late 1670/71, Hooke and Wren presented ideas on arches to the Royal Society including ‘‘the line of an arch supporting a weight assigned,’’ and late in 1671 Hooke presented further information to the Royal

degree and doctorate in mathematics from the University of California, San Diego. Her research interests include abstract algebra, the history of mathematics, and the philosophy of higher education. She has worked on the research staff of the Institute for Defense Analysis and is currently the Chair of the Department of Mathematical, Information and Computer Sciences at Point Loma Nazarene University in San Diego, California, USA. In her spare time she volunteers with the Medical Benevolence Foundation traveling internationally to help strengthen indigenous health care systems in developing nations. Rohr Science Hall 222 Point Loma Nazarene University 3900 Lomaland Drive San Diego, CA 92106 USA email: [email protected]

Figure 2. Wren’s sketches for the design of the St. Paul’s dome (circa 1690). This drawing reproduced courtesy of the British Museum, London ( Trustees of the British Museum).  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Figure 3. William Dickinson’s Sketches for the Dome of St. Paul’s (circa 1696-1702). Reproduced by permission of the Dean and Chapter of St. Paul’s Cathedral and the London Metropolitan Archive. Figure 4. St. Paul’s Cathedral (photo by Maria Zack).

Society about weight-sustaining domes in the shape of a ‘‘cubico-parabolical conoid,’’ a dome that uses the cubic on the positive numbers and rotates it about the vertical axis. By 1675 Hooke understood that the most structurally sound arch was a catenary ‘‘As hangs the flexible line, so but inverted will stand the rigid arch.’’ On June 5, 1675, Hooke writes in his diary ‘‘At Sir Chr. Wren…He [Wren] was making up my principle about arches and altered his module [model] by it.’’ The timing of this diary entry is significant. The design for the postfire St. Paul’s involved a great deal of political wrangling between Wren and the clergy of St. Paul’s. In the spring of 1675 a compromise was reached, and in May of 1675 this compromise design (known as the Warrant design) was approved for construction by King Charles II. However, the King told Wren privately that he could make ‘‘variations, rather ornamental, than essential, as from Time to Time he should see proper,’’ and that the management of the construction was left to Wren. By the summer of 1675, Wren had begun modifying the Warrant design and was laying the foundation for a church that more closely resembled the church he had envisioned. Evidently Hooke and Wren were talking about catenary arches at the precise time that Wren was working on the final (and actual) design of St. Paul’s. In the British Library there is a small drawing dating from approximately 1690 for the triple-dome (see Figure 2). The drawing is in Wren’s own hand, and the

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numerical grids show that Wren was using the cubic to form the central dome of the triple. If you look at the righthand drawing you will see vertical markings. The curve intersects these markings at (1,1) and (2,8), and there is a dot at (3,27) where the curve reaches the buttressing. So certainly in 1690, some mathematical thought about the cubic parabola appears in this stage of the development of the design. The correct equation for the hanging chain was found independently by Christian Huygens, Gottfried Leibniz, and Johann Bernoulli and was published by Leibniz in 1691 in Acta eruditorum. Hooke’s library contained the relevant issues of Acta eruditorum, so he knew that the correct equation was not a cubic parabola but rather what we would call today a hyperbolic cosine. Wren modified the design of the dome several times in the period 1696-1702. As the design evolved, the central dome became more conical in shape, closer to the catenary than the cubic parabola. This change can be seen in a drawing by Wren’s draughtsman William Dickinson (Figure 3). This drawing contains some of the key structural principles of the catenary: the shape of the arch is key to determining its abutment, and that the line of thrust of an arch must stay inside the boundary of the abutment. But the building is not mathematically exact.

Figure 5. The Tower of St. Mary-le-Bow (photo by Maria Zack).

Figure 6. St. Stephen Walbrook (photo by Maria Zack).

For one of the best views in London, climb all 540 stairs to the lantern on top of the dome (Figure 4). Along the way you will be able to see the flying buttresses for the dome that are hidden behind the second floor fac¸ade. The final flights of iron stairs will allow you to walk between the brick cone and the lead outer dome. On the floor of St. Paul’s Cathedral, below the dome, an inscription names Christopher Wren and continues ‘‘Lector Si Monumentum Requiris Circumspice’’ (Reader, if you seek his monument, look around). Wren is buried in the crypt of the church with a very simple marker. As you exit St. Paul’s Cathedral, walk down Cheapside, the main shopping street of medieval London. You will soon reach St. Mary-le-Bow (number 2 on the map). There has been a church on this site since approximately 1090 and a substantial early crypt survives (today there is a cafe´ in the crypt). After the Great Fire, St. Mary’s was one of the first churches rebuilt, and it was the most expensive. The reconstruction of this church involved many people and has a complex history. The steeple at the top of the tower is particularly interesting (Figure 5). The drawing for this tower and steeple was done by Nicholas Hawksmoore (a Wren apprentice) and is inscribed ‘‘as originally intended by Sir Christopher Wren.’’ It is a lovely symmetrical combination of columns, domes, and arches. To be a true

‘‘cockney’’ in London means to have been born within the sound of ‘‘Bow’s bells.’’ After passing St. Mary-le-Bow, turn onto Walbrook Street. You will soon come to St. Stephen Walbrook (number 3 on the map) on the left-hand side of the street. From the front, St. Stephen’s doesn’t look like much, but its interior is filled with light and I find this to be one of the most restful and quiet spots in the city. The Vestry minutes of St. Stephen Walbrook state that ‘‘Dr. Christopher Wren in consideration of his great care and extraordinary pains taken in the contriving and designing of the Church and assisting in the rebuilding the same’’ should be given a gift and invited to dinner (Vestry Minutes, 1648-1699). Wren skillfully uses geometry to impose a centrally planned church on a longitudinal space, turning a very plain box into the most elegant of his parish churches. There is some speculation that working on this church helped Wren to think through some of the features of St. Paul’s, particularly the connection between the aisle of columns and the dome (Figure 6). The church still has its original altar and pulpit but was reconfigured in the 1970s to put the stone Moore altar in the center. You need to look past the Moore altar to imagine the original dark wood altar as the focal point, as Wren designed it.  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Figure 8. Contemporary view of the Monument to the Great Fire (photo by Maria Zack).

Figure 7. Hooke’s Drawing of His Zenith Telescope is Figure 4 from An Attempt to Prove the Motion of the Earth from Observations. Reproduced courtesy of the Guildhall Library, London.

Continue down Walbrook Street and turn left onto Cannon Street. After you cross King William Street (the street that leads to the London Bridge), Fish Street will be on your left, with the enormous pillar that is Hooke’s Monument to the Great Fire (number 4 on the map). (Figure 6). Hooke often used buildings as scientific instruments. As early as 1662 he conducted experiments on gravity by dropping items from the top of Westminster Abbey; experiments from the top of the prefire St. Paul’s Cathedral led to his 1666 paper On Gravity. Certainly the Monument provided another great height for Hooke’s experiments, but its special role was as a telescope designed to measure the parallax and thus ‘‘furnish the Learned with an experimentum crucis to determine between the Tychonick and the Copernican Hypotheses.’’ Hooke measured the path of the star Gamma Draconis because it is bright and it passes directly overhead (near the zenith point) in London every day. This choice simplified the experiment because gravity defines the zenith exactly, so the telescope could be aligned simply by using a plumb bob. In addition, because the star’s light passes through the Earth’s atmosphere perpendicular to the surface of the 82

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Figure 9. The sphere and urn at the top of the Monument to the Great Fire (photo by Maria Zack).

Figure 10. Map with churches: (1) St. Paul’s Cathedral, (2) St. Mary-le-Bow, (3) St. Stephen Walbrook, (4) Monument to the Great Fire (drawing by Lisa Gilbertson based on London maps).

earth, calculations did not need to be adjusted to account for refraction. Hooke described his experiments in An Attempt to Prove the Motion of the Earth from Observations, giving details of the construction, installation, and alignment of his thirty-six foot telescope. To accommodate it, he had to cut a hole through the floor and ceiling of his lodgings at Gresham College (see Figure 7). Hooke began his experiment in July, 1669, but made only four observations (July 6, July 9, August 6, and October 21, 1669), of which he wrote, ‘Tis manifest then by observations…. that there is a sensible parallax of the Earths Orb to the fixt Star in the head of Draco, and consequently a confirmation of the Copernican System against the Ptolomaick and Tichonick. Hooke was conscious of the possibility of experimental error in his measurement and knew that he could obtain more accurate results with a longer telescope. The 1670 City Churches Rebuilding Act provided funds for a monument to ‘‘preserve the memory of this dreadful visitation [the Great Fire of 1666].’’ In designing the Monument, Hooke seized the opportunity to create his desired long focal length (nearly 200 feet) telescope. By 1673, the Monument was under construction. Hooke’s diaries indicate that he was involved in each step of the construction process. The Monument (Figure 8) was completed in 1677 and its use as a scientific instrument began. An underground chamber set in a bed of gravel was the location of the eyepiece of the telescope. The objective lens was mounted 200 feet above, near the top of the pillar inside the ball but below the hinged doors to the flaming urn (see Figure 9). The accuracy of the observations made by this zenith telescope depended on maintaining the alignment of the eyepiece and

objective lens. Unfortunately, the vibrations caused by air currents traveling down the core of the column and from the wheeled traffic passing by the pillar caused a misalignment in the lenses that was greater than the changes in parallax that Hooke was trying to measure. It would be another 165 years before technology advanced sufficiently to measure the parallax. In 1838, Friedrich Bessel computed the parallax for 61 Cygni whose angle of change is much greater than that of Hooke’s Gamma Draconis. For more details about the history of the Monument and Hooke’s experiments there, see my recent paper in the reading list at the end of this article. Both Hooke and Wren saw geometry as a practical art to be used to solve physical problems. In their architecture, their scientific interests are made concrete.

FURTHER READING

Cooper, Michael, A More Beautiful City: Robert Hooke and the Rebuilding of London After the Great Fire, Stroud, Sutton Publishing, 2003. Dorn, Harold and Robert Mark, The Architecture of Christopher Wren, Scientific American, 245(1) (1981), 160–173. Falter, Holger, The Influence of Mathematics on the Development of Structural Form, Nexus II Architecture and Mathematics, Kim Williams, ed., Fucecchio, Edizioni dell’Erba, 1998, 51–64. Gerbino, Anthony and Stephen Johnston, Compass and Rule: Architecture as Mathematical Practice in England, London, Yale University Press, 2009. Heyman, Jaques, Hooke’s Cubico-Parabolical Conoid, Notes and Records of the Royal Society of London, 52(1) (Jan 1998), 39–50. Hooke, Robert, An Attempt to Prove the Motion of the Earth from Observations, London, Royal Society, 1674.

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Hooke, Robert, The Diary of Robert Hooke MA, MD, FRS, 1672–1680,

Wassell, Stephen, The Mathematics of Palladio’s Villas, Nexus II

Henry Robinson and Walter Adams, eds., London, Taylor Francis, 1935.

Architecture and Mathematics, Kim Williams, ed., Fucecchio, Edizioni dell’Erba, 1998, 173–186.

Inwood, Stephen, The Forgotten Genius, the Biography of Robert

Whiteside, Derek, Wren the Mathematician, Notes and Records of the

Hooke 1635–1703, San Francisco, MacAdam/Cage, 2005. Jardine, Lisa, Ingenious Pursuits, New York, Doubleday, 1999. Jardine, Lisa, On a Grander Scale: The Outstanding Life of Sir Christopher Wren, New York, Harper Collins, 2002.

Royal Society, XV (1960), 107–111. Wilson, Robert, Astronomy through the Ages: The Story of the Human Attempt to Understand the Universe, Princeton, Princeton University Press, 1997.

Jeffery, Paul, The City Churches of Sir Christopher Wren, London,

Wren, Christopher, Parentalia: or, Memoirs of the Family of the Wrens

Hambeldon Press, 1996. Kendall, Derek, The City of London Churches a Pictorial Rediscovery,

(heirloom copy), Farnborough, Gregg Press Limited, 1965. Zack, Maria, Robert Hooke’s Fire Monument: Architecture as a

London, Collins and Brown, 1998. Vestry Minutes St. Stephen Walbrook, 1648–1699, Guildhall Library Manuscript MS 594, Volume 2, 128.

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Scientific Instrument, Nexus VII Architecture and Mathematics, Kim

Williams,

117–126.

ed.,

Fucecchio,

Edizioni

dell’Erba,

2008,

Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem ARNO BERGER*

AND

THEODORE P. HILL

he widely known phenomenon called Benford’s Law continues to defy attempts at an easy derivation. This article briefly reviews recurring flaws in ‘‘back-ofthe-envelope’’ explanations of the law, and then analyzes in more detail some of the recently published attempts, many of which replicate an apparently unnoticed error in Feller’s classic 1966 text An Introduction to Probability Theory and Its Applications. Specifically, the claim by Feller and subsequent authors that ‘‘regularity and large spread implies Benford’s Law’’ is fallacious for any reasonable definitions of regularity and spread (measure of dispersion). The fallacy is brought to light by means of concrete examples and a new inequality. As for replacing the wrong assertions by an equally simple explanation which is valid, now—that is a task for the future.

T

It’s All About Digits The eminent logician, mathematician, and philosopher C.S. Peirce once observed [Ga, p.273] that ‘‘in no other

branch of mathematics is it so easy for experts to blunder as in probability theory’’. As the reader as well will see, this is all too true for Benford’s Law, also known as the First-Digit Phenomenon. Benford’s Law, abbreviated henceforth as BL, is one of the gems of statistical folklore. It is the observation that in many collections of numbers, be they mathematical tables, real-life data, or combinations thereof, the leading significant digits are not uniformly distributed, as might be expected, but are heavily skewed toward the smaller digits. More precisely, BL says that the significant digits in many datasets follow a very particular logarithmic distribution. In its most common formulation, the special case of first significant decimal (i.e., base 10) digits, BL reads   ProbðD1 ¼ d1 Þ ¼ log10 1 þ d11 ; for all d1 ¼ 1; . . .; 9; ðBL1Þ

*This author was supported by an NSERC Discovery Grant.

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here D1 denotes the first significant decimal digit, e.g., pffiffiffi D1 ð 2Þ ¼ D1 ð1:414. . .Þ ¼ 1; D1 ðp1 Þ ¼ D1 ð0:3183. . .Þ ¼ 3; D1 ðe p Þ ¼ D1 ð23:14. . .Þ ¼ 2: A crucial part of the content of (BL1), of course, is an appropriate formulation or interpretation of ‘‘Prob’’. For sequences of real numbers or real datasets, for example, Prob usually refers to the proportion (or relative frequency) of entries for which an event such as D1 = 1 occurs, whereas for a random variable, Prob is simply the probability on the underlying probability space. Figure 1 illustrates several of these settings, including mathematical sequences such as the powers of 2, and real-life data from Benford’s original paper as well as recent census statistics. In a form more complete than (BL1), BL is a statement about the joint distribution of all decimal digits: For every natural number n, this version states that D2 ; . . .; Dn Þ ¼ ðd1 ; d2 ; . . .;dn ÞÞ ProbððD1 ; Xn 1 nj ¼ log10 1 þ 10 d j j¼1

ðBL2Þ

holds for all n-tuples ðd1 ; d2 ; . . .; dn Þ, where d1 is an integer in 1; 2; . . .; 9 and where for j [ 1, dj is an integer in 0; 1; . . .; 9. Here D2 ; D3 ; D4 , etc. represent the second, third, fourth, etc. significant decimal digit, so that, for example, pffiffiffi D2 ð 2Þ ¼ 4; D3 ðp1 Þ ¼ 8; D4 ðe p Þ ¼ 4: The ‘‘laws’’ (BL1) and (BL2) were apparently first discovered by polymath S. Newcomb in the 1880s [N]. They were rediscovered by physicist F. Benford [Ben]; Newcomb’s article having been forgotten at the time, they came to be known as Benford’s Law. Today, BL appears in a broad spectrum of mathematics, ranging from differential equations to number theory to statistics. Simultaneously, the applications of BL are mushrooming—from diagnostic tests for mathematical models in biology and finance to fraud detection. For instance, the U.S. Internal Revenue Service uses BL to ferret out suspicious tax returns, political

scientists use it to identify voter fraud, and engineers to detect altered digital images. As Raimi already observed some 25 years ago [R1, p.512], This particular logarithmic distribution of the first digits, while not universal, is so common and yet so surprising at first glance that it has given rise to a varied literature, among the authors of which are mathematicians, statisticians, economists, engineers, physicists, and amateurs. The online database [BH] now contains more than 600 articles on the subject. Many of these articles, including some of the very recent ones, purport to provide easy derivations or proofs of BL. The present article sets out to identify some of the prevalent fallacies in those arguments.

Simple Explanations? Are You Sure? Let’s start with the purely mathematical framework. Many familiar sequences, including the Fibonacci numbers, the powers of 2, and the factorial sequence (n!), all follow BL exactly; in particular, exactly a proportion of log10 2 ¼ 0:3010. . ., that is, approximately 30.1% of all entries of those sequences begin with the decimal digit 1. Similarly, start with any positive number and multiply by 3 repeatedly (i.e., iterate the function x 7! 3x), or multiply alternately by 3 and by 4, or iterate the function x 7! 2x þ 1. Each of these iterations results in a sequence that follows BL exactly, no matter what positive number was chosen in the beginning. Thus, even though many common sequences such as the natural numbers and the primes do not follow BL, those that do are so ubiquitous that many authors have assumed that a simple explanation must exist. Raimi [R1, R2] reviews many of the attempts at such explanations: Some authors simply labeled BL self-evident; thus Benford himself wrote that ‘‘the logarithmic law applies particularly to those outlaw numbers that are without known relationship’’, Goudsmit and Furry opined that it ‘‘is merely the result of our way of writing numbers’’, and likewise Weaver claimed that BL ‘‘is a built-in characteristic of our number system’’. For the more mathematical ones among the back-of-the-envelope derivations, Raimi

AUTHORS

......................................................................... ...... ......................................................................... ARNO BERGER is an Associate Professor

at the University of Alberta. Before coming to Canada, he had held faculty positions in Austria and New Zealand. His main passion in mathematics is dynamical systems theory, so naturally he is also interested in analysis and probability. Department of Mathematical and Statistical Sciences University of Alberta Edmonton, T6G 2G1 Canada e-mail: [email protected]

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THEODORE P. HILL earned his doctorate at University of California, Berkeley. He is now Professor Emeritus at Georgia Tech (after many visiting stints in Costa Rica, The Netherlands, and elsewhere) and is a visiting scholar at California Polytechnic State University. His primary research areas are mathematical probability, especially optimal-stopping theory, fair-division problems – and Benford’s Law.

School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 USA e-mail: [email protected]

Figure 1. Different interpretations of (BL1) for sequences, datasets, and random variables, respectively, and scenarios that may lead to exact conformance with BL.

carefully points out their shortcomings, e.g., Flehinger’s Cesa`ro-summation method, Herzel’s urn model, and the two different fallacies in Logan and Goudsmit’s urn model derivation. In [R1, sec.7], Raimi also explains the basic flaw in Pinkham’s widely cited scale-invariance argument. That is the argument that BL is the only distribution on significant digits that is invariant under changes of scale, meaning that (BL1) and (BL2) remain unchanged if a sequence, dataset, or random variable is multiplied by any positive constant. Raimi credits Knuth [K] for

the discovery that the error is in Pinkham’s implicit assumption that there exists a scale-invariant probability distribution on the positive real numbers, when clearly there is no such distribution. To see this, simply note that for instance multiplying any positive random variable X by 2 doubles the value of its median, and hence X cannot be scale-invariant. To the best of the authors’ knowledge, the first correct proof that BL indeed is the unique scale-invariant probability distribution (and also the unique continuous base-invariant distribution) on the significant digits is in [H2]. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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A closer look at sequences of numbers reveals some surprises that may help explain why correct and quick derivations of BL in a purely mathematical context may be hard to come by. For instance, iterating the function x 7! x 2 þ 1 results in a sequence following BL for (Lebesgue) almost all starting points, but not for all starting points. Thus, if the initial value x1 is chosen randomly from any positive distribution with a density, such as, say, the uniform distribution on (0,1) or an exponential distribution, then the resulting sequence (xn) with xnþ1 ¼ xn2 þ 1 will follow BL exactly with probability 1. But there are exceptional points also. For example, choosing x1 ¼ 9:9496230. . . implies D1 ðxn Þ ¼ 9 for all n, that is, every number xn begins with a decimal digit 9. (See [BBH, exp.4.3] to find out what is special about this remarkable value for x1.) Whether or not the sequence (xn) follows BL when x1 = 0 is still an open problem. All in all, even though it would be highly desirable to have both a rigorous formal proof and a reasonably sound heuristic explanation, it seems unlikely that any quick derivation has much hope of explaining BL mathematically. It is in the realm of real-life data that assertions about easy derivations of BL become especially treacherous. One type of erroneous shortcut in particular continues to propagate, and the remainder of this article is devoted to identifying and illuminating it. A variety of formal mathematical proofs is available for sequences and random variables (see e.g., [BBH, H2]). But in teaching probability and statistics, a correct general explanation of a principle is often as valuable as a detailed formal argument. In his December 2009 column in the IMS Bulletin, UC Berkeley statistics professor T. Speed extols the virtues of derivations in statistics [S]: I think in statistics we need derivations, not proofs. That is, lines of reasoning from some assumptions to a formula, or a procedure, which may or may not have certain properties in a given context, but which, all going well, might provide some insight. For illustration, Speed quotes two examples of the convolution property for the Gamma and Cauchy distributions from the classic 1966 text An Introduction to Probability Theory and Its Applications by W. Feller [Fel]. On page 63, Feller also gave a brief derivation, in Speed’s sense, of BL. For the purposes of this note, a simple and very useful characterization of BL in the stochastic setting can be given in terms of uniform distribution modulo one. Recall that a random variable X is uniformly distributed modulo one, or u.d. mod 1 for short, if the fractional part X mod 1 :¼ X  bXc of X has the same distribution as U (0,1); here b x c denotes, for any real x, the largest integer not larger than x, and U (0,1) is a random variable uniformly distributed on (0,1). In these terms, the promised characterization of BL (see also Figure 2) is (1) A positive random variable X follows BL if and only if log10X is u.d. mod 1. Since Feller has inspired so many who teach probability and statistics today, and since many undergraduate courses 88

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now include a brief introduction to BL, it is not surprising that Feller’s derivation is still in frequent use to ‘‘provide some insight’’ about this phenomenon. For example, a class project report for a 2009 upper-division course in statistics at UC Berkeley [AP1, p.3] said, . . .like the birthday paradox, an explanation [of BL] occurs quickly to those with appropriate mathematical background . . . To a mathematical statistician, Feller’s paragraph says all there is to say . . . Feller’s derivation has been common knowledge in the academic community throughout the last 40 years. The online database [BH] lists about twenty published references since 2000 alone to Feller’s argument (e.g., [AP1, Few]) the crux of which is Feller’s claim (trivially edited) that (2) If the spread of a random variable X is very large, then log10 X will be approximately u.d. mod 1. The implication of (1) and (2) is that all random variables with large spread will approximately follow BL. That sounds quite plausible, but true to C.S. Peirce’s observation, even Feller blundered on Benford’s Law, and he took many other experts with him. Claim (2) is simply false under any reasonable definition of ‘‘spread’’ and any reasonable measure of dispersion, including range, interquartile range, standard deviation and mean difference, no matter how smooth or level a density the random variable X may have. To see this, one does not have to look far. Concretely, no positive uniformly distributed random variable even comes close to following BL, regardless of how large (or small) its spread is. This statement can be quantified explicitly via the following new inequality which is stated in terms of the so-called the Kolmogorov-Smirnov distance dKS ðX; Y Þ between two random variables X and Y, defined as dKS ðX; Y Þ ¼ supx2R PðX  xÞ  PðY  xÞ .

P ROPOSITION 1 ([BER]) For every positive uniformly distributed random variable X, dKS ð log10 X mod 1; U ð0; 1ÞÞ  ¼ 0:1334. . .;

9 þ ln 10 þ 9 ln 9  9 ln ln 10 18 ln 10

and this bound is sharp. There is nothing special about the use of the Kolmogorov-Smirnov distance or of decimal base in this regard; similar universal bounds hold for the Wasserstein distance, for example, and other bases. Likewise, there is nothing special about the choice of the uniform distribution as a source of counterexamples here and below; its usage is solely motivated by the simplicity of the uniform distribution and its role in many applications. For example, if Xa is exponentially distributed with mean a then  X k k PðD1 ðXa Þ ¼ 1Þ ¼ e 10  e 210 k2Z

¼ 0:3296. . . [ log10 2; whenever a is an integer power of 10. Thus in this case as well, it follows immediately from (1) that Xa is not

Figure 2. Uniform (left column) and exponential (center column) random variables do not follow BL as log10X is not uniformly distributed modulo one, see bottom row. However, note that in the exponential case the deviation from BL is quite small.

approaching BL, even though the spread (range, interquartile range, standard deviation, mean difference, etc.) of Xa goes to infinity as a ? ?. How could Feller’s error have persisted in the academic community, among students and experts alike, for over 40 years? Part of the reason, as one colleague put it, is simply that ‘‘Feller, after all, is Feller’’, and Feller’s word on probability has just been taken as gospel. Another reason for the long-lived propagation of the error has apparently been the confusion of (2) with the similar claim (3) If the spread of a random variable X is very large, then X will be approximately u.d. mod 1. For example, [AP1, p.3] cites Feller’s claim (2), but on p. 8 the same article states Feller’s claim as (3). A third possible explanation for the persistence of the error is the common assumption that (3) implies (2). For example, [GD, p.1] states: An elementary new explanation has recently been published, based on the fact that any X whose distribution is ‘‘smooth’’ and ‘‘scattered’’ enough is Benford. The scattering and smoothness of usual data ensures

that log (X) is itself smooth and scattered, which in turn implies the Benford characteristic of X. Now (3) is also intuitive and plausible, but unlike (2), it is often accurate if the distribution is fairly uniform. And if the distribution is not fairly uniform, then without further information, no interesting conclusions at all can be made about the significant digits: most of the values could for instance start with a ‘‘7’’. Now it seems obvious that X has very, very large spread if and only if log X has very large spread, so on the surface (2) and (3) appear to be equivalent. After all, what difference can one tiny extra ‘‘very’’ make? But the obvious again is simply false, as can easily be seen, for instance, when X has a Pareto distribution with parameter 2, that is, PðX [ xÞ ¼ x 2 for all x  1. Then X has infinite variance, whereas the variance of log10X equals 14 ðlog10 eÞ2 and hence is less than 0.05. Thus (2) and (3) are not at all equivalent, and (2) is false under practically any interpretation of ‘‘spread’’. Although (3) is perhaps more accurate than (2), unfortunately it does nothing to explain BL, for the criterion in (1) says that X follows BL if and only if the logarithm of X—and not X itself—is uniformly distributed modulo one. Some authors partially explain the ubiquity

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of BL based on an assumption of a ‘‘large spread on a logarithmic scale’’ (e.g., [AP1, Few, W]), and some, when confronted with the evidence that (2) is false, claim that ‘‘what Feller obviously meant’’ [AP2, italics in original] by spread was log spread, i.e., that when Feller wrote (2) he really meant to say that (4) If log10 X has very large spread, then log10 X will be approximately u.d. mod 1, which is but an unnecessarily convoluted version of (3). They then apply (3) or (4) to conclude that if log10X has large spread, then X approximately follows BL. This avoids Feller’s error (2), but still leaves open the question of why it is reasonable to assume that the logarithm of the spread, as opposed to the spread itself—or, say, the log log spread— should be large. As seen above, those assumptions contain subtle differences, and lead to very different conclusions about the distributions of significant digits. Moreover, via (1) and (3), assuming large spread on a logarithmic scale is equivalent to assuming an approximate conformance with BL. Quite likely, Feller realized this, and in (2) specifically did not hypothesize that the log of the range was large. A related and apparently widespread misconception is that claim (2) or claim (3)—notwithstanding the incorrectness of the former—implies that a larger spread or log spread automatically means better conformance with BL. For example, [W] concludes that ‘‘datasets with large logarithmic spread will naturally follow the law, while datasets with small spread will not’’, and the Conclusion of the study [AP2, p.12] states, On a small stage (18 data-sets) we have checked a theoretical prediction. Not just the literal assertion of Benford’s law — that in a data-set with large spread on a logarithmic scale, the relative frequencies of leading digits will approximately follow the Benford distribution — but the rather more specific prediction that distance from Benford should decrease as that spread increases. In one sense it’s not surprising this works out. But it doesn’t. Distance from BL does not always decrease as the spread increases, regardless of whether the spread is measured on the original scale or on the logarithmic scale. A simple way to see this is as follows: Again, for simplicity, let Y be a random variable uniformly distributed on (0,1), and let X = 10Y and Z = 103Y/2. Then by (1), X follows BL exactly, since log10X = Y, while Z is not close to BL, for 3Y/2 mod 1 is not close to uniform on (0,1). Yet for any reasonable definition of spread, including all those mentioned earlier, the spread of Z is larger than the spread of X, and the spread of log10Z = 3Y/2 is larger than the spread of log10X = Y. Another way to see that the distance from BL does not decrease as the spread increases is contained in the proof of Proposition 1: For XT a random variable uniformly distributed on (0,T), it is shown there that the Kolmogorov-Smirnov distance between log10 XT mod 1 and U (0,1) is a continuous 1-periodic function of log10T. Moreover, when employing a logarithmic scale it is important to keep in mind that what is considered large generally depends on the base of the logarithm. For example, as noted earlier, if Y is uniformly distributed on 90

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(0,1) then X = 10Y is exactly Benford base 10, yet it is not Benford base 2 even though its spread on the log2-scale is log2 10& 3.322 times as large.

Conclusion Classroom experiments based on Feller’s derivation or on an assumption of large spread on a logarithmic scale (e.g., [AP1, AP2, Few, W]) should be used with caution. As alternative, a supplement or teachers might also ask students to compare the significant digits in the first 20-30 articles in tomorrow’s New York Times against BL, thereby testing real-life data against the explanation given in the main theorem in [H2], which, without any assumptions on magnitude of spread, shows that mixing data from different distributions in an unbiased manner leads to exact conformance with BL. Although some experts may still feel that ‘‘like the birthday paradox, there is a simple and standard explanation’’ for BL [AP2, p.6] and that this explanation ‘‘occurs quickly to those with appropriate mathematical background’’, there does not appear to be a simple derivation of BL that both offers a ‘‘correct explanation’’ [AP2, p.7] and satisfies Speed’s goal to provide insight. A broad and often ill-understood phenomenon need not always be reduced to a few theorems. Although many facets of BL now rest on solid ground, there is currently no unified approach that simultaneously explains its appearance in dynamical systems, number theory, statistics, and real-world data. In that sense, most experts seem to agree with [Few] that the ubiquity of BL, especially in real-life data, remains mysterious. ACKNOWLEDGMENT

The authors are grateful to Rachel Fewster, Kent Morrison, and Stan Wagon for excellent suggestions that helped to improve the exposition. REFERENCES

[AP1]

Aldous, D., and Phan, T. (2009), ‘‘When Can One Test an Explanation? Compare and Contrast Benford’s Law and the Fuzzy CLT’’, Class project report dated May 11, 2009, Statistics Department, UC Berkeley; accessed on May 14,

[AP2]

2010, at [BH]. Aldous, D., and Phan, T. (2010), ‘‘When Can One Test an Explanation? Compare and Contrast Benford’s Law and the Fuzzy CLT’’, Preprint dated Jan. 3, 2010, Statistics Department, UC Berkeley; accessed on May 14, 2010, at [BH].

[Ben]

Benford, F. (1938), ‘‘The Law of Anomalous Numbers’’, Proc.

[Ber]

Berger, A. (2010), ‘‘Large Spread Does Not Imply Benford’s

Amer. Philosophical Soc. 78, 551–572. Law’’, Preprint; accessed on May 14, 2010, at http://www. math.ualberta.ca/*aberger/Publications.html. [BBH] Berger, A., Bunimovich, L., and Hill, T.P. (2005), ‘‘Onedimensional Dynamical Systems and Benford’s Law’’, Trans. Amer. Math. Soc. 357, 197–219. [BH]

Berger, A., and Hill, T.P. (2009), Benford Online Bibliography;

[Fel]

Feller, W. (1966), An Introduction to Probability Theory and Its

accessed May 14, 2010, at http://www.benfordonline.net. Applications vol. 2, 2nd ed., J. Wiley, New York.

[Few]

Fewster, R. (2009), ‘‘A Simple Explanation of Benford’s Law’’,

[Ga]

American Statistician 63(1), 20–25. Gardner, M. (1959), ‘‘Mathematical Games: Problems involving questions of probability and ambiguity’’, Scientific

[N]

[P]

American 201, 174–182. [GD]

Newcomb, S. (1881), ‘‘Note on the Frequency of Use of the Different Digits in Natural Numbers’’, Amer. J. Math. 4(1), 39–40. Pinkham, R. (1961), ‘‘On the Distribution of First Significant Digits’’, Annals of Mathematical Statistics 32(4), 1223–1230.

Gauvrit, N., and Delahaye, J.P. (2009), ‘‘Loi de Benford ge´ne´rale’’, Mathe´matiques et sciences humaines 186, 5–15;

[R1]

accessed May 14, 2010, at http://msh.revues.org/document

[R2]

Raimi, R. (1985), ‘‘The First Digit Phenomenon Again’’, Proc.

Raimi, R. (1976), ‘‘The First Digit Problem’’, Amer. Mathematical Monthly 83(7), 521–538.

11034.html.

Amer. Philosophical Soc. 129, 211–219.

[H1]

Hill, T.P. (1995), ‘‘Base-Invariance Implies Benford’s Law’’, Proc. Amer. Math. Soc. 123(3), 887–895.

[S]

Speed, T. (2009), ‘‘You Want Proof?’’, Bull. Inst. Math. Statistics 38, p 11.

[H2]

Hill, T.P. (1995), ‘‘A Statistical Derivation of the Significant-

[W]

Wagon, S. (2010), ‘‘Benford’s Law and Data Spread’’;

[K]

Digit Law’’, Statistical Science 10(4), 354–363.

accessed May 14, 2010, at http://demonstrations.wolfram.

Knuth, D. (1997), The Art of Computer Programming,

com/BenfordsLawAndDataSpread.

pp. 253-264, vol. 2, 3rd ed, Addison-Wesley, Reading, MA.

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

91

Associative Binary Operations and the Pythagorean Theorem DENIS BELL

n a recent article [2], L. Berrone presented a new approach to the Pythagorean Theorem (PT). The idea is to derive the geometric theorem from analytic and algebraic properties, by methods of functional equations. (So we are not dealing with a method that was an option for the ancients!) I thought about Berrone’s ideas, within his context of functional equations. Some pleasant surprises fell out. Then a surprising gift of functional equations back to geometry closed the circle for me – and will close this article. Let a  b denote the hypotenuse of a right triangle with legs a and b. The operation  : ð0; 1Þ2 7!ð0; 1Þ is evidently continuous. It is also (a) Homogeneous (of degree one):

I

ðkxÞ  ðkyÞ ¼ kðx  yÞ;

8k [ 0:

ð1Þ

(b) Reducible: either of the equations a  x ¼ a  y; x  a ¼ y  a implies x = y. (c) Associative. The homogeneity condition states that if the two legs of a right triangle are scaled by a common factor, then the hypotenuse is scaled by the same factor. This is proved in Book VI of Euclid’s Elements. The reducibility of  is easy to prove within the framework of Euclidean geometry. Associativity will be discussed later.

1

This theorem implies that  admits a representation x  y ¼ f 1 ð f ðxÞ þ f ðyÞÞ:

92

THE MATHEMATICAL INTELLIGENCER Ó 2010 Springer Science+Business Media, LLC

Berrone proves the following result in [2]. His argument is based on a deep theorem of J. Acze´l [1, page 256]1

T HEOREM 1 Suppose  is a continuous binary operation on (0,?) that satisfies conditions (a)-(c) above. Then there exists p such that x  y ¼ ðx p þ y p Þ1=p :

ð2Þ

The construction in Figure 1 shows that, for the particular operation  in the Pythagorean case, ð1  1Þ2 ¼ 1  1  1  1 ¼ 2: Hence p = 2 in the representation (2), and the Pythagorean Theorem follows. I was intrigued by the way Berrone’s work brings together the ancient subject of Euclidean geometry and the very different tradition of functional equations. At the same time I was disappointed to see a sophisticated result in the latter area (Acze´l’s theorem) invoked to prove a basic result in the former. It ought to be possible to provide a direct and selfcontained proof of Theorem 1, and so to place Berrone’s approach to PT on an elementary basis. And so it turned out. This search led me to a more general study of associative, homogeneous, binary operations. I will sketch a few of these by-products within the field of functional

P ROOF OF T HEOREM 1 Assume (3) holds (the argument will work equally well under (4)). Consider the function f : N 7! R defined by 1 o1

1o1

f ðnÞ ¼ 1  1      1;

1 1

where  is applied n - 1 times. Then f is strictly increasing by (3) and satisfies

1 1o1 o1o1

f ðnÞ  f ðmÞ ¼ f ðn þ mÞ:

ð6Þ

Furthermore, (1) yields

Figure 1. Evaluation of .

equations. Theorem 2 below is an alternative to Theorem 1 in which continuity and reducibility are replaced by a monotonicity hypothesis that appeared in an earlier work of Bohnenblust [3]. In Theorem 3, I study monotonic binary operations satisfying the condition 1  1 ¼ 1 and characterize these operations according to four possible ‘‘boundary conditions’’ that they can have. But this has left a gap. In order to prove PT from Theorem 1, it is necessary to demonstrate the associativity of the Pythagorean operation. This question, which could not have occurred to Pythagoras or Euclid, is addressed at the end of the paper in terms they would have appreciated. First let me prove Theorem 1. This requires the following preliminary result.

f ðnmÞ ¼ f ðnÞ      f ðnÞ ¼ f ðnÞð1  1      1Þ ¼ f ðnÞf ðmÞ:

f ðn=mÞ ¼

8x [ 0;

ð3Þ

x1\x;

8x [ 0:

ð4Þ

PROOF. We can argue by contradiction that 1  1 6¼ 1. If

¼

1 ½ f ðaÞf ðdÞ  ½ f ðbÞf ðcÞ f ðbÞf ðdÞ

¼

1 f ðadÞ  f ðbcÞ f ðbdÞ

¼

a c  f ðad þ bcÞ ¼f þ ; f ðbdÞ b d

this is false, then (1) yields x  x ¼ xð1  1Þ ¼ x; 8x: In particular, 1  1  2 ¼ 1  2  2: Since  is reducible, we can cancel 1 on the left and 2 on the right to obtain 1 = 2, an absurd conclusion. Hence 1  1 6¼ 1 as claimed. It follows that there exists no a such that a  1 ¼ a, as this would imply a  1  1 ¼ a  1 ) 1  1 ¼ 1: Because the function x 7! x  1 is continuous, the Intermediate-Value Theorem implies that either (3) or (4) holds.

......................................................................... was born in London and earned his doctorate from the University of Warwick. His area of research ordinarily is stochastic analysis. Aside from mathematics, his occupations are spending time with his family, listening to music, and surfing – the web, that is.

DENIS BELL

Department of Mathematics University of North Florida Jacksonville, FL 32224 USA e-mail: [email protected]

f ðnÞ f ðmÞ

ð8Þ

and observe that f is well-defined by (7). Then (6) and (7) extend to Q. Indeed, the extension of (7) is immediate. Using (6)–(8), we have a  c  f ðaÞ  f ðcÞ  f ðaÞf ðdÞ  f ðbÞf ðcÞ   ¼  f f ¼ b d f ðbÞ f ðdÞ f ðbÞf ðdÞ f ðbÞf ðdÞ

following two conditions holds: x1 [ x;

ð7Þ

I will show how to extend the domain of f first to the set of rational numbers, then to the reals, in such a way that the above properties continue to hold. Set

L EMMA Under the hypotheses of Theorem 1, one of the

AUTHOR

ð5Þ

so (6) also holds at rational points. The function f is increasing on Q, since a/b \ c/d implies f ðaÞf ðdÞ ¼ f ðadÞ\f ðbcÞ ¼ f ðbÞf ðcÞ; hence f

a b

¼

c f ðaÞ f ðcÞ : \ ¼f f ðbÞ f ðdÞ d

To extend f to (0,?), define f ðxÞ ¼ supf f ðrÞ : r 2 Q; r  xg: Then f has the multiplicative property f ðxÞf ðyÞ ¼ f ðxyÞ;

8x; y [ 0:

ð9Þ

To see this, choose sequences of rationals rn " x and sn " y such that f ðxÞ ¼ lim f ðrn Þ and f ðyÞ ¼ lim f ðsn Þ. Then f (xy) C f (rnsn) = f (rn)f (sn)?f (x)f (y), thus f (xy) C f (x)f (y). Conversely, let {tn} be a sequence of rationals such that tn " xy and f ðxyÞ ¼ lim f ðtn Þ. Writing tn = rnsn where rn and sn are rationals with rn \ x and sn \ y, we have f (x)f (y) C f (rn)f (sn) = f (tn)?f (xy), implying f (x)f (y) C f (xy). Thus (9) holds. A similar argument shows that f is everywhere nondecreasing. Now it is well-known that condition (9) implies one of the following two situations: either f is a power function x ! 7 x r or f is everywhere discontinuous. The latter case can be ruled out because a monotone function Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

93

cannot be discontinuous on an uncountable set of points. It follows that f (x) = xr for some r [ 0, hence x r  y r ¼ ðx þ yÞr ;

8x; y 2 Q:

ð10Þ

The continuity of  now implies that (10) holds for all x and y. Replacing xr by x and yr by y in (10), yields (2) with p = 1/r. A minor modification of the proof of Theorem 1 (the details will be omitted) gives the same conclusion under alternative hypotheses.

T HEOREM 2 Suppose  is associative, homogeneous, and satisfies the monotonicity condition x  a and

y  b ) x  y  a  b:

1 1 1 b  1   1  [ 0:   1 bn a ba a a

This contradicts the fact that limn#0 1  n ¼ 1  0 ¼ 0: A similar argument shows that 1  y  1; 8y: Since 1  1 ¼ 1 and  is monotone, this implies 1  n ¼ 1 ¼ n  1; 8n  1: It follows that if x B y then  y ¼x xy ¼ x 1 x and if y B x then  xy ¼ y

x 1 y

 ¼ y:

Hence x  y ¼ minfx; yg as claimed. Consider the dual operation  defined by   1 1 1 :  xy ¼ x y

p 1=p

x  y ¼ ðx þ y Þ where p ¼

1

ð11Þ

Suppose further that 1  1 6¼ 1. Then p

Iterating (12), starting with x = a and successively replacing x by bx yields, for all n C 1

1 log2 ð1  1Þ :

Now this result was proved by Bohnenblust [3] in 1940 under the extra assumption that  is commutative. This is just the point that made Berrone’s approach appealing to me initially: Theorem 1 or Theorem 2 seems to work from hypotheses not related to commutativity and end with a conclusion that the relation is commutative. Obviously, the condition 1  1 6¼ 1 is necessary for the conclusion of Theorem 2. What can be said if this condition fails? Let me now address this question. Assume as in Theorem 2 that  is associative and homogeneous, and satisfies (11). It may be extended by standard arguments to [0, ?) 9 [0, ?) preserving these properties, and I will state the answer to the question for the extended operation. The homogeneity of  yields ð1  0Þ2 ¼ 1  0  0 ¼ 1  0: Hence 10 is either 0 or 1, and similarly for 01. So we are down to four cases.

T HEOREM 3 Suppose  is associative, homogeneous, monotonic (11), and idempotent 1  1 ¼ 1. (a) If 1  0 ¼ 1 ¼ 0  1, then x  y ¼ maxfx; y g.2 (b) If 1  0 ¼ 1 and 0  1 ¼ 0, then x  y ¼ x. (c) If 1  0 ¼ 0 and 0  1 ¼ 1, then x  y ¼ y. (d) If 1  0 ¼ 0 ¼ 0  1, then x  y ¼ minfx; y g.

Then it is easy to show that  satisfies all the hypotheses imposed on  in the previous theorems with the exception of its values at the points (1,0) and (0,1). The roles of 0 and 1 at these points are interchanged in passing to the dual, that is 1  0 ¼ 1 () 0  1 ¼ 0; 0  1 ¼ 1 () 1  0 ¼ 0: This observation yields an alternative proof of part (d) of Theorem 3 since, if  satisfies the hypotheses of (d) then  satisfies the hypotheses of (a). Hence x  y = max{x, y} and this implies x  y ¼ minfx; yg. Another side remark: The example  0; x 6¼ y xy ¼ x; x ¼ y which is associative and homogeneous, shows that the monotonicity assumption in Theorem 3 is necessary. I conclude with the problem raised earlier concerning the applicability of these results to Euclidean geometry. Let ab denote the hypotenuse of a right triangle with legs a and b. Recall that in order to prove the Pythagorean Theorem via Theorem 1, one has to know that  is

D

B

The proofs of (a) - (c) are simple and will be omitted. The proof of (d) is as follows. I first prove by contradiction that the function x 7! x  1 is bounded above by 1. Suppose that a  1 ¼ b [ 1 for some a [ 0. Then x  1  b;

2

This case was established in [3].

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THE MATHEMATICAL INTELLIGENCER

z

(x o y) o z

x o (y o z)

8x  a:

yoz

Dividing through by x, composing on the left-hand side with 1, and using (1), (11) and the condition 1  1 ¼ 1 gives 1 b 1 1 ; x x

x

8x  a:

ð12Þ

xoy A

y

Figure 2. Plane thinking.

x C

y

z

z

C

F

y B

l

xo y

E

x

immediately that l ¼ ðx  yÞ  z ¼ x  ðy  zÞ, establishing the associativity of . (This ‘‘box’’ proof of associativity is also given in [2].) It appears that an excursion to 3-dimensional space is required to verify a 2-dimensional law! Another example of this phenomenon is the 3-dimensional proof of Desargues’s Theorem in the plane [4]. Are there other instances where the proof of a geometrical theorem requires a construction in higher dimensions? It would be worthwhile to list some and examine their independence.

yo z

A

D

Figure 3. Thinking inside the box.

REFERENCES

[1] J. Acze´l, Lectures on Functional Equations and their Applications, Academic Press, New York, 1966. [2] L. Berrone, ‘‘The Associativity of the Pythagorean Law’’, Amer-

associative. I have not been able to find an independent proof of this fact via planar figures. One would need to show that the lengths AB and CD in Figure 2 coincide, and it is not clear how to do this without using PT. However, consider the 3-dimensional rectangular box depicted in Figure 3. Since the diagonal AF is the hypotenuse of both the right triangles ACF and AEF, we see

ican Mathematical Monthly 116 (2010), 936-939. [3] F. Bohnenblust, ‘‘An Axiomatic Characterization of Lp-spaces’’. Duke Math. J. 6 (1940), 627–640. [4] ‘‘Desargues’ Theorem’’, Wikipedia article http://en.wikipedia. org/wiki/Desargues’_theorem

Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Review

Osmo Pekonen, Editor

A Disappearing Number by Complicite LONDON: OBERON BOOKS, 2008, 93 PP., £9.99 ISBN: 978-1-84002-830-0 REVIEW OF PERFORMANCE AT LINCOLN CENTER, NEW YORK CITY, 17 JULY 2010, BY MARY W. GRAY

A Feel like writing a review for The Mathematical Intelligence? 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.

â Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected] 96

THE MATHEMATICAL INTELLIGENCER  2011 Springer Science+Business Media, LLC

woman walks out on stage and writes numbers on the board, starting with 1, 2, 3, 4, 5

and moving to sequences of even numbers, and then powers of 2. But soon she moves to the series that was important in engaging the attention of G.H. Hardy when he received an unexpected communication from Ramanujan one morning in 1913: 1 þ 2 þ 3 þ 4 þ 5 þ . . . ¼ 1=12 The mathematician Ruth begins an explanation, but with the expression ‘‘Functional Equation of the Riemann Zeta Function’’ one can feel the unease of the audience, anticipating having to follow the mathematics that is appearing. Not to worry, as that is about the last we see of real mathematics: as another character enters, gaining the audience’s sympathy with ‘‘When I was a child I had a real problem with mathematics…’’ It seems the problem was overcome as he introduces himself as Aninda Rao, string-theory physicist. With a quick explanation of some of the temporal shifts of the play, he then gives a nod to any mathematicians who might be there and an alert to others, ‘‘Mathematics is not a spectator sport.’’ Aninda moves from his location at CERN to

Chennai (Madras), but once we are there the scene shifts to Ramanujan, reciting the result above. The choice of this formula as a catalyst for the exploration of the relationship between Ramanujan and Hardy, as well as of the nature of mathematics, was wise, as the notation is familiar but the result is startling. It also appears to epitomize the collaborators’ dedication to ‘‘pure’’ mathematics. Hardy and probably Ramanujan would have been horrified at Aninda’s enthusiasm over the applicability of modular functions to string theory and the physical world. A Disappearing Number tells the interwoven stories of Ruth and Ramanujan. Ruth’s lover, and subsequently her husband, Al, a futures trader who disclaims all knowledge of or interest in math, wanders into her lecture. However, he confesses to a fascination with infinity—to which Ruth replies, ‘‘It’s just another mathematical concept. No big deal.’’ Although most mathematicians might reply similarly, infinity is the mathematical concept that perhaps most fascinates nonmathematicians, probably including many in the audience. Some might view the play as essentially about infinity and the associations it generates in a variety of characters, contemporary and historical, imaginary and real. The play opened in England in 2007, traveled to Germany, Austria, France, Spain, and Australia before the production at the Lincoln Center Festival in New York City that is reviewed here. Subsequently the play was seen at the 2010 International Congress of Mathematicians in Hyderabad before returning for a revival in London; a highdefinition screening of a live performance has also been seen in a number of theaters worldwide. The Complicite London company, led by Simon McBurney, who directs A Disappearing Number, is noted for exciting contemporary productions of classics as well as creative new work for stage, radio, and film. The combination of video, movement, sound design, music, and text leads to deliberately ambiguous time and space structures. We see Hardy and Ramanujan in Cambridge and encounter snippets of Hardy’s view on mathematics and on war as well as his concern for Ramanujan’s deteriorating health. We experience the grief of Ramanujan’s wife and mother and the tragedy of Ruth’s death on a train as she travels from Chennai on a pilgrimage to Ramanujan’s origins, a journey later made by Al to scatter her ashes. Scenes shift from a Brunel University lecture room, to Harvard, to the Ramada Inn at Heathrow, then to a plane, a taxi, the train. From time to time Al copes with the BT (British Telecom) call centre in Bangalore in an attempt to transfer to his name Ruth’s phone number—ending in 1729, the legendary ‘‘dull’’ number of Hardy, but one of interest to Ramanujan [1]. One critic saw the play as ‘‘numerology as a means of giving substance to the unknown’’—meaning to praise it. More insightful reviewers have claimed to see analogies of love, death, and belonging that do not depend on an understanding of the work of Ramanujan and Hardy. For the most part the scenes that engage real people are quite accurate, or at least in agreement with most references [2], although the river Cauvery gets misplaced. Aside from the key equation, we do hear a bit about partitions, with an explanation that is understandable but that must leave audiences wondering about the significance. The schoolboy

tale of Ramanujan’s inquiry as to whether 0/0 must equal 1, is told. The characters are quite convincing, except possibly for Ramanujan. There is little enough known of him, outside of his work, to say what is lacking in the play’s portrayal, but perhaps it is just that his is really a minor role. Some of the more dramatic aspects of his life are glossed over; for example, the attempted suicide consists merely of the noise of a train. It must be said that the fortuitous nature of the onset and decline of the Ramanujan-Hardy collaboration as well as its extraordinary products defy detailed exposition or analysis in a ninety-minute play. However, it is difficult to imagine a more intriguing and absorbing drama with real mathematics at its center. A Disappearing Number is a truly creative effort to confront a general audience with nontrivial mathematics and the saga of Ramanujan. The Oxford mathematician Marcus du Sautoy was a consultant to the company; his essay introduces the print version of the text and the theater program for the Lincoln Center production. In noting the mock theta function of Ramanujan’s last letter to Hardy, he remarks: ‘‘It is striking in a world dominated by men that a woman [Kathrin Bringmann] has been a key character in illuminating Ramanujan’s work. Complicite’s production also places a woman at the centre of its fascinating mathematical story.’’ It is certainly true that when we see Ruth writing on the board we know that we are in the very near past, not in the 19131919 period of the Hardy-Ramanujan collaboration. The contrast of Hardy’s demand for rigor and Ramanujan’s amazing intuition, which can be seen in their exploration of the nature of proof, might be considered emblematic of a tension between east and west. Marcus du Sautoy reminds us that in an era dominated by the European tradition, Indian mathematics centuries ago investigated the concept of zero and dealt with infinite sums, a heritage expressed in part in the play by the ancestral pride in Aninda’s speaking of the origin of zero. What the play does do, in an understated way, is to contrast present day high-tech India with ancient traditions. Mathematicians, both present-day imaginary and past real, fare better in A Disappearing Number than in most dramatizations of them and their craft. All too often in drama, being a mathematician is a token for strangeness bordering on the bizarre, alienation, or at best inexplicable genius [3, 4, 5, 6, 7]. Rarely do we get any hint of in what the brilliance consists—aside from references to proofs of Fermat’s Last Theorem or the Riemann hypothesis. Although the Poincare´ conjecture seems not yet to be the problem of choice, having seen A Beautiful Mind [4], one can imagine what dramatists might do with Perelman. The last time a drama opened with a woman mathematician at the board, Jill Clayburgh was proving the Snake Lemma in It’s My Turn [8]. There’s not much mathematics in the film beyond that, but it is fairly topical as she is described (by her father) as working on the classification of finite simple groups in the year (1980) usually cited as the date of the solution of the problem. In any case, her character is amazingly (for a mathematician in a film) normal. In Proof [9], of the three mathematicians in the drama, the woman comes off the best, although we never  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

97

learn what her great result is. In contrast her father is based loosely, but negatively, on John Nash, and her lover goes to great lengths to appropriate the family’s mathematical treasure. Although the mathematician heroine in Smilla’s Sense of Snow [10] does seem to carry the ‘‘loner’’ gene, her angry isolation appears to dissolve in the brief moments when she talks of mathematics, and her quest to solve a mystery of a small boy’s death is admirably persistent. Considering the subject of this review and these examples, one might think that women mathematicians, rare though they may be, are treated better in drama. That is, until one sees Agora [11], in which Hypatia, though perhaps heroic (and certainly beautiful), does not come off very well as a mathematician/philosopher; admittedly there is a charming bit when she ‘‘discovers’’ elliptical orbits. At least Agora is better than A Hill on the Dark Side of the Moon [12], from which one reviewer claimed that viewers would go away believing that for a woman (in this case Sonya Kovalevskaya) to be a mathematician she must be ugly, neurotic, and a bad mother. The movie simply ignores her genuine mathematical accomplishments. The mathematics of the most bizarre woman mathematician in fiction, Lisbeth Salander of The Girl Who Played with Fire [13] fame, apparently defied effective transfer to the screen, as nothing is seen in the film of her ‘‘proof’’ of Fermat’s Last Theorem (FLT) or the other mathematical preoccupations present in Stieg Larsson’s book on which the film was based. In A Disappearing Number, Ruth advises Al to read Hardy’s A Mathematician’s Apology [1], excellent advice for an understanding of the true nature and beauty of mathematics. But how to translate the elegance of Hardy’s prose to the stage or screen escapes most authors, who in fact rarely try, whether dealing with real or imaginary mathematicians. Hugh Whitemore’s conveying the character and work of Turing [14] is probably the most successful as a dramatization, whereas the documentary The Proof [15] is stunning in taking us along Wiles’s road to FLT. In A Disappearing Number, Ruth, though certainly not a genius, transmits the joy as well as the fascination and often consuming dedication found in mathematics. The innovative production style of A Disappearing Number certainly is more successful in portraying mathematicians and their work, however sketchily, than, for example, the screen version of The Oxford Murders [16], which is rife with stereotypes and boring—as well as often inaccurate—mathematics. In contrast, in A Disappearing Number mathematicians are real people with real emotions and at least some real mathematics. It may be that this decade’s relative profusion of mathrelated (no matter how poor) dramas might change the public’s view of mathematics. A 1990s episode of the widely distributed American television series Law and Order [17] sees Lenny, the iconic New York City cop, on viewing a murdered art teacher, remarking, ‘‘I could understand it if it were the algebra teacher.’’ In contrast, in a recent episode of Rizzoli & Isles [18], featuring a Boston cop and a medical examiner (both women), familiarity with eip þ 1 ¼ 0 is presented as evidence of intellectual sophistication. 98

THE MATHEMATICAL INTELLIGENCER

Simon McBurney confesses that he is scared of mathematics, but he likes to confront difficult subjects in the theatre. He says, ‘‘I wanted to create a show in which mathematics was absolutely at the center of it.’’ In that he succeeded, making mathematics less mysterious to many to whom it has always been challenging and often a subject to be avoided. It is believable that he came to love mathematics if not completely to understand it—perhaps Al is his avatar and Ruth is that of mathematics. Perhaps the play does not quite reach the level of perfection required by Hardy of mathematicians: ‘‘The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.’’ Nor for ugly dramas about its practitioners, but A Disappearing Number is definitely not ugly. REFERENCES

[1] G.H. Hardy, A Mathematician’s Apology, Cambridge: Cambridge University Press, 1967. [2] R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, New York: Scribner, 1991. [3] T. Koizumi and Y. Ogawa, The Professor and His Beloved Equation (Hakase no aishita suˆshiki), Tokyo: Asmik Ace Entertainment, 2006. [4] A. Goldsman and S. Nasar, A Beautiful Mind, University City: Universal Pictures, 2001. [5] M. Damon and B. Affleck, Good Will Hunting, Los Angeles: Miramax Films, 1998. [6] L. Piedrahita and R. Sopen˜a, Fermat’s Room (La Habitacio´n de Fermat), Barcelona: Notro Films, 2007. [7] N. Falacci and C. Heuton, Numb3rs, Scott Free Productions, CBS Television, 2004. [8] E. Bergstein, It’s My Turn, Burbank: Columbia Pictures, 1980. [9] D. Auburn and R. Miller, Proof, Los Angeles: Miramax Films, 2005. [10] P. Høeg and Ann Biderman, Smilla’s Sense of Snow, Los Angeles: 20th Century Fox, 1997. [11] A. Amena´bar and M. Gil, Agora, Barcelona: Mod Producciones, 2009. [12] S. Pleijel, A Hill on the Dark Side of the Moon, Stockholm: Moviemakers Sweden, 1983. [13] J. Frykberg and Stieg Larsson, The Girl Who Played with Fire (Flickan som lekte med elden), Stockholm: Yellow Bird Films, 2010. [14] H. Whitemore, Breaking the Code, Charlbury, UK: Amber Lane Press Ltd., 1987. [15] S. Singh, The Proof, Boston: WGBH PBS, 1997. [16] J. Guerricaechevaria, A. de la Iglesia, and G. Martinez, The Oxford Murders, Strasbourg: Eurimages, 2008. [17] Dick Wolf, Law and Order, NBC Television, 1990-2010. [18] Tess Gerritsen, Rizzoli & Isles, Warner Horizon Television, 2010.

American University Washington, DC USA e-mail: [email protected]

Osmo Pekonen, Editor

The World of Maria Gaetana Agnesi, Mathematician of God by Massimo Mazzotti BALTIMORE: JOHNS HOPKINS UNIVERSITY PRESS, 2007, 217 PP., US $51.95, ISBN-10: 0801887097, ISBN-13: 978-0801887093 REVIEWED BY J. B. SHANK

magine the practice of mathematics if there were no Fields Medal and no International Mathematical Union. Imagine it also in a world where university mathematics professors taught the ancient quadrivium (arithmetic, geometry, astronomy, and music), and neither possessed nor supervised post-baccalaureate degrees in mathematics. Further imagine the pursuit of mathematics in an environment lacking specialized mathematical journals and the broadly articulated and accepted research agendas characteristic of professionalized mathematical research. What would it mean to be a mathematician and to seriously pursue mathematical research in this environment? This is the large question that Massimo Mazzotti’s beautifully wrought study of Maria Gaetana Agnesi (1718-1799) forces us to confront. Agnesi was an eighteenth-century mathematician and the author of a treatise on the differential and integral calculus (Instituzioni analitiche ad uso della gioventu` italiana, Milan, 1748) that was, for a brief time, a leading introduction to the topic available in both its original Italian and French and English translations. She was also a Milanese aristocrat devoted to the social life of her native city, especially its unique traditions of Catholicism. Italy became famous in the eighteenth century for its learned women, but unlike Laura Bassi, who became a university professor of mathematics in the eighteenth century, and Countess Clelia Borromeo del Grillo, who established her own scientific academy, Agnesi remained essentially an ‘‘amateur’’ mathematician, pursuing her research in conjunction with her ordinary family and social duties. As a woman, Agnesi certainly faced difficult barriers of entry into the male dominated proto-professional scientific networks of the day, but unlike Bassi, or Emilie de Breteuil Marquise du Chaˆtelet, the French translator of Newton’s Principia and the author of a 1740 treatise on physics that Agnesi owned, Maria Gaetana evinced little interest in becoming a publicly acknowledged member of the eighteenth-century scientific community. Her mathematical career, therefore, must be understood in terms of these personal motivations and the largely local historical conditions that generated them. Since, as Mazzotti writes, Agnesi’s mathematical achievements have been deemed ‘‘of little significance within the greater genealogy of modern mathematics,’’ and since ‘‘we do not associate any particular theorem or conceptual advance with her name, only a rather useless curve,’’ modern historians have generally found very little in Agnesi’s career to attract their attention. Few historical accounts of her life or work exist, therefore, and what little biography has been

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written either highlights her identity as an early pioneer for women in mathematics, or celebrates her reputation as a saintly woman who devoted herself to the care of the sick and the poor. She was indeed a pioneering female mathematician in an overwhelmingly male profession, and she was also a devout Catholic who found in eighteenth-century mathematical science a means for pursuing, rather than escaping from, the religious calling that was her ultimate devotion. So who exactly was Maria Gaetana Agnesi, and what kind of history of mathematics can make sense of, and render credit to, her specific scientific pursuits? Mazzotti notes how her idiosyncratic biography tends to make Agnesi appear to moderns as either an ‘‘enigma’’ or a ‘‘curiosity,’’ and the great virtue of his biography is the way that he makes interpretive sense of this curious enigma by reconstructing the precise historical contingencies that produced her life and work. To accomplish this laudable goal Mazzotti makes several historiographical moves that may strike some observers as heterodox. Most glaring to readers of this magazine will be his decision to include so little recognizable mathematical content in his life of Agnesi the mathematician. The early chapters of the book offer a thick description of the social and intellectual milieu that produced Agnesi’s life and work, and the precise mathematical ideas that this environment fostered are then discussed in the book’s penultimate chapters. But even here, the text eschews the extensive and technically elaborated style of mathematical analysis characteristic of much writing in the history of mathematics. How can such a result be called a history of Agnesi’s mathematics and not simply a biography of the person who authored this work? By recognizing the more corporeal and human approach to the history of mathematics that Mazzotti makes the methodological centerpiece of his study. Mathematics is still too often conceived as an otherworldly domain, and when its history is written from this supernatural perspective it becomes an overly ethereal affair, with disembodied ideas begetting disembodied ideas and the contingent factors that actually lead flesh and blood mathematicians to produce their work being pushed to the margins as arbitrary or anecdotal decorations adding narrative drama to an otherwise impersonal story of inexorable mathematical progress. Mazzotti pursues a diametrically opposed historiographical agenda in his book. He sets as his task the project of reconstituting from a very scarce set of archival traces the living fabric that gave birth to Agnesi and her work. Situating her mathematics in relation to her multidimensional social milieu, he also sets out to understand her science as an important, if in no way overdetermined, product of this same historical environment. Mazzotti spent his childhood in Milan, and as he tells us he first learned of Agnesi as a boy when he encountered her ‘‘saintly portrait on the cover of dusty booklets’’ in the paleoChristian Basilica of San Nazaro ‘‘while running up and down the nave under my grandmother’s excessively tolerant gaze.’’ He accordingly has a native’s feel for the neighborhoods, architecture, and culture of the city that dominated every aspect of Agnesi’s life, and from this perspective, and the few documents that survive, he resurrects the Milanese world that gave birth to Agnesi and her mathematics. However, since Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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Mazzotti is also a very broadly trained and learned historian of eighteenth-century science, his study also situates this local case within the general European trends that give it broader significance. The overall result is micro-history at its best, and a history of mathematics that is narrated, as it always should be, through the broader history of the people and places that made this particular science what it is. What do we learn about Agnesi and eighteenth-century mathematics as a result of Mazzotti’s approach? Many of the most illuminating insights center on the social dimension of mathematics as an eighteenth-century learned pursuit. When Agnesi began her mature work around 1740, the modern professionalized practice of mathematical research was in its barest infancy. The networks of experts and expertise that would give birth to the modern mathematical profession were beginning to take shape, developing their specialized methods and practices from the older correspondence networks of the Republic of Letters. Pressures toward scientific specialization and professionalization were also growing as a result of the new academies and state-based scientific institutions that were emerging. But eighteenth-century mathematics and mathematicians were not motivated by anything like the intellectual dynamics of our own time. The history of the differential and integral calculus, Agnesi’s precise mathematical interest, illustrates this point. After its development, largely simultaneously and independently, by Isaac Newton and Gottfried Leibniz around 1680, the calculus was disseminated in a manner characteristic of mathematics overall in early modern Europe. While academies, universities, and the new learned journals of the late seventeenth century played a key role in this development and dissemination, clerics, theology, nonspecialized savants, and the broader world of early modern society played a key role as well. Newton’s work stemmed from his broader scientific research around Cambridge University and his participation in the activities of the newly founded (1661) Royal Society of London. Yet one should not view Newton’s position as already modern and professionally scientific. Clerics of various sorts played a powerful role in each space, for example, and had Newton not successfully manipulated events to avoid this outcome, he would have become an ordained priest of the Anglican church as all chaired professors at Cambridge were required to be. Leibniz was never affiliated with a university and supported his scientific work through service as the court savant to the Duke of LunenbergBrunswick in Saxony. This different social position meant that he secured his broader public reputation as a mathematician by strategically using the networks of the Republic of Letters. For this reason, among others, Leibniz chose to publish his discovery of the calculus in 1684, using the best publication vehicle available to him, a twenty-year-old Latin periodical edited out of Leipzig, the Acta Eruditorum. Yet lest one confuse this with a modern scientific publication, it is worth noting that the same issue that included Leibniz’s method for determining differentials also included articles assessing a leading theological controversy and another evaluating a new study of Greco-Roman antiquities. Newton, believing that his calculus was only useful as a private research tool, and being little inclined toward 100

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mathematical publication in any case, kept his calculus out of the public eye. Yet his university position at Cambridge placed no ‘‘publish or perish’’ imperatives upon him, so his preference not to publish generated no great problem for him. Leibniz, by contrast, actively promoted the dissemination of what he perceived to be his innovative new mathematics, and Johann Bernoulli, a little known physician and university professor in Basel at the time, joined with him soon after in the same campaign. He used the new mathematics to win a name for himself as an esteemed mathematician in the international Republic of Letters. When, in 1691, Bernoulli traveled to Paris and visited with a group of distinguished elites, he was also able to advance his program of mathematical advocacy in a new and influential terrain. Bernoulli’s hosts in France included the Oratorian priest Father Nicolas Malebranche, and the blueblooded aristocrat, the Marquis de l’Hoˆpital. L’Hoˆpital had shocked his peers months earlier by resigning his family’s ancient command in the king’s armies in order to pursue, of all things, advanced work in mathematics. Bernoulli helped Malebranche and l’Hoˆpital to see the value of the new calculus, and its dissemination into France was thus initiated. L’Hoˆpital was made a member of the twenty-five-year-old Acade´mie Royale des Sciences a year after his meeting with Bernoulli, and since the previous summer’s encounter had also produced a kind of patron-client relationship between the two men whereby l’Hoˆpital paid Bernoulli an annual salary of 300 livres (about half the yearly earnings of a professor at the French Colle`ge Royal) in exchange for Bernoulli’s mathematical knowledge, the differential and integral calculus became a part of French academic science, even if it often circulated under l’Hoˆpital’s name. The same relationship also supported l’Hoˆpital’s first ever 1696 textbook in the differential calculus, L’Analyse des infiniment petits pour l’intelligence des lignes courbes, a work that Bernoulli later claimed to have authored. Religious forces also shaped the early development and dissemination of the new calculus in France since Malebranche’s ties to the Oratorian order made his mathematical ideas a centerpiece of Oratorian theology and religious training. Malebranche also organized a circle of mathematicians devoted to the study and application of the new mathematics, and this circle included the royal academicians l’Hoˆpital and Pierre Varignon, and clerics and priests who taught the new mathematics at the many Oratorian colleges in France. By joining in both his life and his writings the secular science of the Royal Academy with the explicitly theological agendas of the Oratorian order, Malebranche further helped to give the new mathematics a broad and important circulation in France. Also instrumental in developing and demonstrating the new calculus was the broader learned public that was starting to take consciousness of itself as its own cultural authority through the practice of reading, letter writing, and urban sociability in salons, coffeehouses, and other urban venues. In France, the calculus became a participant in this new public culture right away, a fact that is illustrated by the 1698 appearance of an introductory text in algebraic analysis, including the rudiments of infinitesimal analysis, in the

primary organ of worldly sociability of the day, Le Mercure galant. The text was accompanied by an announcement that claimed this kind of mathematics to be ‘‘tre`s a` la mode.’’ Louis Carre´, an Oratorian cleric and Malebranche circle member who published the first ever textbook on the integral calculus in 1702, also confirmed the vogue for infinitesimal mathematics by cultivating what his eulogist called ‘‘a secret little empire of women’’ devoted to the study of the new mathematics. The same mathematics was also given even more notoriety through the alliance of one of the leading intellectual lights of the period with it: Bernard le Bovier de Fontenelle. Fontenelle had become famous in 1686 through his book Entretiens sur la pluralite´ des mondes habite´s, a worldly philosophical dialogue that populariuzed Cartesian cosmology through witty exchanges between an urbane learned man and an interested marquise. But he was also a serious student of the new calculus, participating in the Malebranche circle and penning the preface to L’Hoˆpital’s 1696 treatise on the differential calculus. In 1727 he also published his own massive treatise, E´le´ments de la ge´ome´trie de l’infini, devoted to the epistemological foundations of the new science. A reform of the Paris Academy in 1699 further catalyzed this dynamic mathematical reception since it expanded the membership of the Royal Academy, allowing Malebranche, Carre´ and other advocates of the calculus to join the company for the first time. The reform also created a new class of official foreign correspondents that made Newton, Bernoulli, and Leibniz official, if virtual, French academicians. A new public orientation for the academy was also instituted, with Fontenelle serving as the secretary in charge of an annual published volume highlighting the academy’s work and a twice-annual public assembly created where academic science was showcased. The new calculus was further disseminated through these public channels, especially when a set of controversies erupted after 1700 over the validity of the new mathematics. At issue were questions of rigor since the new calculus would not acquire a synthetic proof of its claims until the nineteenth century, and would therefore remain throughout the eighteenth century a powerful and innovative mathematical technique that nevertheless lacked demonstrative validity. For those who defined mathematics precisely in terms of its demonstrative rigor, that made the calculus a rogue intruder upon cherished disciplinary norms. Many spoke vociferously and publicly against the calculus on precisely these grounds, and especially aggressive in this anticalculus camp was the Society of Jesus, which was well represented in Paris and in other French cities as a result of its prestigious and influential network of colleges. As the leading educators of the period, the Jesuits and the Oratorians competed for French college students in early eighteenth-century France, and in 1702 the Society of Jesus further demonstrated its appreciation for the power of the wider public culture by founding its own learned periodical, Me´moires pour servir a` l’histoire des sciences et des beaux arts, more commonly known as the Journal de Trevoux. This journal was edited by the Jesuit professors at the Parisian Colle`ge Louis-le-Grand – Voltaire was one famous alumnus – and like the Acta Eruditorum, or its most direct rival in France, the Journal des savants, the journal addressed itself

to the larger community of the Republic of Letters and was not by any means a specialized theological journal. The socalled ‘‘Affaire des infiniment petits’’ that erupted in the first decade of the eighteenth century and pitted all of these different groups and interests in a public battle over the validity of the calculus illustrates the mathematical culture that this social world generated. The Jesuits had one member, Father Thomas Gouye, in the Royal Academy, and he fought with other academicians on the anticalculus side against Malebranche, his Oratorian allies, Fontenelle, L’Hoˆpital, Varignon, and others. Bernoulli and Leibniz joined in from afar, using journals such as the Journal des savants to air their views. The Jesuits likewise used their own Journal de Trevoux to advance their anticalculus positions. The result was a broadbased public battle that played out in letters, journal articles, academy sessions, textbooks, coffeehouses, religiously affiliated colleges, urbane salons, and everywhere in between. To be a serious mathematician in the eighteenth century meant practicing mathematics within this thoroughly early modern social world. Agnesi came to her mathematics through a similar combination of early modern institutional and social dynamics, and although she was born too late (1718) to be exposed directly to these early struggles over the new calculus, and was a resident of Milan, not Paris, her own path into serious mathematical work followed many of the same patterns illuminated previously. The key players in the French struggle, especially Malebranche and the Oratorians, were especially influential in shaping her work. Mazzoti’s account of Agnesi’s life and work is therefore illuminated through its relationship to this earlier French history. As the first child of an affluent member of the Milanese mercantile class, Agnesi was launched into her intellectual career by the social ambitions of her father, Pietro. Attuned to the new synergies, manifest in Parisian science, that tied public display and urbane sociability with cultural authority, and seeing in this alliance a springboard for his own ambitions to enter the nobility, Pietro began to use his large family (nine daughters and nine sons from three different wives) as instruments for social advancement. In particular, he began to host a series of intellectual soire´es in the 1720s designed to display his family’s culture and refinement to elite Milanese society. Mazzotti opens his book with an account of one of these conversazioni held on 16 July 1727. Two French sophisticates pursuing their Grand Tour in Europe were included among the guests, and together with the usual collection of aristocrats, clerics, city notables, and learned gentlemen and ladies, these French travelers were treated to a gala evening spent amidst the elegant people and artistic appointments (a work by Titian, some drawings by Leonardo) of the Palazzo Agnesi. As the travelers noted in their diaries (the precious source for this history), the evening centered on a variety of entertainments, including musical performances by Maria Gaetana’s two sisters and a series of Latin orations and debates on questions of natural philosophy by Gaetana herself. Agnesi was all of nine years old at the time of this performance. These conversazioni, as Mazzotti shows, were Agnesi’s point of entry into the scientific world of eighteenth-century Europe. Since Pietro was convinced that performances Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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such as these created a golden ticket redeemable for the aristocratic standing he most desired, he spared no expense in preparing for his soire´es. His preparations included providing his ‘‘talent’’ (Maria Gaetana and her sisters) with the rigorous training they needed to perform their assigned duties. Maria accordingly had a stable of very accomplished tutors who worked privately with her to develop her intellect from a very young age. This made her an exceptionally learned woman, with a rare expertise in topics such as mathematics and natural philosophy that were not normally included in a girl’s education. One of the most interesting aspects of Mazzoti’s book, in fact, is his rich demonstration of how Agnesi’s peculiar path into learning produced its ultimate innovations and triumphs. This was especially true in mathematics, since she was introduced to the leading authors and taught mathematical science at the highest level while also being pushed because of her peculiar circumstances to chart an idiosyncratic path through this material. The results of this freedom are evident in Agnesi’s mathematical textbook. But to see the full complexity of her work, one must also see the other major force in her life: religion. As urbane and worldly as the Palazzo Agnesi could be, it was also a proper home with an appropriate respect for traditional religion. Agnesi was accordingly given a rigorous Catholic catechism, one no less strenuous than her mathematical and philosophical education. The two programs were in fact never separated, and since Pietro’s sensibilities leaned toward those wings of the Catholic church that emphasized the harmonies between secular and religious life, he found teachers for his daughters who shared his values. This was not difficult in Milan, as Mazzotti shows in a characteristically rich discussion of the many ‘‘catholicisms’’ that existed in the city. Milan had a long history of charting its own path within the big tent of the Roman Catholic Church, and while religious conservatives were prevalent and powerful, religious reformers were numerous as well. Mazzotti does an especially nice job illuminating the politics that this pluralism produced, and he joins it all into a very persuasive discussion of what he calls the eighteenthcentury ‘‘Catholic Enlightenment’’ in Milan. This ‘‘Enlightenment’’ centered on a union between those Catholics devoted to progressive social and political reform and those devoted to certain progressive theological currents. It was this environment that shaped Agnesi’s self-conception. In the same way that her idiosyncratic scientific education placed her at the cutting edge of some of the most important trends in eighteenth-century science, her religious training oriented her toward equally innovative notions of Catholic devotion. Agnesi’s mathematics finds its most fascinating interpretation when framed in terms of these two intertwined imperatives. Religion and science were entangled from the inception of the infinitesimal calculus in Europe, as the French example showed, and Agnesi’s negotiation of religious and scientific considerations was in many ways indicative of this larger entanglement. Her guiding light was Malebranche and the Oratorian conception of advanced mathematics developed in France in the 1690s. Like these French mathematicians and clerics, Agnesi saw infinitesimal mathematics as a bridge between fallen man and a 102

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transcendent God. Seeing in the ineffable mysteries of the infinitesimal a kind of redemptive link joining the finitude of the human realm with the infinitude of the divinity, she also argued like the Oratorians for a deeply religious justification for the pursuit of advanced mathematics. Agnesi’s treatise on the calculus illustrates these urges. The project was originally inspired by her sense that the existing introductions to the analytical sciences, and especially to infinitesimal calculus, did not bring students to the sublime insights these sciences had to offer. She had cut her intellectual teeth on the Oratorian mathematical works of Charles Reyneau, L’Analyse de´montre´e and La science des grandeurs en ge´ne´rale, which had each been commissioned by Malebranche himself in 1708 and 1714, respectively, as mathematical textbooks designed to develop the theological philosophy of Malebranche’s multivolume De la Recherche de la Ve´rite´(final edition 1711) into an appropriately oriented mathematical curriculum for the Oratorian colleges. Agnesi’s spiritual education under the direction of the Milanese Theatines reinforced this message, and as she came of age she began to see a path for her own mathematical cum spiritual work in this space. Her treatise itself was conceived in the late 1730s, before her twentieth birthday, when she recognized some obscurities and infelicities in l’Hoˆpital’s introduction to the differential calculus in his textbook of 1696. At the same time, she was introduced to the Monk Ramiro Rampinelli, a mathematician and teacher who settled at the Milanese monastery of San Vittore in 1741. Rampinelli never published a single work of mathematics in his life, yet in the mathematical ‘‘profession’’ of the day this in no way prevented him from being a highly esteemed and well connected member of it, one with strong links to the correspondence networks that constituted advanced mathematical research in this period. Rampinelli encouraged Agnesi’s research ambitions, while linking her to other mathematicians who could help her with her work. With his assistance, she succeeded in producing her book, an introduction to all the analytical mathematical sciences including the calculus addressed to young people and to other beginners in the field. Her treatise bears the distinctive mark of its peculiar origin and motivation. Most striking, as Mazzotti discusses in detail, is her focus on pure mathematics alone, and her avoidance of mathematics conceived as a tool for pursuing the physical and empirical sciences. When Agnesi was writing in 1740, the use of differential and integral calculus in celestial and terrestrial mechanics was a forty-year-old scientific fact of life in places such as Paris, and Agnesi studied the work of the second generation of mathematical physicists – Pierre Louis Moreau de Maupertuis, Alexis Claude Clairaut, and Jean le Rond d’Alembert, along with the late work of Johann Bernoulli and his young protege´ Leonhard Euler – as part of her mathematical training. Closer to home, in Padua, where the Bernoullis were influential, Agnesi could find the same brand of physically and empirically oriented mathematical science. Rampinelli’s appointment at Padua to teach advanced mathematics while simultaneously pursuing engineering projects for the state illustrated well this scientific confluence, which was widespread in Europe at the time.

Yet Agnesi rejected this applied understanding of, and motivation for, infinitesimal analysis. Writing to a correspondent on one occasion, she stressed that she had intentionally left out of her treatise certain topics that ‘‘depend on a knowledge of physics’’ and ‘‘are involved in physical matters’’ in order, she declared, ‘‘to avoid going beyond pure analysis, and its applications to geometry.’’ This urge has puzzled modern historians, who see in the alliance between infinitesimal analysis and the physical sciences a fully natural marriage that gives this mathematics its most important reason for existing. As Clifford Truesdell exclaimed in disbelief after a study of Agnesi’s work: ‘‘while learning calculus, she does not wish to study rational mechanics!’’ He also followed other historians in finding both Agnesi’s and Reyneau’s mathematics ‘‘prolix’’ because it does not manifest an engineer’s interest in finding the most economical and efficient mathematical solution to practical empirical problems. Mazzotti helps us to see, however, that Agnesi’s choices were not failures of insight or weaknesses of intellect but rational and coherent outcomes derived from her different intellectual agendas. For her, the reason to pursue advanced mathematics was not to solve knotty problems in hydraulics or celestial mechanics; the goal of mathematics was to achieve the quietude of mind and body that makes one receptive to contemplating God. Such a choice certainly tied mathematical science to Christianity in a strong way, but as Mazzotti also shows, this orientation did not make it by consequence a counter-Enlightenment science. What better training for Enlightenment reform than the disciplinary practices of orderly mathematical inquiry, and since the Catholic Enlighteners did not frame church and progress as enemies, an innovative new approach to advanced mathematics could serve Christianity and Enlightenment science simultaneously. Her treatise, which offered novices a clear and rational foundation for both advanced mathematics and religion, also provided an alternative to other rival pedagogies, such as those of the Jesuits, who anchored reasoned faith in deductive geometry and rational disputation, not in abstract algebraic analysis. In Paris, Jesuits were vigorous opponents of the Leibnizian calculus, for this mathematics, they claimed, lacked firm intellectual foundations, making it a corrupting influence, and not a rational support for Enlightened Christian practice. For someone with Agnesi’s education and religious orientation, by contrast, the very mysteries of the infinitesimal, caught as it was between the finite and the infinite, became a way of uniting the limited capacities of the human mind with the incomprehensible plenitude of God. Within this frame, calculus was not an abomination to divine rationality as the Jesuits argued, but a piece, even a centerpiece, of a well-ordered, if still humble, limited, and human, understanding of God. Agnesi’s later life bears out this understanding of her mathematical work. After the publication of her treatise in 1748, she became a famous celebrity, joining Bassi and others as a leading member of Italy’s famous coterie of learned women. The Academy of Bologna admitted her as a member, and the Paris Academy, which did not admit women, gave her the honor of reviewing her treatise and

sending her an official declaration of approval. Translations of the treatise into French and English soon followed, and her fame became widespread. In the final chapter of the book, one that will be of much interest to gender historians interested in the phenomenon of the Enlightenment femme savante, Mazzotti completes this narrative by showing how Agnesi’s success in the otherwise male world of Enlightenment mathematics drew upon new theories of mind and sex that were dominant in the period. Yet the book’s Epilogue draws a different conclusion about her life, one that does not center on Agnesi’s apotheosis as an Enlightenment femme savante. Throughout her childhood, Maria Gaetana experienced moments of crisis and despair, and before her twenty-first birthday she expressed displeasure with the public limelight her father had placed her in. She dismayed her tutors with talk of entering the convent and ending her program of research and study altogether. Her work in various charitable organizations eased her conscience at first, and by joining intellectual labor with Christian service she sustained her mathematical work throughout her twenties, bringing her treatise into print before her thirtieth birthday. A few years later, however, her father died, and Agnesi seized upon this opportunity to change her life. She gave up all of her intellectual labors and committed herself completely to church service, devoting herself to the teaching of very young children in her parish school and to the care of the sick and infirm. By her sixtieth birthday Agnesi was managing several large and important charitable organizations in the city as well as her family’s estate, and in 1771 she founded her own ‘‘albergo’’ for the sick, using her remaining inheritance to fund the enterprise. In 1783, she moved into a one-room apartment in the albergo so as to be closer to its operations, ridding herself in the process of all but a few remaining possessions. On 9 January 1799, at the age of eighty-one, she died of pneumonia in this same room. Given the full trajectory of Agnesi’s life, can Mazzoti’s study ultimately be described as a history of a life spent in mathematics? From one, overly modernist perspective, Agnesi seems to present us with two lives, and only one spent in science. In this view, she was a youthful and talented mathematician who in her later and more mature years became a devoted servant to church and community. Yet it is the great achievement of Mazzotti’s study to force us to challenge this easy and comfortably modernist understanding. Launched by her father into public life at the center of the new urban intellectual sociability of the eighteenth century, Agnesi found her way to religion and mathematics simultaneously. For a period, she channeled the religious calling that drove everything she did toward advanced mathematical research, but in later life she gave up mathematics in the name of complete religious devotion. Yet in Agnesi’s world, mathematics and religion were entangled in countless fascinating ways, and one could move between mathematics and religion, as she did, without necessarily abandoning the values and goals of either. So did she really leave mathematics behind in the name of religion? Or did she just realign her position within these conjoined vocations? Histories of mathematics such as Mazzotti’s lead us to ask and answer these sorts of deep questions. They may Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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not teach us mathematical lessons from the past or show us how to advance our current mathematical research, but by forcing us out of our comfortable assumptions about what mathematics is and why we do it they may contribute in more valuable ways toward advancing mathematics as a whole. Histories such as Mazzotti’s show us how a different entanglement between mathematics and human life produces a very different reality for each. And by creating a constructive historical vantage point from which to view the ‘‘strange world’’ of mathematics today, they also compel us to ask more profound questions about why we do what

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we do and toward what ends. In the final analysis, this might be the real contribution that the history of mathematics has to offer to mathematicians and mathematical science today. Department of History University of Minnesota Minneapolis, MN 55455 USA e-mail: [email protected]

Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung by Arthur I. Miller NEW YORK, LONDON: W. W. NORTON & COMPANY, 2009, HARDCOVER, £18.99 (US $27.95), XXV + 336 PP., ISBN 978-0-393-06532-9; 2010, PAPERBACK, PUBLISHED AS: 137. JUNG, PAULI, AND THE PURSUIT OF A SCIENTIFIC OBSESSION, £11.99 (US $16.95), ISBN 978-0-393-33864-5 ¨ NTHER NEUMANN REVIEWED BY GU

he ‘‘strange friendship’’ between Wolfgang Pauli (1900–1958), the Nobel Prize winning physicist so influential in the ‘‘quantum revolution’’, and Carl Gustav Jung (1875–1961), famed founder of analytical psychology, has fascinated many people. The correspondence between them has been published [1], and there are now several books and articles dealing with their relationship [2]. The Pauli-Jung friendship is an ideal subject for Arthur I. Miller, professor emeritus at University College London, who has had a long-time interest in the border area between science and art, particularly concerning questions of creativity and imagery. As a student at the City College of New York, Arthur I. Miller took large doses of philosophy in addition to physics. This was the start of a career that would lead him to become a well known historian of science and an acclaimed author. In 1965 he earned a Ph.D. in physics at the Massachusetts Institute of Technology and went on to research in theoretical particle physics. Reading the original Germanlanguage papers written by the giants of twentieth-century physics – scientists such as Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schro¨dinger, and Wolfgang Pauli – drove him to study the role of visual thinking in highly creative research and the importance of the history of ideas. In 1991 Miller moved to England where he became Professor of History and Philosophy of Science at University College London. He is the author not only of academic works but also of several widely acclaimed books for a wider audience, including Einstein, Picasso: Space, Time, and the Beauty That Causes Havoc (2001), nominated for the Pulitzer Prize. The seeds for the extraordinary relationship between the theoretical physicist Wolfgang Pauli and the psychoanalyst Carl Gustav Jung were sown in the first two decades of the twentieth century. These two decades were, both culturally and scientifically, among the richest periods of recent history. Sigmund Freud developed his ideas of the unconscious and psychoanalysis, which were now being popularized, as well as criticized, by Jung. As Miller points out, ‘‘Carl Jung was a celebrity and regarded as the chief rival of the great Sigmund Freud […] He extended the boundaries by using

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dream images to explore the unconscious more deeply than Freud had, probing into the archetypes built into our minds.’’ In 1905 Albert Einstein revolutionized the world of physics with his special theory of relativity, while artists were reassembling notions of reality by delving into cubism and abstract expressionism. ‘‘Classical ways of understanding the world suddenly seemed insufficient. An intellectual tidal wave – the avant-garde – swept across Europe.’’ Wolfgang Pauli was born into this ‘‘ferment of ideas’’. At the age of 21 he burst on the physical scene with a paper on general relativity. His new mentor in Munich, Arnold Sommerfeld, and even Einstein himself were impressed. Pauli was admired as a confident theoretical physicist but also feared as a pitiless critic of every illogical or wooly idea. In 1925 he made a major advance in quantum physics by formulating the famous Pauli exclusion principle (for which he was awarded the 1945 Nobel Prize in Physics). Seven years later, working at the Swiss Federal Institute of Technology Zurich (ETH—Eidgeno¨ssische Technische Hochschule—Zu¨rich), he suffered from depression and hypersensitivity. At his father’s suggestion, the apologist of intellectual rigour sought psychiatric help on the couch of Carl Gustav Jung. As Jung put it: ‘‘When the hard-boiled rationalist […] came to consult me for the first time, he was in such a state of panic that not only he but I myself felt the wind blowing over from the lunatic asylum!’’ And so began a long friendship that is the subject of Arthur I. Miller’s Deciphering the Cosmic Number. Pauli had always nurtured an interest in the irrational as a driving force of scientific creativity. Science, as his mentor Arnold Sommerfeld pointed out, had grown out of mysticism. Inspired by his mentor’s interest in the occult, he immersed himself in the work of two Renaissance thinkers, the German astronomer and mathematician Johannes Kepler (1571–1630) and the English nobleman Robert Fludd (1574–1637), medico and philosopher. Kepler was inspired by the harmonious symmetry of Copernicus’ heliocentric world view and the Pythagorean reverence for number. Kepler’s transition ‘‘From circles to ellipses’’ is treated by Miller, but not emphasized as a revolutionary new ontological concept. For the philosopher of science Ju¨rgen Mittelstraß, professor emeritus at the University of Konstanz, the transition from ‘‘perfect’’ circles to ‘‘imperfect’’ ellipses is not only a change of geometrical figures but ‘‘the momentous abandonment of mathematic-ontological distinctions’’ (‘‘die folgenschwere Preisgabe mathematisch-ontologischer Unterscheidungen’’) [3]. Kepler sought to derive a complete description of the cosmos primarily in terms of mathematics. (One of the best fundamental analyses of the mathematically formulated world due to Galileo Galilei [and Kepler] is given by the phenomenological philosopher and mathematician Edmund Husserl [4].) Fludd, on the other hand, remained rooted in the traditions of mysticism and alchemy. He endeavored to describe the ‘‘true philosophy’’ by means of pictures rather than with ‘‘vulgar mathematics’’. In some respects comparable to mandalas, pictures in the alchemist tradition can be understood as universal forms that depict the whole as made up of opposing parts. Surprisingly for a theoretical (i.e., mathematical) physicist, Pauli had sympathy for Fludd as well as Kepler. But  2011 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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there was another feature of the Kepler/Fludd clash which was very important for Pauli. For Kepler the perfect number was three, but for Fludd four was ‘‘the eternal fountainhead of nature’’. Pauli had been torn between these two numbers in his own work. In order to derive his exclusion principle he had to allow four, rather than three quantum numbers, a break with classical physics, which Bohr admiringly described as ‘‘complete insanity’’. The two numbers, together with another obsessive number, the dimensionless fine structure constant with a value close to 1/137, made frequent appearances in Pauli’s dreams, whose analysis by Jung would play a central role in their relationship. One of the questions is why the prime number 137 repeatedly crops up in quantum physics, in connection with the strength of the electromagnetic force (the number emerges in a combination of the speed of light, the charge of the electron, and Planck’s constant) [5]. The work on the fine structure problem was Arnold Sommerfeld’s primary contribution to atomic physics. His ‘‘brainwave’’ was to apply Einstein’s relativity theory to Bohr’s atomic theory, changing the mass of the electron according to Einstein’s famous equation E = mc2. Miller summarizes the importance of this universal constant as follows: ‘‘A dimensionless number of such fundamental importance had never before appeared in physics. Of course dimensionless numbers had always been present in equations, but never one that was deduced from fundamental constants of nature. Scientists later realized that if the numerical value of the fine structure constant were to differ by a mere 4 percent, almost all carbon and oxygen would be destroyed in every star in the universe and life on our planet would not exist or would be dramatically different.’’ We wouldn’t exist if the fine structure constant were slightly different. This kind of question is religious or philosophical and transcends the means of physics. As pointed out by Terry Eagleton, Professor of Cultural Theory at the University of Manchester, the most fundamental question is the question (e.g., formulated by Gottfried Wilhelm Leibniz [6]): ‘‘Why is there something rather than nothing?’’ [7] This question can also be formulated: ‘‘Where does the cosmos come from?’’ Miller further states in a radio interview that in principle alien intelligent life-forms in other galaxies could likewise find the dimensionless fine structure constant (1/137) as a fundamental cosmic number [8]. In ancient Hebrew, numbers were written with letters, and each letter of the Hebrew alphabet has a number associated with it. The word ‘‘Kabbalah’’ in Hebrew is written with four letters – and we do not wonder at the end of the book that the four Hebrew letters add up to … 137! Thus the mystic number 137 ‘‘continues to fire the imagination of everyone from scientist and mystics to occultists and people from the far-flung edges of society’’ (p. 259). Jung, too, was interested in the occult. Like Freud, he believed that dreams are the key to an individual’s psyche, but in opposition to Freud he viewed them as a portal to a collective unconscious – symbolic notions, called archetypes. (The existence of the unconscious is confirmed by modern neuropsychology, but the question of archetypes as source of human thinking remains controversial.) 106

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Jung’s ideas naturally led him to study the alchemists who were also ‘‘talking in symbols’’, and believed that ultimate wisdom – the philosopher’s stone – would be achieved through a unification of opposite states. He began to incorporate alchemy into his analytical psychology. To avoid mockery, and also to build a more universal theory, he needed to give his quasi-mystical theories some kind of ‘‘scientific’’ footing. In 1932, an opportunity to do so was the less or more chance encounter with Pauli. For Jung, quantum physics became important in his consideration of synchronicity, that is, meaningful coincidences as an acausal connecting principle, which he came to believe formed a link between physics and psychology [9]. More than 20 years ago, Miller came across a book Jung and Pauli had coauthored: The Interpretation of Nature and the Psyche [10]. Pauli re-examined the theories of Kepler (and Fludd) in the light of Jungian psychology, focusing on the role of the irrational in scientific creativity. He argued that the link between sensory experience and the rational concepts that make up a scientific theory is formed by archetypes. Arthur I. Miller’s thoroughly researched book gives an exemplary account of an excursion into the ‘‘no man’s land’’ between physics and psychology and of fruitful interdisciplinarity in the exploration of the human mind and creativity. In the epilogue he states: ‘‘The puzzle of how we reason, how we think – of how we create knowledge from already existing knowledge and how we draw conclusions that go beyond the premises – cannot be solved by logic alone.’’ In an interview the author remarks, ‘‘Although the two men [Pauli and Jung] never came up with answers, the questions they raised, the level of their discussions, and their quest to fold physics and psychology together, merit further consideration. That was one of the reasons I wrote this book.’’ [11] It is no surprise to hear that, when Pauli was dying of pancreatic cancer at the Red Cross Hospital in Zurich in December 1958, Jung was the last person he asked to see. The number of Pauli’s room was 137! This book is also available in German and Italian Translations: Mu¨nchen, Deutsche Verlags-Anstalt, 2011, ¨ bers. Von Hubert Mania, CA, €22.99, 350 pp., ISBN 978-3U 421-04290-3 (German edition); [Milano], Rizzoli, 2009, Trad. di Carlo Capararo, Stefano Galli, 443 PP., ISBN 978-8-81703296-4 (Italian edition).

REFERENCES

[1] Atom and Archetype: The Pauli/Jung Letters, 1932–1958. C[arl] A. Meier (ed.) with the assistance of C. P. Enz and M. Fierz. Transl. from the German by David Roscoe. With an introductory essay by Beverley Zabriskie, Princeton, Princeton University Press, 2001. Originally published as Wolfgang Pauli und C. G. Jung. Ein Briefwechsel, 1932–1958. Hrsg. von C[arl] A. Meier. Unter Mitarbeit von C. P. Enz und M. Fierz, Berlin, SpringerVerlag, 1992. [2] Cf. Suzanne Gieser, The Innermost Kernel. Depth Psychology and Quantum Physics. Wolfgang Pauli’s Dialogue with C. G. Jung, Berlin, Heidelberg, Springer-Verlag, 2005; Harald Atmanspacher, Hans Primas (Eds.), Recasting Reality. Wolfgang Pauli’s

Philosophical Ideas and Contemporary Science, Berlin, Heidel-

[6] ‘‘Principes de la Nature et de la Grace, fonde´s en raison,’’ sec. 7,

berg, Springer-Verlag, 2009; Charles P. Enz, Of Matter and Spirit. Selected Essays, Singapore, Hackensack, NJ, London, World

Gottfried Wilhelm Leibniz, Die philosophischen Schriften. Hrsg. von C[arl] I[mmanuel] Gerhardt [1875–1890], Hildesheim, New

Scientific Publishing, 2009, in particular essay no. 19: ‘‘Wolfgang Pauli – C. G. Jung, a Dialogue over the Boundaries’’. Further two older relevant and interesting books, not listed in Miller’s bibli-

York, Olms, 1978, vol. VI, p. 602. [7] Terry Eagleton, The Meaning of Life, Oxford, New York, Oxford University Press, 2007, p. 2 ff.

ography: Fred Alan Wolf, The Dreaming Universe. A Mind-

[8] Radio interview first broadcast on 2 May 2009 – Arthur I. Miller

Expanding Journey into the Realm Where Psyche and Physics

talking to Gene Heinemeyer about Deciphering the Cosmic Num-

Meet, New York, Simon & Schuster, 1994; Arnold Mindell,

ber (available at the homepage of Arthur I. Miller: http://www.

Quantum Mind. The Edge Between Physics and Psychology, Portland, OR, Lao Tse Press, 2000. [3] Ju¨rgen Mittelstraß, Die Rettung der Pha¨nomene. Ursprung und

arthurimiller.com/books/deciphering-the-cosmic-number). [9] Cf. F. David Peat, Synchronicity: The Bridge Between Matter and

Geschichte eines antiken Forschungsprinzips, Berlin, Walter de Gruyter, 1962, p. 213; cf. Alexandre Koyre´, From the Closed

Revelations of Chance: Synchronicity as Spiritual Experience,

World to the Infinite Universe, Baltimore, Johns Hopkins Press,

[10] Carl G. Jung, ‘‘Synchronicity: An acausal connecting principle’’ and Wolfgang Pauli, ‘‘The Influence of Archetypal Ideas on the

1957, pp. 1–3; Thomas de Padova, Das Weltgeheimnis. Kepler, Galilei und die Vermessung des Himmels, Mu¨nchen, Zu¨rich, Piper, 2009, pp. 276–278. [4] Edmund Husserl, Die Krisis der europa¨ischen Wissenschaften und die transzendentale Pha¨nomenologie. Eine Einleitung in die

Mind, Toronto, New York, Bantam Books, 1987; Roderick Main, Albany, NY, State University of New York Press, 2007.

Scientific Theories of Kepler’’. In: The Interpretation of Nature and the Psyche. Transl. by Priscilla Silz, New York, Pantheon Books, 1955 (Bollingen Series LI). Originally published as Naturerkla¨rung

pha¨nomenologische Philosophie. Hrsg. von Walter Biemel (Hus-

und Psyche, Zu¨rich, Rascher, 1952. [11] ‘‘Why two geniuses delved into the occult,’’ Interview with

serliana, Bd. VI), 2. Aufl., [Den] Haag, Martinus Nijhoff Publishers,

Amanda Gefter, New Scientist, 24 April 2009 (available at the

1962, in particular § 9 (English translation: The Crisis of European

homepage of Arthur I. Miller, see [8]); cf. ‘‘Creativity and intellect:

Sciences and Transcendental Phenomenology. An Introduction to

when great minds meet,’’ Interview with Beatrice Bressan, CERN

Phenomenological Philosophy. Transl., with an introduction, by

Courier, 31 March 2010 (available at the homepage of Arthur I.

David Carr, Evanston, Northwestern University Press, 1970); cf. Gu¨nther Neumann, ‘‘Galilei und der Geist der Neuzeit: Husserls

Miller, see [8], or at the CERN Courier: http://cerncourier.com/ cws/article/cern/42093).

Rekonstruktion der Galileischen Naturwissenschaft in der KrisisSchrift,’’ Pha¨nomenologische Forschungen, Jahrgang, 2001, pp. 259–279. [5] For more details see Michael A. Sherbon, ‘‘Constants of Nature from the Dynamics of Time’’ (5 November 2008) (available at SSRN: http://ssrn.com/abstract=1296854 or at Philpapers: http://philpapers.org/profile/2603).

Martin-Heidegger-Edition Untertaxetweg 90 D-82131 Mu¨nchen-Gauting Germany e-mail: [email protected]

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Introduction to Grid Computing (Numerical Analysis and Scientific Computing Series) by Fre´de´ric Magoule`s, Jie Pan, Kiat-An Tan, and Abhinit Kumar BOCA RATON, FL: CHAPMAN & HALL/CRC, 2009, 334 PP., US$ 79.95, ISBN: 978-1-4200-7406-2 REVIEWED BY JUHA HAATAJA

ublishing topical books on Information and Communication Technology (ICT) is a tricky business, as they tend to be already out of date when they appear. Luckily, the book Introduction to Grid Computing is still mostly relevant as I review it in the spring of 2010, although one can spot signs of impending obsolescence here and there. The book is a compendium describing ICT tools which are used to serve a community of researchers. The term ‘‘grid computing’’ has grown to include a variety of approaches, although there are some common themes. Openness is one of them, referring to both open source software and to the use of interoperable ICT tools to build open services for the researchers involved. The user communities of grids are referred to as ‘‘virtual organizations.’’ The term ‘‘virtual’’ is intended to highlight contrast to the more traditional hierarchy-based organizations. Members of virtual organizations often reside in different countries or institutions that may not have formal cooperation contracts with each other. Grids provide services to widely dispersed groups of users within the virtual organizations. Grid resources are linked together by informal agreements and by joint standards and ICT tools.

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Of course, it would be easier to constitute a grid within a single organization, but then there would not arise the benefits of collaboration between individuals and groups with different backgrounds brought together by a common interest in a scientific problem. What is a grid good for? In a typical application, one runs software to perform simulations of a large-scale mathematical model. Grid technology is especially useful when the computation can be divided into distinct pieces that can be run independently on computers which may be geographically wide apart. The ‘‘free resources,’’ or the time slots when the owner of the computer is not performing computations, can be shared within the community of grid users. Also, grid technology provides tools for managing, sharing, and accessing data produced by researchers in virtual organizations. The main challenge facing the construction of a grid is the complexity of the necessary ICT. It is a hard task to pool together resources from dozens or hundreds of service providers. It is not a simple matter to make the resources readily available and discoverable for the users, or to exclude unauthorized use such as stealing research data or tampering with it. The book contains some simple examples of making computations on a grid, such as solving the heat equation with numerical methods. The examples are quite modest and not really useful for real-world computations, but at least they offer a starting point for exploring the use of grids in the computational sciences. It is apparent that grids are still in the construction phase, and many of their features are only available for expert users. You may need to have your own grid specialist willing to help out. Fortunately for the mathematical reader, the terminology of grids is quite well explained. The book ends with a useful glossary.

CSC, the Finnish ICT Center for Science Keilaranta 14, Box 405 FIN-02101 Espoo Finland e-mail: [email protected]

Mathletics—How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football by Wayne L. Winston PRINCETON UNIVERSITY PRESS, PRINCETON, NJ, 2009, 376 PP. US $29.95, ISBN:978-0-691-13913-5 REVIEWED BY RONALD GOULD

re you an American sports fan? As a kid, did you memorize the backs of every baseball card you ever owned? Are you presently playing in some fantasy league? Was ‘‘football’’ your first word as a child? Do you love sports statistics? If your answer to any or all of these statements is yes, then you might want to read this book. If you are not a fan of American sports of any kind, then this book will probably not interest you. Mathletics was written by a fan of American sports. Wayne Winston loves sports and he clearly loves sports statistics. This book goes beyond all others in presenting the newest developments in sports statistics: the newest efforts and attempts at player evaluation and game analysis for baseball, American football, and basketball, all collected in one place. I did grow up a sports nut. I did memorize the backs of baseball cards, although not every one I owned. As a child I played baseball, football and basketball all the time. I still love these sports today. I remember first reading one of Bill James’ books back in the 1980s; James was one of the first to develop new statistical measures for baseball. He had some interesting ideas, but I did not really care for his writing style. Being a mathematician I needed ‘‘proof ’’ that was not supplied. I did not continue to follow his work seriously, but I was certainly aware that his research (and that of many others) was having a significant impact on a number of Major League Baseball executives from the Boston Red Sox to the Oakland A’s. Over the years I wondered why this had not happened in the other sports. It had; I was just uninformed. Mathletics is divided into four major sections. The first three are statistical developments for baseball, (American) football, and basketball. The fourth section deals with money. This includes the use of sports statistics by gamblers in Las Vegas. Other interesting money- (and sports-) related subjects are also treated. I read Mathletics as someone with a curiosity about the subject, but not a passion to put every number to the test. Others, with more knowledge of these statistical developments, might read the book much differently. Still others, loving these sports but lacking statistical training,

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might have a difficult time following many of the arguments. As a reader with plenty of mathematical background, I found some places where I disagreed with the usefulness of a particular statistic or disputed a conclusion that was drawn. But thinking about how I would have done it was rewarding. For example, I did not really buy the computations attempting to show what the late Ted Williams would hit if he played major league baseball today. I also had some problems with the argument on the parity of competition in various leagues, especially when comparing a 16-game National Football League (NFL) season to an 82-game National Basketball Association (NBA) season. A section I found interesting, but in need of more data to be really convincing, was the analysis of best dollar value per draft position for the National Football League player draft. This is certainly a topic that should interest all NFL teams; I just wanted a larger data set to help convince me. A similar chapter for the National Basketball Association player draft concluded the NBA was more efficient in drafting than the NFL, based on associated salaries for the positions of drafted players. Some sections seem too shallow and do not really draw conclusions or match the expectations you might have based upon the section title. For example, an NBA topic entitled ‘‘Analyzing Team and Individual Matchups’’ is less than three pages long (with two more of tables) and only deals with the San Antonio Spurs versus Dallas Mavericks NBA Western Conference semifinal in 2006. That is a pretty limited set of matchups. I was hoping for much more here. But Winston does statistical analysis for the Dallas Mavericks; he probably had that computation in his desk drawer before he started writing the book. Another example of a disappointing section was ‘‘Are College Basketball Games Fixed?’’ This should be a meanful topic with potentially huge implications. Winston disputes the argument that 5% of the games are fixed. However, he does not draw any conclusions of his own. I wanted something more! On the positive side, I was especially interested in the defensive statistics that have been developed for baseball. A description of the work of John Dewan, author of The Fielding Bible, is presented. Dewan and his colleagues at Baseball Info Solutions watch videos of every major league baseball play, measuring how hard a ball is hit and into what ‘‘zone’’ on the field. Using this information, they measure how many runs a defensive player prevents and how this measure converts to victories. This is certainly not something I can simulate at home in order to test, but the fact that Baseball Info Solutions is a going business with a fair number of employees must mean their conclusions are of value to a number of people in the baseball business. I also found it interesting that Derek Jeter, the all-star shortstop for the Yankees (and certainly a future Hall-ofFamer), is below average, by their measure, as a fielder. Conclusions like that are enough to make a fan like me go on reading! Another striking conclusion was that NBA officials show a small but clear racial bias in calling fouls, an argument I Ó 2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011

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had heard before, but was happy to see included here. Winston also takes on the Rating Percentage Index (RPI) ratings for American college basketball and asserts they are flawed and should be discarded. Further, he concludes that an eight-team playoff would save the Bowl Championship Series (BCS) of college football! Mathletics is not a text book, but it does attempt to explain the reasoning behind many of the statistics it presents. I do not think the author had a firm idea of his audience. There is a detailed explanation of some basic things like correlation coefficients, but regression is treated as well known. Later, the Poisson distribution is taken as something the reader can deal with easily by just following a formula, with no explanation of why this distribution is needed for the problem at hand. Inconsistent levels of explanation are found in a number of places. Despite its flaws, I still enjoyed reading Mathletics. It definitely gave me the overview of modern sports

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statistics I wanted. One could argue that Winston tried to cover too much, or that not all his arguments were convincing, or that his level of explanation varied too much. You cannot argue with his attempt to give the reader as broad a view of the sports statistics revolution as possible. He is very knowledgeable on the subject and passionate about its future. He supplies real Microsoft Office Excelbased data sets for his many examples. There are a number of interesting conclusions drawn. If you do not fear statistics and you do love sports, you might consider giving this book a try. I did and I enjoyed the read.

Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 USA e-mail: [email protected]

Mathematics in Games, Sports, and Gambling—The Games People Play by Ronald J. Gould BOCA RATON, FL: CRC PRESS, 2009, US$ 59.95, 374 PP. ISBN: 9781439801635, ISBN 10: 1439801630 REVIEWED BY WAYNE L. WINSTON

eachers of math love math. The problem is that many of our students don’t. We believe that if only we can get our students interested, they will appreciate the beauty and elegance of mathematics. Ronald Gould has written a book that should go a long way towards turning many college students on to the wonders of mathematics. Games, Sports, and Gambling introduces the reader to many topics in probability, statistics and discrete mathematics through examples from poker, backgammon, Nim, and major league baseball. Many of today’s undergraduates are fascinated by these topics so this book is a natural choice for an underclass seminar in mathematics. The book requires a firm grounding in high-school algebra, but is otherwise totally self-contained. The development of topics is careful and clear, and notation is well chosen in even the more difficult examples. Chapter 1 covers basic probability and combinatorics. The examples are mostly familiar (St. Petersburg Paradox, how many poker-hands, etc.). I would have liked to have seen the brief section on Conditional Expectation expanded. Chapter 2 builds on the concepts of Chapter 1 and covers many interesting topics including the chances of winning at craps, poker-hand probabilities and the famous Monty Hall Paradox. Again, the discussion is always clear and easy to follow. The discussion of backgammon is a bit complex, but very well done. Chapter 3 discusses situations with repeated play. This leads us naturally to the binomial random variable, the normal approximation to the binomial (perfect for determining your chances of surviving a night at the craps tables!), and the Gambler’s Ruin Problem. Chapter 3 also has several sports examples, including a discussion of team winning streaks and hitting streaks. I would have liked to have seen some discussion of Tversky’s ‘‘hot hand’’ research here. Chapter 4 discusses several card ‘‘tricks’’ that provide a justification for introducing the Pigeon Hole Principle, Principle of Inclusion and Exclusion, and some basic topics

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in graph theory and matching theory. Gould does a great job in this chapter of developing these topics in the context of interesting examples. Chapter 5 introduces the reader to several important topics in basic statistics (descriptive statistics, simple linear regression, Simpson’s Paradox, and hypothesis testing) in the context of sports examples (mostly baseball). I would like to have seen more basketball, soccer, and football (maybe hockey?) examples in this chapter. It would be nice to teach students how to use Web Queries to download data from the Internet into Excel, so they could test sports related hypotheses with the most recent available data. I would also have taught the students how to use Excel’s Trend Curve to fit a straight line to data. Chapter 6 continues the discussion of hypothesis testing through more interesting sports examples. (Do batters hit significantly worse against pitchers who throw with the same hand as the batter? They do!) I would have liked some discussion of basic sabermetric concepts such as Runs Created and OPS. A gentle introduction to multiple regression would fit well here. Chapter 7 develops the math needed to understand several well-known games and puzzles such as magic squares, Sudoku, Tower of Hanoi, and the famous Cracker Barrel Peg game. Again, the math is beautifully explained, and the intelligent student will come away with an appreciation of the beauty and elegance of mathematics. Chapter 8 introduces combinatorial games. In these games, the emphasis is on trying to determine which player (based on the current state of the game) can win, and devising a winning strategy. Gould begins with very simple games (for example, if 17 chips are on a table and each player can pick up 1–3 chips, who wins and how?) and quickly goes on to discuss some fairly complicated games: Nim, Northcott’s Game, and Blue-Red Hackenbush. Again, the discussion is well done and teaches the students lots of interesting math (primarily graph theory). In summary, if you are looking for a ‘‘different book’’ to turn students on to the beauty of math, Gould’s book is worthy of your consideration. I believe the first six chapters could be understood by students at just about any school, at any level, but the last two chapters require more sophistication and might be more appropriate for honors seminars.

Kelley School of Business Indiana University Bloomington 107 S. Indiana Ave. Bloomington, IN 47405-7000 USA e-mail: [email protected]

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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2006

University of Vienna, he joined a government commission on technical education in Austrian higher institutions, and was later appointed professor of mechanics and hydraulics at what is now the Czech Technical University in Prague. His textbooks include a Theory of Waves and a Handbook of Mechanics.

ICM2006, Madrid

Antikythera mechanism The Antikythera mechanism is a remarkable arrangement of gears, used for calculations in astronomy. Originating from about the second century BC, it was recovered around 1900 in a shipwreck off the Greek island of Antikythera. Its intricate mechanism has been compared with those of medieval Swiss timepieces dating from 1500 years later.

Curta calculator The Curta calculator, the earliest mechanical pocket calculator, was invented by Curt Herzstark while imprisoned in the Second World War concentration camp at Buchenwald. After the War, he went to Liechtenstein and the first 500 calculators went on sale there in 1948. They can add, subtract, multiply and divide to 11 digits, and can be used to calculate square and cube roots.

ˇek Jakob Gerstner (1756–1832) Frantis Frantisˇek Gerstner was a Bohemian mathematician and engineer. After a period as professor of mathematics at the

In August 2006, the International Congress of Mathematicians, held every four years, took place in Madrid and was attended by over 4500 participants. The Fields Medals were awarded by the King of Spain to Terence Tao, Andrej Okounkov, Wendelin Werner, and Grigory Perelman, the last of whom declined to receive it.

Grigore Moisil (1906–1973) Grigore Moisil is commonly regarded as the father of computer science in Romania. After studying in Bucharest, Paris, and Rome, he returned to Romania to take up teaching posts in Ias¸ i and Bucharest. His research areas were algebra, mathematical logic, and differential equations, and he is remembered in the Łucasiewicz–Moisil algebra.

Snowflake The delicate structure of a snowflake has six-fold rotational symmetry, and no two snowflakes have ever been found that are the same. Their hexagonal form was recognized by the Chinese in the second century and was later investigated by Johannes Kepler and Rene´ Descartes, among others.

Frantisˇek Gerstner Curta calculator

ICM2006 Madrid

Antikythera mechanism

â Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology

Snowflake

The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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Grigore Moisil