The Mathematical Intelligencer Vol 33 No 1 March 2011

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.
II 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.
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 Korper,’’ Jour. fur 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 con- traire, il me vient de la linguistique et plus precisement 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
4, Place Jussieu,
e-mail: [email protected]
2010 Springer Science+Business Media, LLC, Volume 33, Number 1, 2011 1
Note
Young Gauss Meets Dynamical Systems CONSTANTIN P. NICULESCU
MM ost people are convinced that doing mathematics is something like computing sums such as
S ¼ 1þ 2þ 3þ 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 2 þ 3 þ þ 100 100 þ 99 þ 98 þ þ 1
and to sum columns before summing rows. Thus
2S ¼ ð1þ 100Þ þ ð2þ 99Þ þ þ ð100þ 1Þ ¼ 101þ 101þ þ 101 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
100 times
¼ 10100;
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
and the result is the well-known summation formula
a1 þ a2 þ þ an ¼ nða1 þ anÞ
2 : ð2Þ
A similar idea can be used to sum up strings that are not necessarily arithmetic progressions. For example,
n 0

an ¼ 2n1ða0 þ anÞ;
for every arithmetic progression a0;a1; . . .;an:
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
X n
k¼1
ak ¼ f ðnþ 1Þ f ð1Þ ¼ f ðkÞjnþ1 1 :
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,
f ðxÞ ¼ r
2 x2 þ ða1
2 Þx þ C ;
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 Aj j 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
M
f ðkÞdl ¼ a1 þ þ an:
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 : M ! M ; T ðkÞ ¼ n k þ 1:
Indeed, the measure l is invariant under the map T in the sense that
Z
M
f ðT ðkÞÞdl ð3Þ
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
2 THE MATHEMATICAL INTELLIGENCER 2011 Springer Science+Business Media, LLC
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
M
f ðkÞdl ¼ 1
2 C Mj j:
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 $Mf(x)dl, or rather of its expectation,
Eðf Þ ¼ 1
f ðtÞdlðtÞ;
reduces to the computation of $Mg(x)dl. For example, the integral of an odd function over an
interval symmetric about the origin is zero; this corre- sponds 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:
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
0
and
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 measure dx
1þx2 : 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 lnð1þ tan xÞ; the formula (5) becomes
lnð1þ tanðp=4 xÞÞ ¼ lnð1þ tan xÞ þ ln 2
and thus
0
¼ p ln 2
4 Z p=4
Z h
lnð1þ tan h tan xÞdx ¼ h lnðcos hÞ;
for all h [ (-p/2, p/2). In the same manner we obtain the integral formulae
Z p
Z p
Z p=2
f ðsin xÞdx:
There is a relationship between the expectation of a function f and the values of the iterates of f under the action of T,
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 ½0; 1ð Þ: Thus the verification of the invariance formula (3) reduces to the (trivial) case where f is such a characteristic function.
.........................................................................
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.
Romania e-mail: [email protected]
THEOREM 1. (Weyl’s Ergodic Theorem [10]). Suppose that
a[ 0 is irrational. Then
lim N!1
0
f ðtÞdt ð7Þ
for all Riemann integrable functions f : ½0; 1 ! R and all
x 2 ½0; 1: PROOF. 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
gð0Þ ¼ gð1Þ; hð0Þ ¼ hð1Þ and
Z 1
lim N!1
1
N
0 vI ðtÞdt e; R 1
0 vI ðtÞdt þ eÞ: As e[ 0 was arbi- trarily 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.
The convergence provided by Weyl’s ergodic theorem may be very slow.
In fact, we already noticed that
Z 1
dt ¼ Z p=4
0
lnð1þ tan xÞdx ¼ 0:27220. . .;
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. Mane [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
continued fractions he considered the dynamical system consisting of the map

log 2ð Þ 1þ xð Þdx:
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].
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 Appli-
cations, Prentice-Hall. Inc., Upper Saddle River, 2002.
[4] C. F. Gauss, Mathematisches Tagebuch 1796-1814, Akademi-
sche 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?
Math. Ann. 107 (1932), 485-488.
[7] A. Ya. Khinchin, Metrische Kettenbruchprobleme, Compositio
Math. 1 (1935), 361-382.
Verlag, 1987.
[9] P. Strzelecki, On powers of 2, Newsletter European Mathematical
Society No 52, June 2004, pp. 7-8.
[10] H. Weyl, Uber 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.
4 THE MATHEMATICAL INTELLIGENCER
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
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.
‘‘
FF ermat’s last theorem’’ is famous because it is old and easily understood, but it is not particularly interesting. Many,perhapsmost,mathematicianswouldagreewith
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 oneof theworld’s best known mathematicians, alongwith JohnNash,whoachieved famebybeing 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
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 Ein- stein, 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 epsi- lon, however, Wiles solved Fermat in the best possible way. Intense solitary thought produces the best mathematics.
Gel0fand’s List Some of the greatest twentieth-century mathematicians, such as Andre Weil and Atle Selberg, had few joint papers. Others, like Paul Erd}os and I. M. Gel0fand, had many. Erd}os 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 Gel0fand. 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 Gel0fand’s disparaging reply. But the tenth name on the list was not a name, but a description: ‘‘Somewhere in China,’’ said Gel0fand, ‘‘there is a young student, working alone, who understands representation theory.’’
Bers Mafia A traditional form of mathematical collaboration is to join a school. Analogous to the political question, ‘‘Who’s your
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 Ameri- can 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 sci- entifically 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 recommen- dations 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 Rie- mann Hypothesis,’’ at the University of Washington on August 12–15, 1996. According to its website, ‘‘the Ameri- can 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, Mont- gomery, Odlyzko, Sarnak, and Selberg) described ideas for solving the Riemann hypothesis. I asked one of the orga- nizers 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, theAIM 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
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 setM 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 jMj!1
qualðwÞ supfQualðMÞ : M 2Mgð Þ ¼ 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 mobiliza- tion, 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 produc- tively for the greater good, but didn’t create anything. Mathematics, however, requires intense thought. Individual mathematicians do have ‘‘to think all that hard.’’ Mathe- maticians 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 Fur- stenberg in his proof of Szemeredi’s theorem. In this paper, we give the first elementary proof of the theorem of Fur- stenberg 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 unex- pected 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
shows that only three of Tim Gowers’s papers have a co- author. (But Terry Tao responded to this observation by not- ing 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 jour- nalistic 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 Poly- math 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
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)
The Viewpoint column offers mathematicians the
opportunity to write about any issue of interest to the
international mathematical community. Disagreement
chief endorses or accepts responsibility for them.
Viewpoint should be submitted to one of the editors-
in-chief, Chandler Davis or Marjorie Senechal.
TT 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 bal- anced himself on a ladder and juggled: Paul voted a clear ‘nothing,’ and Dave was going to vote ‘something’ before henoticed a safetymat.Dave then concurredwithPaul [1].
We have been developing techniques to analyze so- called ‘‘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 centuries- old 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 environ- mental 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
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 Connecting- Rod 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 ‘‘Poly- math 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.
.........................................................................................................................................................
S DINESH SARVATE was educated first in
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 minor-
ities, into research. One such project
produced a new type of combinatorial
designs, now known as Sarvate-Beam designs.
Department of Mathematics
College of Charleston
Charleston, SC 29424
SUSANNE WETZEL has a Ph.D. in Computer
Science from Saarland University, a Diploma from Karlsruhe University, and also an honor-
ary 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 num-
ber theory, and range from wireless security
and privacy, to biometrics and lattice theory.
Department of Computer Science
Stevens Institute of Technology
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,
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 con- trast, 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 of F9, the ninth Fermat number, 229 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 role 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 Alge- bra of Bounded Analytic Functions’’ by I.J. Schark [13].
Table 1. Comparison of authorship in leading mathematics and computer science journals
Annals of Mathematics (10 recent volumes) Journal of the Association for Computing Machinery
(2 recent volumes)
Number of authors Number of articles % of articles Number of
articles
.........................................................................
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 inner-
city children, and he has been seven times
an invited speaker at the Baseball Hall of Fame in Cooperstown, New York.
Department of Computer Science
e-mail: [email protected]
A footnote in this paper indicated that I.J. Schark was ‘‘a pseudonym for a large group of mathematicians who dis- cussed 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 pub-
lishing?
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.
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 sug- gestion, 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.
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
Are there subgroups working independently? The con- nectivity 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 sixproblems, tobesolved in twosessionsof four and ahalfhours each.Traditionally, the last problem(Problem6) 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 con- taining 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 ‘mini-
Polymath’ 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-trivial- observation-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 shortwindowof time (70personswith 278posts inmore 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 evo- lution 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 incor- poration 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
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 involve- ment 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.)
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 enun- ciating the proof highlighted.
Certainly there are many aspects of graph structure dif- ferentiating 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 thatwehave analyzed in considerable detail, the announcement of the solution and its subsequent vali- dation by other participants, came about in considerably different ways.
Table 3. Ranking of in-degrees and out-degrees of vertices in polymath
project example
I1 I2 I3 I4 I5 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
Figure 1. The graph G(236) describing the state of the IMOq6
Polymath project. Figure 2. The graph G(550) describing the state of the DHJ
Polymath project.
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 corre- sponding 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 out- degrees) 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.’’
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 ameta-descriptionof theproof,weannotate each stepwithan 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 referencewould be a comparison of the number of persons with degrees in mathematics or computer science at various levels. This is not to deny that
Table 4. Comparison of the graphs describing IMOq6 and DHJ
polymaths
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)
between any two posters
Table 5. Concentrations of posts among posters divided into quartiles
Quartile (cumulative
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
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, themost frequently posting are shown inTable 8. The IMOq6 respondents are identifiedasai and theDHJ respondents asbj
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
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.
4. How can we measure or classify the progress in a specific project?
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 profes- sors 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 6. Analysis of the steps of the RSA correctness proof
Step Underlying information Requisite knowledge
for step
Computation of /(n) Knowledge that /(n)
= (p-1)(q-1)
GCD(e, /(n)) = 1
e 9 d : 1 (mod n)
Introductory computer
science course
m : cd (mod n)
Abstract algebra (AA)
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
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 report- ing 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 gen- erate 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.’’
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,
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.
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.
mathematics possible?’’ http://gowers.wordpress.com/2009/01/
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/.
[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 Theo-
rem’’ (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-mini-
polymath-project/.
Table 8. Frequency of posts to the polymaths IMOq6 and DHJ
IMOq6
respondent
Number
Total 130 535
http://terrytao.wordpress.com/2009/07/21/imo-2009-q6-
mini-polymath-project-cont/.
[17] R. Rivest, A. Shamir, and L. Adleman, ‘‘A method for obtaining
digital signatures and public-key cryptosystems,’’ Communica-
tions of the ACM 21(2), 120–126 (1978).
[18] National Science Foundation, SRS Publications and Data,
http://nsf.gov/statistics/degrees/.
reflections, analysis,’’ Terence Tao’s Blog, http://terrytao.
wordpress.com/2009/07/22/imo-2009-q6-mini-polymath-
project-impressions-reflections-analysis/.
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NN 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.
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.
1The 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].
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 founda- tional 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 GottlobFrege and then in the newcenturybyA.N.Whitehead and Bertrand Russell [1910-1913] in Principia mathematica (hereafter, ‘PM’); and the definitions of non-negative integers in axiomatic set theorybyErnst Zermelo (‘ZF’),with adifferent approach taken later by Johann von Neumann (‘NBG’). Parallel with these developments was the advocacy of axi- omatisation 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 somevariant andalternative theoriesofnumberswere 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, ratio- nals 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
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 numbe

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The Mathematical Intelligencer Vol 33 No 1 March 2011