Math a young man's game?

[Owen Byrne]

  I know there have been discussions in this area here:
Owen
http://slate.msn.com/id/2082960/

> *Is Math a Young Man's Game?*
> No. Not every mathematician is washed up at 30.
> By Jordan Ellenberg
> Posted Friday, May 16, 2003, at 7:39 AM PT
>
> Illustration by Mark Alan Stamaty
>
> Last month at MIT, mathematician Grigori Perelman delivered a series 
> of lectures with the innocuous title "Ricci Flow and Geometrization of 
> Three-Manifolds." In the unassuming social universe of mathematics, 
> the equally apt title "I Claim To Be the Winner of a Million-Dollar 
> Prize" would have been considered a bit much. Perelman claims to have 
> proved Thurston's geometrization conjecture, a daring assertion about 
> three-dimensional spaces that implies, among other things, the truth 
> of the century-old Poincaré conjecture. And it's the Poincaré 
> conjecture that, courtesy of the Clay Foundation, carries a 
> million-dollar bounty 
> <http://www.claymath.org/Millennium_Prize_Problems>. If Perelman is 
> correct—and many in the field would bet his way—he's made a major and 
> unexpected breakthrough, brilliantly using the tools of one field to 
> attack a problem in another.
>
> There's only one problem with this story. Perelman is almost 40 years old.
>
> In most people's minds, a 40-year-old man is as likely to be a 
> productive mathematician as he is to be a major league center fielder 
> or an interesting rock musician. Mathematical progress is supposed to 
> occur not through decades of experience and toil but all at once, in a 
> numinous blaze, to a born genius. Think of the young John Nash in /A 
> Beautiful Mind/, discovering the Nash equilibrium in a smoky bar where 
> his less precocious classmates think they're just picking up coeds, or 
> the aged mathematician in /Proof/ who "revolutionized the field twice 
> before he was twenty-two."
>
> It's not hard to see where the stereotype comes from; the history of 
> mathematics is strewn with brilliant young corpses. Evariste Galois, 
> Gotthold Eisenstein, and Niels Abel—mathematicians of such rare 
> importance that their names, like Kafka's, have become adjectives—were 
> all dead by 30. Galois 
> <http://www-gap.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Galois.html> 
> laid down the foundations of modern algebra as a teenager, with enough 
> spare time left over to become a well-known political radical, serve a 
> nine-month jail sentence, and launch an affair with the prison medic's 
> daughter; in connection with this last, he was killed in a duel at the 
> age of 21. The British number theorist G.H. Hardy, in //A 
> Mathematician's Apology/ 
> <http://search.barnesandnoble.com/textbooks/booksearch/isbnInquiry.asp?userid=2VXL2BZ3NV&isbn=0521427061&TXT=Y&itm=1>/, 
> one of the most widely read books about the nature and practice of 
> mathematics, famously wrote: "No mathematician should ever allow 
> himself to forget that mathematics, more than any other art or 
> science, is a young man's game 
> <http://slate.msn.com/id/2082960/sidebar/2082962/>."
>
> For the notion of the inspired moment of mathematical creation, we 
> have Henri Poincaré himself partially to thank. Poincaré 
> <http://www-gap.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Poincare.html> 
> was not only a monumental figure in the mathematics of the late 19^th 
> century but a popular writer on science, creativity, and philosophy. 
> In a famous 1908 lecture at the Société de Psychologie in Paris, he 
> recounted his discovery, at the age of 28, of a principle underlying 
> the theory of automorphic functions.
>
>     Just at this time I left Caen, where I was then living, to go on a
>     geological excursion under the auspices of the school of mines.
>     The changes of travel made me forget my mathematical work. Having
>     reached Countances, we entered an omnibus to go some place or
>     other. At the moment when I put my foot on the step the idea came
>     to me, without anything in my former thoughts seeming to have
>     paved the way for it, that the transformations I had used to
>     define the Fuchsian functions were identical with those of
>     non-Euclidean geometry. I did not verify the idea; I should not
>     have had time, as, upon taking my seat in the omnibus, I went on
>     with a conversation already commenced, but I felt a perfect
>     certainty. On my return to Caen, for conscience's sake I verified
>     the result at my leisure.
>
    Poincaré's story is the most famous contemporary account of
    mathematical creation. (If we throw open the competition to all time
    periods, it must take second place to Newton's apple and Archimedes
    in the tub <http://www.sciencenet.org.uk/Origins/eureka.html>, which
    speak to approximately the same theme.) Poincaré recognizes that
    mathematics requires both moments of illumination and months of
    careful deduction. "[L]ogic and intuition," he writes, "have each
    their necessary role. Each is indispensable." But he can't quite
    conceal his preference for the intuitive leap over the logical slog.

    The youthful genius, the instant of insight: The pictures fuse into
    a romantic vision of the mathematician as a passive conduit for
    inspiration. As Carl Friedrich Gauss wrote of one of his own
    triumphs, "I succeeded, not on account of my painful efforts, but by
    the grace of God. Like a sudden flash of lightning, the riddle
    happened to be solved."

    Every working mathematician feels the truth of the stories Hardy,
    Poincaré, and Gauss tell. And yet: There's Perelman, pushing 40, and
    Andrew Wiles, 41 at the time of the final resolution of Fermat's
    Last Theorem <http://www.pbs.org/wgbh/nova/proof/>. Today one
    doesn't find mathematicians who revolutionize their field—even
    once—before the age of 22.

    What's changed? For one thing, there's simply much more mathematics
    to learn than there was 100 years ago. The undergraduate curriculum
    at Princeton brings students to the state of the art in research—as
    it was around the time of Poincaré's death in 1912. A year of
    backbreaking work in graduate school suffices to turn the clock
    forward to 1950 or so. At the age when a contemporary student first
    opens a current research journal, Galois had already been dead for
    two years (footnote: apologies to Tom Lehrer). In literature, /pace/
    Harold Bloom, it's possible to produce a great work without a deep
    knowledge of the work that went before. Not so in mathematics, not
    any more; maybe, in fact, not ever.

    Poincaré's conjecture is an assertion about certain
    three-dimensional shapes. We say a shape is /simply connected/ if
    any loop drawn in the shape can be pulled closed to a point without
    leaving the shape. The surface of a sphere, for instance, is simply
    connected, but the surface of a doughnut is not; a loop along the
    outer "equator" of the doughnut can't be contracted to a point
    unless it departs, at some moment, from the doughnut's surface. The
    property of being simply connected, the reader will note, doesn't
    depend on how big or small the shape is; and it doesn't change if
    the shape is bent, twisted, or otherwise deformed. You might say the
    property is quite robust; and the study of such robust properties of
    shapes is the mathematical field of topology
    <http://www.math.wayne.edu/%7Errb/topology.html>, which Poincaré
    more or less invented in the late 19^th century. (The reader who's
    seen other nontechnical accounts of the subject will forgive me, I
    hope, for perpetuating the fiction that the whole field of topology
    is actually confined to the study of spheres and doughnuts. There
    are other shapes, I promise: They're just harder to describe.)

    Poincaré was able to prove that if a two-dimensional surface was
    simply connected, it was /automatically/ some bent, twisted, and
    deformed version of the sphere. His conjecture—phrased so loosely
    that I don't advise you to think about it yourself without further
    reading—is that the same is true for three-dimensional shapes.
    Poincaré couldn't have made this conjecture absent his years of
    study of topology, or the earlier theorems he'd carefully proved, or
    the earlier conjectures on the same theme he'd tried out and found
    to be false. But neither could he have made such a bold guess
    without the kind of wild intuition he valued above all. It's only in
    the presence of both conditions—deduction and inspiration, long
    experience and youthful audacity—that new math gets made, as it was
    made by Perelman, and as it was made on the day Poincaré wrote down
    his conjecture. He was 50 years old.