From the graceful pen of Edward Frenkel comes this appreciation of the
prodigious mathematician Alexander Grothendieck:
Frenkel’s summary of Grothendieck’s approach is phrased in distinctly Platonistic
terms:
Grothendieck’s genius was to
recognize that there is a “being” hiding behind a given algebraic equation (or
a system of equations) called a scheme.
The spaces of solutions are mere projections, or shadows of this scheme.
That formulation goes beyond the basic assumptions of
mathematical Realism, adding a quasi-personalistic, quasi-spiritual metaphorical
dimension. We might
epigrammatize the matter thus:
bare-bones Realism (embraced by most mathematicians as a lower bound) is
to the view attributed to Grothendieck, somewhat as deism is to theism.
Frenkel concretely illustrates this (neo-)Platonistic /
Plotinean stance of seeing the
relatively concrete and familiar, as but a pale and lesser projection of some
higher, transcendent, but paradoxically more solid Reality (which dwells in
Platonic heaven, sipping some higher-dimensional nectar), in his justly celebrated
book for the layman:
Imagine a world in which natural
numbers are replaced by vector spaces;
that is, instead of number 1
we have a line, instead of number 2 we have a plane… Addition of numbers is replaced by the
direct sum of vector spaces,
multiplication by their tensor product. The number 3 is a mere
shadow of the 3-dimensional space, reflecting only one attribute of this
space, its dimensionality.
-- Edward Frenkel, Love &
Math (2013), p. 155
That insight is superficially reminiscent of Bertrand Russell’s
definition of, say, the integer three
as being not basic, but the rather the (infinite) set of all triads { {Moe,
Larry, and Curley}; {Huey, Dewey, and Louie}; {the Andrews Sisters}; ..} Russell’s move always struck me as, at
best, an obscurum per obscurius (or even a
clarum per obscurius); at worst, a parlour-trick. Frenkel’s suggestion likely goes
much deeper -- indeed, to a depth I am in no position to assess.
Finally, without the structural richness of Frenkel’s
proposal, but perhaps more accessibly, this from a pair of noted
philosophers-of-science:
The abstract objects of
thought such as “numbers” [or]
“perfectly straight lines” .. are real parts of nature, even though they do not
exist as particular things, but as
the relations or transformations of such particulars.
Morris Cohen & Ernest
Nagel, An Introduction to Logic
and Scientific Method (1934), p. 372
~
Frenkel reminds us that Grothendieck, dramatically and
notoriously, Threw It All Away, resigning his institute and seceding from the
mathematical world, upon learning of its defense funding. Frenkel points indeed to the
possibility that even a subject so “gloriously useless” (as it was seen back in
G.H. Hardy’s day) as number theory, can have crucial cryptographic applications
which, in turn, invite subversion by the intelligence community. He states, indeed, that such
subversion did take place, at the hands of a three-letter Agency too numinous
to name, in the case of Elliptic Curve Cryptography. Yet ironically, that very field was pioneered by a
theoretician politically of the far left, Neal Koblitz. (I knew him slightly when we were both
math majors and antiwar activists at Harvard, and recall him fondly here.)
[
A parallel:
For a highly entertaining, though painful, account of Simon Norton, once Conway’s collaborator, but who
likewise Threw It All Away, see the book by Alex Masters, The Genius in my
Basement (2011) ]
I never met Grothendieck, nor so much as approached the
foothills of his cloud-surmounting summits; but my introduction to mathematics did come via one of
Grothendieck’s students, Robin Hartshorne, here portrayed:
We touch upon Grothendieck tangentially in several
essays; here they are:
[Another parallel:
Grothendieck’s abrupt withdrawal from mathematics, and his sometimes
surly subsequent comments on its practicioners, recalls that more recent
self-reclusion of the Russian topologist Grigori Perelman, who quit his institute in 2005, with the remark
that “I have been disappointed in mathematics and I want to try something else.” Now, to anyone who grasps the
subject, disappointment with mathematics
is almost inconceivable -- it’s like being disappointed with the Universe, or
with penguins, or with life itself. Other
statements he has made suggest that his quarrel is not with math per se but with
certain mathematicians.
For a poignant glimpse of the daily routine of this
semi-recluse, try this:
~
In other cryptographic news … This week’s New Yorker
has a review by Anthony Lane of “The Imitation Game”, which takes us back to
the WWII code-breakers at Bletchley Park, with mathematician Alan Turing at the
center. With his welcome
skeptical eye, Lane punctures the movie’s superstar approach to the material:
No word is breathed, for instance,
of the Polish cryptographers who did much of the heavy lifting on the
project before Turing came on the
scene. As for the cracking of
codes, it is shrunk to single, Oh-my-God epiphany, triggered by a comment in a
pub.
That last motif is strikingly reminiscent of the trumped-up
Eureka moment ascribed to Gauss in the movie “Die Vermessung der Welt” (which
we reviewed here), whose insights into differential geometry were supposedly engendered -- in a flash, on the spot -- by a buxom mädchen
handing him an apple from the knowledge-tree.
Another warning-sign:
The role of Turing’s Comely Female Sidekick (de rigueur in Hollywood these days) is
played by Keira Knightley, an overactress whose pornographic approach to
proximity to great men of science, in the movie “A Dangerous Method”, we
earlier had occasion to denounce.
~
[Late-evening update]
And now, fellow devotees of that supreme queen of the noösphere, math --
now that we have all stuffed our tummies with turkey, and are sprawled upon the
couch: as we are probably not just
now engaged in settling the Hodge Conjecture (and making but scant progress if
we are), let us rather refresh the neurons with a bit of mathematical
merriment:
[Update, 30 November 2014] Further thoughts.
(1) Integers as shadows
That epigram, that petite
phrase, “The number 3 is a mere shadow of the 3-dimensional
space”, continues to intrigue.
Further thoughts on the subject here:
The point being that Frenkel/Grothendieck are here proceeding
in the less-familiar opposite direction.
(2) “Goodbye to All That”
The catchphrase “I Threw it All Away” is from Bob Dylan. An earlier more familiar
expression is “Goodbye to All That”, the title of a memoir by the poet Robert
Graves.
Somewhat in the spirit of the Stith-Thompson index of folkloric motifs, to I Threw It All Away (TaleType #1729a), we add:
To fascinate a public beyond the circle of connoisseurs, a
mathematician (or physicist, or chess-player) needs to sport some quirk, some
handle onto which the layman can hang his attention. That is massively true of Turing, first because of his role
in the thrillingly clandestine -- and militarily crucial -- cryptographic
factory at Bletchley park; and
more recently -- and less relevantly, from any mathematical standpoint -- by
reason of his personal predilections (held in common with such extra-scientific
figures as Ἀντίνοος and the Baron de Charlus) which at present
have reached an apogee of public celebrity. On a more modest scale (no blockbuster biographies, no
movies) the careers of Emmy Noether or Ada Lovelace come to mind.
For anyone not a member of the identity-groups in question,
such chance affiliations are mathematically and philosophically boring. But not so a figure like
Groethendieck, who reached intellectual levels that most of the rest of us can
only pant and sigh for -- then threw it all away, in a contemptuous
gesture. For that is a
challenge to all of us, whatever our private identities; it calls into question the very value
of the ‘it’ for which we so painfully and vainly strive.
A well-known example from chess: Bobby Fischer, who withdrew from the sport at the top of his
game.
Or, in the sixteenth century, the pioneering anatomist Vesalius,
who, having published the book that settled his fame forever, and still a young
man, left the field and became a simple physician to a valetudinarian Emperor.
A grey-area variant of this motif is found in the case of
Simon Norton. True, he
voluntarily withdrew from the field;
but in view of his later pointless eccentricities, we cannot avoid the
suspicion by then he had largely Lost It.
That variant motif forms the spine of Rebecca Goldstein’s
fascinating mathematico-philosophical novel, The Mind-Body Problem.
And while Goldstein manages to craft a good read out of the
tragedy, in the typical case of gradually fading powers, it is just sad. Thus Lagrange, and other victims of Oligophrenia mathematica tardiva, chronicled in the appendix to this
essay:
Other examples of burnout:
Newton had a nervous breakdown a few years after publishing
his Principia, and never did real science again.
Russell seems to have fried many of his math neurons in the
course of writing his Principia
(memo to prodigies: Don’t try to
write a Tenth Symphony, and don’t write a book called Principia). He did much interesting
philosophical work after that, but mostly with other areas of his capacious
brain.
Somewhat in the spirit of the Stith-Thompson index of folkloric motifs, to I Threw It All Away (TaleType #1729a), we add:
* Taken all away
(#1729b): Persecution took them out of the game,
at the height of their powers: Galois (permanently) and, for those who
considered that he was hounded to death, Turing. Temporarily: André Weil and Neil Koblitz (during their imprisonment
-- though both managed to use their ‘time inside’ more profitably
mathematically than most of us do
with all the free time in the world).
* Would throw it
all away, if had to do it over again (#1729c):
Wolfgang Pauli in 1925:
"At the moment, physics is again terribly confused. In any case, it is difficult for me, and I wish I had been a movie comedian or something of the sort, and had never heard of physics."
"At the moment, physics is again terribly confused. In any case, it is difficult for me, and I wish I had been a movie comedian or something of the sort, and had never heard of physics."
Schrödinger, to Niels Bohr:
“If all this damned quantum jumping
were really here to stay, then I should be sorry I ever got involved with
quantum physics.”
[quoted in: -- Roger Penrose, The Road to Reality (2004), p. 516]
More such anecdotes here:
* Should throw it all away, if that’s how they feel (#1729d): String theorists who have wound up at the dead-end of the
Landscape picture. (“The
reductionist voyage that has taken physics so far has come to an end. Since that is what they believe, I
can’t understand why they don’t take up something else -- macramé, for example.” More here.)
