Thursday, November 27, 2014

Alexander Grothendieck


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 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:




~
Sotie : 
le mathématicien  et la conspiration Riemann
~



[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.


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 ντίνοος and the Baron de Charlus) which at present have reached an apogree 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."


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.)

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