Saturday, November 23, 2013

Gradus ad Parnassum

We are familiar with the genre of spiritual (auto)biography.
First, our young enlightenee-to-be  experiences little but Unordnung und frühes Leid;  then, groping and grappling with shadowly intuited truths; then at some point there supervenes something supernatural -- most starkly, in the form of angelic intervention. 
Thus, Muhammad of Mecca, “enwrapped” (a detail telling in its biographical specificity -- this is not just all made-up -- it’s like the detail of Jesus doodling with his finger in the sand), alternately sweating and shivering in his Cave of Retreat, at last is confronted with an Archangel, who (after some preliminaries which it would delicious to retell, but which space does not permit), says:  Iqra’! (“Read!” -- or rather, “Recite!” or “Repeat after me!”)  and reveals the Qur’an.  
Likewise the future Saint Augustine.  He led a misspent youth, at one point sinking to the actual infamy of stealing pears (!);  until one day, a unseen voice cried out: 

Tolle, lege!
(‘Pick up [the Bible] and read!’)

After these interventions, it is pretty much smooth sailing for our Chosen Ones (one of the epithets of Muhammad,  Mustafa, means precisely ‘chosen’), who never look back.


And now we come to the intellectual autobiography -- specifically, the mathematical memoir -- of Edward Frenkel (Френкель, Эдвард, de son vrai nom):  Love and Math.

He too grew up in somewhat unpromising circumstances, well outside of Moscow, which for a Soviet of the time  was as cruel a fate as living far from Paris, for the French.  Jewish to boot, which meant that, so far from being called (here in a secular sense summoned, rather than that of ‘having a calling’), he was actually turned away at the gate, and later (not taking nyet for an answer) had literally to scale a fence and sneak past armed guards to reach the seminar rooms of that sanctum sanctorum, Moscow State University.
(There is some takeaway here for idealistic educators:  You can paint the classrooms with colors as bright as you like, but ultimately it comes down to student capability, and motivation.)

Now, all that high adventure is fine preparation for an actor, or a novelist, or a revolutionary, but is not especially helpful to gain a grounding in the principles and arcana of contemporary mathematics:  a path that has risen at an ever-increasing pitch, since antiquity, and branched into perplexing byways, before the blessed consilience  of synthesis, such as Cartesian geometry, the Erlangen Program, and latterly the Langlands Program, forged new anastomoses, reknitting the whole thing.  Yet at the age of sixteen, when most of us are just learning to shave (or looking forward to needing to -- meanwhile, these pesky pimples), he somehow comes to the attention of a world-class mathematician, who refers him to the special care of one of the archangels of the field, Prof. Dmitry Fuchs.  Fuchs hands him an article from the forefront of breaking research, and says:  Tolle, lege.  (Or, one supposes, принять! читать!)   And the next thing you know, our shaveling is attending the legendary evening seminars of that god among men, I.M. Gelfand -- the Wiener Kreis of Soviet mathematics --  understanding everything, and swiftly publishing a research breakthrough of his own.  By the time he is twenty-one (barely old enough to vote, when I was that age), he has been summoned to Harvard.

(For anecdotal evidence about how hard this stuff is, even for people who have been doing math their whole life, try this:  Oligophrenia mathematica.)

Now, if you have never yourself grappled with research-forefront mathematics, you will have no idea how extraordinary, almost preposterous, that account is.  The epiphany-stories of Muhammad and of Augustine, which theophobes will dismiss out of hand as «miraculous» (as though the presence of a miracle itself suffices to spoil the tale, like a fly in the soup), are humdrum by comparison.

For, both those chosen were presented with texts in a language they already knew (Arabic and Latin respectively), and which  moreover  had been composed specifically to be received by the masses (with imperfect understanding, it may be, but getting the gist and the uplift).  The Qur'an, indeed, helpfully mentions that it has been revealed «in plain Arabic».   Whereas the Fuchsian manuscript presupposes millennia of progressively more successful wrestling, with abstruse insights, by the finest minds on earth.
So:  Either Professor Frenkel is embellishing just a bit, or rather compressing, in retrospect, or else this scene indeed was:  a miracle.  For, for anyone else, that manuscript would have been a book  of seven times seven seals.


Frenkel's heart is in the right place.   He has joined with such popularizers as Stanislas Dehaene, in suggesting that more or less everyone has la bosse des maths,  the little darlings need merely be placed into the right pot and watered, and they will bud and blossom.   The invariant come-on is a pointing to results «beautiful and elegant».   On the very first page of his book, attempting to explain the public indifference or actual aversion to what they imagine to constitute math, Frenkel writes:

What if at school  you had to take an 'art class' in which you were only taught how to paint a fence? ... While the paintings of the great masters are readily available, the math of the great masters  is locked away.

True, and nicely observed. But such beauty and such elegance are perceptible only to the mind prepared -- otherwise it is like playing Bach to a baby.

The suggestion that one can chug one's way to the top of this particular ethereal Parnassus, simply with hard work and the right attitude (I think I can, I think I can), fits in well with the myth of the Little Engine that Could, that I and my playmates were brought up on, pluckily chugging uphill.  Whereas in practice, the brave little engine makes it only as far as the first false-peak, never ascending the Ladder of Abstraction that lies beyond; while one of your company  suddenly sprouts wings, and is halfway up the slope.
The position that we are all Gausses in nuce, if only we were given half a  chance, likewise fits in well with the anti-innatist, doggedly/dogmatically environmentalist political-correctitudes of our own day.   Yet I am here not really plunking for either side of that false dichotomy.   Yes, both are necessary, sweat-equity and the right genes;  but beyond that, something mysterious ... Call it Grace.

Well;  bless him.  May his infinite series never fail to converge, may his commutators ever commute.  For the rest of us, we must be content with a Pisgah-glimpse.  And to reconcile ourselves to the following refractory, diamond-hard truth:

Not that many are even called,
and precious few are chosen.

~     ~     ~

[A note to my readers, puzzled  perhaps  by a sudden change in punctuation-style.   My word-processor, for reasons best known to itself, between the hour at which I posted the beginning of this essay, and a moment ago when I posted the rest, has suddenly and inexplicably switched from American-style quotation-marks  to the angled version favored in France (or, in reverse order, in Germany).  Apparently the software has been favored with some sort of epiphany, to which I myself am not privy.
Perhaps, as the day wears on, the keyboard will begin printing in Cyrillic.  And yea, I shall be baffled thereby, and sore afraid.
Then strange symbols, and equations, will begin creeping in:  and I shall shake, in fear and trembling.
But then a voice from on high rings out --


And all will be well.]


Further reading:
Mathematical autobiography: 
            André Weil
            Neal Koblitz
Psychologia mathematica:  Invention and Insight 

No comments:

Post a Comment