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.)
(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.
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 --
TOLLE -- LEGE !
~
Further reading:
Mathematical autobiography:
Psychologia mathematica: Invention and Insight
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