Monday, March 18, 2013

De Stultitiâ (updated)

A first fact should surprise us, or rather,  would surprise us if we were not so used to it.  How does it happen that there are people who do not understand mathematics?
-- Henri Poincaré

A major study (NIH grant #44-9035- E) has finally produced statistically reliable results.  As it turns out, a majority of those so handicapped  were dropped on their heads as infants, by careless nursemaids;  the remainder had abused narcotics; and a few folks are just plumb dumb.

Whatever the etiology, this affliction has a scientific name:

Oligophrenia mathematica

(Horresco referens...)

Anthem of the Oligophreniacs:  If I Only Had a Brain

Everyone who has felt himself reach his mathematical frontier, whether at long division  or out somewhere beyond the calculus, must know something of the helpless resentment engendered by the hidden beauty of the abstract.
-- Charles Gillispie, The Edge of Objectivity (1960), p. 188

A lifetime of patient toil  at last has led me to this sad but plain conclusion:
I am a very, very, very  stupid man.
Not across the board, of course, but where it counts -- algebraic geometry, say, or QFT.   And there is truly nothing that I can do about it now -- no more than the proverbial leopard, blushing furiously over his spots.

Technically, the affliction is known as oligophrenia mathematica.  This condition is incurable, though it may be treated symptomatically by getting a doctorate in mathematics from a good school, which helps takes the edge off the more embarrassing symptoms.
It is not for lack of opportunity, or lack of trying.  Indeed, I was enrolled from an early age in a sort of remedial mathematical Head Start:  Gamow’s Mr Tompkins and Abbot’s Flatland  beneath the Christmas tree, along with an erector set for spatial reasoning;  a father who, though no James Mill, at least was comfortable with things like uranium or the complex plane, and did not conceal them from his offspring .  And then in junior high, the New Math -- later much decried, but exactly what a lad needs  should he wish ever to graduate beyond manning a cash-register.
And yet the finest minds at Harvard could do nothing with me, eager but unteachable, as I sat round-eyed at the feet of the COLOSSI -- yea, mighty Gleason, and e’en Quine.  MIT was able to spoonfeed me (a nestling with gaping beak) the rudiments  of Special Relativity and EM :  but no fruits were to bloom upon that grateful ground.  When I later applied to grad school, to obtain what I still think of as a "remedial Ph.D.", I did not see the letters of recommendation that Andrew Gleason was kind enough to write on my behalf, but they probably ran something like this:  “I could make nothing out of this sow’s-ear.  Take him, take him off my hands -- for simple pity !”

The much-put-upon master, administering a well-deserved thrashing to one who *simply will not learn*

At Berkeley, Goldschmidt and Chern  labored in vain to impress anything into my head.   Much later, in a very different context, I actually worked for Goldschmidt, in a cliffside eyrie  packed with glittering mathematicians.   He naturally did not remember me, and I  for sheer shame  did not evoke that earlier connection, which in happier circumstances  we would have chuckled over: 
            “I was the one who sat at the back; 
             I was the dunce of your class.”

Doktorand Justice (sed haud doctorandus) at Berkeley

Forever, alas, must I relinquish the vision, of ever seeing anything like this in print:

Justice’s “Little” Theorem.   Let J be a Justice manifold embedded in DBJ-space;  and P be a Penguin-functor from J into a lattice of monostichs.  We know by the Justice Lemma that the Trinitarian minimal index of such a functor  must be infinite …

[Update:  I have recently scored a triumph in physics  that partly makes up for that lack.]

What, then, is to be done?  For neither may I relinquish nor forget.  Yea, for I have stood on Pisgah, beneath the lowering clouds; and glimpsed bright Canaan, though destined never to tread it;  and the sight shines still in memory.

Perhaps, like that simple Sister who, falling ill with leprosy herself, founded and tended a leper-colony, I might minister to those similarly afflicted.  Offering chatty little -- tatty little anecdotes, about Category Theory or the UMT.  We could meet in church basements, on Tuesday nights.  “Hi, I’m Bob, and I’m a moron.”

Koncentrating kitteh, realleh trying very hard

~      ~      ~

The lamentations above would be no more than maudlin, did they not in fact point to something true about mathematics -- or rather, about the accessibility of mathematics to the human mind.  (This is a different, and a lesser, question, from that of the truths of mathematics themselves.  We discuss the distinction here.)   For the fact is, the stuff is just plain damn hard.  For anyone. 
That even something so basic as counting is unintuitive for the average man, may be seen from the French expressions “aujourd’hui en huit”, meaning… seven days from now; or quatre-vingts-onze ‘four twenties eleven’, for 91  (the parent language, Latin, had done it better;  but it was all too much for the simple Frenchmen to retain in their heads).   Yet mathematical travails persist  far up the totem-pole of experience and ability.

I first had a hint of this  at the end of my undergraduate career.   I had applied, and been accepted, to the mathematics Ph.D. programs at Berkeley and at Stanford;  and, hedging my bets, to a Master’s in something-or-other (I forget what) at the University of Washington, where a woman whom I imagined to be my girlfriend (there I labored under a misconception) was already pursuing graduate studies of her own. 
I was tired.  And in the guise of seeking advice -- in reality, an excuse, an alibi, an out -- I “asked” Professor Loomis what he thought of the idea of taking a year’s respite, and going back to math later.  (I’d actually done something comparable at the end of High School:  Applied to colleges; they said Yes; then I said No, and went to Europe instead -- in what, in retrospect, was a very fortunate move.)  To my surprise, he got a pained expression around the eyes, wincing at the evocation of memories -- he who was a blazing star in the mathematical firmament (particularly among Harvard undergraduates, since he was the Loomis of Loomis&Sternberg, that pons asinorum of all Charles-side math majors) -- and said, with a weary sigh, that he wouldn’t advise it.  And why not? I inquired.  “Because it’s just -- so -- … difficult …”

Compare the testimony of a leading algebraic topologist:

Algebraic topology is a strange and … bewildering field.  The tools used  sometimes look weird, even those that are applied to simple problems.  … The whole field changes radically over every ten-year period, and someone who has been away from it for any length of time  might not understand a single word  if he tries to read a paper.
-- Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 98


Still baffled, the great French mathematician enquires:

And further:  How is error possible in mathematics?
-- Henri Poincaré, quoted in James Newman, ed., The World of Mathematics, p. 2041

The very first time that I trod upon Berkeley math department spaces, as a first-semester graduate student, a sign emblazoned the wall, purporting to be counsel for sectionmen (TA’s), read (poetically)

Insist upon the horror   of the slightest   error


Vergebens, dass ihr ringsum wissenschaftlich schweift:
Ein jeder lernt nur, was er lernen kann.
-- Mephistopheles

Other such testimony, from the eponym of the Mordell conjecture (which resisted assaults for over sixty years) :

I, speaking as a professional mathematician  who has struggled with mathematics  most of his life, would most certainly agree that every aspect of mathematics bristles with difficulties.  …
I am very conscious of many unsuccessful efforts to comprehend fully  and to obtain a mastery of some subjects which have a special interest for me.  My mind seems incapable of absorbing them.  There are a great many loose ends  which I have never been able to tie up. …
It is not easy to concentrate at fixed hours upon difficult mathematics … The brain refuses to function, and one can neither understand nor do anything. …
I have a poor memory, and cannot remember many of my results or proofs, let alone prove them again.
-- Louis J. Mordell, Reflections of a Mathematician (1959), p. 10, 12, 14

From an expert in Hilbert Space:

In 1953  I laboriously proved that the infinite-dimensional case is different;  in that case there exists a commutator C with ||1 - C||^2 =< 0.97.  I keep returning to the subject, but nothing happens.
-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p.

This anecdote is even more discouraged than might be apparent to the layman:  to prove something laboriously is, in the first place, no cause for pride at hard work rewarded:  you always want the final proof to be elegant, and  ideally  to be easy  once you know what to do (things like Cantor’s Diagonal Argument).   Worse, nobody wants to work in infinite dimensions and then come up with a paltry decimal as an approximate upper bound for anything.  (Put all the venom you can into pronouncing that word “decimal”. --  Freudians will further note a remembered echo in Halmos' final choice of words.)


~  Posthumous Endorsement ~
"If I were alive today, and in the mood for a mystery,
this is what I'd be reading: "
(My name is Paul Halmos, and I approved this message.)
~         ~

It behoves a man -- if he is to call himself a man -- to grasp at least a smidgen of Category Theory  before he is gathered to his ancestors  (who, for their part, had of course  not the least idea),  if only enough to decide that it wasn’t all it was cracked up to be after all (such as happens with Hegel and Nietzsche and Sartre, should you survive adolescence  uncorrupted by these). 
Now  I had virtually resigned myself to tumbling head-foremost into the intellectual equivalent of a pauper’s grave, unknown and unlamented, without ever having so much as a Pisgah-glimpse of that fair land.   But just recently, by luck --a possible reprieve, as I happened upon an unusually user-friendly book, Lawvere & Schanuel, Conceptual Mathematics (1997).   Of course, they may manage to remain so user-friendly at the cost of leaving out the hard stuff;  a glance at the index reveals, disturbingly, that they do not reference either adjoints or ultraproducts, two terms that crop up repeatedly in other reading.

But now I read this, which sends me back to square minus-zero :

To assert that topoi correspond to theories  is not to deny that certain topoi may be viewed as models for our logic.  Models may be described syntactically by “diagrams”, so theories may be said to include models.  Our point is that, in general, topoi may be viewed as theories.  In particular, some topoi which arise semantically  are better understood as theories than as models.  Thus we regard topos theory as the “algebraic” form of this higher-order intuitionistic logic.
-- Michael Fourman,  in the introductory first paragraph of “The Logic of Topoi”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 1054

Problem is, I don’t begin to understand that paragraph; worse:  the way that is phrased, I don’t even want to understand it.

A bit of rhetorical analysis:

Usually, in technical writing, you put "quotes" around a word  only if it is
   (a) a bit of informal language that you are permitting yourself, in just this one instance, for convenience;
  (b) a technical term too new to have achieved widespread familiarity, and you are apologising to your reader;
  (c )  a technical term from some earlier stage of the science, which is now discredited.

Neither “diagram” nor “algebraic” falls into this category.  (“Diagram” the less so  as it is one of the primitives of Category Theory, which is the mother of Topos Theory.)
Yet the science of this gentleman has proceeded so far  that such childish terms must drop by the wayside…

[Update June 2014]  I happened to reread this last section just now, and found my then-remarks quite stupid.  I would delete it, or apologize for posting it, except that, after all, the point of this essay is to share with you the wrenching sense of what it feels like to be this stupid, so that you will stop moaning over orphaned teddy-bears in Borriobooola and sending them all your aluminum pull-tabs  and instead contribute to a Riemann-related charity such as the Dr Justice Retirement Yacht Fund (all contributions tax-deductible for citizens of Antarctica, donations in Swiss francs only please).
However, the good or at least less-than-awful news is that, upon mature reflection, the quoted passage about theories and topoi seems much more straightforward and intriguing, then it did back then;  sparking perhaps even   a glimmer of understanding.   In fact, I’d like to retire from my day-job right now and study this subject on my, um, yacht


[Appendix]  Notable laments from other suffering oligophreniacs:

I told him once the story of a surgeon  who said that, if he ever reached the Eternal Throne, he would come armed with a cancerous bone, and ask the Almighty what He had to say about it.  Freud’s reply was:  “If I were to find myself in a similar situation, my chief reproach to the Almighty  would be that he had not given me a better brain."
-- Ernest Jones, Freud: the Formative Years (1953), p. 35

This is not mere posturing.   It is a real problem:  We are   many of us  given just enough smarts to detect a problem, but not to solve it.  Such, alas, when all is said and done, may well have been the case with Freud.

Freud specifically confessed to oligophrenia mathematica:

“I have very restricted capacities or talents.  None at all for the natural sciences;  nothing for mathematics; nothing for anything quantitative.”
-- Ernest Jones, Freud: Years of Maturity (1955), p. 397


Differing from inborn oligophrenia, is what we shall call Mathematical exhaustion, a condition afflicting some of the very best mathematical minds.
The best-known instance of this is Bertrand Russell, who wound up mathematically depleted by his long intense labors on Principia Mathematica, and who thereafter stuck to more general philosophical topics.

James Newman quotes him:

“Neither of us alone could have written the book;  even together … the effort was so severe  that  at the end  we both turned aside from mathematial logic  with a kind of nausea.”

Likewise Lagrange in 1781, after his return to Paris:

Mathematicians thronged to meet him, and to show him every honor, but they were dismayed to find him distracted, melancholy, and indifferent to his surroundings.  Worse still -- his taste for mathematics had gone!  The years of activity had told; and Lagrange was mathematically worn out.
-- Herbert Turnbull, The Great Mathematicians (1929); collected in James Newman, The World of Mathematics, vol. I., p. 154

And this plaint, from the very-brainy  J. M. Keynes:

Anyone who has ever attempted  pure scientific or philosophical thought,  knows how one can hold a problem momentarily  in one’s mind, and apply all ones powers of concentration  to piercing through it,
and how it will dissolve,     and escape …
and you find that  what  you  are   surveying
is a
    blank ……………….
-- John Maynard Keynes, from a biography of Newton, quoted in James R. Newman, ed. World of Mathematics (1956), p. 278

1969.  According to Oskar Morgenstern, Gödel confessed that, after the late 1960s, he could no longer understand the work of younger logicians.
-- Hao Wang,  Reflections on Kurt Gödel (1987), p. xxv

At this point, my sense that I control my subject matter  ends.
-- Thomas Kuhn, “Logic of Discovery or Psychology of Research?”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 21

Thus far we have offered only  the perspective of the minus habentes.   But what of their poor, much-put-upon  magister?
Here, Eric Temple Bell, translating Gauss (the Great) as teacher:
This winter I am giving two courses of lectures, to three students:  of whom  one  is only moderately prepared;  the other less than moderately;  and the third lacks both preparation and ability.  Such are the burdens of a mathematical calling.

Pupil’s plaint

Regrets, Professor Unrat;
I cannot prove the lemma.
Nor should you put us to the test
so soon in the ack emma.

[Second appendix]  For a sotie on this theme, click here:

[Third appendix]

A usually Paris-based mathematician and satirist writes:

René Thom -- a Field medalist  and a great mathematician -- acknowledged that he left pure mathematics because he was oppresssed by Mr Grothendiek’s “crushing technical superiority.”
-- David Berlinski,  “Inside the Mathematical Mind ” (2007), collected in :  The Deniable Darwin (2009), p. 499

[Fourth appendix]  Although nescient to an extent that beggars description, I did manage to do alright -- not great, but alright -- in Harvard’s undergraduate pons asinorum, Math 55 (1966-67):  the which course, I now learn, is fabled, with its own Wikipedia entry.  Here we learn:

In the class of 1970, only 20 of the 75 students who began the class finished it due to its difficulty.  Similar drop-out rates were true for the class of 1976: "Seventy started it, 20 finished it, and only 10 understood it."

I actually did understand it, but that was thanks to Andrew Gleason, who taught it in alternate years.  The other years it was taught by Lynn Loomis and Schlomo Sternberg.  I subsequently took Real Analysis from the former, and learned little;  and a very etherial meta-eka-hyper-dynamics from the latter, where I understood not a word.  (I was too ashamed to admit this;  but one day, a couple of months into the course, one fellow did speak up:  “I’m just not understanding any of this!” 
I leaned over to the student next to me and whispered:  “Who’s that?” -- “That’s Professor So&So,” he replied.)

[Fifth appendix]   From a lecture by Professor Messing in Princeton, March 2002, in allusion to Robert Langlands of the Princeton Institute for Advanced Studies:

Langlands, with characteristic humility, wrote:  The virtue of Dieudonné theory  is that, for mathematicians of middling ability, it lets you translate difficult problems in abelian varieties  into straightforward problems in linear algebra.

I can well believe this.  I personally attended Professor Langlands IAS lecture series (autumn 1999);  in the first of these he stated that he'd wanted to be a physicist, but physics was "too difficult", so he had to settle for being a humble mathematics professor at the Institute for Advanced Studies.   Nor was this a pose;  his whole manner is that of straightforward humility, very … Canadian. 

John Conway, one of the most brilliantly elflike of mathematicians, confessed to a similar trajectory, during his time at Cambridge.  In a lecture at Princeton University (17 XI 1999) he confessed:

I studied Quantum Mechanics with Dirac. Quantum Mechanics is hard to understand, even when you can answer the questions on the exams.  And I couldn't answer the questions on the exams anymore."

After his lecture, I went up with him to his office, and the sense that his almost Franciscan combination of humility and playfulness  was confirmed.   His small and crowded office (Princeton has an enormous endowment;  this is the way they treat their stars?)  was a four-dimensional playpen (of which, with my limited vision, I could perceive only three) of mathematical mind-toys, stacked up here and there, hanging from the ceiling, projecting from the walls …

[Sixth appendix]

Alex Masters, The Genius in my Basement  (2011)  [a book that treats of the  downfall of Simon Norton, once a key collaborator of John Conway]:

At Trinity College in Cambridge, there was a man who scored a double first in his undergraduate degree, took his PhD in the flash of an eye, and still gave up in despair  and became a tuba player.

How much more poignant, with the humble, doleful tuba, rather than piano or violin!
[Seventh appendix]   On a related note:
In the moving coda to her splendid examination of the meteoric careers (or perhaps, fireworks-like, including the eventual fizzle) of a pair of brilliant (counter-)Freudians, In the Freud Archives,  Janet Maslin, all passion spent, has a last interview with the man she has so closely followed, and not-unsympathetically depicted (and who would later sue her), Jeffrey Masson.   He says this, he says that;  and then she says:

“You know, as you’ve been talking, I’ve had the feeling that you’re bored with what you’re saying.”

The prodigy, his own passion likewise depleted, concedes that this is so.   He has seen farther than others, and has already said, and re-said, all that he has to say.  And then adds, fatefully:

“For people who are truly smart, like Bob Goldman, there really isn’t much they want to do, or can do.  The truly smart people seem to do less and less.  It’s terrible.  The dull have taken over.”

This chilling observation put me in mind of my best friend in high school, Ted Franklin, whose gifts in math and physics  lay far beyond my own.   I had an excursus in Europe before following him to Harvard, shortly after which time he had dropped out.   I did get to room (informally) with his erstwhile roommate, the brilliant math major David Collins -- who, however, himself promptly parachuted out of the university -- leaving the undergraduate mathematics to the tortoise, myself.  Both of them went on to quixotic social projects, and then I lost track.   Somehow neither one managed to stay sufficiently interested  to wind up doing anything intellectually really interesting.   They thought of themselves as rebels, but in that way they were more like Edwardians.   Leaving the field to us dullards.

How valid Masson’s plaint may be, in psychology or philology (two fields in which he greatly distinguished himself), I do not know.   But enormously smart people are doing deeper things than ever before, in mathematics.   They are not bored.

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