Adventures in Juvenile Lexicography

Katherine Nelson, in Keith Nelson, ed., Children’s Language (1978), p. 66,  offered the following glimpse into the orismological instincts of budding lexicographers.  Asked a “What is it?” question about the word tiger (“a large, fierce, flesh-eating animal (Panthera tigris)…” to you and me), the tots responded:

(1) “at the zoo”
(2) “animal”

(3) “it’s like a lion:

(4) “lives in the jungle and runs a lot”

(5) “animal with stripes and it eats a lot of things”

(6) “to run”

(7) “someone growls”

(8) “hair on its head”

I then polled our son (aet. su. 4 years,  3 months), who offered this:

   “It’s something that is big, and it eats people, and it runs around in the jungle.”

His assessment of her other examples:

apple: “it’s juicy; it’s big; it’s round”

car:  “Something that’s big, and not so tall -- it’s this tall [shows with his arms];  it can kill someone that stays in front of it, if it’s moving.”

coat: “It’s something that keeps you warm, is big and sort of smooth, and has little furry stuff”  [Note:  Our family was at that time facing an Edmonton winter]

bed:  “It has legs, or doesn’t, and it has a pillow, and it stands up on its posts”.
[Note: That first idea in the definiens, at once oddly precise and maddeningly vague, probably meant:  “Prototypically a bed has legs (the ones you see in books), but ours doesn’t” -- the family, indeed, then in exile and furniture-poor, slept on a mattress on the floor.]

Striking  is this repeated note of ‘big’, present in every definition except the last:  extending even to the humble apple -- even a baby is bigger than an apple, let alone a robust four-year-old.   But in light of the lad’s subsequent specialization in differential geometry, an explanatory hypothesis presents itself.  What may well have struck him was the apple’s unabashed convexity -- round, not like a thin dime, but round all around:  having everywhere positive and (roughly) constant Riemannian curvature, as he would no doubt rephrase the definition upon more mature reflection.  Such an apperception of an apple was indeed the Eureka moment of the founder of differential geometry, Carl Friedrich Gauss, as depicted in the movie “Die Vermessung der Welt”.

"Die Vermessung der Welt"

 Gauss saw things in terms of sheets embedded in three-space, the natural abstraction from his experience as a land-surveyor.  Riemann shifted the view to that interior to the space.

-- John Derbyshire, Prime Obsession (2004)

In an earlier essay,  we exulted (as sober man of science) and lamented (as blogging satirist) that there exists very little indeed by way of “math porn” (in the non-sexual journalistic sense), as against “physics porn”.  However, we did come across a (rather pale and marginal) example of that seldom-met genre, and now share it with you here.

I recently attended a screening of the 2012 German film “Die Vermessung der Welt” (literally geo-metry, in the etymological Greek sense) presented to a select circle of the deutschgesinnt.  What moved me to tear myself away momentarily from my usual daytime occupation of hand-to-hand rooftop combat with our adversaries, was the fact that the film was billed as a sort of dual biography of two scientific figures very much worth biographizing:  Alexander von Humboldt, the naturalist brother of the philologist Wilhelm, and Carl Gauss, probably the greatest mathematician who ever lived.   We watched it off a DVD;  had I known that the original theatrical version was 3-D, I would have lowered my expectations accordingly.  (Actually, 3-D could be put to very good use in the exploration of the Gaussian geometry of manifolds, but that was not its use here.)

The movie begins, as all Gauss sagas must, with the tale of how the young schoolboy, given a pensum  along with his fellows  of reckoning up the sum of the integers from one to a hundred, shot back an answer instanter, by finding a clever shortcut, rather than, as John von Neumann would have done, simply adding the series instantly in his head.  (That’s a joke.)   Our eighth-grade algebra class was regaled with this (and wisely so;  it’s one of the few things I remember), and our son, in the Princeton Friends School at a tender age, was instructed in the same as well.   But in the film, the anecdote was given what I take to be a possibly Germanic twist, for the scene opens with the explicitly filmed rhythmic  

    thwack,

                     thwack,

                                      thwack

of a supple and vicious-looking cane upon the bared buttocks of a lad of around nine;  graphic enough as it was, but probably even more disquieting in a theatre, with Surroundsound and 3-D.   Somehow, this sequence alone marked the movie out as not of American provenience -- here, you might be sent to prison for even watching it.  (Later, after Gauss has solved the arithmetic problem, the scene is repeated with Gauss as victim, for any viewers who didn’t manage to come to climax during the first sequence.)

So:  a rather pornographic presentation of what was in reality an utterly asexual and indeed incorporeal milestone in the annals of mathematical awakening.   But as long as we’re here, let us dwell -- as the film alas did not -- for just a moment  on the math part.

That sequence 1 + 2 + 3 + … + 100   equals, as it happens, 5050.  That fact is of no mathematical interest whatsoever, but belongs rather to the Museum of Particular Results.  (We presented a jolly fable of this notion here.   Be sure to click on that essay, it’s full of woodchucks.)   Of marginally more interest is the shortcut found by young Gauss:  pair the outermost integers in turn and you get 50 × 101.   That is clever enough;  but at the lowest level, it might be simply one of an unrelated jumble of dodges used by a Calculating Idiot-Savant, and thus belong to the Museum of Particular Tricks, just one step up from brute-force addition.  A significant step up from this recognizes that the trick is (with some tiny extra cleverness) generalizable to any sequence 1 + 2 + … + n.   Now it has risen to the level of a general trick, and thus belongs to the Museum of Particular Algorithms.   But then this finding generalizes to the idea of summation-formulas überhaupt:  for instance the sum of the squares of the first n integers, or the cubes, or any power.   The resulting infinite collection of formulas belongs to the Museum of Particular Strokes of Genius.   Striving to generalize these, you eventually wind up with Analytic Number Theory, and its sought-after crown jewel, the Riemann Hypothesis;  which is where things stand today.

None of this is even hinted at in the movie;  but really, such a development is the only reason to treasure that Gaussian anecdote:  otherwise the whole thing can seem a mere transient bit of precious cleverosity -- as it did (in the film’s telling) to Gauss’s schoolfellows, who give him a beating for his trick, and no doubt to the bulk of the audience.   And this is the “math porn” aspect of the presentation:  Even in the absence of any mathematical understanding whatsoever, we spectators are nonetheless supposed to be tremendously impressed with young Gauss, who is presented as a romantic and tragic figure, his attraction being thus, not Gaussian, but Byronic.

A superb mathematician -- and you can take that to the bank!

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

http://www.linguasacrapublishing.com/justice.html

*

~

Thus far, the perspective is that of male narcissism.   That stance applies as well to the portrayal of von Humboldt, as he goes stalking about the Amazon in his seven-league boots, freeing slaves as he goes, and making immortal discoveries.   Both portraits involve a certain taste of algolagnia (in the naturalist’s case, it involves his naked back and an electric eel -- all for science, you understand).   There is nothing explicitly homoerotic, though perhaps a touch of a repressed version of that, by implication, when von Humboldt goes apeshit upon discovering his handsome French traveling companion  dallying with a local squaw.

Subsequently, the movie tosses a bouquet in the direction of unearned autogynophilia as well, in the incident of young-man Gauss, still all sturm-und-drangy, brought wisdom from the Tree of Knowledge by a chance remark of a comely though uneducated Fräulein  posing Evelike with an apple.

Mathematically, the scene will almost certainly have soared over most of the audience’s heads.   Gauss chats about measuring the Earth (Vermessung der Welt) by adding up triangles;  the lass objects that the Earth is not flat …. (not a Euclidean surface, as we say in the trade) … portentous pause … Gauss, reflecting, says, Well, you’d need lots of leeetle weeentsy triangles (infinitessimal, mathematicae linguâ).  She sensuously/attentively pares the apple;  and the penny drops, the scales fall from his eyes, and he rushes off to scribble calculations.

What just happened -- and the viewer may well be excused for having missed it -- is that Gauss has (apple-prompted, like Newton and gravity) just discovered Gaussian curvature, differential geometry, and much of modern mathematics.  This episode will be utterly opaque to anyone coming fresh to the movie;  apparently, we are expected to have read the book, as with the Harry Potter movies (the latter of which were incoherent, and would have baffled anyone who hadn’t already read the series).  Which, indeed, the director had cause to suppose, since the movie is based upon a novel that was a humongous German bestseller.

*     *     *

~ Commercial break ~

For a mini-movie of our own, try this:

The Penitent and the Impenitent Thief

We now return you to your regularly scheduled essay.

*     *     *

The depiction of the wisdom-from-the-mouth-of-babes Mädchen is likewise Byronic -- namely it recalls Lord Byron’s daughter, Ada the countess of Lovelace, an associate of Babbage.  She has been exalted by those who go hunting in history for neglected heroines;  a summary can be found here:

http://en.wikipedia.org/wiki/Ada_Byron#Controversy_over_extent_of_contributions

In fact, though, Ada is a reasonably admirable and realistic role-model for girls, since she did hang around a really smart guy and did work hard and did achieve some understanding if not any actual original results, which is all that most of us can ever hope to do.  Gauss, by contrast, is no role-model at all, for anyone, since none of us have been born with his genius, which is almost unexampled in history.  Indeed, for any actual stellar mathematician, his example is yet worse, since he was notorious for hoarding results.  Hopeful young mathematicians would make the pilgrimmage to Göttingen to present their results (much as young Gauss himself is shown as doing, in a singularly infructuous interview with Immanuel Kant), only to be told that he himself had discovered those results long ago, and had them in his drawer, but had never bothered to publish them. (His dismissal of Bolyai in this regard  is notorious.)

Lagniappe:  Mathematically inclined lasses seeking ipsigeneric role-models would do better to follow Noether, Kovalevskaya, Julia Robinson, or Ingrid Daubechies.  Though, once you reach that level, you have come to realize that pure mathematics is entirely genderless, and even (so we have argued here and there in this series of essays) extraspecific.

Note:  Eventually, after an hour or so, weary of its pieties, and disinclined to take in  yet another sex scene (Memo to directors:  That is not why moviegoers flock to a film about mathematicians and scientists), I walked out.  So maybe I missed some dazzling final mathematical exposition.  But I doubt it.

~

Lesen Sie die Geschichte  spesenfrei !

“Der Detektiv und das Mädlein”

~

[Footnote:   For another psychologically attuned analysis of movies, click here.]

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

Ich bin so dumm, du bist so dumm,

wir wollen sterben gehen, kumm!

 

-- Ein Esel, einem anderen Esel;

per Christian Morgenstern

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

A sample attendee would be Einstein’s collaborator Infeld, who confessed in his autobiography:

The diagnosis was: “Geistig minderwertig.”  I was feeble-minded.  My mental level was depreciated below the level required of the Austrian soldier.  One of the symptoms, according to the report, was that I had a smooth tongue, without lines.  The name for it as “Idiotenzunge”.

-- Leopold Infeld, Quest (1941, 1965), p. 81

Indeed, his buddy Albert might tag alone as well:

Learning differential geometry  was not an easy task for Einstein.  The spirit of the subject was alien to the intuitive physical arguments.  … “In all my life, I have never struggled so hard.”
-- Kip Thorne, Black Holes & Times Warps (1994), p. 114

~      ~      ~

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: "

I Don't Do Divorce Cases

Murphy on the Mount.

(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

[Appendix]

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:
http://worldofdrjustice.blogspot.com/2011/02/excelsior.html

[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 Malcolm, 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.

[Post scriptum:  Title minus the ablative hat, zwecks stringmatch search:  De Stultitia.  It's Latin.  It means '(a treatise) on stupidity'].

[Eighth appendix]  Yet further  candid confessions  from top-flight math-mavens:

 [Oppenheimer’s] first lecture at Caltech  in the spring of 1930  was a tour de force -- powerful, elegant, insightful.  When the lecture was over and the room had emptied, Richard Tolman, the chemist-turned-physicist  who by now was a close friend, remained behind  to bring him down to earth.  “Well, Robert,” he said, “that was beautiful, but I didn’t understand a damned word.”
-- Kip Thorne, Black Holes & Times Warps (1994), p.188

I am out of sympathy with the extreme formalism of the Peano-Russell school … My repeated efforts to master their involved symbolism  have invariably resulted in helpless confusion and despair.

Tobias Dantzig, Number, the Language of Science (1930; 4^th edn. 1959)

To this day I cannot read “how to” instructions in printed form.  Psychologically, these are indigestible for me.

Stanislaw Ulam, Adventures of a Mathematician (1976), p. 53

A psychiatrist was no help.  Neither was Niels Bohr, who asked [Robert Oppenheimer] whether his problems with theoretical studies were mathematical or physical.

“I don’t know,” Oppenheimer admitted.

“That’s bad,” Bohr said flatly.

-- Daniel Kevles, The Physicists (1978, repr. 1979), p. 217

My own confession is even more embarrassing.  If my memory is correct, a few years ago, when I first worked on the problems of this paper, I found some interesting examples.  Now I find myself unable to recall either the examples or their proofs.

-- Saul Kripke, “Is There a Problem about Substitutional Quantification?”,

in: Evans & McDowell, eds., Truth and Meaning (1976), p. 401

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