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.”

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 …

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,

-- 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

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).

*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 G____ödel__(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]

[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.

[Post scriptum: Title minus the ablative hat,

[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,

-- 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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