We have, to our lasting pleasure and instruction, been
re-reading some of the works and class-notes of our former teacher Andrew
Gleason, several of whose classes we took as an undergraduate. For better or worse, my own path --
like some ill-behaved series -- soon diverged, away from mathematics; yet
eventually re-united in a sort of spectral intersection, when, in the wake of the
toppling of those cloud-crowned towers, I spent several months helping out at a
mathematical think-tank in Princeton, a sort of members-only affair, and was
surprised and pleased to find his name on the list of coworkers of that
Institute. It is a matter of
lasting sadness that I was never able to show my appreciation for his keen and
patient teaching, whether by proving any theorem of my own, or contributing to
any sort of Festschrift; and now
he has left this vale forever.
Assuming, however, that he still keeps up with the field, we offer at any rate

__this__, the roster of essays in which strands of his thought have been woven into my own.1921 - 2008 |

~ ~ ~

Andrew
Gleason’s brother was Henry Gleason, a linguist whose introductory text
was a standard, and which I read without awareness of the fraternal
relation. As befits the favored Gleason genes, Andrew himself was
linguistically percipient. I’ll quote a couple of his semantic obiter
dicta, since these are peripheral to our mathematical essays and are
not cited there.

~

In
observing a fine distinction, he is wont, not simply to do so tacitly,
but to note the fact explicitly. For S a metric space:

If we refer to the

*subspace*T, we are focusing our attention primarily on T itself. If we refer to the*subset*T, we are considering T in relation to S.
-- Andrew Gleason,

*Fundamentals of Abstract Analysis*(1966), p 226
…
a rotation of the plane about the origin through an angle θ. (More
precisely, through an angle of measure θ. We often ignore this
distinction to reduce the complexity of our sentences.)

-- Andrew Gleason,

__Fundamentals of Abstract Analysis__(1966), p. 335
Andrew Gleason,

*Fundamentals of Abstract Analysis*(1966), p. 3: "Note that "1 + 4" is not a command to execute a certain addition, it is a notation for the result." Here one might harbor reservations; but in a more substantive context, he makes the necessary distinction: on p. 83, he distinguishes a*formal*infinite series (which exists simply by definition, and is an omega-tuple) from its*sum*if this exists, which is a single scalar. This distinction is reminiscent of the*actio*/*actum*dichotomy, familiar from linguistics.
Andrew Gleason,

*Fundamentals of Abstract Analysis*(1966), p. 61: "There has been a tendency to reserve the word*theorem*to describe results of importance; lesser statements are called*propositions*."
In this he is followed by others:

I. Stewart & D. Tall,

*Foundations of Mathematics*(1977), p. 138: "We prefer to use the word 'proposition' to describe ordinary run-of-the-mill results, reserving the word 'theorem' for something more important."

More original and far more important conceptually is his (proposed) distinction between

*postulate*and*axiom*; for which see the essay Axiomatics in its Element,*in fine*.
~

He is quick to notice ambiguity:

Andrew Gleason,

*Fundamentals of Abstract Analysis*(1966), p. 3: "We have then renounced the use of the word*equals*in any other context than identity. In classical geometry the word is used to mean variously 'has the same length as', 'has the same area as', or 'has the same volume as'; all such uses we forego."
Defining (p. 239) a subset as

*nowhere dense*in a metric space iff the interior of its closure is empty, he adds: “Such a set was formerly said to be*nondense*. Since there is a danger of confusing*nondense*with the much weaker*not dense*, the term*nowhere dense*is preferable.”
More subtly, he notices (p. 247) that calling a function

*uniformly continuous*on a subset E of a metric space S is used in two different senses, depending on whether, for each point of E, the function preserves nearness to other points*of E*, or (a stronger condition) of all of S.
It
is only because mathematics itself is incomparably capable of
precision, that we can pry apart such semantic entwinements. Its
study, accordingly, is commended to all budding semanticians.

~ ~ ~

I
took three courses from Gleason -- thrice what I would take from anyone
else --: Advanced Calculus, Group Theory, and Complex Analysis.

The first -- the celebrated Math 55
-- was very much a course for math majors in the preferred Harvard
mold, with great reverence for generality and abstraction. That ethos
-- along with an implicit Platonism -- I absorbed in that class, though
such agendas were never discussed explicitly. The resulting heady
culture I have described

__here__.
The two-year sequence for (pure-)math majors, of which the first was Math 11 (described

__here__) ran in parallel to a three-year sequence of Beginning, Intermediate, Advanced, covering roughly the same material though from a less ethereal standpoint, aimed at -- whom? I didn’t know anyone who took those courses (with one very eccentric exception, the “Bah” man, who as a freshman went straight to the Advanced; tell y’bout’m sometime, mebbe). Engineers, truck drivers? Who knows.
I
adored the course, and the man who taught it. (He taught it every
other year, the alternate years being taught half by Lynn Loomis, half
by Shlomo Sternberg.) And it was, to a degree I did not realize at the time, a

*pons asinorum*. I had come from an almost purely literary and linguistic background, and turned my path towards the sciences in a spirit of genuine humility (and initially, as a chemistry major, even of penance); so far as I recall, I was reasonably satisfied with the B+ I received in that class. Yet a couple of years later, talking with my friend and fellow math-major Haynes Miller (who had graded the homework for Math 55 one year), I casually mentioned the fact, and his eyes bugged out: “You got a*bee-*in Math 55?!?” And then, settling back into emotionless objectivity, already losing interest in my existence, stated something that shocked me, but which was in fact a fact:**plus**
"Your career is over."

~

The
one-semester, undergraduate-level course in group theory was -- despite
the reigning Cantabridgian abstraction-fetish -- by far the most
concrete and hands-on of any math course I met. It was there that I
encountered the art of Escher, and learned its appeal to mathematicians.

Complex
analysis, alas, is just a blur, for by then the campus had been thrust
into political turmoil. Most professors chose to roll with the
punches; and Gleason, with a shrug of helplessness, announced that the
course would be pass/fail. Somehow it felt like giving up; of the
courses I took that year, in that atmosphere, none really worked out. --
Quine (in his Introduction to Logic -- Phil 140) took the opposite
approach, defiantly not only maintaining grades but posting

*weekly*statistics ranking each individual student (I almost wrote “pupil”) on his precise rung of the ladder of his superiors and inferiors: a kind of local schoolmasterly application of the principle of the Scala naturae.
~

A note on

__Fundamentals of Abstract Analysis__, Gleason’s (unfortunately) only book.
It
is still in print, though it’ll set you back eighty-nine smackers on
Amazon. (Cheap at the price; my own publisher brought out

__The Semantics of Form in Arabic__at over two hundred bucks). On Amazon, there is only one review, from someone in the Philippines who writes, surprisingly, “Fundamentals of Abstract Analysis was our prescribed text in my Advanced Calculus class.”
Now, unlike many go-getters
who assign their own books, and although his advanced calculus class
came only a couple of years after the book was first published, Gleason
did not assign his own. We used his handwritten (sic -- longhand)
mimeographed notes, alongside the austere, elegant, and not very
pedagogical Loomis & Sternberg text (more like a

*pons angelorum*, with cherubim tumbling off on every side). He occasionally referred wistfully to “FAA”, as he always called it, like a nickname for a loved but not very successful child; one gathers it had not flown off the shelves. And it isn’t really a calculus course, but rather the skeleton key to all the courses you will later take, focussing on ideas and (to a lesser extent) techniques. It is a very good book.
~

(For further portraits of mathematicians -- as opposed to mathematics itself -- click here.)

Update 10 March 2013]
Today I re-read my notes from Math 55 (Fall ’66 to Spring ’67) -- alas,
without profit. One phrase that
kept popping up was “director space”. Apparently Gleason was not widely accompanied in his
enthusiasm for this term, alas;
Google search:

No results found for

**"director space" "andrew gleason"***Ubi sunt*…

[Footnote 2]
Our bow to Gleason’s semantic precisionism is not by way of fetishizing fine distinctions. These, per se, are a dime a
dozen. I recently attempted to
read Bourbaki’s

__Théorie des Ensembles__(1964), and was immediately met with a flurry of linguistic precisionism:
(p. 15)

**Par abus de langage**, la*valeur f(x)*de*f*pour I s’appelle aussi*image de x par f*.
(p. 16)

**Par abus de langage**, on écrit aussi*f(-1)(x)*au lieu de*f(-1) ({x}).*
The problem is, these nitpickings come before anything
whatsoever of substance has been introduced: we have as yet no reason to even

*care*about such prickly niceties. Gleason, by contrast, mentions his terminological observations as dessert to the meat.
Compare our assessment concerning axiomatization

*a priori*vs*a**posteriori.*
[Update 14 Dec 2013]

That is, it is a statement that one needs -- perhaps,
indeed, at every turn, so that in a sense it may even be fundamental -- but
whose truth is utterly unsurprising, and whose proof involves no interesting
insights or techniques.
During the time he paused at the blackboard, he probably (like John von Neumann summing an infinite
series in his head) instantly proved the proposition in his own unconscious.

Since I lacked any spark of mathematical creativity (this
sad fact only became apparent to me later), though otherwise technically
proficient, I seldom participated in the classroom in any active way, even to
ask a question. I sat towards the
back, took copious notes, and tried to follow the arguments as best I could. Yet one day, in Gleason’s undergraduate
Introduction to Group Theory class, something puzzled me and I did speak up. The group operation is required,
by fiat, to satisfy an Associative Law -- but how, in the actual case before us
now, did we know that the operation in question really did associate, in every
case?

My shy query did not, we may say, turn out to open up new pathways
for research in mathematics; the
great professor did not gape and slap his forehead and cry out “My God! This casts abstract algebra in an
entirely new light!”; but nor --
and this was more surprising -- did I receive, in this instance, a satisfactory
reply. For Gleason,
interrupted at the blackboard, suspended amid his lecture chalk in hand, found

*the question itself*… puzzling. He shrugged, he really didn’t know where to begin. “It’s …*obvious*,” he said finally, giving up on me, and, brushing the dust from his sleeve, resumed the lesson.
Now, this hapless anecdote -- which, for shame, I have never
mentioned previously to anyone, before this very date -- bids fair on the face
of it to be booked beneath the scarlet rubric of

*Oligophrenia mathematica*, which I have treated at sorrowful length in the essay “__De Stultitiâ__”. And yet some recent reading frames the matter more sharply, and recalled the anecdote to mind.
The first was an article about matrix groups like GL

_{n}, which did not assume that matrix multiplication is associative, yet nor did it bother to prove it in any straightforward calculational way (this can be done, but is very messy), but said that*since the matrices represent linear operations*on a vector space, their associativity follows from the associativity of composition of the operations that underlie them. Now, that is a thought with some content.
The second passage, which really nails the matter, comes
from Tim Gowers’ lucid and insight-packed introduction to his collection of
articles surveying all of mathematics.
He observes:

The associative law [says],
informally, that “brackets do not matter”. However, while it shows that we can write x * y * z without
fear of ambiguity, it does not show quite so obviously that we can write a * b
* c * d * e, for example. How do
we know that, just because the positions of brackets do not matter when you
have three objects, they do not matter when you have more than three?

Many mathematics students go
happily through university without noticing that this is a problem.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 73
So, to return to the tableau in which Gleason is frozen in
stupefaction behind the lectern, while the dunce of the class blushes and
desires to be mapped to the empty-set

*instanter*: It is obvious, to any practiced hand,*how to prove*the proposition (by induction, one supposes, on the length of the string), but the proposition itself, as Gowers observes, does require such proof, since from the simple “(a * b) * c = a * (b * c)” we are now asserting an analogue (which, even to state, requires some symbological ingenuity) for an*infinity*of cases.
Now, Gleason himself was perfectly familiar with all this,
so how did he not recognize that what I had asked was not actually such a stupid
question? And the answer is
now plain: It was not a

*stupid*question, but it was a*boring*question, in a precise sense, which Gowers addresses on the same page, in light of this very example. In the course of a straightforward précis of the meanings of the terms*theorem*,*proposition*,*lemma*, and*corollary*, he puts forward this epigrammatic distinction:
A

**proposition**is a bit like a theorem, but it tends to be slightly “boring”.
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