Andrew Gleason (in memoriam)

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

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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.

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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.

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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."

(Incidentally, the sentence-cadence here  is quite Quinean.)

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.

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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-plus 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:

"Your career is over."

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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.

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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.

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(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] 

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”.

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.