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.

~

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."

(Incidentally, the sentence-cadence here  is quite Quinean. Quine and Gleason had offices rather near each other, at Harvard.  Seldom, and with trepidation, but not never, did I tread either.)

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-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."

~

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] 

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.

Axiomatics in its Element

[Further thoughts along the lines of the thought-themes treated in the essay that begins here.]

John Locke, An Essay Concerning Human Understanding (1690), §IV.ii.8:

The necessity of this intuitive knowledge, in each step of scientifical or demonstrative reasoning, gave occasion, I imagine, to that mistaken axiom, that all reasoning was ex praecognitis et praeconcessis;  which  how far it is mistaken, I shall have occasion to show  more at large, where I come to consider proposititions, and particularly those propositions, which are called maxims;  and to show that ‘tis by a mistake, that they are supposed to be the foundations of all our knowledge and reasonings.

Quite otherwise is the role of axiomatics in Set Theory.  Here, the axioms are once again not simply arbitrary -- they always have some prior empirical motivation.  But the consequences of assuming any new axiomatization  are initially quite obscure, and have turned out in many cases  to be enormous (or even fatal).  Hence  taking on a new axiom, or retwiddling an old one, is more akin to creating a new universe for exploration, than tidying up our formal description of the old universe.

In physics, if your axiomatization results in predictions refuted by experience -- so much the worse for that axiomatization.  Its role is not truly foundational, it’s mostly just there for show.

In logic and set theory, the case is much more complex, requiring qualitatively deeper criteria for analysis.  You wind up with logically distinct and often incompatible theories of these utterly fundamental subjects.   (Or, such we had taken them to be.  If there exists no canonical description, is their foundational character impugned?)

The proliferation of spaces (finite-dimensional and otherwise) and of surfaces or, more generally, topological objects within them, is all good clean fun -- like discovering new fauna in newly-explored islands.   But to have incompatible characterizations of logic is more like being suddenly troubled by what actually counts as a life-form.  In our own time, the biologist has had actual glimpses of such worries, in the shape of empirical discoveries not known to earlier centuries, such as viruses (alive? not alive?), and of a continuum of creaturely objects of uncertain individuation:  ant colonies; slime molds; clonal species of certain trees and fungi; and, on another dimension, organisms that divide by mitosis, or that clone themselves;  not to mention the newer view that individual genes are the fundamental vital entity, organisms being just their carrying-cases.   Now add further discoveries, such as Quantum Cats -- separate and independent twins, yet joined in an Einstein-Podolsky-Rosen way, so that they are actually not independent at all.   Or the planet in Solaris, a single organism;  and  distant cousin, the self-aware Oort Cloud.  Or:  certain subsets of integers, like the fateful one in the TV series “Lost”, with malevolent propensities of their own, and measurable biological effects.  -- The consequences for our conception of mathematical reality are quite as drastic as that.

The result is highly uncomfortable for a Platonist.  It seems as though we are coming after all  to the sterile game of creating arbitrary universes at will -- lifeless and pointless entities, with no connection to the Real (or the divine) -- a nightmare of sterile atheism.  Yet here we see one who did not flinch.  John Dawson (Logical Dilemmas, p. 175) quotes Gödel 1946 on the subject:

He asserted that, even if some new axiom ‘had no intrinsic necessity at all”, its truth might come to be accepted inductively  [dbj: Note, actual “truth”, not mere “usefulness”], on the basis of its “verifiable’ consequences” (those demonstrable without the new axiom, whose proofs by means of the new axiom are considerably simpler and easier to discover”).  Indeed, he declared, “There might exist axioms  so abundant in their verifiable consequences, shedding so much light upon the whole discipline, and furnishing such powerful methods for solving given problems… that they would have to be assumed… in the same sense as any well-established physical theory.”

The style of reasoning here strikes me as theological, quite in the tradition of those views that saw the universe, in all its parts, as having been created just so, for the well-being of God’s favorite creature, Man.  To find such an axiom as that, must indeed strike one as finding a kind of key (like that proverbial watch encountered on the empty strand, left there by the Watchmaker…)

            For:  The invention of the vacuum-cleaner certainly added spring to the housewife’s step;  but we do not therefrom conclude as to the nature of The Vacuum.    By contrast, the mathematical lawfulness of physical phenomena, in a myriad of ways, has indeed impressed observers as saying something about the Universe itself.  In similar fashion, should some axiom (with its associated methods) prove to open up, at a stroke, whole vistas of the already otherwise-intuited invisible world, we would feel it had been left there for us:  that it is as real as rocks.

~

In Modern Philosophy (1994; p.99) the philosopher and theist Roger Scruton  lists some familiar foundational principles of Truth Theory -- things like “If the sentence ‘p’ is true, then so is the sentence ‘ “p” is true’; and vice versa -- but calls these platitudes.  Very nice.  Putting axiomatics in its place.

Another way of putting them in their place  is to observe that no determinate character of ‘axiomaticity’ inheres in any one of them -- a salient illustration of Quine’s point in “Two Dogmas of Empiricism”.  Thus, from a very straightforward and standard textbook, with no philosophical (let alone relativist)  axe to grind:

What is axiom and what is theorem  is often an arbitrary choice, since one is often able to derive each logically from the other.

Sometimes one can go a step further, and make all the axioms of a mathematical system into theorems  by basing the system entirely upon some other system.  Thus, analytical geometry makes it possible to base all of plane geometry upon the properties of the real numbers.  This is also possible with R itself, throwing it back onto modern set theory.

-- Creighton Buck, Advanced Calculus (1956, 3^rd edn. 1978), p. 57

You sketch my hand,  I'll sketch yours

~

Andrew Gleason, by contrast, acutely proposes a distinction between postulates and axioms,  restoring lustre to the latter.  The distinction only makes sense  on a Realist account of mathematical truth :

When we leave the domain of abstract configurations, what we call postulates  take on a different significance.  In the abstract domain  we are in charge; we can frame postulates as we please  and simply exclude from consideration  configurations which fail to satisfy them.  But we cannot take this attitude toward concepts which have any sort of independent existence.  What are appropriately called postulates in the context of configurations  become axioms when we deal with independent conceptions.  They are axioms because they are accepted as true, or at least granted for purposes of argument, for intutitive reasons.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Such a conceptual armature  in effect brings mathematics into comparison with both theology and physics, while increasing the contrast with both postmodernism and finger-painting.

Gleason is not here reporting settled usage, but putting forth a semantic proposal, one of répartition or desynonymization.  As he states earlier, anent the defining properties for an ordered set:

The conditions appearing in the definition are called the axioms or postulates for an ordered set.  The word axiom has long carried the connotation of being self-evident, but it is hard to find a sense in which these conditions are self-evident.  The word postulate (from Latin postulare, to demand) seems more appropriate.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 59

It is, incidentally, refreshing, to hear a mathematician who helped settle one of the Hilbert Problems  freely concede that a (not specially complex) set of condititions  is not, in fact, self-evident.
(In the classroom, he was very much like that:  No trace, either of arrogance or of false humility.)

~

The establishing of first principles  is not a matter for math and science only;  the widest field for its application is the law.  And as we saw in the case of mathematics and of physics, so too in law, the axioms do not historically arise first:

Primitive law is made up of simple, precise, detailed rules  for definite narrowly-defined situations.  It has no general principles.

-- Roscoe Pound, An Introduction to the Philosophy of Law (1922, 1954)

In time, such atomistic empricism is systematized.  (For a fable along these lines, consult our parable of the mathematizing woodchuck.)

In place of detailed rules, precisely determining what shall take place  upon a precisely detailed state of facts,  reliance is had upon general premises, for judicial and juristic reasoning.  These legal principles, as we call them, are made use of to supply new rules [and] to interpret old ones.

-- id.

Spinoza

Once a structured body of legal principles has arisen, judgments may be arrived at more geometrico --  though there will always be a residue of hard cases:

Judicial treatment of a controversy  is a measuring of it by a rule  in order to reach a universal solution for a class of causes, of which the cause in hand is but an example.

Administrative treatment of a situation  is a disposition of it  as a unique occurrence.

-- id.