Saturday, December 14, 2013

Theorems, Propositions, Dumb Questions, Unspoken Assumptions

In an earlier essay (Andrew Gleason:  in Memoriam) we fondly recalled our favorite teacher from Harvard.   The incident below was not included;  but now, owing to recent events, it can be declassified.  We take you back to the year 1969 …


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 “Mein Gott!" (mathematicians revert to German when suitably moved), "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, grimaced, he really didn’t know where to begin.  “It’s … obvious,” he said at last, 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 GLn, 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 messy, and quite unilluminating), 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

And later, in an article that is a masterpiece of step-by-step exposition, Gowers shows how quaternions can be represented as ordinary matrices, and adds:

As an immediate corollary, we have a proof of a fact mentioned earlier:  that quaternionic multiplication is associative.  Why?  Because matrix multiplication is associative.  (And that is true because the composition of functions is associative.)
-- “Quaternions, Octonions, and Normed Division Algebras”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 277


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 this1, 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,  Gleason probably (like John von Neumann summing an infinite series in his head) instantly proved the proposition in his own unconscious.

Footnote 1:   Gleason was in fact especially sensitive to such quasi-linguistic matters of hidden assumptions.  Thus, in his text Fundamentals of Abstract Analysis (1966), he remarks that the direct-product procedure is strictly speaking not associative, but that there exists a natural bijection among the various possibilities, so that we speak simply of “the” direct product of a roster of spaces, par abus de langage.


A more recent example of my posing a question which left the teacher speechless, apparently as being unanswerably dumb, happened a couple of years ago.  A visiting combinatorialist, scholar-in-residence at the Cryptological Museum, gave a public talk about Stirling numbers of the second kind.   By doing this, that, and the other thing, you can find all sorts of pretty geometric patterns popping out at you in Pascal’s triangle and whatnot.   Since I am these days but infrequently in the audience of a combinatorialist (since moving from Princeton, my mathematical surroundings have become quite impoverished -- really an algebraic social-worker should stop by with some charitable Ideals on Wheels), it seemed a good occasion to pose a question that has always bothered me:  what is the point of “perfect numbers” (those that are equal to the sum of their prime dividers)?  

“It soon becomes obvious why prime numbers are of prime importance:  they are used for many purposes other than in the study of their own properties, and they jump out at you even when you’re not looking for them, in physics or wherever; they are part of the woodwork of the world.   Also, on a more intuitive or metaphorical level, they are the evident “building blocks” of all the integers, the way the atomic elements are the building blocks of all the molecules.   But -- “perfect” numbers.  The definition seems so arbitrary.  Why study them?  What are they good for?”

Instead of instructing the curious groundling by giving examples of their usefulness, or their inevitability, or even saying “An explanation exists but it would be way over your head, you peasant” (which, while impolite, would actually be somewhat informative), or “You’re quite right, they are purely recreational”, he simply looked blank.  The question evidently made no sense to him;  it was as though I had asked whether the value of pi were the same on the dark side of the moon, or under all gravitational conditions. (Thus, not exactly a dumb question, more like a crank question,  of the sort which is likely to spring from the lips of the unemployed middle-aged men in raincoats who wander into lecture halls in hopes of a donut and to get in out of the cold, and who have their own private but quite definite opinions about whether a circle can indeed be squared or whether, rather, it might not be square in fact already, only They don’t want you to realize this;  the lecturer’s only defense is to decline to be drawn into debate.)  And yet, mathematicians might be characterized as people to whom such questions make a lot of sense, are even fundamental.
Perhaps, though, combinatorialists less than other specialties.   There does seem to be a fair amount of pointless ingenuity in what some of them do, but then I’m no judge of it.  

However!  Once again, Gowers to the rescue, to clarify the sort of issues that are at stake.  On the next page of that same Introduction to mathematical terminology, he defines (or explains) the notion of definition.
Mathematical definitions are generally what linguists call stipulative definitions, essentially just rewordings or abbreviations.  “Definitions like this,” Gowers comments, “are mere definitions of convenience”.   Yet, just as in the case of the taxonomic definitions of philology or biology (Indo-European; crustacean), where the really useful ones reflect a significant amount of research and analysis leading up to them, so in mathematics;  and indeed, Gowers reveals, in some of its branches, even moreso:

Some mathematicians will tell you that the main aim of their research is to find the right definition, after which their whole area will be illuminated.  Yes, they will have to write proofs, but if the definition is the one they are looking for, then these proofs will be fairly straightforward.
-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 73

The author does not instance such cases where the definitions do much or most of the real work; these would have been fascinating to hear;  but I imagine he has in mind as precedents  such things as the homotopy groups, easy to define but the devil to calculate, versus homology groups, fiendish to define and visualize  but easy to compute; or the Generalized Stokes Theorem, which by our own day possesses a brief proof once all your definitional ducks are in a row -- but oh!  what ducks!
He does, though, go on to provide an example of a definition that might appeal to the readers of “brain teasers” in the Sunday papers, but which does no work at all -- one so distasteful, it is distressing even to write it down:

A number is called palindromic if its representation in base 10 is a palindrome.

Such ontological excrescences are even more thewless than “perfect” numbers, since at least the latter are independent of their inscriptional base.   (You can think of the writing of one of God’s own integers in any base  as representing a tragic demotion from the Platonic sphere, sort of like a soul’s being incarnated in the body of a frog.) 
At that point I almost skipped on to the following page, so little do I wish to learn the least thing about such concocted objects;  but Gowers goes in an interesting direction with this.  One might ask:  How many primes are palindromes?  There are some, although, in a well-defined sense, “not many” (even if there are infinitely many), examples being 919, 929, followed only much later by 10310.   And thus the question:  Are there infinitely many?  (Once you have more than about seventeen of something, that is the first question a mathematician asks:  They don’t like sequences of integers that go on for a bit and then just stop.)   The answer would be boring either way;  but unlike the “boring” propositions alluded to earlier, it would be the very Dickens to prove or disprove (and thus not worth the candle).  For,

It can be shown quite easily that  the number of palindromic numbers less than n is in the region of  n, which is a very small fraction indeed.   It is notoriously hard to prove results about primes in sparse sets like this.
-- id., p. 75

And in any such endeavor, the “definition” of palindromic would be of no help at all, since it is “so artificial that there seems to be no way of using it in a detailed way in a mathematical proof.” (p. 76).  And that same infirmity of the beginning definition insures that the bare answer, whatever it might be, would be uninteresting per se (although, as Gowers points out, there might be a much more general conjecture with no original connection to “palindromes”, which would be interesting and which might turn out to settle the result for palindromes as well):  for, unlike prime numbers, palindromes, being irremediably notation-dependent, do not form part of the Furniture of the Universe.  (For that concept, consult the series of essays begun here.)  That was what I had been trying to get at my my question to the itinerant combitorialist, and which meant nothing to him;  perhaps he is not Platonistically inclined.


While we’re on the subject, let us consider further the question of definition in mathematics.

Re Hilbert’s approach to the axiomatization of geometry:

Rather than defining points or lines at the outset  and then postulating axioms that are assumed to be valid for them, a point and a line were not directly defined, except as entities that satisfy the axioms postulated by the system.
-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

This is not quite so radical or ‘post-modernist’ as it might sound, since traditional grammar recognizes many analogous cases in natural language, under the rubric of syncategorematic.   It is a relative notion, with a sliding scale;  but analysis will suggest that a very large set of words and multiword expressions (as, the use of a word in an idiom, especially in an opaque idiom) partake of some degree of syncategorematicity.   However, in the particular perspective of mathematics, this idea harmonizes especially well with a logicist or formalist approach to the subject:

The use of undefined concepts  and the concomitant conception of axioms as implicit definitions  gave enormous impetus to the view of geometry as a purely logical system.
-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

Again, this is much less disorienting and self-bootstrapping than it may seem, since -- outside, indeed, of formal contexts -- virtually all of natural language works exactly like that; and not only expressions like whereas, the moreso as, French ne, German doch, which wear their syncategorematicity on their (empty) sleeves, either,  but plain words like bunny.   You do not learn to use such words on the basis of a definition, formal or informal -- however much it might please linguistic philosophers to invent a terminus technicus “ostensive definition”.   For, as we have seen in our discussions and parables related to matters Quinean, these don’t really work, not logically;  they work pragmatically, to the extent that they do, because (since we are all molded from the same clay; or  if you prefer, since our bloodlines have all been subjected to the rigors of Natural Selection) we are all cut to the same cloth.  (To the extent that some individuals fall outside the innate cognitive norms, they fail to acquire the same semantics that the rest of us do:  or else, like some gifted and industrious autists, they acquire this only by dint of an artificial study, like someone learning Sumerian logographics.)   Thus, the following Onomastic Primal Scene does not actually obtain in any real nursery:

That, Timmy” (pointing -- but at or towards what?) “is a rabbit (noun count, singular).   And by this -- attend now, and please do not misunderstand me -- I do not intend to indicate the entire scene embracing carrots and furballs and playpen and binky (who left that there?) etc., let alone the cosmos as a whole (after all, one has to point somewhere), whether by itself or considered as but one flaky layer in the whole baclava-like complexus known as the multiverse;  but only the, er, furball-related entity.   And by this, I do not mean, so much, (although I do not literally not mean it, either), a pointlike or infinitessimal space-time slice of a leporiform trajectory along the world-sheet, nor a “thickened” (perceptually available) neighborhood of the same;  nor a sort of puddle of rabbit-stuff, undifferentiated from the rest of the puddle; nor a concrete instantiation of the Platonic Form, ‘Rabbit’;  nor a subobject in the Category Leporidae;  nor an agnostically structured pointset consisting of Undetached Rabbit Parts (although I sort of mean that, since, at some point, once you have hacked the poor critter to bits and scattered its disjecta membra over the face of the earth to be eaten by vermin and recycled as independent atoms, at some point, we can no longer confidently say, “That is a rabbit”, in the sense of noun count, singular),  nor -- well, dash it all, I mean just Fluffy, okay?  And by the way it looks like Fluffy wants a cuddle or something, because she is sprintzing the wood-shavings in a semantophobic panic.”


An extension of this linguistic thought-thread  can be appreciated here:

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