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