That question is about as interesting as the nature of our knowledge of elephants. We are interested in the zoology
of elephants, not in the specificities of classroom biology lessons,
or the economics of zoos. We are uninterested in each blind man’s
subjective and partial report upon the individual organs of these
splendid creatures.
Mathematical
knowledge, like pachydermal knowledge, is imperfect knowledge of
something real that exists independently of us. By contrast, just which
images we manage to form of these objects are
very much dependent upon ourselves – and to that extent, of interest only to unemployed social workers.
As our former math teacher put it:
Mathematics has a real content which transcends the inadequacies of our efforts to formalize it.
-- Andrew Gleason, Fundamentals of Abstract Analysis (1966), p. v.
Or, from a philosopher:
For
most mathematicians most of the time, having a feel for what is
evident is important, but it is also enough: There is no further need
for a theory of what that feel is.
-- Shaughan Lavine, Understanding the Infinite (1994)
*
Actually,
it should not be deduced from the above, that I am somehow dissing our
big friends the elephants. A Theory of Elephants is considerably more
promising than that idle grail of the physicists, a Theory of
Everything.
The outstanding problem in the Theory of Elephants is the
ontological status of BABAR –
THE KING !!!!!
 |
| His Majesty ... The King !!! |
An agnostic or non-Realist version, wilfully po-faced:
Our deductivist proposes clean
answers to philosophical questions.
What is mathematics about?
Nothing … What is mathematical knowledge? It is knowledge of what follows from what. Mathematical knowledge is logical knowledge.
-- Stewart Shapiro, Thinking about
Mathematics (2000), p. 150
*
*
There is a systematic ambiguity (roughly that of actio versus actum) to the term mathematics:
(I) The praxis of mathematizing. This is a human pastime, comparable to needlework or basketball.
(II)
The truths of mathematics. Or, more or less synonymously: The real
(though invisible) world. These, in themselves, bear no dependency upon
human practice or to any species whatever; they existed before we were born.
The
above may count as a polemical reformulation of roughly the dichotomy
in the title of Hao Wang’s fine essay, “The Theory and Practice of
Mathematics”.
*
In
1950, Raymond Wilder gave an address, “The Cultural Basis of
Mathematics”, reprinted in various places, and later wrote a whole book
on the subject, Mathematics as a Cultural System (1981). Bien-pensant commentors treat these with grave respect; but the notion is practically nonsense. For, if we take mathematics
in sense (II) – the only sense of interest to us here – that is like
saying “The cultural basis of elephants”: there is none. There is a
cultural basis of circus stunts, of mouse- and peanut-myths, of Dumbo,
but not of elephants themselves. Their basis is their own four feet.
Why should the culture of mathematics (necessarily, sensu
I) – that is, the foibles of mathematicians – retain our attention?
The purely “human side” of mathematicians is generally less interesting
than that of country music stars. A lot of mathematicians are pretty
Asperger’s, frankly.
(For a poignant illustration of this, read The Genius in My Basement.)
There
is, we grant, a certain interest in the sociology of mathematics, or
in biographies of the great mathematicians. Intellectually, it is on a
level with gossip about the off-court antics of basketball stars. Fun,
but of no mathematical
(or basketball) interest. It’s just a way for the mind to chew gum
while it’s too exhausted to do anything substantial. To get real, do
math (or play basketball).
Not
to come down too hard on the small geeky community that does follow
the doings of math and physics whizzes; I number myself among them.
It would even be neat if, instead of collecting baseball cards, people
collected mathematician cards (“Trajea two Steven Smales for a John
Milnor!”) . -- By “people”, I here mean “eight-year olds”.
*
More
interesting is the purported “reduction of mathematics to logic”. It
is not initially clear, however, to what extent this program, if
successful on its own terms, would enlighten us as to
mathematics-sensu-(II), as opposed to the sense-(I) territory of our own
mathematical formulations and formalizations (these being, after all,
largely for mere convenience). It might be more along the lines of the
demonstration of the equivalence of the Heisenberg-style
matrix-mechanics formulation with that of the Schroedinger-style wave
formulation, of quantum mechanics. That feat didn't tell us all that
much about the actual phenomena of physics, apart from the fact that
the world is a many-splendored thing, and can be described --
blind-man-fondling-elephant-fashion -- in a variety of ways. It’s more
like deciding whether today’s symposium shall be conducted in English
or in French.
*
That said --
We argued
here that the axiomatic method is cognitively post-hoc, and that
only
in cases where (as with the Euclidean axioms) their positing is
transparently motivated by our experience of the sensible world, is a
top-down, axiomatic presentation
pedagogically sound.
Thus similarly in physics:
In lecture after lecture, and essay after essay, Einstein
began, not with an introduction to the subject at hand, but with an
overview of how he’d arrived at that subject, or of how scientists in
general arrive
at subjects in general. … For Einstein himself, the results of science
had become incomprehensible without an understanding of the processes
that led to them.
Richard Panek, The Invisible Century (2004), p. 153-4
C'est exact; and
the farther physics wanders from our human experience, and the father
math develops beyond anything the world has seen before, the more
necessary such a psycho-cognitive ladder does become.
*
Something
like the dichotomy outlined above must have been behind André Weil’s
tart remark, in “History of Mathematics” (reprinted in Collected Works v. III as (1978b)):
Some
universities have established chairs for “the history and philosophy
of mathematics”; it is hard for me to imagine what those two have in
common.
For: the one is situated and contingent, the other timeless and beyond place.
*
Footnote:
These remarks about mathematics
apply mutatis mutandis to the Deity. Deliberately confusing the distinction between truth and
praxis, Karen Armstrong wrote a book -- a minor best-seller -- with the
impudent title A History of God.
(At least she put A, not The; probably saved herself an
extra millennium in Purgatory right there.)
*
Lakatos’ classic dialectical-dialogue Proofs and
Refutations (you see the Hegel-style paradox already in the title), though
focussing on the (as he persuasively argues, in the course of a detailed case-study spanning many decades) micro-level mess of actual mathematical progress, is yet Realist at
its core: the subtitle is “The
Logic of Mathematical Discovery”, not “The Sociology of ‘Mathematical’ Invention”. We quoted him in another
context thus:
As far as naïve classification is
concerned, nominalists are close to the truth when claiming that the only thing
that polyhedra have in common is
their name. But after a few
centuries of proofs and refutations, as the theory of polyhedra develops, and
theoretical classification replaces naïve classification, the balance changes in favour of the
realist.
-- Imre Lakatos, Proofs and
Refutations (1976), p. 92
In an appendix to the main work, he offers a Hegelian
formulation, one which (by the time the reader has progressed this far) has a certain
paradoxical piquancy:
Mathematics, this product of human
activity, ‘alienates itself’ [in the sense of Hegel and Marx] from the human
activity, which has been producing it.
It becomes a living, growing organism, that acquires a certain autonomy [emphasis in original] from the
activity that produced it. … The activity of human mathematicians,
as it appears in history, is only a fumbling realisation of the wonderful
dialectic of mathematical ideas.
-- Imre Lakatos, Proofs and Refutations (1976), p. 146
Plato, in his Paradise, smiles.
