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)
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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 !!!!!
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| His Majesty ... The King !!! |
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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 relation to human practice or to any species whatever.
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”.
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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). These are treated 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.
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”.
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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.
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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
Exacte; 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.
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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.
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Footnote: These remarks about mathematics apply equally 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.)
Postscript: This essay is extended here:
http://worldofdrjustice.blogspot.com/2015/03/on-nature-of-mathematical-knowledge.html
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