Date: Tue, 19 May 1998 09:25:47 +0100
Subject: Math/Metaphysics


In one of H's articles concerning Maths, there are numerous claims
about the discipline, e.g., about views concerning its characteristic
axiomatization, about views concerning its grounding on ultimate, first
principles, i.e., concerning its having a complete/comprehensive
(Euclidean) edifice, albeit, yet-to-be formulated, resting on self-evident
axioms, axioms from which, in accord with some finite set of rules of
inference, all the 'true' theorems of math, and no 'false' statements of
math, would be derived.

I submit that most of these views are now archaic, that they are of
historical or pedagogical interest only. It was in light of Godel's and
Godel-related results -- not only his incompleteability results, but his
independence results also concerning the Ax of Choice (and its various
equivalents) -- that these views came to be shelved.

The aspiration and, more importantly, the conceptions of math which
engender it, for such a complete/comprehensive axiomatic edifice have
long been abandoned. This abandonment was not inspired because
such an edifice would be too difficult to erect (Whitehead and Russell
thought they had satsifactorily accomplished it with their PM), but
because of the logical impossibility of constructing such an edifice. In
addition, more recently, many philosophers have tried to formulate
views concerning math which negatively address Quine's and others'
view that math 'objects' must exist because they are "indispensable," in
some sense, to a natural scientific account of "the world." Such recent
views aim to divorce math from this apparent indispensability.

In short, neither is the Kantian view, that there is only as much genuine
science as there is math, a committment which necessarily characterizes
contemporary views of math/science, but nor is the view that theorems
in math are true, in any philosophically respectable sense.

Thus, I find that many of H's views apropos of maths adumbrate an
interpretation of an archaic 'understanding' of math and its relations to
the natural sciences. This is not to claim that these views are without any
philosophical value; but that H's views are, it seems to me, much more
difficult to defend in light of the revolution in math.

(There is, of course, a problem here about the dissemination of so-
called 'math knowledge'. Whose understanding of math/science is
targeted by H? We might ask the same question were a philosopher to
advance an interpretation of our CURRENT "understanding of science,"
but illustrate this understanding by reference to a 'widespread
understanding' which countenanced phlogiston, ether, Vulcan, fluxions,
and 'quantities of motion'. Whose understanding is targeted? Or is the
suggestion here of a 'division of 'epistemological' labour' simply out of
place'?)   

On a final note, during a guest presentation at Harvard, Bertrand Russell
was asked by the famous algebraist Saunders McClane how Russell
might modify his position concerning math in light of Godel's results. It
was clear to the moderator, and the audience, that Russell was either
not familiar with these results or did not understand their profoundly
revolutionary significance. The moderator intervened and asked for any
other questions. Perhaps H was also not as up-to-date on the revolution
in math which was occurring from roughly 1935 on? 

Any help here? 
jim


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