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Date: Mon, 7 Nov 1994 11:28:48 -0700
From: Randall Holmes
To: podnieks@mii.lu.lv, qed@mcs.anl.gov
Subject: Semantics
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I found Dr. Podnieks's article very interesting, without being budged
in the least from a fully self-conscious mathematical "platonism". I
firmly maintain that mathematical objects are real, in no sense
products of human (or any other) minds, and that this fact matters.
Dr. Podnieks had interesting remarks to make about the psychology of
mathematical intuition and the development of mathematical theories
and their relation to the sciences; most of these are perfectly
comprehensible from a platonist standpoint.
The one thing which I found rather forced was the idea that the
natural numbers are in any sense a product of human psychology; also,
while one might be able to distinguish two theories of the natural
numbers in history, neither of them is likely to be exactly
first-order arithmetic; I suspect that the ancient theory would be
more limited than first-order, if it could be identified precisely at
all, while, since second-order arithmetic was proposed before
first-order arithmetic, there would never have been any time when
first-order arithmetic was actually the dominant model.
This is a philosophical topic which I find fascinating; I'm not
certain what its relevance to QED may be. I suppose that our scruples
on the reality of the objects of mathematics may have something to do
with the kinds of root theories we adopt? Also, the operational
definition in a recent posting of the dichotomy between Platonism and
formalism in terms of what kinds of objects we take ourselves to be
manipulating was instructive.
--Randall Holmes