FOM: sharp boundaries/tameness
Alexander R. Pruss
ap85 at georgetown.edu
Thu Feb 14 23:42:45 EST 2002
1. Certainly there are interesting theories of objects in general. Take,
say, Aristotle's _Metaphysics_ Z and _Categories_, Aquinas's _De Ente et
Essentia_, Leibniz's _Monadology_, Kant's _First Critique_, and tons of 20th
century stuff. The theories make claims that are indubitably significant,
if true. They offer thought-out arguments for the claims. Now, you may say
that the arguments in all of these cases are fallacious and the claims
false, but that is a substantial claim that would evidently require rather a
lot of work to substantiate. And of course even if there were nothing
significant to be said about objects in general, that fact would itself be
something significant--hence in any case, there is something significant to
be said about objects in general.
2. It is of course true that the vast majority of mathematicians do not work
on well-orderings of the reals--there isn't, I assume, all that much to be
said about them as such, and the topic is rather narrow. But there is a
whole slew of topics on which the vast majority of mathematicians does not
work. (There are even tons of topics on which the total number of
mathematicians currently seriously working is under 50--indeed, I suspect
that quite a significant number of mathematicians work in well-defined
subfields in which no more than about 50 mathematicians seriously work.)
But to cite another non-constructive example, there no doubt are significant
numbers of mathematicians using Hahn-Banach mappings (MathSci find 1180
papers containing "Hahn-Banach" somewhere in the MR entry). Are they using
them ineliminably? Maybe not--but it is not obvious that they don't.
3. Occasionally one nit-picks at some thesis, as I do in this case, because
of a conviction that the thesis _cannot_ be given a satisfactory rendition,
and this will come out when one starts giving obvious counterexamples.
Alex Pruss
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