[FOM] John Baez on David Corfield's book
mjmurphy
4mjmu at rogers.com
Mon Sep 29 22:37:58 EDT 2003
Neil Tennant wrote:
> It remains to be seen whether the further philosophizing about what, to
> FOM, are more peripheral or less basic issues, might cause the FOM-er to
> reassess what counts as central, important, and basic. And there is still
> the prospect that whatever neglected notion N is philosophized about in
> Corfield's fashion can have its interest explained entirely in terms of
> those notions that are currently taken as central, important and basic in
> FOM.
>
> To give just one example: Suppose one were to ask why certain mathematical
> concepts or results strike mathematicians as elegant. It might turn out
> that the notion "E is elegant" can be analyzed in terms of (or the
> appearance of elegance explained in terms of, or reduced to) certain local
> and global features of logical syntax, trains of definitions, etc.
>
> Neil Tennant
----
Neil,
So what happens if mathematical "elegance" can be "explained" or "reduced"
to logical syntax, trains of definitions, and "those notions that are
currently taken as central, important and basic in
FOM"?
For example, one possibility is that the FOM concepts etc. can simply take
the place of the other bits of math.
Examples of this can be found in the history of science. One set of
concepts eliminates and replaces the other. It is also the kind of thing
the Churchland's seem to be talking about when they say that intentional
terms like "belief" can be reduced to physicalist terms: one day when
cognitive science pans out we will stop talking about what we believe, even
in OL, and start talking of our neural twitchings. Presumably, if F.O.M. is
set theory, then all the rest of math really is too, and the only reason
everyone is not doing set theory has to do with pragmatic convenience.
However, if we could all do the steps really really fast, we could replicate
any math result in set theory and wouldn't need to teach anything else.
The other possibility is to say, in the manner of Carnap that, no, reduced
terms are not eliminated in this fashion (due to the slippery relation
between observational and theoretical terms in his system). But in that
case what has the reduction accomplished? You arrange bits of math in
relation to set theory in a hierarchy with set theory at the bottom. That's
pretty, but what's the point?
That is, in the first case the foundationalist is a kind of philosophical
imperialist, absorbing other mathematical vocabularies into itself. What is
happening in the second case? And why is it good?
Cheers,
M.J.Murphy
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