FOM: music versus a couple of facts
cxm7 at po.cwru.edu
Mon Mar 2 14:30:59 EST 1998
Friedman endorses a quote from Feferman 7:15PM 1/16/98:
>>... the notion of topos is a relatively sophisticated mathematical
>>notion which assumes understanding of the notion of category and that in
>>turn assumes understanding of notions of collection and function. ...
>>Thus there is both a logical and psychological
>>priority for the latter notions to the former. 'Logical' because what a
>>topos is requires a definition in order to work with it and prove theorems
>>about it, and this definition ultimately returns to the notions of
>>collection (class, set, or whatever word you prefer) and function
>>(or operation). 'Psychological' because you can't understand what a topos
>>is unless you have some understanding of those notions. Just writing down
>>the "axioms" for a topos does not provide that understanding.
This is a fairly straight repetition of arguments by Feferman in
1977 (Feferman, S. 1977: "Categorical foundations and foundations of
category theory" in R. Butts and J. Hintikka eds. International Congress of
Logic, Methodology, and Philosophy of Science. Boston: D. Reidel, 149-69.)
But Feferman himself refuted these claims in the very same article,
when he said:
"I realize that workers in category theory are so at home in their subject
that they find it more natural to think in categorical rather than
set-theoretic terms, but I would liken this to not needing to hear once one
has learned to compose music" [page 153].
That is the truth. We can, and do, think categorically. Many set
theorists find this unsettling (like the thought of Beethoven composing
after he was deaf) but it goes on. Serious arguments against categorical
foundations will have to come to grips with the plain facts that these
foundations are logically possible, and psychologically suited to some people.
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