FOM: What is f.o.m., briefly?
Robert Black
Robert.Black at nottingham.ac.uk
Tue Oct 2 16:52:00 EDT 2001
My problem with Peter Schuster's one-liner is not 'a priori' but
rather 'synthetic'. I take it as uncontroversial that Kant's
explication of 'synthetic' (or 'analytic') is no longer viable
(because the subject/predicate theory of logic it presupposes was
shot out of the water in 1879 by Frege). The only modern notion of
'synthetic' that I know that gets off the ground is the Fregean
notion that, roughly, synthetic knowledge is knowledge that goes
beyond logical knowledge. But it's not at all obvious that
mathematical knowledge goes beyond logical knowledge because:
1) Whether there is a philosophically significant distinction between
mathematics and logic and if so where it falls is up for grabs (is
'second-order logic' logic etc)?
2) Even if you've decided which truths are logical truths, you've got
to decide whether or not the the fact that a logical truth is a
logical truth (rather than just true) is a further logical truth
(e.g. you could plausibly claim that the logical truths of
first-order logic are all the logical truths there are, and for each
of them the fact that it is a logical truth is not itself a logical
truth but a truth of set theory - because the semantics of
first-order logic is phrased in set theory - and ditto [perhaps in
some ways even more plausibly] for second-order logic).
3) There are still a few logicists around who believe that all
mathematical truths are logical truths.
4) There are other people around who are not logicists but still
believe that all mathematical knowledge is logical knowledge (e.g.
Hartry Field, or any sort of 'if-then-ist' who thinks that all that
mathematicians know is what theorems follow logically from what
axioms).
So Schuster's suggestion seems to me to presuppose answers to a lot
of very open questions.
Robert
>When one is asked to give an explanation as brief as
>possible, could one perhaps reduce foundations of
>mathematics to the question whether synthetic knowledge
>a priori is possible---and, if so, which, how, etc.?
>
--
Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD
tel. 0115-951 5845
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