FOM: What is f.o.m., briefly?

[Robert Black]

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