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

[Peter Schuster]

In reaction to the reactions to my posting 

>> >> Date: Thu, 27 Sep 2001 12:09:01 +0200 (MET DST)
>> >>
>> >> 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.? 

let me try to explain---this time in other words---what I had in mind. 

Here we go: 

Could one explain the discussion about "the right" foundations of 
mathematics as a discussion about what is "to be" accepted as given 
in mathematics (e.g., numbers, infinity, continuum, sets, categories, 
cardinals, axioms, and the like) in addition to the maybe less doubtful 
logical/analytic items (notwithstanding the fact that a particular choice 
of the former has a certain impact on the latter)? 

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There are also two notes to be made, of a comparatively marginal character. 

First, I should stress that I wrote 

"synthetic knowledge a priori is possible", 

instead of perhaps writing 

"synthetic knowledge is a priori possible" 

or, presumably equivalently to the latter but not to the former, 

"synthetic knowledge is possible a priori". 

Next, when I said that "it is both possible and impossible" is a somewhat 
tautological answer to every question like the one mentioned above,  
I did not mean the formal character of this conjunction, which indeed 
is rather that of a contradiction. What I wanted to say is that as long 
there is no commonly accepted answer, and I doubt that there is any, 
it is in order to view the item whose possibility is put under scrutiny 
as possible---but also, and at the same time, as impossible. Needless to 
say, there is little substance in any such statements when they are viewed 
as such. Moreover, "it is either impossible or else possible" is no more a 
tautology than anything like the law of bivalence, the law of the excluded 
middle, etc.