FOM: Foundationalism

[JoeShipman@aol.com]

In a message dated 9/12/98 8:31:48 AM Eastern Daylight Time,
J.P.Mayberry at bristol.ac.uk writes:

<< ... there seems to me to be 
> a clear sense of "foundationalism" in which to discredit 
>foundationalism in mathematics is to discredit mathematics itself. 	
>... it seems to me obvious that mathematics not only 
> has foundations, but cannot do without them. This is because 
> mathematics, unlike any of the other sciences, deals in rigorous proof 
> and exact definition.
>... When you have arrived at 
> propositions that don't require, or admit of, proof, or at concepts 
> that don't require, or admit of, definition, then you have reached 
> bedrock, and there the genuine foundations of mathematics are to be 
> found.
> ... 	So I say that the idea of "mathematics without 
> foundations", if the term "foundations" is understood in the sense I 
> have just given (and that seems to me to be its most basic sense), is 
> simply a contradiction in terms.

This is obviously right.  Why doesn't someone challenge those "anti-
foundationalists" who are actually mathematicians to explicate their proofs to
the point where their "real" foundations become apparent?  Obviously they will
have to stop at some point with some definitions and propositions that they
won't justify further, and it will be interesting to see what these are.  For
those anti-foundationalists who are not mathematicians, there is no hope of
anything like this so they can be safely ignored by f.o.m. researchers.

-- Joe Shipman