FOM: Foundationalism

Colin Mclarty cxm7 at
Sat Sep 12 16:06:43 EDT 1998

	Joe Shipman first quotes John Mayberry:

>>... 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.

	And then Joe writes:

>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.

	These are compellingly put, and I believe Reuben Hersh does
too little justice to the motives behnd such thinking throughout the
history of math. And yet I am not sure whether John and Joe really 
mean quite what I think they do. 

	Do either of you mean to say that Descartes did no mathematics?
Certainly he could not have traced any of his claims in geometry back to 
statements that he considered so fundamental as to have no proofs. He 
(in principle) pushes them back to claims that he clearly sees are true. 
But he knew such a claim can well have further proof. And of course all 
kinds of people routinely believe that they "clearly see to be true" 
all kinds of claims. That cannot be what you mean by mathematics.

	Neither could Newton in the calculus. Is that not mathematics?

	To take one well studied case, what about Riemann in complex
analysis? He proved many theorems that depended on the Dirichlet
Principle, for example the Riemann-Roch theorem. He knew he had no 
rigorous proof of this Principle. And he would have liked to. He
certainly did not consider it bedrock. He even knew that the 
Dirichlet Principle as he stated it was wrong--it implied some 
obvious falsehoods.

	His proofs also used intuitive claims about the topology of
surfaces. He wanted these claims better established, and extended
to higher dimensions, but he did not do it.

	We cannot say Riemann actually proved conditionals such
as "If the Dirichlet princple is true and claims A,B,C.. are true 
of topology of the plane, than the Riemann-Roch statement follows".
We cannot say that, because he knew the Dirichlet principle is not
true, and he would have been hard pressed to say exactly what
claims A,B, and C were.

	Was Riemann's complex analysis not mathematics? 


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