[FOM] The lure of the infinite

joeshipman@aol.com joeshipman at aol.com
Thu Feb 16 19:48:25 EST 2006

I think one point Friedman made properly frames the issue as far as 
mathematical practice is concerned, and is the main point that needs to 
be addressed. I restate it here with my own emphasis:

** For any natural axiomatic basis for mathematical reasoning, there is 
a natural mathematical theorem of exactly the same logical strength. **

Here the first "natural" refers to philosophical, metamathematical, and 
other non-mathematical criteria; and the second "natural" refers to 
criteria internal to mathematics.

In my opinion, the empirical suppport for this statement is strong and 
getting stronger. I would like to know what people think about

1) what reasons there might be to doubt this statement

2) if the statement is true, what consequences follow for mathematical 

3) if the statement is true, what consequences follow for the 
philosophy and foundations of mathematics

To get the ball rolling:

1) I don't think that the mathematical theorems known to exceed ZFC are 
very "natural" yet compared with the many mathematical theorems that 
hit weaker systems than ZFC "on the nose"

2) Mathematicians ought familiarize themselves with systems of higher 
logical strength than they are used to using and understand the ways in 
which they add more *mathematical* power

3) Since it is much easier to identify "philosphical stopping places" 
than "mathematical stopping places", those who would give certain 
systems of lesser logical strength a more privileged position need to 
take a position on what mathematicians "should make of" mathematical 
statements that require systems of higher logical strength. In 
particular, should they regard such statements as 
true-or-false-but-we'll-never-know-which, or neither-true-nor-false, or 
is there some other attitude they should take?

-- Joe Shipman

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