FOM: The REAL Foundations Of Mathematics (FOM)

Steve Stevenson steve at
Fri Sep 10 08:36:48 EDT 1999

John Pais writes:
 > <much snipped>

 >RFOM3. Is it true that "Reverse Mathematics does NOT produce new
 >results in mathematics (unlike set theory, model theory, and
 >computability theory).  It merely takes an existing mathematical
 >result and proves it equivalent to some logical axiom.  This is like
 >stirring a pot of stew and having the pieces [theorem] adhere to the
 >sides of the pot at some level [corresponding to the level of proof
 >theoretic strength].  Nothing NEW is ever added to the pot in terms
 >of new mathematical results."

It seems to me that these discussions miss one point of foundations:
the characterization of the various elements. It may be just as new
(and wonderful) to know two old things are related in new and perhaps
surprising ways as it is to have some totally new concept emerge. This
nattering about someone else's area is not as important as another
seems unproductive --- gleefully enforced by our deans and presidents
and funding agencies --- but unproductive, none the less.

My take in watching all this: it's like the 20s and 30s when people
were looking for *the* foundation: logicists, set theorists,
etc. Ultimately, won't all this work show that "mathematics" is
categorical at the foundation --- that all these metaphysical
assumptions lead to a class of structures that are inherently equivalent?

Isn't that a good question for foundations of both mathematics in general
and the subset I'm interested in: computing?


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