FOM: The REAL Foundations Of Mathematics (FOM)

[John Pais]

soare at gargoyle.cs.uchicago.edu.cs wrote:

> TO:     fom subscribers
> FROM:   Robert Soare, University  of Chicago
> TITLE:  WHAT'S HOT AND WHAT'S NOT IN FOM and Reverse Mathematics
> DATE:   September 9, 1999
> ----------------------------------------------------------------------------------
> NOTE:   This message is  being delivered to you directly (i.e. "exfom")
>         rather than by the normal method "infom" (via the Moderator)
>         because the Moderator has apparently removed my name from the fom
>         list of subscribers without my permission or approval.
> ========================================================================
>

I thank Bob Soare for posting exfom his views, since he hasn't had an opportunity to fully
respond to the unsolicited attack and characterization of his field and its workers
launched by Steve Simpson and supported by Harvey Friedman. Furthermore, I strongly
disagree with Neil Tennant in his discription of Soare's recent posting as "spamming." In
fact I have found Soare's postings to be a welcome counterpoint to the party line of
Simpson and Friedman. Particularly interesting to me is the excerpt below from Soare,
challenging the status and significance of Reverse Mathematics regarding its historical
place and future impact in real foundations of mathematics--more properly and broadly
conceived. I would very much like to hear the opinions of other members of this forum on
the following questions raised by Soare in the excerpt below:

RFOM1. Is it true that "Reverse Mathematics is a very SMALL field"?

RFOM2. Is it true that "there is NO evidence that Godel or Hilbert would have been
interested in Reverse Mathematics at all"?

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."

Thanks,
John Pais


>
> The REAL Foundations Of Mathematics (FOM)
>
> Friedman wrote on fom Sept 1, 1999 that "it is crucial for
> mathematical logic to return to its roots in the foundations of
> mathematics."  Does he mean that we should return to the genuine roots
> of Foundations of Mathematics (FOM upper case) under Hilbert, Godel,
> and others, or does he mean that we should return to Reverse
> Mathematics which Friedman invented two decades ago?  Reverse
> Mathematics is a very SMALL field.  Those working primarily in the
> field include only Friedman, Simpson and their students, with only a
> couple of exceptions.  There is NO evidence that Godel or Hilbert
> would have been interested in Reverse Mathematics at all.  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.  For example, if this
> principle had been adopted after Pythagoras, our present knowledge of
> mathematics would be restricted to a fraction of Greek mathematics,
> with nothing new in the last 2,000 years.  Maybe this is what they
> mean by "reverse" mathematics.
>
> CONCLUSION.
>
> It is time for Reverse Mathematics to add something *new* to
> mathematics, or to get off the pot.
>
> -----------------------------------------------------------------------