[FOM] Simpson on Tymoczkoism

Stephen G Simpson simpson at math.psu.edu
Thu Sep 25 10:48:14 EDT 2003

A.P. Hazen Wed, 24 Sep 2003 18:01:15 +1000 writes:

 > "Foundationalism" is term of art among philosophers, contrasting
 > with "coherentism".  It denotes a certain (family of)
 > epistemological theory (-ies) according to which the propositions
 > that can, rationally, be believed are organized into a well-founded
 > hierarchy, [...]

OK.  Thank you for explaining what philosophers mean by
"foundationalism".  This kind of explanation helps to bridge the gaps
among researchers in the various disciplines represented here on the
FOM list -- philosophers, mathematicians, computer scienctists, et al.

Under this notion of "foundationalism", I am certainly a
foundationalist.  Indeed, I firmly insist that the best method of
organizing mathematics -- the "gold standard" -- is the orthodox,
rigorous method, involving axioms, definitions, lemmas, theorems, and

Furthermore, a look at the contemporary mathematical literature
reveals that the definition-theorem-proof methodology continues to
pervade contemporary mathematical practice.  This methodology is a key
component of the high standard of rigor which continues to prevail in
mathematics textbooks, mathematics research articles, mathematics
Ph. D. theses, etc.  In this sense, virtually the entire mathematics
community -- core and non-core -- is foundationalist.

It does not seem reasonable for authors like Tymoczko and Hersh, and
perhaps Corfield, to ignore or downplay the rigorous
definition-theorem-proof methodology, yet still claim to be in touch
with "the pulse of contemporary mathematics."

 > an interest in, and conviction of the philosophical value of, FoM
 > in the sense of this forum does not commit one to foundationalism.

I disagree with this comment.  It seems to me that f.o.m. (foundations
of mathematics) certainly requires some sort of "foundationalist"
outlook.  See also my little essay


 > Stewart Shapiro's excellent "Foundations without Foundationalism"
 > makes this point in its title

Obviously the bare phrase "foundations without foundationalism"
doesn't make any kind of point.  Could you please explain the point

 > well-organized survey of results about and issues relating to
 > Higher Order Logic, 

Shapiro's views on both higher-order logic and "foundationalism" were
discussed in detail here on the FOM list, back in the Golden Age.

 > it contains a discussion of why the study of "foundations" is
 > independent of any commitment to "foundationalism."

This seems paradoxical on its face.  Could you please explain it here?

In particular, you have contrasted the "foundationalist" viewpoint
with an alternative viewpoint called "coherentism".  Could you please
explain what "coherentism" is, and what a "coherentist" mathematical
practice or view of mathematics would look like?  This may help to
clarify the remaining issues regarding "foundationalism".

 >      Simpson criticizes Tymoczko for
 > 	"never acknowledg[ing] that, in the present historical
 > 	  era, the orthodox and almost universally accepted
 > 	  explication of mathematical rigor is formal
 > 	  provability in ZFC, Zermelo-Frankel set
 > 	  theory with the axiom of choice"
 >    This seems to me to be only approximately true. [...]

It is 98 or 99 percent true.  Furthermore, the thrust of contemporary
f.o.m. research cannot be understood without grasping this aspect of
contemporary mathematical practice.


Stephen G. Simpson
Professor of Mathematics
Pennsylvania State University

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