[FOM] John Baez on David Corfield's book

Stephen G Simpson simpson at math.psu.edu
Mon Sep 29 17:09:15 EDT 2003

David Corfield, Sat, 27 Sep 2003 16:44:22 +0100 writes:

 > I don't recall using the word 'foundationalism'. The 'foundationalist
 > filter' I object to is [...]

Aren't you evading the point here?  Even if you didn't use the word
"foundationalism", you certainly used the derivative word

The question remains, do you agree or not with my point that the vast
majority of mathematicians choose to organize their mathematical work
in a foundationalist manner, with rigorous definitions, theorems,
proofs, etc?  And, do you agree or not that the purpose of
f.o.m. research is to analyze this logical, hierarchical structure?

 > the conception common to the majority of philosophers of
 > mathematics that the ONLY aspects of mathematics of philosophical
 > interest can be detected by proof theory, model theory, set theory,
 > recursion theory. 

This seems remarkable.  Is this really a common conception among
philosophers of mathematics?  Have any of them actually said this, or
is this merely an inference on your part?  Aren't you putting words
into their mouths?

 > Notice that this does NOT imply that these theories are of no
 > philosophical interest.

OK, good.  Thank you for that ambigous concession, in the form of a
double negative.  Maybe you are not unrelentingly hostile to
f.o.m. after all.

However, you are still attacking f.o.m. in ways that seem highly
dubious.  For example:

 > I meant f.o.m. research has no bearing on choice between rigorously
 > defined concepts.

Again, I have to disagree.  It seems to me that f.o.m. research has
often had a decisive influence on choices among rigorously defined

An example is the pervasive concept of topological space, i.e., an
ordered pair (X,t) where X is a nonempty set and t is a collection of
subsets of X containing X and the empty set and closed under unions
and finite intersections.  This concept occurs very frequently in the
20th and 21st century mathematical literature, especially in textbooks
for advanced mathematics students.  Now, my point is that the history
of this concept of topological space seems to show that it developed
hand-in-hand with set-theoretical f.o.m. as pioneered by
f.o.m. researchers such as Zermelo.  If set-theoretical f.o.m. were
not in vogue, then mathematicians would surely have chosen some other

Don't you agree?

Another example is the history of the acceptance of the rigorous
concept of real numbers as Dedekind cuts, pioneered by Dedekind's
f.o.m. research as reported in his monograph `Was sind und was sollen
die Zahlen.'

Don't you agree?

Anyway, ...

Maybe I missed it in the flurry of postings today, but I think we are
still waiting for someone to present an example of a piece of
non-foundational mathematics that is of philosophical interest.  I'm
not asserting that there is no such thing, but surely it is desirable
to have at least one good example on the table.

In particular, I would like to know what philosophical questions are
addressed.  Are they "real" or "core" philosophical questions, or are
they merely peripheral ones?  And, how compelling are the answers to
these questions?


Stephen G. Simpson
Professor of Mathematics
Penn State University

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