FOM: anti-foundationalism

[Stephen G Simpson]

Response to Stewart Shapiro 22 Apr 1999 00:48:30.

Fair enough, I'll stop goading you to post to FOM.  Everybody has lots
of commitments.  I can well understand that the FOM list may not be a
top priority.  I want to thank you for participating, to whatever
extent you can.

Referring to finitistic reductionism a la my paper ``Partial
realizations of Hilbert's program''
<http://www.math.psu.edu/simpson/papers/hilbert/>, you ask

 > Steve says that one of his goals is to "justify at least a
 > significant fragment of mathematics".  If I can be permitted a
 > question, what is his view of the parts of mathematics that do not
 > get justified this way?  I presume that he will not follow Weyl and
 > "sacrifice the rest".

This research based on reverse mathematics has helped to delineate
which parts of mathematics are and are not finitistically reducible,
in a sense that is closely related to Hilbert's program.  In my view,
the parts that aren't finitistically reducible are still mathematics
(i.e., I don't want to expell them from the kingdom), but it may be
scientifically important to distinguish them from the finitistically
reducible parts, because that may speak to their relevance to the
physical world, etc.  The exact scientific implications of this
distinction should be a subject of further f.o.m. research.

A by-product of existing f.o.m. research is that it helps us to
evaluate claims such as the one that you made, that Weyl's program
would ``cripple mathematics''.  It seems to me that we can't really
know any such thing until we have examined which parts of mathematics
go through in Weyl's system.  A tentative conclusion that I draw from
reverse mathematics etc is that a huge amount of what's usually called
``core'' mathematics (algebra, analysis, geometry, etc) *does* go
through in Weyl's system.  The parts of standard mathematics that are
known *not* to go through tend to be more set-theoretical, use ordinal
numbers in various ways, etc.  (Much more detail is in my book
``Subsystems of Second Order Arithmetic''
<http://www.math.psu.edu/simpson/sosoa/>.)  This kind of information
puts a very different light on your anti-foundationalist remarks.

 > 1.  Is it possible (or desirable) to provide a single absolutely
 > secure foundation for all of mathematics?

My feeling is, probably not, because the all-inclusive phrase ``all of
mathematics'' leaves room for people to do all kinds of heterogeneous
things: new large cardinal axioms, alternative logics, NF, combinatory
logic, category theory, etc etc.  Who knows what people will come up
with in the future?  Also, we would need to consider various kinds of
mathematical heuristics, non-rigorous approaches, etc.  A single
foundation for all of this stuff is hard to imagine.  If it were done,
it would probably be very weak or inconclusive.

To me it seems much more fruitful to delimit what we are talking
about.  We can start with a foundational scheme (e.g. Weyl's system
or the modern variant ACA_0) and ask how much of mathematics it
justifies.  Or, we can start with a portion of mathematics and ask
what kind of foundation is needed to justify it.  This kind of
f.o.m. activity has been very fruitful.  It leads to all kinds of
philosophically/foundationally interesting information.

 > 2.  Is it desirable to determine whether a given chunk of
 > mathematics can be reduced to, or dervied from, the otherwise
 > founded on, a given foundation?

Yes, and I'm glad you agree.

One problem I have with the anti-foundationalism of your book, not to
mention some works of some other academic philosophers, is that I
imagine young people may be misled.  Think about a graduate student
with foundational/philsophical/mathematical inclinations, in the
process of deciding whether to pursue f.o.m. as a career.  If he hears
a chorus of academic philosophers telling him (inaccurately) that the
programs of the great f.o.m. thinkers have failed miserably and
foundationalism is dead, what is he to conclude?  Have you considered
whether your anti-foundationalism makes sense in that light?

-- Steve