FOM: quasi-empiricism and anti-foundationalism

[Stephen G Simpson]

Harvey Friedman writes:
 > I admit to not having followed carefully the Hersh/Simpson, ...

Well, you haven't missed much.  It's hard to get Hersh to discuss any
cutting-edge issues in f.o.m.  He keeps harping on hackneyed
half-truths such as "Hilbert's program was killed by Goedel" and
"logicism was killed by the Russell paradox".  I'm trying to draw him
into a discussion of specific f.o.m. issues that are philosophically
relevant, e.g. finitistic reductionism and logicist motivations of the
axioms of set theory, but it's difficult.  I don't know if I can get
anywhere with him.  He's a hard case, because with his book etc he has
a huge investment in philosophy of the "humanist" variety.  In
addition, he seems to feel that philosophy of mathematics is a
playground where one is not subject to constraints of rigor and
precision.

 > I propose that the relevant people carefully clarify their positions
 > according to the following scheme.

As usual, your "scheme" raises a huge number of fascinating issues.  I
certainly can't do them justice in one e-mail.

 > 1. Is there a meaningful concept of absolute certainty, beyond
 > which one cannot go?

Well, I would say that the general scientific standard of certainty is
as follows: Something is certain when all the evidence confirms it,
and none contradicts it.  In mathematics, rigorous proof is an
important part of the evidence.  In some parts of mathematics,
rigorous proof is the only evidence.

I am talking about "certainty".  You are talking about "absolute
certainty" -- I'm not sure what this would be beyond what I said
above.

 > And is this ever realized in mathematics? If it is sometimes
 > realized in mathematics, then where in mathematics is it realized? 

Sure, a great deal of mathematics is certain in the sense I described
above.

 > Do you know of a candidate for a piece of mathematics that is not
 > absolutely certain?

I would look for candidates in two areas: (1) pure math results where
the proof is very long and complicated and non-conceptual and
therefore subject to error; (2) applied math results which are subject
to experimental confirmation.  I think it often happens in fluid
mechanics that a result is theoretically predicted but not confirmed
by experiment, so the predictions have to be refined, revised, etc.
Indeed, there are huge gray areas of mathematical and statistical
modeling that are plagued by unreliability: climate models, ecology
models, statistical polling, etc etc.

Both (1) and (2) would seem to be good topics for the
quasi-empiricists to look at, but so far they don't seem to have
accomplished much.  They seem to be more interested in bashing f.o.m.

Another interesting source of candidates might be mathematical results
proved using large cardinals.  Here I think the quasi-empiricists
would be at a loss; traditional f.o.m. techniques would be most
relevant.

 > 2. How does the level of certainty in mathematics compare with the
 > level of certainty in other disciplines such as physics?

I think a lot of pure math is at a significantly higher level of
certainty than other sciences, due to high standards of rigor in pure
math.  This philosophically interesting state of affairs is at least
partly an outcome of f.o.m. research early in the century.  Hersh and
other quasi-empiricists don't want to acknowledge this.

 > 3. Does the work in f.o.m. using formal systems and their
 > relationships bear on issues of absolute certainty or relative
 > certainty in mathematics?

Sure.  F.o.m. research growing from Hilbert's program, logicism
and set theory are very relevant as models suggesting the limits
of what can be expected from rigorous proof.

 > Is there any significance of formal systems such as ZFC for the
 > philosophy of mathematics?

Of course.  But Hersh and other quasi-empiricists don't admit this.

 > 4. Does Hersh's writings convey any judgment on the value of formal
 > systems and the extensive work on them in f.o.m.?

I think his basic line is that formal systems are an interesting
specialized topic or branch of pure mathematics, of interest to
specialists in that branch, but largely irrelevant or harmful to
philosophy of mathematics.

 > what guidance does Hersh give to this extensive literature? 

Hersh gives virtually no guidance to the literature of f.o.m.  or
mathematical logic.  He has read a little bit of Frege, Russell,
Hilbert, and Brouwer, but he cites mostly secondary sources and
textbooks in philosophy of mathematics.  I don't think he has absorbed
the technical details of any significant achievements in f.o.m.

 > 5. What is foundationalism?

Hersh says that foundationalism is a grab-bag of logicism, formalism,
and intuitionism, lumped together by virtue of the fact that all of
these schools pursued certainty.  

I interpret Hersh's attack on foundationalism as an attack on a
subject near and dear to my heart, f.o.m.

 > To what extent is ZFC a formalization of mathematics?

Hersh says that no formal system is a good model of any significant
feature of mathematics.  I'm not sure whether he is familiar with how
mathematics is formalized in ZFC specifically, or any other formal
system for that matter.

 > If one believes that ZFC is, in some appropriate sense, a
 > formalization of mathematics, then is one a foundationalist?

Hersh in his book says that foundationalism is dead.  Perhaps he
wasn't aware of the existence of f.o.m. as a lively field of research,
before he joined the FOM list.

 > If one believes that ZFC is, in some appropriate sense, a
 > formalization of mathematics, then is one an anti-humanist?

Hersh divides philosophy of mathematics into two camps,
foundationalism and humanism.  According to Hersh, the humanists are
the good guys and the foundationalists are the villains.  There are a
few other minor camps too, but foundationalism is Hersh's bogeyman.

 > If one believes that ZFC is, in some appropriate sense, a
 > formalization of mathematics, then is one a fascist?

Well, it's not as stark as that, but yes, Hersh's chapter on politics
amounts to something like that.

 > I would like to make a list of books and papers (and perhaps other
 > forms of publication) that are of this type -
 > ...
 > ***where there at least appears to be a claim or the strong
 > suggestion of the insignificance of at least some work done in
 > mainline f.o.m.***
 
Well, you are familiar with Macintyre's short paper "The Strength of
Weak Systems" and Sacks' short paper "Remarks Against Foundations".
And there is the book edited by Tymoczko.

How about Kreisel's writings?  The problem is that there is so much,
and so badly written ....

 > After the list gets ripe enough, I pledge to review all the items
 > for the fom list.

Wow!  I'm eagerly anticipating this ....

-- Steve