FOM: real issues?

[Harvey Friedman]

I admit to not having followed carefully the Hersh/Simpson, and secondarily
the Hersh/Davis/Shipman et al, correspondence on the fom. I have certainly
casually read all of the relevant postings.

I propose that the relevant people carefully clarify their positions
according to the following scheme. Why should you? Well, I think a
clarification along these lines would be useful for those people on the fom
who, like I, have *not* carefully studied these exchanges in detail.

I also call for an attempt to create a comprehensive list of publications
in a certain category as you shall see at the end of this posting.
**********

1. Is there a meaningful concept of absolute certainty, beyond which one
cannot go? And is this ever realized in mathematics? If it is sometimes
realized in mathematics, then where in mathematics is it realized? Do you
know of a candidate for a piece of mathematics that is not absolutely
certain? If so, what is the oldest and/or most elementary example of a
piece of mathematics that is not absolutely certain? E.g., is the "fact"
that there is no one-one correspondence between {1,2,3} and {1,2,3,4}
absolutely certain? And, e.g., is the "fact" that, for all positive
integers n, there is no one-one correspondence between {1,...,n} and
{1,...,n+1} absolutely certain?

2. How does the level of certainty in mathematics compare with the level of
certainty in other disciplines such as physics? E.g., is standard
elementary school mathematics more certain than standard elementary school
science?

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

4. Does Hersh's writings convey any judgment on the value of formal systems
and the extensive work on them in f.o.m.? If so, what judgment is conveyed
to his readers? If the answer to 3 is yes, what guidance does Hersh give to
this extensive literature? Is Hersh under any obligation to give such
guidance?

5. What is foundationalism? To what extent is ZFC a formalization of
mathematics? If one believes that ZFC is, in some appropriate sense, a
formalization of mathematics, then is one a foundationalist? If one
believes that ZFC is, in some appropriate sense, a formalization of
mathematics, then is one an anti-humanist? If one believes that ZFC is, in
some appropriate sense, a formalization of mathematics, then is one a
fascist?
***********

It seems apparent that at least some people think that Hersh's book "What
is mathematics, really?" does claim or the strong suggestion of the
insignificance of at least some substantial work done in mainline f.o.m.

I would like to make a list of books and papers (and perhaps other forms of
publication) that are of this type - with the help of the fom. I have a
short list I will share with you later, but I would like to see postings on
the fom mentioning such books and papers. I.e.,

***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.***

Here are some general categories to look for:

1. Writings by mathematicians who got interested in f.o.m. and/or p.o.m.
Possibly Hersh, etc.
2. Writings by mathematicians who remain disinterested in f.o.m. and/or
p.o.m. Possibly Bourbaki, etc.
3. Writings by mathematical logicians who remain disinterested in f.o.m.
and/or p.o.m. Possibly MacIntyre, Sacks, etc.
4. Writings by mathematical logicians who got interested in f.o.m. and/or
p.o.m.
5. Writings by specialists in f.o.m. Possibly Feferman, etc.
6. Writings by philosophers who got interested in f.o.m. and/or p.o.m.
Possibly Wittgenstein, etc.

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

Of course, having been a professional f.o.m.-er for over 30 years, I have
heard and heard about all kinds of oral remarks which never made their way
into print. But I think there may be more than enough to chew on that is in
print (or e-mail). Of course, anybody who wishes to come out in the open
and express their critical views about mainline f.o.m. is doing quite a
service.