[FOM] Wittgenstein Inspired Skepticism

[Harvey Friedman]

On Sun, Mar 5, 2017 at 3:16 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> I do not believe that ZFC is the "undisputed foundations of mathematics." We
> have skeptics right here on FOM.  Nik Weaver doesn't find the usual
> justification for the cumulative hierarchy convincing, for example.  The
> fact that there is so much discussion and debate about the consistency of PA
> among working mathematicians also shows that just because something ("PA is
> consistent") is an established theorem of ZFC, it won't necessarily be
> considered as settling the matter.

ZFC is clearly the currently undisputed foundation of mathematics in
the following specific sense. Let me start with an anecdote. Charles
Fefferman once served (at least) a term as principal Editor of the
Annals of Mathematics. We did have an occasion to talk in person about
f.o.m. some and the matter came up of what he viewed the standards of
publication were for the Annals of Mathematics. On the issue of
correctness, he brought up the standard that the proof must be readily
formalizable in ZFC, and any use of additional axioms needs to be put
in as an hypothesis to an implication.

Of course, this does not mean to imply that all reasonable
mathematician agree that every theorem of ZFC is true, or has even
been appropriately established. Sure, there are finitists,
constructivists, predicativists, and other creatures among the
mathematicians. But I am only talking about the acceptance as an
established result by the general community as reflected by top
Journals.

The clear idea of the math community is that those who are skeptical
about the validity of ZFC have a clear outlet not in the form of
contesting standards at major Journals, but rather as an opportunity
to minimize the ZFC axioms needed for a certain proof or rework a
certain development to minimize the ZFC axioms used - and call
attention to it.

There is no significant level of debate about the consistency of
systems like PA or FST = finite set theory. This appears in a tiny
fragment of the math community with essentially no one listening,
except for very strange people like me who like to alternate between
looking for new consistency proofs of PA or FST, and making fun of the
relevant people.

> I agree that the challenge is to answer the specific concern, but since
> there are so many varieties of doubts out there, I think that those of us
> who are conversant with f.o.m. do well to have a "triage" plan, to quickly
> home in on where the sticking point is.  For this purpose, I believe that
> starting with Turing machines is, nowadays, a better opening gambit than
> starting with ZFC.  This way, you'll immediately establish a foundation on
> which you can agree (unless you really do meet a Wittgenstein-inspired
> skeptic) and can build up from there.
>
> You could, perhaps, instead start with ZFC and work your way down, but I
> don't think this works too well.  Most mathematicians have only a very hazy
> understanding of what ZFC is and couldn't tell you the axioms, so you kind
> of have to explain formal logic and Turing machines anyway.
>
In the last two paragraphs above, I am not sure what "problem" we are
trying to solve. For almost all mathematicians, they are using a tiny
fragment of ZFC, and never have experienced any doubts about the truth
of what they are doing, at least to the very small extent to which
they think about truth.

This does raise the following interesting f.o.m. question, which I
really should try to do better with now that I think of it.

What is a really good simple relatively weak fragment of ZFC that
mathematicians would easily recognize that what they do and what they
are interested can readily be formalized in that weak fragment? I am
thinking that this should be approached based on the choice of area of
math. Different systems for different branches of math.

A crude answer that seems to work extremely well, but should be fine
tuned considerably according to area, is Zermelo Set Theory.
Essentially ZFC without Replacement. = ZC.

Incidentally, the FLT situation is extremely illuminating for these
issues. There was NEVER an issue in the mathematics community about
the correctness of the FLT proof after it was corrected for a
technical error. I think there would be an issue if the experts
actually thought that the proof really dependended on Grothendieck
Universes. But the experts knew right away that that could be replaced
by readily grasped readily limited pieces of the mathematical
universe. I.e., ZC was enough.

Now ZC is an enormously powerful system compared to what we are
talking about triggering any skepticism. Yet as strong as ZC is, and
fragments climbing up a few levels of the cumulative hierarchy, I
NEVER saw any skepticism whatsoever from the math community concerning
whether FLT had been established, or whether FLT was true. The
commonly referred to proofs still use several levels beyond omega of
the cumulative hierarchy. See work of Colin McLarty.

Harvey Friedman