[FOM] More Friedman/Baez

[Harvey Friedman]

More from https://golem.ph.utexas.edu/category/

Friedman:

I would greatly prefer it if people would use phrases like

foundations of applied mathematics foundations of mathematical physics
foundations of algebraic topology foundations of category theory
foundations of geometry (now this is something I have some unfinished
work on) etcetera

than

foundations of mathematics.

I would also be nice if a few things were generally acknowledged:

That by several orders of magnitudes, the most highly developed,
deepest, successful, rich, foundations of anything is the standard
foundations for mathematics.

1 is not because people who have developed it are so great (some of
them are), but because foundations of mathematics is MUCH EASIER than
foundations of physical science, music, etcetera. It is the first, and
needs to be the first, being so comparatively trivial.

Despite the fact that real f.o.m. is comparatively trivial, in order
to make real progress in the foundations of practically anything else,
especially physical science, music, etcetera, the many many long
fought painful lessons that were learned and enormous hurdles that
were overcome to get to real f.o.m., things that at a higher level we
still struggle with and get nowhere, or almost nowhere - promise to
eventually help jump start the developments to be in these other
foundational efforts. Absorbing at a deep level what worked and didn’t
work, and why it took so long to get where we are (roughly 1920 and
beyond) could save hundreds of years in wandering around the desert
when trying to seriously develop real foundations of other subjects.

Of course, I am not talking about directly applying standard f.o.m.,
although some of that is certainly going to be illuminating. I’m
talking about absorbing the genuine f.o.m. experience.

In fact, I have found various kinds of physicists much more interested
in standard f.o.m. than mathematicians. I think it may have something
to do with how top physicists realize that at some fundamental level,
they are completely lost foundationally. But my impressions here are
not to be trusted.

So I am content if it is generally recognized that some of the
greatest and most powerful intellectual structures ever created (or
discovered), and some of the most shocking revelations ever revealed
have come out of standard f.o.m. And the expectation is that there
will be yet much more powerful and yet much more shocking things to
come in standard f.o.m.

Baez:

“Where can I read a nice overview of the results you’re alluding to?”

There are a number of quite different things that you could be referring to.

A. Taking the view that there is an objective reality (V,epsilon), and
getting truth values of various well known problems. Well, the well
known problems here are going to be intensely set theoretic, led by CH
(continuum hypothesis). In my biased opinion, this is not going well
by the grandiose standards I apply to everything. Of course, there is
a credible community that legitimately draws salaries, and many are
very clever. There is Solovay, Woodin, Martin, and the philosopher
Koellner, in no particular order. Woodin already has written for the
Notices. If I remember right, he retracted some of the positions he
took there. There are also seriously knowledgeable scholars of this
line, notably Aki Kanamori. Steel, also in the inner circle, has
broken with the objective reality of (V,epsilon), causing a deep
fracture. Woodin’s student, Hamkins, has broken even further away,
tending to treat models of ZFC as an ordinary mathematical subject
like the study of finite groups. I would recommend that you contact
these guys to see about materials and the possibility of writing for
the Notices. If I remember, there are a number of articles in the
Stanford Encyclopedia by Koellner, who works very closely with Woodin.
That is probably the best source for this line of taking (V,epsilon)
to be the preferred objective reality, with statements like the CH
being matter of fact.

B. Focusing on whether ZFC is sufficient to do all that we really care
about in normal mathematics. The idea here is that normal mathematics
is very far from being intensely set theoretic, and the crucial issue
since 1930 is whether ZFC is limited for normal mathematics As a
working approximation, I am talking about perfectly natural thematic
arithmetic sentences - with an emphasis on Pi-0-1 sentences. At least
one person now believes that a revolutionary sea change is brewing in
various senses after a very long incremental effort. The goal here is
to find examples that are sufficiently shocking to other than
mathematical logicians (they are beyond being shocked) so as to create
a feeling of being profoundly disturbed. I don’t have a good feel
about how profoundly disturbing these brewing developments will be to
the typical reflective mathematician. But I have no doubt that by the
end of the century, this will have infected the whole of pure
mathematics, and will be profoundly disturbing to everybody.
Unfortunately, I won’t be here in 2100 to see this.

Harvey Friedman