[FOM] Coherence:Friedman/Cody/Roberts

[Harvey Friedman]

More from https://plus.google.com/110536551627130071099/posts/6TiKLxjSCnu

Cody:

Come on, Harvey, your argument boils down to: if ZFC was good enough
for Gödel, it should be good enough for everyone! You can do better
than that. We were starting to have a constructive discussion here.

Friedman:

What argument for what?

Cody:

The argument by historical fiat can be used to deny legitimacy of any
radically new point of view. See e.g. schemes in algebraic geometry,
distributions in PDEs etc.

Friedman:

Before you or anyone call something foundations for mathematics, I
suggest that you at least attempt to make philosophically coherent
presentations. Maybe you can make a decent attempt at this, and I hope
you will!

Cody:

I'd direct your attention to the fact that the first paper on
incompleteness explicitly refers to Russel and Whitehead's theory,
which is not "standard f.o.m" by any means, so maybe ZFC wasn't the
instant zeitgeist you seem to suggest it was.

Friedman:

We are into almost 100 years of a model of what a foundation for
mathematics is, which of course I reexamine every day. So we need to
apply higher standards in 2016 than we did at various points during
its growing pains when we had no clear idea at all of what it was
going to look like.

Cody:

I'd by overjoyed by some vigorous attention applied to finding
decision procedures for fragments of advanced mathematics. I've always
believed the the "left as exercise to the reader" proofs in category
theory were ripe examples, and I guess Gowers is working on
undergraduate level proofs for topology. Of course, there are much
more developed examples with software like Sage and Gap, or things
like Kenzo for homology.

Friedman:

There are many genuine f.o.m. research programs lurking here,
including from category theory. I don't know anything about Sage and
Gap and Kenzo. Maybe you can give a two page overview of what kinds of
statements are being decided automatically by them.This is not any
kind of alternative foundations for mathematics, but rather something
that fits squarely into the standard f.o.m. tradition. I hope that
doesn't make it uninteresting. 

David Roberts

There we go again with philosophical coherence. Define it for us
please, and not in such a way that, surprise, only ZFC (or relatives)
can possibly satisfy it; I don't believe that there is a unique
philosophically coherent system on which to base mathematics. This
would be a more amazing theorem, if it could be elevated to such
status, than incompleteness.

Friedman:

I think there is a theorem to the effect that any two natural systems
that interpret PA are comparable under interpretation. Also in the
suitable sense, their Pi-0-1 theorems are comparable. And stronger
things. I wrote about this on the n-category cafe in response to John
Baez. Here "natural" is much weaker than "philosophically coherent".

Also I wrote on the n-category cafe I responded to Eugene Lerman when
he asked me what foundations of X is. My answer is relevant there.

There is a whole big Western philosophy community that is in the
business of writing philosophically coherent papers, and I have yet to
see them carefully define what that means. So this is not going to be
easy to do. It is also not easy to define "musically coherent in the
context of classical piano performance". 

Arnaud Spiwack:

Man! This is entertaining!

http://media.riffsy.com/images/7260ad7de2d77fa3b597026aee82b391/raw

Roberts:

If you cannot say what philosophical coherence is, how are proponents
of non-ZFC foundational systems supposed to fulfil your demand for
such an explanation? If responses I've seen are merely code for "*I*
don't find that philosophically coherent", then the issue of
communication is worse than previously suspected, since I've seen
plenty of explanations that seem to have a good deal of thought behind
them, and which make sense.

Comparisons to music are fraught with risk: the European musical scene
that rejected Beethoven's late string quartets became after a few
decades the movement that would have called them too conservative. Is
it not possible we are seeing a similar shift in attitudes here? For
some people, Lawvere's ETCS, so derided on the fom mailing list in the
late 90s, is just another set theory and doesn't go far enough.

Friedman:

I claim that everybody on the discussion has a good working idea of
what philosophical coherence means without the difficult and important
work of analyzing what it means. I'm just not in a position to do
this. There is a whole philosophy community who is more in a position
to do this, and certainly does it in a practical sense all through
their Western curriculum

This is sort of like asking what a rigorous proof is. Sure, we have
formalisms, but as a practical matter, a working mathematician must
use instinct in order to actually create rigorous proofs and also
evaluate whether someone else has given a rigorous proof. Of course,
the formalisms are important as final arbiters, when issues really
arise. ZFC has been around for so long and is so natural, that just
about everybody knows without much thought and very reliably whether
they have stayed within ZFC.

I don't think there is any serious communication problem along the
lines that you are worried about. There does seem to be other
disagreements, intellectual value conflicts, or whatever. Here is how
I see it.

All of you definitely agree that I can give an explanation of standard
f.o.m. from first principles that relies only on the audience having
some sort of reasonably competent basic mathematical instincts, with
essentially no special mathematical knowledge or experience. Putting
together prescribed things into a set, where those things might
themselves be sets, and using words like not, if then, there exists,
equal, element of, and so forth, these are IQ matters. Especially in
the finite world. I have some theorems about going from the finite
world to the infinite world here, and experience with this sort of
thing would make my explanation of standard f.o.m. from first
principles arguably more compelling than everybody else's (maybe).

So there is definitely the recognition out there that making standard
f.o.m. part of you depends on nothing but a raw brain, and perhaps can
be absorbed while in the womb.

I have been looking for a similar fundamental explanation from first
principles for ANY purported foundation for mathematics, and consider
that the first standard for getting into the foundations game.

I have been using this harsh sieve and asking people to conform to
this standard in open forums such as the FOM and the Baez (and now
Cody) sites.

Nobody seems to want to take this up for a number of varying reasons.

1. They know what I mean, but they recognize that this is hard to do
and would require some serious thought. They think they could do this,
but it would not be worth their time given their other priorities.

2. The know what I mean, but they recognize that they don't know how
to do this, because their understanding of seriously alternative
foundations is simply too closely tied with some special mathematical
situations that are not compelling from first principle thinking.

3. Most think that this is not an important criteria for a seriously
alternative foundation for mathematics. I suspect that some minority
do think that it is, but more in the public relations sense, not in
any deep intellectual sense.

So there you have my assessment of the so called "communications problem".

When I explain standard f.o.m. from first principles, I don't tell you
to go learn some complex variables or algebraic topology or finite
group theory or differential equations. I present it from first
principles, and YOU pick something YOU like to think about, ANYWHERE
in math, and see how this works for YOU. Now that's general purpose.
Ideally, you may need very modest doses of this, but the main point is
that you get to pick if you want a reference point and see it in
action.

When you are willing and able to do this for your alternative
foundations for mathematics, you have a seat at the foundational
table. You may have to vastly improve on what you have before you can
get that seat at the table.

Actually, at this point, I have no idea how far you are from a seat at
the table. Maybe you are not too far, and I can lift you up and carry
to over there and sit you down. Maybe you are essentially there and
are missing maybe two sentences from somebody like me.

Harvey Friedman