[FOM] ZFC Foundations for Mathematics

[Harvey Friedman]

I think I agree with every single sentence of the lengthy Chow's reply
to Katz, https://cs.nyu.edu/pipermail/fom/2020-January/021929.html

One may ask why one wants to have a foundation for all of mathematics,
satisfying, say, Maddy's criteria, in the first place.

This is something that is basically considered obvious for those with
what one may call "philosophical sensibility" or "general intellectual
outlook" for lack of a better analysis.

It is tied to the more general "processing of intellectual structures"
that has proved so important to human affairs

In general, it appears that the power of our intellectual structures
is tied to the extent that we have at least something resembling a
foundation underneath them.

A particularly well known example of an intellectual structure way
outside of mathematics is our legal system - and it is divided into
many legal subsystems. Just like in mathematics, there are general
legal principles that apply to all of the legal subsystems, and there
are more special principles that apply to the various subsystems.

Coming back to mathematics, I want to point out just what is gained by
having a credible general foundation for all of mathematical activity.
Notice that I have now raised the stakes, perhaps, not so noticeably.
I said "mathematical activity" not "mathematics".

There are all kinds of crucially important "mathematical activities"
that ZFC does not provide a foundation for, and that we really don't
have a foundation for. And in some cases, there are important
mathematical activities that ZFC does not provide a foundation for but
there are some attempts, perhaps some credible, that are, to some
extent at least, effective.

So I think it would be helpful to state some main consequences of us
having the highly effective and powerful foundation ZFC for
mathematics - in addition to matters of philosophical coherence,
general intellectual interest, etcetera.

1. A clear way of being able to join the question of great general
intellectual interest - if not of interest to a typical core
mathematician - of whether mathematics is at least sound in the sense
that there is no mathematical proof of a contradiction. Without
something like ZFC, this question cannot even be mathematically or
effectively joined. Core mathematicians may find the question
disgustingly stupid and irrelevant, and consistency of mathematics is
so far beyond reproach that this is a silly ridiculous question. But
this is not generally the attitude outside the mathematical community.

In fact in the general intellectual community, there is far more
interest in whether mathematics is sound at least in the sense of not
leading to a contradiction, than there is in any other fixed
mathematical problem or issue.

2. A clear way of being able to join the issue of just what kinds of
mathematical questions cannot be resolved using current mathematical
practice. Without something like ZFC there is no credible way to join
this issue.

3. Now that the claimed examples under 2 are moving so rapidly into
those being woven inextricably in the very fabric of current
mathematical thinking and practice, the power of ZFC as a universal
foundation comes to the fore.

Harvey Friedman

On Thu, Jan 16, 2020 at 11:00 PM Timothy Y. Chow
<tchow at math.princeton.edu> wrote:
>
> M. Katz wrote:
>
> > It has been assumed for about 70 years that in homological algebra and
> > certain subfields of algebraic geometry, category theory is a more
> > appropriate framework than set theory.
> [...]
> > Moreover, attempts by set theorists to challenge the conclusion that
> > category theory is a more appropriate foundation in these fields can
> > only lead to such attempts being discredited.
> [...]
> > To retain credibility, arguments in favor of ZFC or such as foundation
> > should be grounded in a choice of field where such an argument is
> > proposed, and preferably made by experts in such fields together with
> > set theorists.
> [...]
> > However, I think that ostrich insistence on ZFC as universal foundation
> > for all of mathematics is no longer credible, and may have the opposite
> > effect as far as grants are concerned.
>
> I think that Maddy's terminology, and her distinctions between different
> roles that a "foundation" for mathematics might play, are helpful here.
>
> Also, as I've said before on FOM, I think that it's important to
> distinguish between "foundation of X" where X is some subfield of
> mathematics, and "foundation of mathematics."
>
> In the 20th century, there was a shift from varieties to schemes in
> algebraic geometry.  This is regarded by practitioners of algebraic
> geometry as a change in the "foundations of algebraic geometry."
> Categories, of course, play an essential role in scheme-theoretic
> foundations (and its descendants).
>
> In my opinion, the shift from varieties to schemes should *not* be
> regarded as a shift from "set theory" to "category theory" in the
> "foundations of algebraic geometry," much less in the "foundations of
> mathematics."  When practitioners think about the "foundations of X," they
> are generally *not* thinking about drilling down to the foundations of all
> of mathematics, or trying to build up the entire theory from absolutely
> primitive philosophical concepts.  They are drilling down only as far as
> they need to for the purposes of furthering their research in X.  In
> Maddy's language, they are concerned with Essential Guidance.  They
> couldn't care less about whether their new foundations provide a Generous
> Arena that is generous enough to encompass some other branch of
> mathematics that they don't care about.  They also couldn't care less
> about providing an adequate Metamathematical Corral, unless for some
> reason metamathematical concerns are directly relevant to their next
> research paper in X.  Conversely, if someone comes along and insists on
> defining categories in terms of sets, the general attitude will be, "Fine,
> do whatever you want, as long as I can have my categories and reason with
> them the way I want to reason with them."  Turf wars between category
> theory and set theory are of little interest to them, again, unless their
> next paper actually requires them to pay attention to such niceties.
>
> This kind of attitude toward "foundations of X" can, for the most part,
> coexist peacefully with a variety of views of the "foundations of
> mathematics."  In particular, I don't share M. Katz's pessimistic view
> about set theory getting discredited if someone insists on set theory as a
> foundation of mathematics.  As I said above, if you want to insist on
> using set theory as a Generous Arena for *all* mathematics, and you claim
> that your set-theoretic framework can support everything that the
> algebraic geometer wants to say about categories and topoi and schemes and
> so forth, the algebraic geometer is unlikely to complain.  Just don't
> commit the faux pas of saying that you're proposing set theory as a
> "foundation for algebraic geometry" in a way that suggests that they're
> going to have to abandon their talk of categories and whatnot.
>
> I think that the real clashes come not when some particular subfield X
> makes some assertion about the "foundations of X" but when a broader claim
> is made, about something (say, category theory) being a complete
> substitute for set theory for the foundations of *mathematics*.  If and
> when such a claim is made, then I think one can rightly ask if the
> alternative foundation provides not just better Essential Guidance for
> some subfield X, but is also well equipped to shoulder the burden of other
> tasks that a foundation for all mathematics is often expected to
> shoulder---a Generous Arena and Shared Standard for *all* of mathematics,
> and a Metamathematical Corral that gives a clear explanation for why we
> need not worry about logical paradoxes arising.
>
> The vibe that I sometimes get from proponents of alternative foundations
> is that they don't see the need for Generous Arena and Metamathematical
> Corral.  They may talk about a foundation for "mathematics" but secretly
> they really only care about their own subfield X, and so it doesn't bother
> them that their proposed Arena is awkward for other mathematicians.  They
> also, by and large, don't care too much about metamathematics, and I think
> that part of this is that from their point of view, mathematics is
> *obviously* sound, so worrying about its soundness is silly philosophical
> nonsense.  The feeling that mathematics is obviously sound is what, in my
> opinion, underlies Penrose's notorious arguments about Goedel's theorems.
> I also think that it explains Voevodsky's notorious lecture on whether PA
> might be inconsistent.  In the last part of that lecture, he talks about
> "reliable" mathematics in a way that, to me at least, betrays his feeling
> that the mathematics that *he* is interested in is obviously "reliable"
> and so there is surely some way to formalize that obvious fact.
>
> A cavalier attitude toward Generous Arena and Metamathematical Corral is
> one thing that annoys those with a traditional view of foundations, who
> feel that the hard-won insights of the early 20th century are being
> slighted.  So again, I don't agree with M. Katz that ZFC as a universal
> foundation for all mathematics is not credible, if we understand that
> claim rightly.  The claim isn't that for every subfield X of mathematics,
> we must explicitly use raw set theory for the "foundations of X" and
> eschew any defined notions.  The claim, rather, is that set theory is
> still the most convincing candidate when it comes to Generous Arena,
> Shared Standard, and Metamathematical Corral for mathematics considered as
> a whole.  In my view, this remains a highly credible claim, and it is not
> threatened by the success of any particular "foundation of X" that one
> might devise for particular subfields X.
>
> Tim
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> https://cs.nyu.edu/mailman/listinfo/fom