FOM: set theoretic foundations

[Harvey Friedman]

Reply to Tragesser, 7:44AM 2/26/98.

>Fruitful interaction of mathematical
>domains as no justfication for SET as
>fundamental (indeed,  more as grounds
>for being suspcious of SET).

Set theory is obviously fundamental. At the elementary level, putting a few
objects together into a whole unit is arguably the most fundamental of all
mathematical concepts.

>The
>need for an explanation of the empirical
>"fact" of near ubiquitous reducibility
>to SET.

I addressed this in my posting of 3:40AM 2/26/98. As I said before, normal
mathematical moves are trivially set theoretic, so what's so surprising?
Often one wants a structure satisfying certain conditions, with no
particular ad hoc features. So one makes the obvious set theoretic
construction, and then introduces constant symbols as I have discussed.
There is nothing surprising here. When mathematicians "get rigorous" they
recast things so that the set theoretic aspect becomes totally transparent.

>Boolos's Conjecture
>as the only promising avenue for investigat-
>ing the foundational significance of SET.

You don't even state this till the end. What you call B's Conj. is simply a
concise variant of what I am telling you. And it is not due to Boolos or
me. It's standard stuff in f.o.m. that has been around before we were born.

>        My rhetoric -- well mocked by Tennant --
>was not so much responding to Maddy/Tennant
>as to a kind of dogmatic/ideological barrier
>I've long experienced from Generation ZF+Xer
>foundationalists.

What is a Generation ZF+Xer foundationalist?

>        I want below to give the only sensible
>defense I've heard for the fundamental character
>of ZF+X;  I heard it from George Boolos.

The fundamental character of V(0), V(1), ... V(w), V(w+1), V(w+2), ...,
V(w+w) is obvious to anyone who understands them. Of course, there are
issues at deep levels but not at the level at which you are conversing on
the fom.

>        That set theory does not have a strong
>and coherent sense,  that it's basically a
>product of tinkering,  and that it has not
>been re-produced from a clear and distinct
>idea that allows us to see why it is so fundamental,
>does suggest that the Generation ZF+Xer
>foundationalists are (were?--are there any?)
>driven not by serious thought but
>by some combination of ideology and convenience.

Set theory obviously has a strong and coherent sense, has many times been
reproduced from clear and distinct ideas that make it obvious that it is
completely fundamental. This is especially clear of the set theory that
supports V(0), V(1), ... V(w), V(w+1), V(w+2), ...; e.g., Zermelo set
theory. The work of Russell, Zermelo, Godel, and others is obviously based
on serious thought. You should emulate them.

This is not to say that I haven't been interested in reaxiomatizing set
theory in a way that the axioms look even more fundamental, and the large
cardinal axioms also look more fundamental. This stuff is on my website,
and I plan to discuss it on the fom.

>The
>"picture" of the cumulative hierarchy is so blurry,
>so mathematically, logically and philosophically
>problematic,that one cannot imagine "it" having
>(a right to) foundational authority over the clearer,
>significantly less problematic, "picture"
>of the finite ordinal numbers.

Is V(0) blurry? Is V(1) blurry? Is V(2) blurry? Is V(w) blurry? Is V(w+1)
blurry? Is V(w+2) blurry? And blurry compared to what? Perhaps you want to
claim that these get significantly more blurry than the natural numbers
under <. This is a defensible position. But locally the picture is clear,
and one is iterating along the "finite ordinal numbers." You should
emphasize just how non-blurry the short initial segments of the cumulative
hierarchy are! That's the interesting point. Not how it may be less clear
than the natural numbers under <.

>Even if it is thought that the cumulative
>hierarchy is an expansion on "set of" as
>something like a (Neo-)Platonic Form,  and
>that the cumulative hierarchy picture is defective
>simply because we have not sufficiently
>plumbed this form,  we are still owed an
>EXPLANATION of why this Form ("set-of") is
>fundamental to mathematics,  always and forever
>(or at all).

Because when mathematicians "get rigorous" they get very set theoretic.
What is so surprising about this? After all, a set of few objects is,
arguably, the most fundamental mathematical idea.

>        I am not in the least moved -- and deeply
>wonder why anyone is -- by the reputed "empirical"
>fact that every piece of mathematics known can
>be coded in ZF+X (for some well-considered X's).

The coding is immediate as soon as one tries to "get rigorous." There is
nothing surprising about it.

>        I do not find this "marvellous";  rather I
>find it suspicious.

Its marvellous how such a few fundamental principles about the fundamental
notion of set support all of accepted mathematical reasoning. There is
nothing suspicious about this. It has been well known since the early part
of the century.

>This "empirical law" needs
>an accounting,  an explanation: What is there
>about mathematics that they can be buried
>in SET,  and what is it about SET that makes
>them their Earth (Dust)?

I don't think it is properly viewed as empirical, in the sense that
measurements of the US population is empirical. It is an analysis of the
conceptual moves inherent in "getting rigorous."

>Could it be that the standards of
>mathematicians are so much
>lower than those of theoretical physicists?

The standards of mathematicians are hugely higher than those of theoretical
physicists when it comes to foundational matters. F.o.m. is incomparably
more powerful, coherent, and convincing than any foundations for even
classical mechanics - let alone quantum mechanics, relativity, continuum
mechanics, cosmology,  etcetera.

>[1] There is no parity between fundamental physics
>and SET.

The foundations of mathematics is incomparably more effective and
convincing than any foundations of physics.

>I dare say that for a rather great
>range of mathematics,  mathematicians gain nothing
>by modelling their conceptions in SET;

The modelling is trivial and immediate. What is gained is a form of
absolute rigor which is of the greatest intellectual significance.

>worse,
>they likely lose touch with their subject.

As I have said in this and earlier postings, this is completely false. One
uses constant symbols, so that there is no change. One has simply added
something that supports a form of absolute rigor. And this is precious.

> There
>is no serious Botanist whose imagination and
>understanding is not going to be fruitfully
>furthered by learning as much quantum mechanics
>as they can manage,  and not just "in principle"
>or out of physicalist/reductionist convictions,
>but because quantum mechanical facts/principles
>have immediate cash value for understanding a
>thousand processes they encounter daily.

I doubt this. How many Botanists have any useable understanding of quantum
mechanics? Do you have an interesting application of quantum mechanics to
botany?

>I
>see no evidence of this kind of turn around wrt
>principles and facts of set theory (sufficient to
>take set theory as fundamental).

People in various areas of mathematics outside logic routinely bump up
against very deep set theoretic matters when they work in rather full
generality. E.g., discontinuous homomorphisms between separable Banach
Algebras, Whitehead groups, Fubini's theorem for arbitrary sets, Ramsey
theorems for uncountable sets, probability measures on all sets of reals,
etcetera. These matters are known to be deeply tied up with modern topics
in set theory. This is incomparably closer to mathematics outside set
theory than quantum mechanics is to botany.

>But I do not see here
>any justification [Shipman seems to] for
>set theory in its role as the great unifier
>of domains,  as undwrwriting or even as
>explaining these fruitful "interactions".

See above. And wait for some of my future postings about recent work.

>... so in general
>mathematical conceptions ARE NOT
>instructively reflected in SET.

You should instead emphasize just how set theoretic mathematical
conceptions are, when thought of rigorously.

>show us exactly what those translations'
>are worth to any arbitrary piece of
>mathematics.

Set theoretic f.o.m. would already be of extreme importance if the
formalization paid absolutely no dividends to the practice of mathematics.
It is vitally important, interesting, satisfying, etcetera, that there is a
spectacularly successful and coherent f.o.m. And right now, this is done
only through set theory.

>        I have by the way an analogous
>problem with the sorts of compression results
>of Feferman and Simpson: exactly
>what are we learning about mathematics?
>What is the mathematical signifiance of such
>results?

Any crank can say of any result about anything, "what is the xxxx
significance of these results?" You can fill in the xxxx. The results of
reverse mathematics have an obvious interest to anyone with general
intellectual competence. It gives us a fundamental classification scheme
with surprisingly few equivalence classes. This kind of situation is always
greatly interesting.

>most any research
>program in mathematics where the
>aim is not just to secure "truths",  but
>to understanding something (e.g.,  Charles
>Feferman in pursuit of "uncertainty"
>phenomena dropping out of Fourier analysis)
>would likely not be furthered,  but be
>hampered by (so that its fundamental "facts"
>are not adequately coded or represented by,
>but rather distorted by)
>such PRA,  PR,  ZF reductions.

I suspect that if you were more familiar with such reductions, you would
see that no relevant kind of distortion occurs. In any case, the matter is
irrelevant to the purposes served by such reductions. Are you familiar with
those purposes? By the way, you mispelled the name of my good friend, who,
incidentally, is a strong supporter of f.o.m. His first papers were in
model theory.

>That is,  yes, it is surely important to know
>the conditions under which, say,
>the Bolzano-Weierstrass theorem,  Ramsey's
>Theorem(s) for coloring,  the theorem that
>every countable field of characteristic 0
>has a transcendence base..  But it does
>seem that once one knows this, then it
>becomes deeply important to know (but it is
>deeply important anyhow) how
>these theorems are inequivalent,--
>the exact senses in which they cannot
>be substituted for one another salve
>veritate.

What results do you have in mind? What conjectures do you wish to make?
What questions do you have?

>        I'll have to save Boolos' Conjecture
>for another occasion;  but it was  this
>(told to me in conversations circa 1990):
>        "An intuitive proof [=a proof wanting
>in some respects in formal-logical rigor]
>of a theorem is [based on an] an intuition
>that it is demonstrable in ZF+X."

This is something of a variation on what I have been saying. It is ancient,
and not due to Boolos or me.

Let me close with some advice to you. You often mention Tennant and Maddy,
both of whom I know well, especially through e-mail. Both of these people
are sufficiently technically competent in order to address many aspects of
f.o.m. and ph.o.m. in a productive manner. And this is partly due to the
fact that they know what they know and don't know, technically, and ask and
learn what they need to know, technically, in order to maintain their work
on f.o.m. and ph.o.m. at a coherent, relevant, and productive level. This
is not easy. It does mean that they don't throw around technical ideas that
they don't really understand, or  drop names and use jargon that they hope
their readers will be impressed with, or make summary judgments of delicate
issues without documentation. It does mean that they also ask a lot of
questions of a lot of experts, and exercise restraint before coming to
conclusions.

I suggest that you emulate Tennant and Maddy.