[FOM] Foundations/Philosophy

Harvey Friedman friedman at math.ohio-state.edu
Fri Sep 19 03:12:53 EDT 2003

Reply to Slater.

On 9/18/03 12:09 AM, "Hartley Slater" <slaterbh at cyllene.uwa.edu.au> wrote:

> Martin Davis has written to me:
>> I would encourage you to post explaining why you think that the
>> set-theoretic foundation for mathematics with which most advanced
>> mathematics textbooks begin, is no longer relevant.

Slater did not take up the Davis challenge in what he wrote.

The set theoretic foundations of mathematics is still the only workable
universal foundations for mathematics that we have. Other candidates are
either philosophically incoherent, or are naturally mutually interpretable
with the set theoretic foundations.

For example, all known forms of categorical universal foundations that have
been clearly put into a form that is both workable and philosophically
coherent, are naturally mutually interpretable with the set theoretic
foundations. In fact, they overtly borrow from set theoretic foundations.

By "universal" I meant to exclude exclusionary views of mathematics (which
are extremely interesting) such as intuitionism, or finitism, or views that
mathematics is only number theory and geometry and real/complex analysis,
etc. But even here, there are appropriate versions cast as fragments of the
set theoretic foundations. So once again, set theoretic foundations proves
its incredible (and unique) versatility and power. E.g., intuitionistic set
theory, and also finite set theory.

> That was because I expressed to him my disappointment at what seemed
> to me a shallow interest in foundational issues amonst members on the
> list, despite it being 'FOM'.

Whatever disappointment you might have is overshadowed by my disappointment
in seeing such a statement.

Perhaps you do not understand or accept or take into account the difference
between the (usual) methodologies in foundations and the (usual)
methodologies in philosophy.

>I had in mind here centrally FOM
> reactions to Benacceraf's points about 'what numbers cannot be'.  In
> discussions on the list earlier this year, which I took part in,
> these points, and similar ones about real numbers, were taken to be
> irrelevant by many other correspondents.

When this discussion came up, I engaged, as usual, in the usual methodology
of foundations, and not the usual methodology of philosophy.

I wrote down some (apparently) new formal systems of set theory in which the
real field is taken as primitive, discussed how mathematics would be
formalized in this system, and gave some sharp conservative extension
results - which if I remember had to be rather carefully drawn. Also,
Andreas Blass discussed another approach.

The usual methodology in foundations is to use insights from philosophical
criticism/discussion/argumentation in order to
develop/discover/invent/create new subjects.

By subject, I mean a systematically organized body of knowledge, with clear
standards for the drawing of conclusions, etc.

I don't use the word "science" here, because that word has some specific
connotations surrounding experimentation. I.e., most scientists do not
accept mathematics as science.

So here is a clear example. I acted on points about "what numbers cannot be"
by seeing what one can do in the way of foundations of mathematics that is
not subject to the same criticism. This does not mean that I endorse any
kind of negative attitude towards the usual way numbers are treated in set
theory, on this basis. It also does not mean that I endorse the usual way
numbers are treated in set theory, on this basis.

I take instead the attitude that anything can be criticized or defended -
imaginatively and unimaginatively. And any such criticism or defense can be
criticized or defended - imaginatively and unimaginatively.

And that this process can be iterated.

So engaging in that iteration for its own sake is not the focus of
foundations. The focus is on what new subjects arise from analyzing
philosophical criticisms and defenses - even iterated.

The great power and scope of foundations is that a surprisingly varied array
of fruitful, satisfying, illuminating, exciting, deep, subjects flow
naturally and prolifically out of virtually any competent philosophical

In contrast, the philosopher would normally act on points about "what
numbers cannot be" by, say, continuing with "what numbers could be" or "what
numbers really are". The philosopher will take sides on various related
issues, write papers defending and criticizing various positions, and ride
various waves of fashion concerning prevailing attitudes towards such
issues, with no expectation of any "resolution". Just giving it their best
shot, hoping that their views and criticisms will stand the test of time.

Obviously, you can tell which methodology I prefer to work under, and what I
think are the relative prospects for doing things that have a definite, or
even definitive, permanent value. Kurt Godel, at least partly, operated
under this methodology.

But I am not at odds with the philosopher. I have to be philosophically
perceptive and engaged in order to create philosophically motivated
subjects. And philosopher's arguing interminably - even running around in
circles with dubious progress - just gives me more material to work with.

And I don't have to put up with "running around in circles with dubious

I do regret not having had the time and the opportunities to do this kind of
thing in geometry, statistics and physics. I may yet do something serious
about this.

Of course, foundations presently has no established position in academic
life. E.g., 

1. Philosophers do not normally engage in the development of new subjects
(in the sense we are talking about).
2. Mathematicians do not normally engage in philosophical thinking (in the
sense we are talking about).

What remains to be seen is whether/how foundations will survive.

Over the next few years, as quite a number of my major pieces of work come
to fruition and are published, I will begin tackling this practical problem
in the preceding paragraph.

Harvey Friedman 

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