FOM: "crises" and crises

[Harvey Friedman]

This is a response to Lou van den Dries, 09:18 PM 10/14/97 -0500.

>Okay, "crisis" is not quite the right expression, and i regret
>having dropped that term.  

You seem to hold a hardened view that says something like this:  

"no really important core mathematician today can be seriously interested in
foundations of mathematics at this time," 

and wish to explain why the present time is different from previous times.
You apparently concede that these great mathematicians from the past,
Hilbert, Poincare, Brouwer, Weyl, von Neumann, were genuinely interested in
FOM (right?). You picked the word "crisis" because that would make it easy
to explain what is so different today, since nobody today thinks there is
presently a crisis in foundations of mathematics. 

Now that you have retreated from the word "crisis," it would be useful to
get clearer about some aspects of your position. I think you know my
positions very well. So let me ask you outright: to what extent do you agree
with the following:

"no really important active core mathematician today is or can be seriously
interested in FOM at this time."

And to what extent do you agree/disagree with the following?

"the general public, the general academic community, the general
science/engineering community, is more interested in FOM than pure mathematics."

And also:

"the general public, the general academic community, the general
science/engineering community, has minimal interest in pure mathematics
independently of its applications to other fields."

And also:

"the general public, the general academic community, the general
science/engineering community, has minimal interest in FOM independently of
its applications to other fields."

Once we get our positions out in the open on these questions, I think we can
focus more effectively on where we disagree, and why we disagree. 

>(It smells of those silly paradoxes,
>and i certainly didn't have that in mind at all.)

Does "silly" mean uninteresting? 

In any case, the paradoxes did represent a crisis - not for the working
mathematician and the practice of mathematics itself - but for our
understanding of the general laws of thought. E.g., it is compelling that
there should be a property that holds of exactly the properties that obey
any given well defined condition, and this leads to an apparent
contradiction. There are many paradoxes, and some books seeking to classify
them. Some other paradoxes are mere variants on this one, whereas others
involve apparently genuinely new elements.

In the general laws of thought, properties obviously play a central role,
and the rules whereby one creates properties appears to be an essential
issue. This is very very far from being adequately resolved or understood,
and is at the heart of a considerable amount of work done in FOM, in
Philosophy Departments, and in Computer Science Departments. Lou - are you
interested in "the general laws of thought"? And what do you think of the
discussion of the paradoxes in Godel's article, "Russell's Mathematical Logic"?

Interest in the general laws of thought goes back to antiquity, and appears
to be of the greatest fascination to a great number of people, including, of
course, people thinking about artificial intelligence and related topics in
computer science. It is, of course, extremely difficult to do something
definitive in this vein. 

So, again I think it useful to ask you to what extent you agree/disagree
with the following, in order to get a clearer sense of what your position is. 

"the general public, the general academic community, the general
science/engineering community, is more interested in the paradoxes than in
pure mathematics."

>Perhaps a better
>way of saying what was on my mind is this: At the time the now
>prevalent way of thinking about mathematical objects and correct
>way of reasoning about them had not yet consolidated itself, and
>of course several of the leading mathematicians had strong ideas
>about it and got involved. If I am not mistaken there wasn't even
>a consensus yet what to make of the various non-euclidean geometries,
>or the relation of geometry to physical reality. 

First of all, this does not address what I said about von Neumann in the
last e-mail: 

"And then there's von Neumann. The later part of his intellectual life was
spent on various foundational topics, including FOM. Again, crisis was not
the guiding principle of his foundational work in computing, reliability of
components, self reproducing automata, game theory, etcetera. There is no
doubt from his writings and from people who knew him that he would have been
extremely fascinated by modern work in FOM."

His choice of foundational topics and his foundational approach cannot be
"explained away" by you in these terms. 

Secondly, let's take one of the most intriguing statements in the Poincare
essay I cited:

"... that this induction is only possible if the same operation can be
repeated indefinitely. That is why the theory of chess can never become a
science, for the different moves of the same piece are limited and do not
resemble each other."

Now, Lou, how on earth can you explain Poincare's interest in this matter -
can chess become a science - in your terms? And, as I said in the earlier
e-mail, this question still has deep ramifications today, not only in FOM,
but in artificial intelligence and the specific project of chess playing
machines, which has attracted a great deal of attention. Which reminds me.
It would be nice to know to what extent you agree/disagree on:

"the general public, the general academic community, the general
science/engineering community, is more interested in chess playing programs
and/or artificial intelligence than in pure mathematics."

Finally, let me say that to this day, there is no consensus on "what to make
of the various non-euclidean geometries, or the relation of geometry to
physical reality." Issues such as "is physical reality continuous?" are as
controversial as they have ever been. Even a really appropriate
understanding of the subtleties of Zeno's Paradox still eludes us. 

In fact, there are even more foundational issues today that are of obvious
interest. Crisis - no. That will have to wait, apparently, for the project
that I talked about (see below). For instance, there is the new issue of
what the role of computers is, can be, or should be, in proofs. Witness such
topics as "why should I believe my computer?" Are you interested in that?

>All sorts of
>philosophical preconceptions were gradually put overboard (and perhaps
>others were imported in the process), and this took some time to
>play itself out, say from about 1870-1930.  

I have no doubt that we are now operating on a considerable number of
philosophical preconceptions now which will have to be put overboard. I
expect some of these preconceptions will be exposed by FOM. Some of them
have already. Such as various forms of the doctrine that "concrete problems
have concrete solutions". Or even the view that, to quote you, "the now
prevalent way of thinking about mathematical objects and correct way of
reasoning about them" is not subject to greatly useful and essential expansion. 

And there is the philosophical preconception that you are intimately
involved with on a daily basis. That "nothing useful for mathematics can
come out of the predicate calculus, since that is a philosophical
development." You use predicate calculus every day to grossly simplify
proofs, to find new proofs, to state things in proper generality, etcetera,
all through real algebraic geometry and other fields.

>And yes, I have read all
>those books and articles people like Poincare, Hilbert, Weyl and so
>on wrote on such "foundational matters". I must confess I considered it
>as light reading, for the most part, and it never impressed me as
>much as some other things they did. Some of what they said made a
>lot of sense, and was taken over by those who came after them,
>some of it is perhaps better forgotten.

There are two separate reasons for it being comparatively light reading.
First of all, it is so refreshing to most people to read something that is
of clear interest and is in very familiar terms. FOM concerns matters that
are very basic to our general perceptions of the world. I.e., we are in a
sense "genetically engineered" to understand the objects of concern to FOM.
We are born with a kind of primitive understanding of collections, counting,
shapes, elemental reasoning, and the like. Fundamental discussion of such
are naturally very human friendly. 

A two year old may grasp the difference between a triangle and a circle, or
the difference between 1 and 2 things; but rarely do they have a taste for
integral solutions to quadratic equations (even in one variable). 

A second reason that it is comparatively light reading is that these great
mathematicians we cited did not have the tools and machinery to do
definitive things in FOM. This took a number of other people, who are more
closely associated with FOM, to develop. I'm thinking of Russell, Zermelo,
Frankel, Godel, Gentzen, Turing, Tarski, Kleene, Cohen, and others. It is
well known how impressed von Neumann was with Godel, and how he arranged for
a permanent position for him at the IAS. 

I'm sure that you do not find Godel's papers "light reading." Do you have
any doubt that these great mathematicians would have published Godel's
results if they had been the discoverers? Then the reading wouldn't be so light.

You do acknowledge that our great hero of FOM, Godel, succeeded
spectacularly where all of these people tread and failed (or didn't have the
imagination to try). What would you say about mathematicians who think that 

"Godel's work in the 1930's represents the most impressive mathematical
acheivment of the 20-th century"?

By the way, this view is consistent with the view that this work is not
actually mathematics: it is certainly viewed as mathematical.

>Your idea of reviving foundations by creating a "permanent crisis"
>is amusing. Well, nobody is stopping you from trying. But in my
>opinion it is a grand delusion.

Now Lou, if you are going to do some cute name calling, and want to be fair,
you really should get more specific. I think you must hold one or both of
these viewpoints:

1) that the program I outline will never be carried out; or
2) if carried out, will not create a crisis of the kind I indicate.

I really don't want to called delusional without some specificity on your
part. Then we discuss the matter intelligently and civilly. 

>PS Leading mathematicians of our time also write about broad issues
>concerning math and its relations to science and society. A good way
>to find out is to take a look at Collected Works, things like that.
>As I mentioned before (since Atiyah's name came up), volume 1 of
>his Collected Works contains several quite thoughtful essays of
>this kind. (I liked his Bakerian lecture where he sketches for a
>general educated audience the preeminent role of geometric thinking
>and alludes indirectly to cohomology when talking about the various
>kinds of holes in mathematical spaces; he even manages to give a
>very rough idea of the Weil conjectures in this connection. Maybe
>Steve would like this, as he seems very eager to find out what all
>this is about.) 

To date, my experience in this regard has been disappointing. For example, I
was the representative of mathematical logic to the AMS Centennial
Celebration in Providence in 1988, where they recruited about one American,
generally under 40 or 45, to talk to a general mathematical audience about
each major field of mathematics (under some reasonable subdivision). I got
to about 60% of the talks, as I had to do last minute preprations on mine. 

I must say that I was disappointed, although many of these speakers are
friends of mine, and I might have missed their talk. With few exceptions,
there was no serious effort to make any general intellectual points that
could be understand by the general mathematical community. And, with few
exceptions, they were nowhere near stating anything of general intellectual
interest. The talks had the usual deep presuppositions of the
appropriateness of very complicated setups in which they work, with no
appropriate explanations as to their special status. And, Lou, these people
- except for me - are the creme of the creme of the creme of American
mathematics - people, except for me, that you respect in the highest
possible terms. 

Come to think of it, I have this volume right here in my home library! It is
called "Mathematics into the 21st Century," AMS Centennial Publications,
Volume II, Felix Browder, editor, 1992. Symposium Speakers: Aschbacher,
Caffarelli, Diaconis, Fefferman, Freedman, Friedman, Gross, Harris, Howe,
Jones, Kac, Majda, Peskin, Sullivan, Tarjan, Thurston, Uhlenbeck, Witten. 

I quote from the Introduction of the Proceedings:

I. The talks should cover as many as possible of the most important central
directions of contemporary mathematical research.
II. As far as we could choose, the speakers should be individuals of stature
in these directions who have done their principal work in the United States
and who are likely to be principal contributors afer the year 2000.
III. The central topics of the talks should include not only pure
mathematics in its classical forms but also the development of the rapidly
developing interaction of sophisticated mathematics with front line areas in
science and engineering -- in physics, fluid dynamics, computational
science, biology, statistics, and computer science.
IV. The talks ought to have an expository intent to make it possible for a
broad audience of mathematicians and mathematical practitioners to
understand as much as possible of the spirit of what mathematics ahas
accomplished in the last fifty years and of what it hopes to accomplish in
the next fifty years.

Lou - let me ask you if you could pick one of the articles in the
Proceedings as an (especially good) example of IV. 

Let me close this response by an indication of what topics I wish to write
about in the future - in addition to continuing our interesting interchange.

1. Careful analysis of section 8 of A. MacIntyre's "Strength of weak systems."
2. Careful analysis of Sol Feferman's "Does mathematics need new axioms?"
3. Analysis of Atiyah's general essays in Vol. 1 of his Collected Works.