[FOM] V.I. Arnold lecture

[Harvey Friedman]

I would like to comment on some aspects of Arnold's lecture - the 
lecture that I put on the FOM e-mail list 9/12/02, and which can be 
found at http://pauli.uni-muenster.de/~munsteg/arnold.html

I first saw this lecture text when my colleague Dave Goss put it up 
on the OSU Math Dept e-mail.

As I have already indicated, there are some things that I like very 
much about this lecture, and a number of things that I don't.

>Mathematics is a part of physics.

What I like about this is that it shows a determination to try to put 
mathematics into a firm place in the universe of ideas. Conventional 
wisdom is that this is far easier to do with physics than with any 
notion of mathematics independent of physics or physical systems. 
Hence Arnold's strategy to borrow from the position of physics in the 
universe of ideas.

What I don't like about this is its hidden meaning. Notice that 
Arnold forgoes an opportunity to start giving, or at least clearly 
hint at an imaginative and creative organizational structure for 
mathematics. He could have delineated "physically motivated 
mathematics" or "physical mathematics" as a branch or superbranch of 
mathematics. I do think he means something broader than "mathematical 
physics".

To say that "mathematics is a part of physics" conveys the suggestion 
that the evaluation of a piece of "mathematics" or "mathematical 
thought" is to be in terms of what good it does for physics, or what 
physical interpretation it has.

We are all familiar with mathematical thought which is highly valued 
by various people for various reasons, but which clearly is not 
intended to say anything about physics or physical systems. Arnold is 
suggesting that this mathematical thought is either not mathematics, 
or as mathematics, it is either bad or weak or even worthless.

I would be more comfortable with this if he were to

i) delineate this kind of mathematical thought ("physical 
mathematics"), and explain how it differs from other types of 
mathematical thought, and also explain how it is to be evaluated;

ii) make a justification of the place of physics in the universe of 
ideas, no matter how obvious that appears to be to Arnold and many 
others;

iii) defend such a justification against the skepticism of hard nosed 
experimental physicists and engineers;

iv) put up a challenge to others to give a justification for 
mathematical thought that is not "physical mathematics".

Some major issues arise when trying to do this. E.g., in ii), how 
broad a notion of "physics" does one intend? Remember "mathematics is 
a branch of physics". Does physics include engineering, or even 
economics?

There are lots of disagreements and lost opportunities for 
interactive discussion. For instance, I know of one very famous 
analyst who has contacts with physicists, who told me the following. 
That he talked to a famous experimental physicist who claimed that in 
something like a hundred years, there have been almost no examples of 
theoretical physics having any impact on physics, and that this 
famous experimental physicist is by no means alone among experimental 
physicists in saying this.

I have no doubt that one can find famous theoretical physicists who 
think quite analogously that in something like a hundred years, there 
are almost no examples of mathematics having any impact on 
theoretical physics, and that he/she is by no means alone among 
theoretical physicists in this view.

It would be interesting to have Arnold, such a theoretical physicist, 
and such an experimental physicist in extended discussions over an 
extended period of time, trying to reconcile and attack each other's 
views of the universe of ideas.

***********************************

I would like to go through this Arnold lecture, line by line, and 
give some more of my thoughts, but right now I just have the time to 
do this for his opening sentence. Perhaps I am saying enough to start 
(provoke) a discussion.

There are very serious issues raised by the Arnold lecture, even if 
he comes nowhere near being able to deal with these issues in any 
convincing way. This is, of course, an impossible task in the space 
of one lecture.

But I want to jump the gun and say something about how foundations of 
mathematics (f.o.m.) fits into this picture.

I regard foundations of mathematics as a very coherent and 
identifiable kind of mathematical thought that is in no reasonable 
way "part of physics", and is certainly not motivated by physics or 
physical systems. So under Arnold's conception, f.o.m. is not 
mathematics.

Actually, I don't have a problem with the view that "f.o.m. is not a 
branch of mathematics" or even "f.o.m. is not mathematics" provided 
that at the same time, it is also recognized that

i) f.o.m. is mathematical thought of the highest order;
ii) f.o.m. sits between mathematics and philosophy.

What I particularly like about items i) and ii) are their explanatory power.

Item i) explains why

a) such a disproportionately large number of the universally 
recognized gigantic figures of 20th century intellectual life were 
known, or at least partly known, for their contributions to f.o.m.;

Item ii) explains why

b) f.o.m. is in such a difficult position in contemporary academic life.

By b), I mean that it awkwardly sits between two academic communities 
that have grown apart, and are growing farther apart as we speak: the 
mathematical community, and the philosophical community.

By a), I offer the following illustrative anecdote. TIME-LIFE books 
put together a volume where they, among other things, list "the most 
influential people of the 20th century". This is divided into five 
groups, "Leaders and Revolutionaries", "Artists and Entertainers", 
"Builders and Titans", "Scientists and Thinkers", "Great minds of the 
century", 20 people each. They claimed an involved and impressive 
process of outside output to come up with these names.

Fully 3 out of 20, or an amazing 15%, of the "Scientists and 
Thinkers" are very much in and about f.o.m. Here is the TIME-LIFE 
list:

The Wright Brothers
Albert Einstein
Ludwig Wittgenstein
Sigmund Freud
Leo Baekeland
Alexander Fleming
Philo T. Farnsworth
Jean Piaget
Kurt Godel
Robert Goddard
Edwin Hubble
Enrico Fermi
John Maynard Keynes
Alan Turing
William Shockley
James Watson and Francis Crick
Jonas Salk
Rachel Carson
The Leakeys
Tim Berners-Lee

I am referring to Kurt Godel, Alan Turing, and Ludwig Wittgenstein.

How many of these 20 are very much in and about mathematics under 
Arnold's conception?

Godel is Ultimate Professor of f.o.m. Turing's early model of 
computation, before the development of computer science, is real 
f.o.m. It is foundations of a mathematical concept.

There is a rivalry between Bertrand Russell and Ludwig Wittgenstein, 
and W is right now ahead of R. In more ordinary times, I think, R 
will be ahead of W. I would rather see R on this list than W - and by 
my merely saying this, a lot of philosophers are going to have an 
even lower opinion of me than they do now (smile). Although W is 
clearly in and about f.o.m., (e.g., his famous "Remarks on the 
Foundations of Mathematics"), R is an absolute f.o.m. icon, with 
Russell's paradox, his type theory, and the book he wrote while in 
jail during WWI.

I close with a major challenge to f.o.m. that I see is implicit in my 
reading of Arnold's lecture.

F.o.m. has been remarkably successful in gaining an understanding of 
at least a certain kind of mathematical thought. It is not clear just 
how much f.o.m. has to say, in particular, about the kind of 
mathematical thought that Arnold values. This is a difficult, but by 
no means hopeless challenge.