[FOM] Re: [FOM] V.I. Arnold lecture

[Lucas Wiman]

Harvey Friedman writes:
 > 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 had a similar discussion with a friend of mine who is an experimental 
physicist.  He made that claim, and I was very skeptical.  I pointed out 
the gravitational bending of light, the existence of black holes, the 
discovery of the positron, the observation of the Josephson effect, and 
the discovery of a Bose-Einstein condensate.  His response was something 
like "yeah, and that's all they've done in more that a hundred years.  
Besides, a lot of  these predictions were done by experimentalists."

I found the whole conversation somewhat childish.  The point of physics 
is to develop a satisfactory theory of physical phenomena, and so the 
theory is the end result.  If a theory doesn't predict totally new 
phenomena, then that's not particularly surprising.  When it does, that 
shows you're on the right track, but if it doesn't, then you may still 
be on the right track.

 > 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.

Probably so.  The analogue for math would be a mathematician who thinks 
physics has not substantially influenced mathematics in the last 100 
years.  Once again, that's demonstrably false.

 From the lecture:  "But teaching ideals to students who have never seen 
a hypocycloid is as ridiculous as teaching addition of fractions to 
children who have never cut (at least mentally) a cake or an apple into 
equal parts. No wonder that the children will prefer to add a numerator 
to a numerator and a denominator to a denominator. "

This doesn't seem right to me.  I have a very good grasp of ideals 
(apparently not, according to Arnold and Poincar\'e), but I had to look 
up what a hypocycloid is.  Ideals were developed to explain certain 
concepts of algebraic number theory, and I think the best introduction 
to them is in this context (algebraic extensions of the rationals).

"This is all the definition there is. The so-called "axioms" are in fact 
just (obvious) properties of groups of transformations. What 
axiomatisators call "abstract groups" are just groups of transformations 
of various sets considered up to isomorphisms (which are one-to-one 
mappings preserving the operations). As Cayley proved, there are no 
"more abstract" groups in the world. So why do the algebraists keep on 
tormenting students with the abstract definition? "

They keep on doing this because it is simplifies life.  I agree that 
it's bad pedagogy to start out with the definition of a group, and then 
tell students later that it was based upon sets of transformations.  Yet 
what's the point of thinking of the integers as a set of transformations 
on the integers?  The fundamental thing here is addition, not the fact 
that each number can be viewed as a function (as Arnold would have it).  
That's very bad pedagogy indeed!

"An "abstract" smooth manifold is a smooth submanifold of a Euclidean 
space considered up to a diffeomorphism. There are no "more abstract" 
finite-dimensional smooth manifolds in the world (Whitney's theorem). 
Why do we keep on tormenting students with the abstract definition? 
Would it not be better to prove them the theorem about the explicit 
classification of closed two-dimensional manifolds (surfaces)?"

We define abstract manifolds because we like to do things like put Lie 
groups on them.  It would be somewhat cumbersome to think of GL(n) as a 
subset of R^(n^2).  Furthermore, the classification of two surfaces is 
usually done after the introduction of manifolds in most textbooks I've 
seen.  This is a straw man.

"The theorem of classification of surfaces is a top-class mathematical 
achievement, comparable with the discovery of America or X-rays. This is 
a genuine discovery of mathematical natural science and it is even 
difficult to say whether the fact itself is more attributable to physics 
or to mathematics. In its significance for both the applications and the 
development of correct Weltanschauung it by far surpasses such 
"achievements" of mathematics as the proof of Fermat's last theorem or 
the proof of the fact that any sufficiently large whole number can be 
represented as a sum of three prime numbers."

Ok, I think we can safely say that Arnold doesn't like number theory.  
Personally, I think number theory is a really beautiful subject, and is 
hardly that abstract.  It's only as abstract as the proofs need it to 
be.  The theorems of elementary number theory present interesting and 
very concrete properties of the natural numbers.  This perhaps isn't the 
public's cup of tea, but I hardly consider that reason to stop doing 
something I enjoy.  It has also produced some very powerful encryption 
and factoring methods.  These have genuine applications.

Back to Dr. Friedman's posting:

 > 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.

I think he is merely saying that mathematics is essentially an 
experimental science.  I think this is to some extent the case.  
Well-chosen examples are often the key to solving a problem.  Yet since 
mathematics requires some kind of certainty about an infinite class of 
cases, we end up finding logical deductions to show something--i.e. 
proofs.

Of course proofs are really the cornerstone of "my" kind of mathematics, 
but not Arnold's.  To me good mathematics is an array of beautiful 
proofs (and beautiful theorems are often defined by their proofs).  To 
Arnold, mathematics is a collection of facts.  This is the fundamental 
aspect of his lecture:  axiomatizers are just finding new ways of 
stating the obvious, while the real work has been done by somebody else.

 > 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.

This is a very interesting thought.  I've always thought of f.o.m. as a 
branch of philosophy for the following reason:  theorems in f.o.m. are 
"interesting", but theorems in regular math are "beautiful".  I don't 
have the same kind of emotional connection to f.o.m. that I feel to 
regular mathematics.  Philosophy and f.o.m. are plainly related, and I 
find philosophy "interesting", not "beautiful".  Hence (for me at 
least), f.o.m. is much closer to philosophy than math.  Certainly no 
disagreement on (i).

 > 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'm actually surprised that R didn't make the list.  In non-academic 
circles, he's certainly better known than W for his philosophy books for 
the layman, his anti-war activism, and his strong atheism (and the drama 
surrounding his appointment to the City College of NY).  I'm hardly 
qualified to comment on either of their philosophies, but R was 
certainly the clearer writer of the two!

 > 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.

F.o.m has already made some headway into Arnold's kind of math, with the 
application of descriptive set theory and model theory to "regular" 
math.  The fact that a polynomial endomorphism of C^n is surjective if 
it is injective would surely be admitted into Arnold's brand of 
mathematics, and it has a beautiful model theoretic proof.  My guess is 
that this will only continue.

- Lucas Wiman