FOM: ReplyToAnand/Baldin/Mattes

[Josef Mattes]

On Sat, 25 Oct 1997, Harvey Friedman wrote:
>
>Mattes writes
>
>
>>I think this seriously underestimates the consequences of the
>
>>work of Grothendieck et al. For example, these ideas are at the very
>
>>foundation of 'noncommutative geometry' and 'quantum groups' with
>
>>deep and important relations to measure theory, von Neumann algebras,
>
>>knots, string theory, etc. 
>
>>In another message we read that the foundational approach regards the
>
>>unity of human knowledge as paramount. Well, nocommutative geometry
>seems
>
>>to be doing quite well in this respect: 1.) They arose out of
>statistical
>
>>mechanics 2.) Connes came up with a new, elegant model unifying
>gravity
>
>>with the other fundamental forces of nature. Surely the problem of
>fitting
>
>>gravity into the rest of physics is of importance even on a general
>
>>intellectual scale?
>
>
>This begs the following question. To what extent are the "new-fangled"
>approaches to physics being pushed by certain mathematicians and very
>theoretical physicists being genuinely accepted by the physics
>community as "real physics" subject to the usual classical criteia of
>"experimental confirmation" that is so ingrained in physics of the
>past? Or ratheris it a kind of pure mathematics that has physical
>interpretations? I don't have any opinion on the matter, and am
>certainly not competent to judge this myself. However, I do think I am
>aware that this is a controversial issue that hasn't been fleshed out
>properly. 
>

I'm not sure how strict your criteria are. As far as I know it took a
while until general relativity was considered firmly established by
experiment. Would you have counted as intellectually important when it
was first proposed? What about string theory: It has been mentioned by
others in this discussion that there is no experimental evidence for it
and physicists' opinion seems divided: Witten thinks very highly of it,
Hawkings does not. 
In any case, if Connes' model does not satisfy your criteria, what about
the following (a concise reference for all this is Drinfeld's talk at the 
ICM 1986): 
One of the main sources of quantum groups were integrable
models in statistical mechanics. As far as I know, this is considered
"real physics", see e.g Baxter's book on exactly solved models. In
addition, quantum groups serve as hidden symmetries in conformal field
theory, which is connected not only to statistical mechanics but also to
string theory. This extends into the attempts to put quantum field theory
on an axiomatic basis [Atiyah, Segal etc.], including  relations to knot
theory. etc. 
In addition, noncomutative geometry in general is related to Penrose
tilings, Julia sets, etc. etc. (see Connes' book and also my original
message). Quite impressive for a
theory that is barely 20 years old, isn't it? And one major underlying
theme is: Take a space, forget about the point set but look at the algebra
of functions instead. A simple, powerful, wideranging, i.e. a   
foundational idea. 


Also, even relativity theory has the appearance of confusing
>nonsense to most people, and James Ax has had a longstanding interest
>in making some coherent sense out of this, too. 
>

Who is 'most people'? Couldn't you equally well say that most people find
even mildly advanced mathematics confusing nonsense? In any case, 
I did not exactly find it helpful to try to explain it in an axiomatic
fashion. Whenever I manage to get something across it is because I draw
pictures and start from things that are close to experience, not from
those that are low in the "hierarchy of concepts" that Simpson talks
about.

>
>Of course, I have to admit that the appropriate exposition on
>Foundations of Mathematics has not yet been written - but the
>appropriate results exist, and some of us can write such a exposition
>right now.  
>

Appropriate for what? In any case, I hope you will write it soon.

>
>But the main point I wish to emphasize is that we need to maintain high
>standards for conceptually clear foundational expositions - otherwise
>we really don't have anything which is penetrable by the general
>intellectual community.
>

I doubt that the problems of the intellectual community with
mathematics stem from a lack of clarity of the concepts in mathematics. I
rather think they come about because a.) many of our important concepts
are quite abstract and b.) even if they are not, we like to hide them
behind lots of technicalities.

>
>Mattes writes
>
>
>>Goedel's theorem has been quoted as an example of being of interest
>to
>
>>non-mathematicians. I just wonder, how much interest would there be if
>it
>
>>were not a statement about mathematics? Without interest in
>
>>mathematics/computation, who would be interested in Goedel?
>
>
>
>analysis of proofs is far from a completed topic, but what has already
>been achieved through Gottlob Frege, Godel, and others regarding
>predicate calculus and formal set theory, stands tall with the greatest
>intellectual achievments of all time. A similar kind of intellectual
>activity later produced the first programming languages and their
>implementation - and this stands tall with the greatest engineering
>achievements of all time. Such is the power of the FOM outlook and
>style of thinking. No other way of thinking about things is even
>remotely as powerful. 

This does not seem to answer my question. It goes without saying that I
have no intention of denying
the importance of Goedel's results. But it seems to me that they are
considered important because they are statements about mathematics and
computers, which in turn are considered important. In contrast, cosmology
for example is important because it is about nature (if you want to
appreciate it you just have to look at a clear night sky).
Several contributions made the point that it connects to algorithms: How
should the importance of Goedel's work have been explained to the 
nonexpert before the advent of computers? By explaining
formalization of mathematics? Or was it not foundational then?


But there are other important aproaches to
>intellectual life - I grant this. 
>

But none is even remotely as powerful??

>
>Why do you hate sets so much? Did a large set fall on your head? At the
>present time, they are the only vehicle we have right now for a smooth,
>coherent foundation for mathematics. It sounds like you want to
>question the cosmic importance of this foundation. Right? Or question
>the need for a foundation at all? Or question whether or not it is a
>foundation? Or question in what sense is it a foundation? Or what?

I don't hate sets. In fact they I find them quite interesting. But there
is more to mathematics than sets. To me it seems that set theory
has enriched mathematics, not reduced it (in the same manner as Cartesian
coordinates have led to a fruitful interaction of geometry and algebra
rather than to the reduction of one to the other). In the language of
Simpson I suppose I am one of those who 'just don't get it'.

Maybe you can put it this way: I don't see the point of a foundation
without a building. A building without foundation might be shaky, but at
least you have a roof over your head. Which, after all, is what you want. 

Josef Mattes