FOM: ReplyToAnand

[Harvey Friedman]

Again, you (and some of your collealgues) seek to minimize the special
importance and status of Foundations of Mathematics by denying the rather
obvious glaring distinctions between it and fundamental mathematics.

I will respond in detail in three quite different ways. The first two ways
I am committed to do anyways, and they are:
	1. I will be writing a number of little essays that go into some
depth, on a single specific FOM development; e.g., the completeness theorem.
	2. I will be compiling some substantial lists of foundational
developments and foundational issues, with a short sentence or two about
each. I expect that some people on the fom will want to hear about them in
greater depth.

These two plans have nothing to do specifically with your and some your
colleagues continuing attempts to minimize the special importance and
status of FOM. For people who "don't get it" this still could be quite
illustrative and educational. For people who "do get it" I hope they find
something new in what I say, even when I revisit old, familiar topics.

Thirdly, I will, somewhat reluctantly, continue to carry on this argument
with some of you and your collegues who seek to minimize the special
importance and status of FOM. (A later e-mail will go into your
reversations in detail.) ***However, I have been getting reports that
people are frustrated with the lack of constructive intellectual content in
these attempts to minimize, and would rather see some content.*** The best
way for you and some of your colleagues to handle this may be to follow my
lead in discucssing matters of genuine foundational interest. Can you do
this for us? It should be clear, generally understandable, of obvious
interest, usefully accurate and detailed, foundational, etcetera.

I want to emphasize that I do not expect you to concede that the special
importance and status of FOM is "greater than" that of fundamental
mathematics. You may well be much more interested in core mathematics than
you are in FOM; core mathematicians are also much more interested in core
mathematics than in FOM. However, I do expect that I can get you to
understand the special importance and status of FOM - even though you wish
to wear the hat of a core mathematician much more prominently. I similarly
expect the theoretical physicist to be much more interested in theoretical
physics than in FOM. And so on, throughout the various disciplines. All of
this is expected. However, I do expect that for specialists in the
overwhelming majority of disciplines outside mathematics, there will be
more interest in and understanding of FOM than in/of core mathematics.

I hate to be repetitive, but sometimes I find telling quotes that are much
stronger than anything I am saying - from people you would least expect. I
would like your reaction to the Morriss Kline quote from Mathematics from
Ancient to Modern Times, Chapter 51, pp. 1182:

"By far the most profound activity of twentieth-century mathematics has
been the research on the foundations."
*********************

Every time I see one of these attempts to minimize the special importance
and status of FOM, I contact some key people who you would expect to have
sympathy with your point of view - including people who are not on fom.
This time, I got the clearest indication I have ever gotten, in no
uncertain terms, that these attempts are completely out of touch with the
high level mathematical community and even more so with the general
scientific community. As expected, all of my views on the matter were
completely confirmed. I also got feedback about the difficulties in stating
Falting's theorem in terms of genus in general intellectual terms - which
are considerable, as I expected. It would be interesting to see if you or
your colleagues can recognize the serious difficulties and overcome them.
And, since FLT is so much in the news, it was fairly easy for my contacts
to talk to people about FLT versus, say, Godel's work - even just the
Completeness Theorem. The feedback about FLT versus Godel that I received
has again completely confirmed my position. The literate scientific public
has "no idea why mathematicians are interested in FLT" but are quite
interested in and respectful about the sociology surrounding it. This is
quite different with respect to most of Godel's work. People generally have
quite a clear idea as to why people are interested in Godel's work.

Now it is not my style to water down subtle matters by talking about "how
many people can or will understand it" or by appealing to authorities that
you would respect. On the contrary, I would much prefer to talk about the
issues directly. But you and some of your colleagues leave me no choice.

The reason I mention authority is simply this. You and your colleagues are
aware that the support for genuine FOM - and mathematical logic as a whole
- is not all that strong right now in the mathematics community. You
believe that it is the lack of connection with core mathematics that is the
cause of this, and you seek to remedy this by emphasizing and developing
these connections. But in your desire to be recognized as REAL
mathematicians, you misread the attitude of the high level mathematics
community towards FOM.

The attitude of the high level mathematics community towards FOM is one of
great respect. However, the attitude is also that not much of significance
has happened in FOM for a long time. This is partly true, partly false, and
also is colored by the fact that so much happened that is so spectacular in
the 1930's, that the standards for FOM are sometimes unrealistically high.

And whereas the high level mathematics community recognizes the obvious
special importance and status of FOM for academics as a whole - they don't
necessarily think that it is of comparable priority in Mathematics
Departments to core mathematics. As a general intellectual, I naturally
think that this is short sighted, but understandable. In a period of
shrinking resources, people tend to stick to their main preoccupations. To
put it extremely, although I know that mathematicians have the greatest
respect for Beethoven's 5th Symphony, I would not expect them to commit
resources to musicians. And - frankly - they do recognize a difference
between Beethoven's 5th Symphony and Falting's theorem.

As I have indicated in earlier e-mail to fom, the ideal way to support
genuine FOM in Universities should be as part of a wider subject called
Foundational Studies. But this isn't going to happen until suitable
foundational expositions of classical subjects have been completed. In the
interim, the proper support for genuine FOM will have to rest with the
Mathematics and Philosophy communities. Joint efforts by these communities
make the most sense. However, there is at this time very little
communication between these two communities - probably less so than in the
past. The bottom line is that there will continue to be some support -
mainly from the mathematics community - of genuine FOM. I think that it
will grow as more successes in FOM are acheived and exposited in general
intellectual terms.