FOM: F.O.M./math comparison
Harvey Friedman
friedman at math.ohio-state.edu
Wed Dec 31 01:44:17 EST 1997
I am Harvey M. Friedman, and my title/affiliation is Distinguished
University Professor of Mathematics, Philosophy, Computer Science, and
Music, Ohio State University. My research interests include foundations of
mathematics, mathematical logic, philosophy of mathematics, computational
complexity, and computer assisted music performance.
There are now far more subscribers on this list than when I had previously
made many important points concerning distinguishing features of FOM as a
subject. So I am going to repeat many of these in the clearest possible
terms, and I apologize for those of you that have very good memories of
what I wrote earlier.
Furthermore, I am going to carry on the discussion of distinguishing
features of FOM - particularly as compared to pure mathematics - on many
fronts, including reviewing various volumes of papers that have appeared
over the years that represent more or less the "best" mathematicians of our
time. I caution that these will not be real reviews in the usual sense of
the word, but only reviews with regard to particular comparisons made
between high level FOM and pure mathematics. E.g., I will be "reviewing"
the entire Proceedings of the Centennial Celebration of the AMS, ISBN
0-8218-0167-8, 1992. The meeting took place in 1988. The articles are by:
Michael Aschbacher, Luis Caffarelli, Persi Diaconis, Charles Fefferman,
Michael Freedman, Harvey Friedman, Benedict Gross, Joe Harris, Roger Howe,
Vaughn Jones, Victor Kac, Andrew Majda, Charles Peskin, Dennis Sullivan,
Karen Uhlenbeck, Edward Witten.
I consider this whole matter of vital importance for a number of reasons.
Firstly, recognition of the distinguishing features of FOM is required in
order to develop FOM at a high level. One must constantly refer to the big
picture items being tackled by FOM. In fact, in a nutshell, one of the
distinguishing features of contemporary FOM as opposed to contemporary pure
mathematics is that the former is occupied with a direct assualt on genuine
big picture intellectual issues of completely transparent general interest.
A detailed discussion of what these "genuine big picture intellectual
issues of completely transparent general interest" are, what is known about
them, and what the prospects are for future knowledge, will appear later in
my series of postive postings.
I argue that, in contrast, contemporary pure mathematics is not occupied
with a direct assault on genuine big picture intellectual issues of
completely transparent general interest. It is generally practiced far more
as a sport or art than a search for knowledge. I may, as I develop these
ideas further, armed with the experience of defending them against attacks
on this fom list, actually come to the conclusion that: **contemporary pure
mathematics is (practiced as) an art/sport and not (primarily) a subject at
all!!** However, physics, chemistry, biology, statistics, psychology,
sociology, law, anthropology, finance, economics, geology, geophysics,
computer science, electrical engineering, mechanical engineering,
engineering mechanics, mathematical modeling, chemical engineering, genetic
engineering, medicine, ..., and foundations of mathematics, are subjects in
the normal use of the term.
In any case, I will argue that contemporary pure mathematics is not
occupied with a direct assault on genuine big picture intellectual issues
of completely transparent general interest until either a) nobody can or
will effectively argue against me anymore; or b) I am shown to be wrong, or
at least partly wrong, by explicit examples; i.e., explicit examples of
direct assault on big picture intellectual issues of completely transparent
general interest by pure mathematicians. In case of b), I will then concede
that I am partly wrong, and then seek to make clear the extent to which I
am right.
Secondly, most mathematical logic and FOM reserach is currently supported
by mathematics departments. As I discussed in my positive posting 11:F.O.M.
& Math Logic of 5:47AM 12/14/97, mathematical logic and FOM is in danger of
being marginalized in mathematics departments. I said that I believe this
is taking place partly becuase of the move away from FOM in the
mathematical logic community, which makes it comparatively difficult to
explain the content of the research being supported. The natural tendency
of mathematicians to lend their support more freely to things more readily
understood and appreciated by them, combined with shrinking resources,
makes for a very vulnerable situation.
A revitalization of FOM in the mathematical logic community would have a
very positive effect in this regard. And there is a history of news
coverage in the scientific and wider press about FOM which isn't going to
be matched in parts of mathematical logic remote from Foundations and in
*most* of pure mathematics. Even the news coverage of FLT - which was very
extensive - didn't really cover substantive ideas. Press coverage on FOM,
on the other hand, has covered substantial ideas. That is the nature of
these great ideas. And this will continue.
But again this revitalization of FOM in the mathematical logic community
depends on first recognizing and appreciating the distinguishing features
of FOM, not only as opposed to pure mathematics, but also as opposed to
general mathematical logic.
Thirdly, there are a few people - only very few - on the fom who seek to
minimize the special importance and status of FOM by adopting rather
extreme forms of the typical bias one meets among the average pure
mathematician. My experience has been that this typical bias is not really
typical of pure mathematicians operating at the highest levels - although
it certainly does occur there too. I very much appreciate these fom-ers
continued interest in what's happening on the fom, despite their views
which are so uncharacteristic of most of the subscribers on the fom.
But I think it is vital for someone - looks like I'm mainly nominated for
this - to go into these dissident views carefully and ferret out what is
behind them and see if there is any real substance to them. My overall
assessment is that these dissident views are based on an entirely different
approach to intellectual life - one that is completely out of tune with all
aspects of intellectual life outside of the pure mathematics community.
Pure mathematicians are frequently so completely mesmerized by the special
features of pure mathematics that it adversely affects their judgement.
They begin to look for these same special features, present in such
abundance in pure mathematics, in everything else - and when they don't
find it, they become blind to the virtues of everything else. There is no
question that pure mathematics does have some very special features that
almost nothing else has - and I will discuss this also in detail at a later
time. However, it is important to recognize that these features are not
highly valued or even understood by the wider intellectual community. And
in fact, pure mathematics also has some special features - related to
art/sport - that are are poorly valued by the wider intellectual community.
My basic message for the dissidents is to perform the following thought
experiment as often as possible: What are you trying to accomplish in
simple terms? What information are you seeking in simple terms? Why do you
care in simple terms? Why would anyone else care in simple terms? How would
you get anyone else to care in simple terms? Who could you get to care in
simple terms? I think that you might gravitate to the art/sport model of
pure mathematics when contemplating these questions. Otherwise, I think you
are in intellectual trouble.
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