FOM: Golden Age/FOM plans
Harvey Friedman
friedman at math.ohio-state.edu
Wed Oct 22 12:15:53 EDT 1997
Looking at the growth of the list of subscribers to FOM (currently 62),
there is evidently a great deal of interest in the foundations of
mathematics and the vital intellectual issues surrounding it. As a
consequence, I, for one, am poised to make a growing committment to
actively participate in FOM. Of course, everyone knows that I speak for
noone other than myself, and there is never going to be any kind of
standard FOM view.
My intentions are to contribute what I can in the following ways:
1. I will write personal reviews of the major articles and books relating
to the foundations of mathematics. This may include lesser ones if there
are major provacative points in them. Two examples: Angus MacIntyre's "The
Strength of Weak Systems", in P.Weingartner and G.Shurz (eds.) Proc. 11th
Wittgenstein Symp. 1986, 1987, focusing on pp. 56-57; and Sol Feferman's
"Does Mathematics need new Axioms?", preprint. Also a book by Penelope
Maddy is coming out soon on Naturalist Foundations of Mathematics. If any
of you have seen articles or books that you might want to see me review,
please contact me at friedman at math.ohio-state.edu.
2. I will discuss, from my perspective, what has been accomplished in the
foundations of mathematics; and also what has not yet been accomplished in
the foundations of mathematics that I feel can realistically be
accomplished. I am planning to discuss a wide range of problems and
projects of significance for the foundations of mathematics, which are at
various levels of formality. Many of you know that I have been doing this
in person on an individual basis for many years.
3. I have been articulating a strong view that the general area of
mathematical logic has suffered greatly by overspecialization and
compartmentalization, with a gross overemphasis on pure technique without
regard to any higher intellectual purpose. On the other hand, I have always
conceded that periods of great technocracy are sometimes essential for the
proper development of a subject. I freely admit that a number of things
that I do simply could not be done without a significant technical
background.
Nevertheless, we are now at a point in mathematical logic where the higher
intellectual purposes can now be effectively addressed, at an unprecedented
level, armed with a considerable amount of technical machinery and
experience.
I will be reviewing what is happening in mathematical logic with emphasis
on the issue of its connections or lack of connections with higher
intellectual purposes. There will be a special focus on missed
opportunities, and candid discussion of the issues of overspecialization
and compartmentalization. I am convinced that overspecialization and
compartmentalization will eventually destroy mathematical logic as a viable
area within mathematics, and also threatens to destroy the support for pure
mathematics as a whole.
4. I have been expressing concern that students are forced into
overspecialization at an early stage in their career before they can grasp
the higher intellectual issues. This is a direct result of the minimal role
that higher intellectual issues have in the ongoing research of the
faculty. There is a tendency that with each generation there is a
decreasing awareness of the higher intellectual issues - and the proper
connection between higher intellectual issues and research becomes more and
more remote.
Thus there is a vicious circle where overspecialized faculty advise
overspecialized students who become even more overspecialized faculty who
advise even more overspecialized students, etcetera. I want to talk about
what can be done to break this vicious circle. Foundations of mathematics
can be expected to play a leading role. In particular, there is a different
way to teach mathematical logic, based on genuinely foundational
organizational schemes for mathematical logic.
5. There has been discussion of "fundamental concepts" of mathematics. I
have my own view on this matter. I believe that there is a tremendously
selective notion of fundamental concept - where the concept must speak
clearly to everyone as intellectuals. However, most people would think that
under such a strong criterion, nearly nothing of nontrivial substance in
mathematics can be reconstructed. I claim that this is completely false. So
I want to write about a radical reconstruction of at least classical
mathematics from this point of view. This requires altered presentation of
material, new theorems, even the development of new subareas. But all of
this is highly motivated - to give a general intellectual's account of much
of mathematics.
6. I want to discuss other major foundational programs. For instance, it is
clear that in applications of mathematics to other subjects, one must
ultimately ccnfront our finiteness - as intellectuals, as physical beings,
as observers and measurers - so that finite precision, as opposed to
infinite precision, assumes paramount importance. This emphasis on finite
precision should begin with mathematics itself. So there is the program of
giving a finitary treatment of basic mathematics in a systematic, readily
useful, and intelligible way. For instance, what does a treatment of
functions of one complex variable look like in strictly finitary terms?
Does this necessarily degenerate into an ugly morass? I don't think so. In
fact, I believe this can be done appropriately, and that it may spawn a
number of new subareas where new kinds of estimates have to be made - and
the whole development will look quite interesting.
****************************
At the moment, I see three higher intellectual purposes that mathematical
logicians should turn to.
a. Foundations of Mathematics.
b. Direct applications to mathematics.
c. Foundational Studies in general.
For practical reaons, I think that the FOM group needs to have some sort of
limitation of scope, and so I, for one, will not focus on c. There may come
a time when it is appropriate to have a group in Foundational Studies, and
I hope to have a role in its setup. And as I have said before, I think the
ultimate future of Foundations of Mathematics rests with Foundational
Studies in general, which is a context in which Foundations of Mathematics
has the special role of being the most highly developed Foundations - one
that serves as a model of what can be accomplished.
But for the FOM group, I, personally, would like to focus on a). However,
there is the contentious issue on the table that direct applications to
mathematics - of the kind with which there have been recent successes - is
somehow to be construed as genuine Foundations of Mathematics, in a
legitimate use of this term. As long as this issue is on the table, it
seems entirely appropriate for people to discuss direct applications to
mathematics (as they are doing) as long as it is presented as foundations
of mathematics, in perhaps some alternative sense. I would prefer a serious
articulation of this nonstandard view of foundations of mathematics, rather
than simply a declaration that there is such - in particular, I would
prefer to hear what is not foundations of mathematics in order to make
clarifying contrasts.
Now I for one do not think that these direct applications to mathematics
discussed by Anand, Dave, and Lou, and the mathematical logic that supports
them - at least in their present form - constitute Foundations of
Mathematics in any accepted sense of the term; i.e., as long as the phrase
"Foundations of Mathematics" is used in remotely the same way as it has
been consistently used for at least one hundred and fifty years. And this
normal way of using the phrase "Foundations of Mathematics" is entirely
analogous to the way "Foundations of X" has also been used, for any field
X, for at least one hundred and fifty years - and arguably for, say, two
thousand years. Moreover, there are excellent ways to sharply distinguish
genuine "Foundations of mathematics" with these other pieces of
"Fundamental Mathematics." ***This will be the subject of another e-mailing
called: Foundations??***
This episode - the controversy over the appropriate meaning of "Foundations
of Mathematics" - is typical of how I personally want to handle controversy
in FOM. E.g., I talked the matter through with someone I thought that Lou,
Anand, and Dave would think would agree with or have considerable sympathy
with their point of view. In this case I might have heard some real support
for their viewpoint on the matter, but I detected none. In fact, I was
explicitly told that the typical mathematician in their Department of
Mathematics holds a view much closer to mine than to theirs on this issue.
And that the level of respect for the work of Kurt Godel as science is
immeasurably high - much higher than that of themselves or their
colleagues.
We now appear to be in a Golden Age with regard to a) and b). In both
cases, things are now possible because of this prior technical development.
However, one should not even begin to think that a large percentage of the
technical development has, or even will have any substantial bearing on a)
or b) or any other higher intellectual purpose. It's simply not true.
Nor should one think that any technical advance has an equal chance as any
other technical advance of becoming relevant to a) or b) or any higher
intellectual purpose.
On the contrary, upon choosing a particular problem to work on, one should
have its place clearly in mind in the general intellectual landscape. This
kind of judgment is not arbitrary - although at this time it is not subject
to precise formal characterization. The quality of this kind of judgment is
what separates the truly great thinkers from others. For the higher levels
of intellectual achievement, one must go far beyond technique. The
technique must be under such complete control, that it is second nature -
that it is effortless, and always subservient to a higher intellectual
purpose.
Many pure mathematicians are quite familiar with how this applies within
pure mathematics, and how it separates the truly great mathematician from
the rest. However, for people dedicated to Foundations of Mathematics, it
is neither necessary nor sufficient to look for placement of ideas in the
***general MATHEMATICAL landscape***. Its the placement in the ***general
INTELLECTUAL landscape*** that is important.
In fact, this is the distinguishing feature of Foundational Studies in
general - that one speaks to the general intellectual community - the
general world of ideas. There is no place for the slavish acceptance of the
importance or significance of a particular development or kind of problem
that is fashionable in a particular field. Its relationship to the whole
intellectual community is to be continually evaluated and reevaluated on a
case by case basis, in consultation with people in other fields - even with
people in other walks of life. Normally what happens is that one or more
aspects of the fashion do have connections with something globally
fundamental - but not usually in the exact form in which specialists
normally cast them (for each other). They have to be recast in more
fundamental terms. Often this requires further developments that would not
be conceived of by specialists.
The problem with pure mathematics and the wider intellectual community at
this time is that this recasting of the current fashions in globally
fundamental terms has not taken place - or at least not systematically
enough to create any real understanding on the part of the general
intellectual community. Instead, today there is great confusion among the
scientific public - or even within the mathematical sciences - as to the
appropriate placement of pure mathematics in the general intellectual
landscape.
Which brings me back to the phrase "Golden Age." There can be no doubt that
during the last part of the 19th century up through the 1930's, that
Foundations of Mathematics was in a period where great revelations
regarding questions of the highest possible general intellectual
significance were being successfully addressed on a continuing basis
replete with stirring surprises. Although great advances were made in pure
mathematics during this period, many thoughtful people would agree that
this development was singular, in that its extraordinary meaning and
significance in the general world of ideas was incomparably higher. For
example, the applied mathematician and historian Morris Kline, Professor of
Mathematics, Courant Institute of Mathematical Sciences, New York
University, writes the following in his epic Mathematical Thought from
Ancient to Modern Times, Oxford University Press, 1972, Chapter 51, The
Foundations of Mathematics, p. 1182:
"By far the most profound activity of twentieth-century mathematics has
been the research on the Foundations."
After a long period of primarily technical development, I think the time is
now ripe for another Golden Age for Foundations of Mathematics. And by
talking to applied model theorists like Anand, Dave, Lou, as well as some
interested mathematicians, they may be poised for a first Golden Age for
"Mathematical Applications of Mathematical Logic." We shall see what
emerges. However one distinction is clear: "Mathematical Applications of
Mathematical Logic" seeks to speak to the pure mathematicians. "Foundations
of Mathematics" seeks to speak to everyone. As long as the former speaks
primarily to pure mathematicians and the latter speaks to everyone, it is
absurd to pretend that there is no genuine and fundamental distinction.
HMF
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