FOM: Kazhdan's abstract

[Harvey Friedman]

This is Kazhdan's abstract for his talk at the Millenium Conference at
Hebrew University, December 30, 1999, (see
http://hug.phys.huji.ac.il/winterschool/ ), which I took from

http://hug.phys.huji.ac.il/winterschool/Lecture3_6/lecture0.htm

This certainly provides some insight into the persepctive of the core
mathematician (as represented by a well known representation theorist).

Abstract

At the turn of the century, it is interesting to pose and to compare the
mathematics as it it was in the beginning of the century with the
contemporary mathematics. What is new in the mathematics of 20th century?
Of course there is an infinite number of new results and any attempt to
list the most important results is bound to be incomplete.  So let us ask
the question differently: What are the basic changes in the mathematical
intuition, what questions are natural for us but could not be imagined in
the beginning of the century?  There are some areas in which there was an
immense progress but where we do not find a drastic change of mathematical
intuition.  For example I think that the words of Poincare "Analysis
profits by geometric considerations, as it profits by the problems it is
obliged to solve in order to satisfy the requirements of physics ", written
at the turn of the century, describe well the contemporary understanding of
analysis.  Therefore I will not talk about the development of Analysis in
this century but choose topics which represent for me the basic shifts of
the mathematical perspective.  Of course I can only present my personal
views and different mathematicians will see the mathematical landscape in a
completely different light. But I prefer to start not from a discussion of
particular mathematical achievements with an analysis of the old question:"
How is mathematics possible?".  One of the possible interpretations of this
question is "Why we, mathematicians, are able to perform our work?".  In
the beginning of this century many considered this problem in the light of
one of the main themes of the 19th century mathematics -"make mathematics
rigorous".  Correspondingly they interpreted the Kantian question as the
directive "set up the formalism of mathematics reasoning and prove that
such a reasoning does not lead to contradictions" and "show that any
question can be resolved".  This assurance of the possibility to solve any
problem would be very important since it would tell us that our efforts are
never in vain.  The development of mathematics in the 20-th century
banished any hope for such a "naive" understanding. Godel has shown that we
can never be sure that our framework, our chosen system of axioms does not
lead to a contradiction.  Moreover, we now know that any [sufficiently
rich] system of axioms is incomplete.  In other words if we are working in
a framework of a sufficiently rich system of axioms then either our system
leads to a contradiction or we meet statements about
which we will never have anything to say.  That is, we will neither be able
to prove these statements nor to disprove them nor to show that we cannot
neither to prove nor disprove them. On the first glance, Godel has signed
the death sentence for mathematics.  One would expect the unsolvable
questions to jump on us in big numbers. As a result we would never be sure
that it make sense to try seriously to solve difficult problems and
mathematics would come to a halt. Fortunately, the reality is very
different.  Aside from some very specific areas, we never run into
questions which we can not settle and even in these areas we are able to
prove that the questions we can
answer are "independent"-that is we know that we can neither prove nor
disprove these statements. So the old question "How is mathematics
possible?" has now a new interpretation. We ask "what is the mechanism
which leads us to ask only "meaningful questions" the questions which can
be resolved. How are we able to get around the Godel's theorem? I do not
think that any one has even an inkling of where to look for an answer. But
I think that the existence of such a way to "cheat" Godel's theorem is
related to the experience, which tells us that it is easier to solve a more
general problem than a specific one. You see there is a big difference
between generalizations in mathematics and generalizations in social
studies.  In the case of social studies we pay for any generalization by
the acceptance of an increasing number of counterexamples. In contrast in
mathematics, where exceptions are not accepted, an existence of a
sufficiently general statement to which we can not find counterexamples is
a strong indication that the statement is provable. [For example many
people thought that the Fermat conjecture could neither be proved nor
disproved nor shown to be undecidable.  But immediately after Fray realized
that the Fermat conjecture follows from a much more general Taniyama-Weil
conjecture it became "clear" that that the Fermat's conjecture will be
solved.]  We can also ask :"How is mathematics possible ?" or "Why does not
mathematics split in a number of unrelated disciplines?" When one reads
writing from the turn of the century one sees the explosion of mathematics
was seen the main problem which could destroy the unity of mathematics.
Already then was no
mathematician who could follow all the developments and mathematics could
become a bunch of unrelated disciplines. Poincare writes: "An attempt is
made to cut it in pieces --to specialize.  Too great a movement in this
direction constitutes a serious obstacle to the progress of science." How
could the unity be preserved?  A "formal" interpretation of the first
question :"How is mathematics possible ?" represents a very specific
understanding of the structure of mathematics whereby the logical structure
takes on primary importance.  This interpretation suggests one answer to
the second question-why mathematics does not become a bunch of unrelated
disciplines, what is responsible for the unity of mathematics? For example,
Hilbert thought that the main uniting force comes from the common structure
the logic of proofs.  On the other hand Poincare who considered the
"economy of thoughts", the development of new conceptual approaches as the
backbone of mathematics expressed the hope that the unity of mathematics
will be preserved by unexpected concurrence that mathematics progresses.
We see now that both Hilbert and Poincare are right-the mathematics was
able to preserved the unity during a multi facet development of the
20-century and this unity is due both to-the structural clarity and the
immense number of unexpected connections between different areas of
mathematics.  Actually the question :"How is mathematics possible ?" was
asked already by Kant who understood it as a question :"  How is
mathematics possible?"  Kant saw
the existence of mathematics as a proof for the existence of pure
intuition.  Mathematics, which for Kant was reduced to the Euclidean
geometry, was considered as an outcome of the innate ability to see
unmitigated truth. Such an understanding of mathematics does not correspond
to an every day experience which teaches that some statements which are
"intuitively clear" to one mathematician could be "counter intuitive" to
another. As Poincare has beautifully described in his article "Mathematical
discovery" the unexpected immediate illumination comes after a long and
often seemingly unproductive work.  In other words, the mathematical
intuition is not a natural phenomena, is not given by birth, but is
developed through our life.  I do not want to discuss here philo-genesis of
mathematical intuition but want to concentrate on the onto-genesis, on the
way the mathematical intuition was developing in this century.  As before I
can only present my personal position.  I think that the most drastic
change in mathematics intuition came from the development of the algebra.
In the end of previous century, it was possible to subdivide mathematics to
Algebra and Analysis, which contained Geometry, and these two areas were
almost independent.  In the end of this century we find ourselves in the
position when the majority of achievements in Analysis and Geometry are, at
least partially, based on the development of an algebraic intuition.  It is
very characteristic that such a brilliant mathematician as Pontriagin
dropped mathematics after the appearance of the post-war French school,
which was based on new algebraic intuition.  This new understanding that
the analysis of different algebraic structures is central for the
development of mathematics found the most striking expression in the
development of the category theory. I do not think that it would be
possible to explain the basics of the category theory to any mathematician
of the last century.  The reason is that the theory of categories is "too
simple".  This theory, which originated in the forties, is based on a
drastic shift of perspective: instead of studying the logic of the
properties of mathematical objects the category theory studies the logic of
relations. Maybe the category theory
is the first serious extension of Aristotelian logic. In Aristotelian logic
all the statements are "absolutely trivial" but in spite of this triviality
Aristotelian logic is the backbone of all sciences. Analogously all the
basic statements of category theory are absolutely trivial but this logic
of relations is a basis for a big chunk of modern mathematics.  It is very
indicative that the first paper in the category theory was rejected by a
first-rate mathematical journal for the lack of content.  How does this new
way of thinking change mathematical reality?  Of course, it is impossible
to describe the full picture while standing on one foot but I can give two
applications of this new way of thinking.  The first application is the
possibility for a construction of "ideal" objects which are completely
defined in terms of their relations with the previously known objects.  The
second advantage coming from the category theory is a possibility to see
familiar mathematical objects as a "materializations " or, if you wish,
shades of the more elaborate and structured objects.  For example, a lot of
the recent progress in representation theory is based on the understanding
that, in a number of cases, functions are "materializations" of more
elaborated algebra-geometric objects.  The third topic I want to discuss is
the change of the structure of the interrelation between mathematics and
physics.  There were two different stages of this change. In the first
stage, which started already in the beginning of this century, physicists
realized that they need mathematics not only as a tool to solve their
problems, but also as a language to formulate laws of physics.  Both
relativity theory and quantum mechanics rely on "modern" mathematics for
the formulation of "physical" reality.  There is no way to explain some of
the most basic problems of the contemporary physics to people who do not
have an extensive mathematical background. But in this first stage, we
still find a familiar structure of the relation between mathematics and
physics when mathematics is used by physicists as a tool for the
formulation and solution of their problems.  The second stage, which
started 20 years ago, brought the reversal of their roles.  In the last two
decades of this century, we have an increasing number of examples of
applications of physics to mathematics.  These applications are primarily
in the form of conjectures, which relate mathematical problems that were
viewed by mathematicians as having nothing in common.  How is this
possible?  In many physical theories, there is a way to express physical
quantities in terms of a functional integral. Since the functional integral
does not have a rigorous definition such expressions do not have any
meaning for mathematicians. Imagine now that the problem we consider
depends on a parameter [say energy] and in the case when the energy is
either very large or very small there is way to approximate the
corresponding functional integrals by conventional mathematical
expressions. These conventional expressions for the case of small and large
energy are very different and we obtain two different rigorous expressions
for the physical quantities -one from the analysis of the case when the
energy is large and the other when it is small. From the point of view of
physics, both expressions are specialization of the original functional
integral.  Therefore "physical intuition" implies that these two different
expressions coincide. On the other hand there is no obvious mathematical
explanation for such a coincidence.  The existence of mathematics
consequences of physical theories leads to the situation where mathematics
plays a role of experimental physics for some branches of theoretical
physics. It became either impossible or too expensive to check the validity
of some physical theories by experiment.  Instead the validity of a
physical theory is "confirmed" by the correctness of the mathematical
predictions which can be deduced from this theory.
The last topic I want to discuss is the appearance of computer science.  As
a result of this development, mathematicians realized that it is not
sufficient to ask whether a particular problem is solvable, but one should
also inquire whether it can be solved in a reasonable amount of time.
Computer scientists defined "reasonable" questions as such questions that
you can check the correctness of an answer in a short [=polynomial] time.
On the other hand, one can consider a more restricted group of questions
which could be solved in a short time.  The basic problem of the computer
science is whether these two groups are really different, whether P-NP. On
the first glance it is "clear" that P-NP, that there are many ways to ask
"reasonable" questions which is difficult to solve.  But as we have already
discussed, mathematical problems have a strong tendency to be solved in a
relatively short time. really if a solution to a particular mathematical
problem would take an exponentially long time we would never be able to
solve such a problem. So either P=NP or we, mathematicians are somehow able
to choose very special "solvable" questions.  Therefore we can restate the
question "  Why we, mathematicians, are able to perform our work?".  In a
stronger form.  We ask what is the mechanism which leads us to ask
questions which can be solved and can be solved "in real time".