Previous post: The value of insights and the identity of the author.
This is a partial reply to a comment by vznvzn in Gowers's blog.
Combinatorics is most resolutely not "a new way of
looking at mathematics". It is very old, definitely known for hundreds
years. Perhaps, it was known in the ancient Babylon already.
And Erdős is not a "contrarian". His work belongs to the
most widely practiced tradition in analysis. As a crude approximation, one can
say that this tradition originates in the calculus of Leibniz, which is quite
different from the calculus of Newton. Even most of mathematicians are not
aware of the difference between the Leibniz calculus and the Newton calculus.
This is not surprising at all, since only the Leibniz calculus is taught
nowadays.
It is the Grothendieck's way of looking at mathematics, the one which I
advocate, which is new. This new, conceptual, way of doing mathematics immediately met strong resistance.
And in some cases its opponents won. For example, the early work of
Grothedieck in functional analysis had no influence till analysts managed to
translate part of his ideas into their standard language. It seems that only quite
recently some of analysts realized that a lot was lost in this translation, and
done a better translation, closer to the spirit of the original work of
Grothendieck.
Another example is provided by the invasion of this new style and even some
technical concepts developed in this style into the analysis of several complex
variables. This was intolerable for the classical complex analysts, and they
started to stress problems about which it was more or less clear that they can
be approached by familiar methods. They succeeded, and already in the
1970ies a prominent representative of the classical school, W. Rudin, was able
proudly say that Grothendieck's methods (he was more specific) disappeared into
background. He did not publish his opinion at the time, but attempted to insult
a prominent representative of the new style, A. Borel by such statements. A quarter
of century (or more) later he told this story in an autobiographical book. (W.
Rudin is a good mathematician and the author of several exceptionally good
books, but A. Borel was a brilliant mathematician.)
Now we are observing a much broader attempt, apparently led by T. Gowers, to
eliminate the conceptual way of doing mathematics completely. At the very
least T. Gowers is the face of this movement for the mathematical
public. After this T. Gowers envisions an elimination of the mathematics
itself by relegating it to computers. It looks like the second step is the one
most dear to his heart (see the discussion in his blog about a year ago). It
seems that combinatorics is much more amenable to the computerization
(although I don't believe that even this is possible) than the conceptual
mathematics.
Actually, it is not hard to believe that computers can efficiently
produce proofs of a wide class of theorem (the proofs will be unreadable to
humans, but still some will consider them as proofs). But for the conceptual
mathematics it is the definition, and not the proofs, which is important. The
conceptual mathematics is looking for new definitions interesting to humans.
The proof and theorems serve as a stimulus for work and as a necessary testing
ground for new definitions. If a new definition does not help to prove new
theorems or to simplify the proofs of old ones, it is not interesting for
humans.
There is only one way to get rid of the conceptual mathematics, namely, the
Wigner shift of the second kind. The new generation should be told that combinatorics
is new, that it is the field to work in, and very soon we will see the
young people only the ones doing combinatorics. Since mathematics is to a huge
extent "a young people’s game", such a shift can be
accomplished very quickly.
P.S. It is worth to note that there are two branches of combinatorics, and one of them is already belongs to the conceptual mathematics. Some people (like D. Zeilberger) are intentionally ignoring this to promote the non-conceptual kind.
Next post: D. Zeilberger's Opinions 1 and 62.
