regarding evolution of mathematical areas

[Harvey Friedman]

In the course of some correspondence I came up with some ideas
concerning the evolution of mathematical areas - form a general
Foundational Perspective.

It seemed to me that what I wrote might be of interest to the FOM
readership. Below is from the actual correspondence, lightly edited.

There is an abrupt jump from the obviously foundationally fundamental
to the sensibly
technical that is generally much too large a jump to represent any kind of
major advance in foundations of mathematics. Of course the number of
major advances in foundations of mathematics is indeed very limited
but they are very powerfully influential.

I can illustrate how this viewpoint manifests itself in modern
mathematical logic, where major foundational advances arise from time
to time.

The set theory community was for a limited period of time, rather
stronger foundationally than other parts of math logic, with Cohen and
the fallout. Model theory has gotten stronger, not
foundationally, but in terms of the mission started by A. Robinson to
forge itself as a useful tool in core math. Recursion theory has
slowly but surely emerged from a rather limited period to become much
more foundational with the
ascent of RM. Proof theory has become more foundational with the
ascent of the Tangible Incompleteness, and the Interpretation Theory.

The main problem with areas of math that have been born out of or just
been energized by epic breaking
legendary advances - be it geometry/topology or mathematical physics
or computation theory or math logic - is that they uncritically latch on certain
natural formalisms emanating out of those epic developments, and dig
in to specialized issues and don't rethink
and refine those formalisms. They proceed based on the hidden
assumption that these formalisms are fully appropriate. In fact these formalisms
can almost always be vastly improved ushering in new overarching
foundational developments. And so the process continues.

Most mathematical subjects are in intermediate stages where the
developments do not cause any rethinking of the fundamentals or
generate any legendary advances of obvious permanence or deliver
tangible penetrating messages.

That doesn't mean I am against the ongoing developments during these
inevitable intermediate stages. On the contrary, they are totally
necessary and do provide useful technical background information and
techniques
that greatly facilitate in bringing the new truly path breaking ideas
into fruition.

But the major practical challenge that needs to be addressed is how
the environment can facilitate the emergence of the badly needed
legendary advances? There is a tendency for areas during intermediate
stages to actually suppress the development of path breaking ideas.

I see major obstacles in the way mathematical areas are taught and the
reward systems that are in place that I believe serve to make
legendary foundational advances even rarer than they could be
otherwise.

The countervailing view is that legendary foundational advances are
totally unpredictable and have little to do with environmental factors
and that any attempt to alter the environment in this regard is
pointless.

Harvey Friedman