[FOM] good math

[Harvey Friedman]

Thanks to Nik Weaver for the link

http://arxiv.org/pdf/math/0702396v1.pdf

which is a very well written and very interesting paper by Terrence
Tao, What is Good Mathematics?, written in 2007.

It describes many important aspects of existing mathematical culture,
and then focuses on a very active and well respected part of
mathematics with many deep connections with diverse parts of
mathematics, surrounding Szemeredi's theorem on arithmetic
progressions.

Terry is extremely positive about current mathematical culture, and
emphasizes a great diversity of values. There is no indication there
of any need for any kind of reform. I think differently. I am very
skeptical of both the mathematics and philosophy communities. They
both have almost opposite drawbacks and limitations. A discussion of
this in the future should be of keen interest to FOM.

But even in the realm of mainstream thinking in math, (my thinking is
far from mainstream in math), there are wars boiling below the
surface. I think that these wars cannot be understood or negotiated or
resolved without going deeper into the very essence of mathematical
thought. This is all wrapped up in "general intellectual interest",
"foundational exposition", "unified foundational thinking", and
related ideas.

Within mainstream mathematical thinking, there is definitely a strong
strain of devaluation - sometimes rising to the level of contempt -
for the combinatorial. This can of course range from mild - "not my
cup of tea" - to the bizarre. Witness the following bizarre
interchange between me and a Fields Medalist FM who will remain
unnamed:

HMF. I would like to tell you about a theorem involving a concrete
kind of graph. You recall what a graph is?

FM. No, not really.

HMF. Well, the simplest definition of a graph is a set with an
irreflexive symmetric ...

FM's eyes roll, breaking eye contact with HMF. I start again,

HMF. It's a set of elements called vertices, and there is an
irreflexive symmetric relation on the vertices called the adjacency
relation. Ok?

FM. I don't know what a graph is, and I never want to know what a graph is.

So this was the end of my effort to talk about a theorem involving
graphs. FM obviously felt he knew enough about graphs to know that he
never wanted to know what a graph is. Obviously very interesting...

This FM clearly had a view of mathematics that was oriented toward the
algebraic and the geometric, and anything not immediately directly
connected with algebra or geometry was regarded as utterly irrelevant
to the point of actually causing pain.

You might say "well, this is a bizarre exception - there must have
been something personal".

OK, I have to admit the possibility that there could have been
something extra intellectual going on here.

BUT, I have experienced the same kind of thing repeatedly over the
years with both mainstream mathematicians and mainstream philosophers,
involving different topics - and, yet, it is usually more polite.

1. As I said, combinatorial topics presented to mathematicians from
almost any other areas.

2. Most of pure mathematics presented to mainstream applied mathematicians.

3. Almost anything in mathematical logic presented to almost any
mathematician outside logic, with the exception of some work in model
theory that is being applied to certain parts of algebra and geometry.

4. Almost anything in genuine foundations of mathematics presented to
almost any mathematician, including most scholars from mathematical
logic.

But there are many many other examples.

I am optimistic about a grand unification of overall purpose in
mathematics, but I don't think that this can come about without
properly putting mathematics into the much wider context of general
intellectual activity. Mathematics is but one vitally important
component of general intellectual activity.

I hope to be able to say more about exactly what I have in mind, in the future.

Harvey Friedman
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