[FOM] Re: pom/fom. directions

[Harvey Friedman]

Reply to Williamson.

On 10/22/03 7:28 AM, "Jon Williamson" <jon.williamson at kcl.ac.uk> wrote:

> Is this the point of dispute?
> 
> Corfield thinks that philosophers of maths should study and understand
> maths;
> 
> Friedman thinks that philosophers of maths should only study those bits of
> maths of general intellectual interest, or of traditional philosophical
> interest:

No. I would not characterize the "dispute" (if there is one) in this way.

Before starting on a restatement of my position, let me first say that my
notion of general intellectual interest does not correspond to items that
have been successfully or partly successfully popularized. I mean something
quite different than that - but not unrelated - as I shall explain below.

This would be closer to my point of view:

Friedman thinks that philosophers of mathematics should concentrate on those
parts of mathematics for which it is promising to recast them so that they
attain (a higher level of) general intellectual interest, and/or relate to
clear philosophical issues.

If some part of mathematics is only understood by insiders, and philosophers
are able to rework it so that it is understood in a serious way by other
communities, then that is an achievement. The philosopher is likely to have
uncovered some fundamental conceptual thread that has remained buried in the
work - at least to outsiders.

As a test case, I believe that the standard texts used at the advanced
undergraduate and early graduate level in mathematics are opaque to the
preponderance of philosophers. The same is true - perhaps even more so - for
physics. Obviously the further away philosophers are from philosophy of
mathematics and philosophy of science, the more opaque they look.

So a suggested goal for philosophers of mathematics is to exposit this level
of material in such a way that its import is clear to a general
philosophical audience, as well as what the major results actually assert.

I don't believe that this has been done properly even for mathematical logic
and f.o.m. beyond the level of basic material from the 1930's. Even with
regard to the basic material from the 1930's, it isn't really fully
effective for a general philosophical audience; e.g., the philosophically
clearest formulations and proofs, and various missing clarifying results,
haven't been given.

Expositions of mathematical logic move quickly away from fundamental issues,
into slick presentations of material known to be essential for a
professional career. This is a feature that is also present in the
expositions of core mathematics.

This will be difficult - but not impossible - for most philosophers to pull
off, as it will require some real imagination - and probably luck - to get
appropriate help from mathematicians.

> The difficulties I can see with Friedman's view are these:
> 
> 1. it is hard to see how much of post-1930s FOM has been of general
> intellectual interest. In contrast, there have been a number of topics in
> recent maths of general intellectual interest, e.g. nonlinear dynamics and
> the general interest in chaos theory; number theory and the general interest
> in cryptography and in Fermat's last theorem,; many areas of recent maths
> and the general interest in fundamental physics.

By general intellectual interest, I don't mean "having been popularized".

There are matters of general intellectual interest where good progress is
being made in f.o.m., post 1930's, for which general popularization is
premature, and perhaps counterproductive. But the gii is very much there.

There are also some results of general intellectual interest that could be
or have been popularized, post 1930's, at least to some extent.

I think it would be illuminating to compare the general intellectual
interest of items I am going to list that are post 1930's, with the topics
that you mention. 

In general, the communities which are represented by your list are far more
well connected with the press, and the number of people working in them are
far larger, than in f.o.m., even if you include the whole of mathematical
logic. 

At least for now, I will only discuss work that is 1970 or earlier.

1. 1951, 1959. A. Tarski. Decision procedure and completeness of axioms for
elementary Euclidean geometry, real algebra, and complex algebra. A complete
general solution to all high school plane geometry problems, as well as all
high school algebra problems involving real numbers, but not integers.

2. 1963. PJ Cohen. The unprovability of the axiom of choice from ZF, and the
unprovability of the continuum hypothesis from ZFC. The gii is clear from a
lot of directions. Furthermore, it is closely allied to 1930's developments
that have a very well received gii. For very wide audiences, there is the
story about how the 'rules of the game' that mathematicians have accepted
for so long are shown to be insufficient for settling some well known
problems  - raising the issue of an appropriate expansion of the 'rules of
the game', in the one subject where it was thought that the 'rules of game'
will always be the same - mathematics.

3. 1970. Y. Matiyasevich, J. Robinson, M. Davis, H. Putnam. No algorithm for
solving all Diophantine equations. Again, closely allied to 1930's
developments that have a very well received gii. I.e., Turing's model of
computation, and its robustness. Also, a timeless odyssey in mathematics
called Diophantine equations.

I would be interested to see just what specific results you list are
comparable in gii to these three. I am sure that you can make a case for
some of them. It would be interesting to evaluate the case that you can
make. 

One interesting comparison would be between 3 and FLT.

a. FLT and subsequent offshoots are of incomparably greater interest among
number theorists than 3. Also among mathematicians generally, FLT and
subsequent offshoots are of greater interest than 3.

b. However, in terms of gii, 3 ranks significantly higher than FLT and its
subsequent offshoots. Popular accounts of FLT and subsequent developments,
even accounts to scholars outside mathematics, rely heavily on the
historical and personal challenge, along with the fact that a lot of other
mathematics gets used. However, such an account does not deal directly with
the intellectual content of FLT itself. 3 has the totally honest tie in with
great events from the 1930's of long recognized gii - models of computation
and impossibility theorems.

> 2. how should one delimit `traditional philosophical interest'? The
> philosophical interests of the Pythagorean school? Of Plato? Of Kant? In any
> case, why be so backward-looking?

I don't care. Anything that makes clear philosophical sense. By
"traditional" I just meant "makes clear philosophical sense by traditional
philosophical standards".

Harvey Friedman