[FOM] Formalism/Platonism

[Harvey Friedman]

My overall view towards the isms is very much like

"Philosophy of Mathematics: Why Nothing Works", in Putnam 1994: Words
and Life (Harvard UP), pp. 499-512. Ed. James Conant.

See http://www.cs.nyu.edu/pipermail/fom/2000-June/004199.html

I don't personally see any point in throwing my head in the ring in order to
get it chopped off like everybody before me.

For me, the productive questions are oblique, and very exciting.

1. What observable consequences ensue from adopting any of the usual isms,
or various modifications of the usual isms?

2. What kinds of exact findings would bear on the merits of various isms?

Item 1 is particularly important because of the following observed fact.
Many mathematicians have very strong formalist leanings that come out very
explicitly in discussions - e.g., my unnamed correspondent mentioned in
previous postings - and also many mathematicians have very strong Platonist
leanings that come out very explicitly in discussions. I would surmise,
without investigation yet, that the majority of mathematicians take a view
somewhat closer to Naturalism than to formalism or Platonism. I.e.,

"whatever attitudes and preoccupations that helps mathematics get on with
its main traditional goals in as smooth and nondisruptive a manner as
possible, is to be adopted".

This helps explain why they run away so hard and so fast from any kind of
mathematics that even has a whiff of having logical difficulties about it.
(As people may surmise from reading this list, I am slowly but surely
succeeding in trapping them so they can't run away. Stay tuned.)

So if there are no observable consequences from taking one point of view or
another, then what is the fuss about?

It can be argued that even if there are no such observable consequences,
then it is still a very important philosophical issue.

This is certainly a defensible position, but the problem is that this itself
is unconvincing - no more convincing than the actual arguments for/against
formalism/Platonism.

For example, it can also be argued that any declaration of the form

A. "any first order sentence about such and such structure has a definite
truth value"

is itself meaningless. This is also a very defensible position. Trying to
explain just what A means seems singularly difficult.

The most obvious attempt at explaining A is to replace it with

B. "any first order sentence about such and such structure will have its
truth value known someday".

However, most Platonists categorically reject that A is the same as B. Many
argue that they have grave doubts about B, but not, of course, about A.

In making such declarations, I am not sure that people have really thought
through some fundamental issues that are hidden. E.g., a first order
sentence can have arbitrarily long length, and therefore never even be
comprehended by any human mind - and are there other minds at work? And so
forth. 

So given the inherent difficulty in explaining the very meaning of various
positions that are taken on such matters, the issue of

*what observable consequences does the taking of such positions have?*

takes on added significance.

As I mentioned earlier, and Steel mentioned even earlier than that,
apparently there is an observable consequence of the taking of positions or
the making of declarations regarding formalism/Platonism.

These consequences are in the area of attitudes towards mathematical
investigations. More specifically,

a. What research does one engage in? This may not be applicable.
b. What research does one get interested in?
c. What research does one value?

So for the people on the FOM who hold various isms, I think we would all
like to hear about their a-c.

Putnam is right. I know of no position taken regarding the
formalist/Platonist divide that isn't riddled with difficulties and
unconvincing pronouncements.

Recall 2 from the beginning of this posting:

2. What kinds of exact findings would bear on the merits of various isms?

This is obviously closely related to a-c. I am not attracted to any ism, but
I am attracted to 1 and 2.

The paradigm case that we should all keep in mind for 2 is the Godel first
and second incompleteness theorem.

It is now clear that Godel's incompleteness theorems have been used for and
against formalism, as well as for and against Platonism.

It is hard to imagine any in depth discussion of f.o.m. that does not in
some way on some side invoke the Godel incompleteness theorems. If you
witness such a discussion not invoking them, then this should raise
suspicion about the competence of the participants.

Harvey Friedman