[FOM] Platonism and Formalism

[Harvey Friedman]

Reply to Tennant.

On 9/28/03 9:15 AM, "Neil Tennant" <neilt at mercutio.cohums.ohio-state.edu>
wrote:

> Do you mean by "more valued consequences" that these consequences are
> *true* ?

No. Because they are stronger and more interesting and simpler,
mathematically.
 
> Is the Platonist in difficulties here because s/he wants to show that
> these arithmetic consequences are *true* ?

The Platonist is uncomfortable at least partly because the Platonist might
be interested in persuading others to adopt their point of view.
 
> Are these consequences valued because they are *true* ?

Because people might be impressed as to their intrinsic interest and how
they were proved, and how they demonstrably can't be proved.
  
> Are these results more satisfying because they are *true* ?

Because they are stronger and more interesting and simpler.
> 
> I take it that an affirmative answer to any of these questions implies an
> affirmative answer to each of them. Likewise with negative answers.
> 
> If you give a negative answer, however, then there is no difficulty (as
> far as I can tell) for the Platonist.

Consider the reaction of non Platonists to Platonists in light of such
results. Specifically, the reaction of non Platonists to Platonists, as
Platonists try to convince non Platonists of their point of view.

Their best argument, or at least one of their very strong arguments, to
convince non Platonists, would be the startling consequences in V(9) of
their most powerful principles that they would claim are absolutely true,
and which cannot be proved without accepting their most powerful principles.

Yet even more startling consequences of a similar nature in V(9) would come
out of principles that they would claim are absolutely false. And of course
the Platonist isn't in a position to prove or refute these more startling
consequences in V(9). This tends to undermine their argument, if perhaps not
in the eyes of some of them, certainly in the eyes of many anti Platonists
and anti realists.

If Tennant is a Platonist and this doesn't make him uncomfortable, then that
is a fact of the matter. However, I believe that it makes it harder to
convert non Platonists to Platonism. This is also a fact of the matter,
about which I may be RIGHT or I may be WRONG.

In addition to Platonism, there are various shades of realism. I am not any
kind of expert in these "isms", as I work with the standard methodology of
f.o.m., rather than with the standard methodology of philosophy. I was quite
interested in reading the account in

Hilary Putnam, Philosophy of Mathematics: Why nothing works, in the Words
and Life volume, edited by Conant.

Perhaps we can direct this thread into something more productive.

1. What consequences does it have if one adopts Platonism or rejects
Platonism? 

2. Same question for various shades of realism. And for other "isms".

3. By consequences, I mean, especially, what kind of mathematical or logical
research one does, or what kind of mathematical or logical results one
values or most values, or what kind of mathematical or logical questions one
asks?

4. What kind of mathematical or logical results would one look for to
bolster Platonism, anti Platonism, various kinds of realism, various kinds
of anti realism, various kinds of formalism, etc.?

5. If it is the case that an "ism" has no consequences or effects of a
mathematical or logical nature as in 3, or even "isms" generally have no
consequences or effects of a mathematical or logical nature as in 3, then
how does one make a case for the general intellectual interest of work on
"isms"? Or does one take work on "isms" as obviously of general intellectual
interest, independently of such consequences?

To give some orientation to this, it is obvious that what the anti Platonist
or anti realist would most want to have is an inconsistency in, say, ZFC + a
measurable cardinal, or much better yet, an inconsistency in ZFC, and of
course, the weakest set theory or even arithmetic (PA and fragments!!),
possible. However, there seems to be no sign of this, and probably no one is
even specifically attempting this.

Harvey Friedman