[FOM] Platonism and Formalism

[Neil Tennant]

On Fri, 26 Sep 2003, Harvey Friedman wrote:
 
> On the one hand, it should comfort Platonists a lot that there will be
> essential and valued uses of large cardinal principles, in the realm of
> V(9), that they accept, or are very inclined to accept, like various
> standard large cardinal axioms in the ZFC context.
> 
> HOWEVER, the results should also show that even sharper and cleaner and more
> valued consequences of a similar general nature are made, in the realm of
> V(9), from various large cardinal principles that are firmly rejected by
> Platonists ­ on the grounds that they outright contradict the axiom of
> choice. 

Do you mean by "more valued consequences" that these consequences are
*true* ?
 
> It has been known for some time that from the point of view of arithmetic
> consequences, the strongest principles that have been studied are
> inconsistent with the axiom of choice, and so the Platonists cannot argue
> that their arithmetic consequences are true by appealing to the truth of
> these principles (since they contradict the axiom of choice).

Is the Platonist in difficulties here because s/he wants to show that
these arithmetic consequences are *true* ?

> I suspect that this difference in strength will be clearly manifest in
> various specific valued consequences that they have for V(9), ...

Are these consequences valued because they are *true* ?

> where the
> axioms inconsistent with the axiom of choice are substantially more
> effective for deriving sharper and more satisfying results for V(9) than
> axioms that are believed to be consistent with the axiom of choice.

Are these results more satisfying because they are *true* ?

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.

If you give an affirmative answer, then the Platonist is likely to inquire
after your grounds for believing in the truth of the results that are
provable only by using large cardinal principles inconsistent with the
axiom of choice. Why should the Platonist share your own belief, in this
case, in the truth of the results thus proved?

Observe now that nothing in the foregoing questions (or in their possible
answers) really rests on the essential content of Platonism as a
philosophy of mathematical objects and mathematical truth. It rests only
on whether one wishes to assert the axiom of choice, regardless of one's
philosophical outlook. Hence, in all of the foregoing---my quotations
from what you wrote, and my questions in light of those quotations---one
can write "proponent of [believer in/asserter of] the axiom of choice"
instead of "Platonist".

It would seem, then, that (in the event of an affirmative answer to my
questions) your results will bear on Platonism only if you can indicate
why it might be the case that the Platonist, and only the Platonist, would
have independent reasons for wishing to assert the results whose proof
requires the use of principles inconsistent with the axiom of choice.

Neil Tennant