[FOM] Foundational Issues: Friedman/Carneiro

[tchow]

I think I understand your position now and have no major additional
comments, but I have a couple of minor comments.

Mario Carneiro wrote:
> On Thu, Apr 14, 2016 at 10:20 AM, tchow <tchow at alum.mit.edu> wrote:
>> What you don't want to do is to attach the word "exists"
>> to a large finite object inside a platonic realm.
[...]
>> At the same time, you *do* allow the use of phrases such as "the 
>> natural
>> numbers exist" but only in the sense of operator overloading:
[...]
> This is a fair characterization. The first "exists" is the concern of a
> philosopher, the second "exists" is the concern of a mathematician. As
> such, they don't overlap very much and it's generally clear which is 
> which.

I don't think the disambiguation is as trivial as you make it out to be,
since when it comes to small finite numbers, the only sense in which you
allow them to "exist" is in the unabbreviated sense.  Therefore if I say
that 2^2 exists, and 3^3 exists, and 4^4 exists, and so forth, then at 
some
fuzzy point there is a transition from the unabbreviated "exists" to the
abbreviated "exists."  For a platonist, there is no transition, but for
you, there is.

For a less trivial example, consider the schema "P is a proof of the
Riemann Hypothesis in PA."  If we insert a feasibly-sized value of P
then this is a statement that you regard as objectively true or false,
right?  But the statement "The Riemann Hypothesis is provable in PA"
is not objectively true or false.  The introduction of the quantifier
has forced a switch from the unabbreviated "exists" to the abbreviated
"exists".  This is clear once you get used to the operator overloading
but it's surely confusing to most people.

> Finally, I should also be clear that I am not asserting that the 
> platonic
> realm does not exist in reality, only that we have zero evidence that 
> it
> does exist. (And I don't think this is controversial.)

"Controversial" may be the wrong word, but certainly you will not get 
any
consensus about what it even *means* for "the platonic realm to exist in
reality", let alone any consensus about what would constitute "evidence"
for or against such a statement, and let alone whether we have such
evidence.

Tim Chow