[FOM] Foundational Issues: Friedman/Carneiro

[Mario Carneiro]

On Fri, Apr 8, 2016 at 1:29 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> Mario Carneiro wrote:
>
> I think we have a difference of understanding of the word "exists" here.
>> When I say "the natural numbers exist", I mean no more or less than "? N,
>> (0 in N /\ ? n in N, n+1 in N) is a theorem in the theory of interest".
>> That's a finite statement about a finite string, with a finite proof
>> (assuming I am not making a false claim). It's all well and good if you
>> want to think there is some world out there where you can meet the actual
>> natural numbers, but it literally has no bearing on the statement. I
>> suppose you could call this "anti-platonism", although that makes it sound
>> as if I assert the *non*existence of platonic objects, when instead my
>> position is closer to "who cares".
>>
>
> Just for clarification, do you believe that there is "some world out there
> where you can meet actual finite strings"?
>
> Only a diehard skeptic would deny that there is a world out there where we
> can meet ink attached to paper, or fluorescent pixels on a screen, or talc
> clinging to slate, but that of course is not the same as meeting a "finite
> string."  A string, on the face of it, is a platonic object.  Is your
> position on the existence of strings close to "who cares"?
>

Yes, that's right. Evidence of some world out there with mathematical
objects of any nature is dubious, and my general approach is "who cares" /
"it's beside the point". If pressed on whether finite strings have any
better representation in this mythical land than other objects, I would say
no, there is no reason to discriminate against infinite sets, seeing as it
is all in our heads anyway.

Taking this to its logical extreme we are faced with questions like "does 1
exist?". There is certainly a concept of one that exists, and is maintained
relatively unchanged from one person to the next, so that in some sense it
can be seen to have objective existence. But this more pragmatic kind of
existence is much more tenuous, one that an eradication of humanity would
snuff out, rather than the platonic view that we are just visiting a world
beyond ourselves, that would live on without us.

The rubber meets the road when you actually start writing proofs, and at
this stage you are limited by your short human lifetime and the limited
patience of your reviewers ;) . So here we certainly have an "ultrafinite"
constraint. But the objects themselves, the things we purport to describe,
can be as crazy as our imaginations can take us. Furthermore, this still
does not preclude the idea that math is "discovered" instead of invented,
because the "matters of fact" (as Baez says), of the type "is P a proof of
A in the theory T" are objectively true or false, and "is statement A true
in theory T" is also usually an objective question with an answer (although
the edges on this are a little fuzzy as the proofs or disproofs get longer
or disappear entirely).

Mario
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