[FOM] Foundational Issues: Friedman/Carneiro

[Timothy Y. Chow]

On Fri, 8 Apr 2016, Mario Carneiro wrote:
> 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

I thought I followed you up to this point, but here I don't understand how 
you arrive at the claim that a statement such as

1. P is a proof of A in the theory T

is "objectively true or false."  I can see how you could claim that

2. All working mathematicians agree that P is a proof of A in the theory T

is objectively true or false.  But 2 is much closer to

3. P is *known to be* a proof of A in the theory T

than to 1.  So maybe you mean that 1 is synonymous with something like

4. If presented with the appropriate sensory stimuli, the mathematical 
community would arrive at a stable consensus that P is a proof of A in the 
theory T

But 4 makes assertions about a somewhat nebulous hypothetical scenario and 
so it's not quite clear to me that such a counterfactual conditional is 
necessarily "objectively true or false".

Tim