[FOM] Is mathematical realism compatible with classical reasoning?

[Timothy Y. Chow]

Andre Kornell wrote:
> so the question is whether we are more confident in the validity of 
> classical reasoning for the mathematical universe, or more confident 
> that the truth predicate is mathematical.

It seems obvious to me that we are more confident in the former.  In fact 
the statement that the truth predicate is *not* mathematical is almost a 
dogma nowadays.  While I don't think this dogma should be accepted 
unquestioningly, I remain amazed that you regard its negation as more 
certain than the validity of classical reasoning.  But I don't have any 
more arguments to offer you, since you and I diverge so far on this point.

> In my mind, the validity of classical reasoning for the mathematical 
> universe is the principle that the conclusion of any classical 
> mathematical proof that appeals exclusively to logical axioms is true. 
> In any finitary formalization of this principle, truth is necessarily a 
> predicate, as we cannot simply form a conjunction of all the sentences 
> that have proofs. Thus, if we have no truth predicate for the 
> mathematical universe, then we cannot express the validity of classical 
> reasoning for the mathematical universe as a general principle.

This looks like an argument that it cannot be expressed as a 
*mathematical* general principle, but I don't see why we can't take the 
point of view that the truth predicate is non-mathematical and that the 
validity of classical reasoning for the mathematical universe is a 
non-mathematical general principle.  If we are not required to be 
"mathematical" (whatever that means) or "finitary" (whatever *that* means) 
when stating non-mathematical general principles, then for example I don't 
see why we can't form an infinitary conjunction.

Tim