[FOM] Is mathematical realism compatible with classical reasoning?
Timothy Y. Chow
tchow at alum.mit.edu
Wed Aug 2 22:47:39 EDT 2017
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.
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