[FOM] Is mathematical realism compatible with classical reasoning?
joeshipman at aol.com
Thu Aug 3 00:52:35 EDT 2017
The problem here is in not specifying the language. We are, presumably, talking about applying a "truth predicate" to sentential objects, describable as finite strings in a finite alphabet, which excludes "infinite conjunctions". One can postulate an unproblematic criterion for well-formedness and talk about a "truth predicate" if this criterion looks like the ones we have seen in different logics. But for this to comport with the common sense around the word "truth", we need to either give the linguistic terms semantics, or have an intended mathematical model of the language.
If your language is the language of arithmetic, I can define a truth predicate for you using set theory, after we agree on an informal description of which "arithmetic" we are talking about (integer, rational, real, complex, etc.). But if the language is the language of set theory, you had better tell me what you think "set" means before I will attempt to define a truth predicate for the language.
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> On Aug 2, 2017, at 10:47 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> 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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