[FOM] Is mathematical realism compatible with classical reasoning?
andre.kornell at gmail.com
Thu Jul 27 20:27:35 EDT 2017
Tim writes: In fact, even the desire for "is true" to be expressible
as a mathematical predicate seems to be something that a mathematical
realist would not necessarily have.
The use of classical reasoning in mathematical realism is justified on
the basis that each mathematical sentence is either true or false. My
impression is that truth is typically taken to be a nonmathematical
predicate, so that there is a uniform sense in which a sentence may be
true or false. Then, it is possible to state and argue the validity of
classical reasoning for the mathematical universe as a general
principle. The crux of the argument in my FOM posting is that if
classical reasoning is valid for this truth predicate also, then it
appears to have all the features of a mathematical predicate.
I agree that mathematical realism does not require that there be a
uniform sense in which a sentence may be true or false. Truth may be
treated as a trivial operator, so that the truth of each sentence is
taken to be equivalent to that sentence. In this variant of
mathematical realism, the validity of classical reasoning for the
mathematical universe cannot be expressed as a general principle,
mathematically or otherwise. Moreover, the metamathematical study of
the mathematical universe becomes difficult to justify whatever our
choice of logic. What is the foundational significance of ZFC from a
realist position if it is not that the theorems or at least the axioms
of ZFC are true of the mathematical universe?
On Wed, Jul 26, 2017 at 11:55 AM, tchow <tchow at alum.mit.edu> wrote:
> Colin McLarty wrote:
>> The idea that "we should reason entirely in terms of propositions that can
>> be verified in principle by" any specified set of procedures is usually
>> considered a kind of verificationism and thus not realism. It seems that
>> you are actually questioning whether mathematical verificationism (using
>> your specified means of verification) is compatible with classical logic.
> Yes. In fact, even the desire for "is true" to be expressible as a
> mathematical predicate seems to be something that a mathematical realist
> would not necessarily have. According to most forms of realism, reality
> exists independently of us, and if we find ourselves having trouble
> describing reality linguistically, then (so to speak) that's our problem,
> not reality's problem. Language, even foundational language, is not seen as
> defining or creating reality; it's just putting into words (as best as we
> can) what we know and/or believe about reality.
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