[FOM] Is mathematical realism compatible with classical reasoning?

[Colin McLarty]

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.

Colin

On Tue, Jul 25, 2017 at 12:46 AM, Andre Kornell <andre.kornell at gmail.com>
wrote:

> Dear FOM,
>
> I've been considering an approach to foundations that generalizes
> finitism to infinitary settings. If we have in mind some exhaustive
> notion of possible procedure on mathematical objects, then I suppose
> we should reason entirely in terms of propositions that can be
> verified in principle by such procedures. I have posted my notes on
> this approach to the arXiv ( https://arxiv.org/pdf/1704.08155.pdf ),
> but I think that most FOM readers will not have time to look at them
> closely, so I thought I might stimulate a bit of discussion with a
> question.
>
> Is mathematical realism compatible with the use of classical logic in
> foundational metamathematics?
>
> Suppose that abstract mathematical objects exist, that mathematical
> sentences are objective propositions in their literal sense, that
> mathematical formulas are closed under the formation rules of
> classical first-order logic, and that classical reasoning is valid for
> mathematical sentences. The meaning of mathematical sentences, or any
> other class of syntactic object, is given by specifying when such a
> sentence is true, i. e., by a truth predicate. By Tarski’s
> undefinability theorem, this truth predicate cannot be mathematical in
> exactly the sense that we’ve been using. The usual response to this
> difficulty is to introduce a metalanguage in which the truth predicate
> of the original structure can be expressed, and to claim that we are
> engaged in a kind of higher-level mathematics. The obvious critique of
> this response is that in a foundational context it is appropriate to
> work with an inclusive notion of mathematical predicate. If classical
> logic is valid for the class of first-order sentences generated by the
> truth predicate together with the mathematical predicates, then what
> feature of the truth predicate precludes it from being itself a
> mathematical predicate? If classical logic is not valid for the class
> of sentences obtained by adding the truth predicate, then what
> justifies the use of classical logic in foundational metamathematics?
>
> We respond to Tarski’s theorem by working with a hierarchy of
> languages, and we respond to Russell’s paradox by working with a
> hierarchy of sets, and in both cases I think we are arresting the
> hierarchy prematurely. If we can include a next level of predicates or
> a next rank of sets, then the foundational imperative to include all
> mathematical predicates and all mathematical objects requires that we
> do so. Since we continue the hierarchy through stages at which
> classical reasoning is valid, classical reasoning should be not be
> valid for the hierarchy as a whole, as the relevant paradox indicates.
> Nevertheless, it is the the hierarchy as a whole that interests us.
>
> I would be grateful for any feedback from readers who do get a chance
> to look at my notes. This draft does need more work, but I worry about
> spending too much time on the wrong thing. The bibliography is
> absolutely bare-bones, and I am hopeful that readers will suggest
> references, including their own work, which they feel would be
> appropriate.
>
>
> Andre Kornell
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