[FOM] Is mathematical realism compatible with classical reasoning?

[Andre Kornell]

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