[FOM] Is mathematical realism compatible with classical reasoning?

[Andre Kornell]

Tim writes: In fact the statement that the truth predicate is *not*
mathematical is almost a dogma nowadays.

I'm not certain of the majority position among avowed realists. The
narrative that I recall from my time as a graduate student is that "we can
always discuss truth in the metalanguage". My understanding of this slogan
is that there is a hierarchy of truth predicates, that it is always
possible to express the validity of classical reasoning for a particular
language in a more expressive language. In particular, at no level of this
hierarchy have we stopped doing mathematics; metamathematics is
mathematics. I accept this narrative. I just want to go beyond making
claims at specific levels of this hierarchy to making claims about the
hierarchy as a whole.

Perhaps another FOM reader can suggest a good account of the modern realist
position toward Tarski's undefinability theorem?

Tim writes: This looks like an argument that [the truth predicate for
mathematical universe] 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.

This branch of the discussion concerns the case that we have no truth
predicate for the mathematical universe, mathematical or otherwise. I do
not see a way to express the validity of classical reasoning without using
the truth predicate. Indeed, an infinitary agent can simply form the
conjunction of all mathematical sentences provable just from classical
logic. However, such an agent can surely also simply form the disjunction
of the partial truth predicates, yielding a truth predicate for the
mathematical universe. This discussion of infinitary conjunctions and
disjunctions is certainly cursory, but already it distracts from the basic,
practical question: can you explain what it means for classical logic to be
valid for the mathematical universe without appealing to the notion of
truth as a predicate?

Andre


On Wed, Aug 2, 2017 at 7:47 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.
>
> Tim
>
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