[FOM] Is mathematical realism compatible with classical reasoning?
tchow at alum.mit.edu
Sun Aug 6 17:17:41 EDT 2017
Andre Kornell wrote:
> 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
> always discuss truth in the metalanguage". My understanding of this
> is that there is a hierarchy of truth predicates, that it is always
> possible to express the validity of classical reasoning for a
> language in a more expressive language. In particular, at no level of
> 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.
I thought it was tolerably clear what I meant by saying that the current
"dogma" is that the truth predicate is not mathematical, but if you wish
to challenge that by pointing out that "metamathematics is mathematics"
then I am forced to speak more pedantically. The "dogma" is that if you
want to *fix* a formal definition for "all mathematical discourse" and
then also wish to speak of a truth predicate for "all mathematical
discourse," then that truth predicate cannot itself be part of "all
mathematical discourse." This does not, of course, preclude us from
fixing some circumscribed domain, introducing a truth predicate for that
domain, and then declaring that our truth predicate is "mathematical,"
so long as we refrain from claiming that the initially circumscribed
domain comprises all mathematics. But if you want to talk about "the
hierarchy as a whole" and wish to claim that the "hierarchy as a whole"
is supposed to comprise *all mathematical discourse* and then you want
to talk about a truth predicate for hierarchy as a whole, then the
standard view is that if the hierarchy is formalizable in the usual
ways, then the truth predicate for the hierarchy will not be
formalizable inside the hierarchy, and hence by your own declaration
will not be mathematical.
Again, I'm not really trying to defend this dogma. Alternatives are
possible, e.g., Kripke's "Outline of a Theory of Truth." I'm only
saying that it surprises me that you would claim that the *negation* of
this view is highly certain.
> This branch of the discussion concerns the case that we have no truth
> predicate for the mathematical universe, mathematical or otherwise. I
> not see a way to express the validity of classical reasoning without
> 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
> 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
> 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?
Why is that the basic, practical question? I'm happy to appeal to the
notion of truth as a predicate. I'm just saying that the truth
predicate is going to be *non-mathematical*. Why do you say that this
is not possible?
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