[FOM] Is mathematical realism compatible with classical reasoning?

tchow 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 
> 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.

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 
> 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?

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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