[FOM] Is mathematical realism compatible with classical reasoning?
Patrik Eklund
peklund at cs.umu.se
Mon Aug 7 00:20:57 EDT 2017
This discussion is turning into a very strange discussion.
If we start to use things like "majority position" as de facto
justifications of mathematical rightfulness, what are we really doing?
Philosophy is ok, and even new branches of philosophy (whatever they
are) are ok, but philosophy is not mathematics. Metamathematics is
mathematics, but as soon as we realize there may be meta of meta, and
meta of meta of meta, and so on, we wonder about fons et origo, and how
to start from there. Set theory needs logic, and logic needs set theory
to kick off. After a while they develop hand in hand and become
intertwined. This is pretty much where we were one hundred years ago.
That intertwining didn't per se justify self-referentiality, since on
the set theoretic side it remained strictly forbidden. On the logical
side it was allowed.
The problem is about types. For sets we really don't have them, and in
logic we mix terms and sentences, and we mix truth and provability.
Type theory tries to fix this but they make the same mistake as is done
is this debate. They reject the idea that metamathematics must be
mathematics. They want a meta that is the first and fundamental meta of
everything. What happens in such approaches is that something eventually
goes wrong, and then something cute is brought in from the outside as a
rescue, but it is not called another meta, but it lives its own life as
a nice trick.
The editorial board of FOM accepts and rejects postings also in this
discussion, but what is the standpoint of FOM and the editorial board on
these question other that steering the debate by accepting and rejecting
postings?
Best,
Patrik
On 2017-08-07 00:17, tchow wrote:
> 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?
>
> Tim
>
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