[FOM] Is mathematical realism compatible with classical reasoning?
Andre Kornell
andre.kornell at gmail.com
Mon Jul 31 08:54:01 EDT 2017
Tim writes:
Appearances can be deceiving. In this case, we have a mathematical
proof that the truth predicate is not a mathematical predicate. What
stronger reason could we have for rejecting the hypothesis that this
truth predicate---"appearances" notwithstanding---than a mathematical
proof?
Broadly, Tarski's theorem obtains a contradiction from assumptions
about finitary mathematics, assumptions about first-order logic, and
assumptions about the truth predicate. The first class of assumptions
is fairly innocuous, 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. If
classical reasoning is valid for the truth predicate also, then it is
a definite predicate on mathematical objects that is subject to
mathematical methods of inquiry. Particularly from a finitist
perspective, the mathematical nature of such a predicate appears more
certain than the validity of classical reasoning.
I haven't specified what it means for an object or a predicate to be
mathematical, so I am not really arguing for a definite proposition. I
am arguing against the use of classical reasoning in metamathematics
on the basis that it conflicts with the intuition that the
mathematical universe should be maximally inclusive in various ways.
However, if we define a mathematical object to be a pure set, and we
define a mathematical predicate to be a definite predicate on pure
sets, that is, a predicate that is necessarily true or false for each
pure set, then evidently the truth predicate cannot be definite, so
the use of classical logic for the truth predicate is not justified.
Tim writes:
Why can it not be expressed as a non-mathematical general principle?
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.
Andre
On Fri, Jul 28, 2017 at 7:28 PM, tchow <tchow at alum.mit.edu> wrote:
> Andre writes:
>>
>> The crux of the argument in my FOM posting is that if
>> classical reasoning is valid for this truth predicate also, then it
>> appears to have all the features of a mathematical predicate.
>
>
> Appearances can be deceiving. In this case, we have a mathematical
> proof that the truth predicate is not a mathematical predicate. What
> stronger reason could we have for rejecting the hypothesis that this
> truth predicate---"appearances" notwithstanding---than a mathematical
> proof?
>
>> I agree that mathematical realism does not require that there be a
>> uniform sense in which a sentence may be true or false. Truth may be
>> treated as a trivial operator, so that the truth of each sentence is
>> taken to be equivalent to that sentence. In this variant of
>> mathematical realism, the validity of classical reasoning for the
>> mathematical universe cannot be expressed as a general principle,
>> mathematically or otherwise.
>
>
> Why can it not be expressed as a non-mathematical general principle?
>
> Tim
>
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