[FOM] Is mathematical realism compatible with classical reasoning?

Patrik Eklund peklund at cs.umu.se
Thu Jul 27 02:23:21 EDT 2017

Hi Andre,

I don't use the word 'defect'. Foundations are split about allowing Liar 
style constructions or not. Hilbert and Bernays were pretty soft in 
their critique in GdMII. Gödel's fame was in the making, and there was 
no opposition. Your machine view relates to "a Turing machine accepts a 
Turing machine". The Church-Post-Turing-Kleene thesis remains unsolved 
basically because they don't share a common metalanguage. It is about 
finite/infinite, but it is even more about recursion. I don't think we 
have recursion in its most appropriate form.

Russell's Paradox revealed a defect in naive set theory. The Liar 
Paradox revealed a defect in naive logic. In my view, the Incompleteness 
Theorem is a Incompleteness Paradox, but very few think so. Gödel is the 
hero. Hilbert was declared a loser. Time will change that, I believe.

Good luck with your work!


On 2017-07-26 14:46, Andre Kornell wrote:
> Patrik writes: ...and to me it is very clear that I cannot precisely
> say what a sentence in a language is by using the same language to say
> so.
> Hi Patrik,
> In a universal register machine, we have an object that codes a
> program that determines whether or not its input codes a program, and
> we have an object that codes a program that determines whether its
> input codes a program that halts. I think a universal register machine
> is an adequate model of finitist mathematics, so I interpret these
> objects to be an example of a language and its semantics being
> precisely specified within that same language. What do you see to be
> the defects of this example that are absent in the category theoretic
> approach?
> Best,
> Andre
> On Tue, Jul 25, 2017 at 10:45 PM, Patrik Eklund <peklund at cs.umu.se> 
> wrote:
>> Hi Andre,
>> I won't write many lines, but if you want to communicate, I will be 
>> happy to
>> do so.
>> My postings under various mailing lists have pointed out the need to 
>> be
>> constructive about expressions and sentences, as a basis for a more 
>> formal
>> and constructive definition of Logic, and to me it is very clear that 
>> I
>> cannot precisely say what a sentence in a language is by using the 
>> same
>> language to say so. This makes me a partner of the Liar, and I do not 
>> want
>> to be a partner of the Liar. I prefer to try out Category Theory as a
>> metamachinery based on which I can formally construct various things. 
>> Set
>> Theory is then meta to Category Theory, and Category Theory is meta to
>> Logic. This way I can not only be constructive, but also applicable in 
>> real
>> world problems.
>> There are many constructions in your note that are potentially 
>> categorical.
>> For instance, your P is used like P(P(...)...), so you basically need 
>> some
>> "flattening" argumentation for this to work properly.
>> You will find some of our views under www.glioc.com, and our 
>> publication
>> lists are found on the internet.
>> Best,
>> Patrik
>> On 2017-07-25 07:46, Andre Kornell wrote:
>>> Dear FOM,
>>> I've been considering an approach to foundations that generalizes
>>> finitism to infinitary settings. If we have in mind some exhaustive
>>> notion of possible procedure on mathematical objects, then I suppose
>>> we should reason entirely in terms of propositions that can be
>>> verified in principle by such procedures. I have posted my notes on
>>> this approach to the arXiv ( https://arxiv.org/pdf/1704.08155.pdf ),
>>> but I think that most FOM readers will not have time to look at them
>>> closely, so I thought I might stimulate a bit of discussion with a
>>> question.
>>> Is mathematical realism compatible with the use of classical logic in
>>> foundational metamathematics?
>>> Suppose that abstract mathematical objects exist, that mathematical
>>> sentences are objective propositions in their literal sense, that
>>> mathematical formulas are closed under the formation rules of
>>> classical first-order logic, and that classical reasoning is valid 
>>> for
>>> mathematical sentences. The meaning of mathematical sentences, or any
>>> other class of syntactic object, is given by specifying when such a
>>> sentence is true, i. e., by a truth predicate. By Tarski’s
>>> undefinability theorem, this truth predicate cannot be mathematical 
>>> in
>>> exactly the sense that we’ve been using. The usual response to this
>>> difficulty is to introduce a metalanguage in which the truth 
>>> predicate
>>> of the original structure can be expressed, and to claim that we are
>>> engaged in a kind of higher-level mathematics. The obvious critique 
>>> of
>>> this response is that in a foundational context it is appropriate to
>>> work with an inclusive notion of mathematical predicate. If classical
>>> logic is valid for the class of first-order sentences generated by 
>>> the
>>> truth predicate together with the mathematical predicates, then what
>>> feature of the truth predicate precludes it from being itself a
>>> mathematical predicate? If classical logic is not valid for the class
>>> of sentences obtained by adding the truth predicate, then what
>>> justifies the use of classical logic in foundational metamathematics?
>>> We respond to Tarski’s theorem by working with a hierarchy of
>>> languages, and we respond to Russell’s paradox by working with a
>>> hierarchy of sets, and in both cases I think we are arresting the
>>> hierarchy prematurely. If we can include a next level of predicates 
>>> or
>>> a next rank of sets, then the foundational imperative to include all
>>> mathematical predicates and all mathematical objects requires that we
>>> do so. Since we continue the hierarchy through stages at which
>>> classical reasoning is valid, classical reasoning should be not be
>>> valid for the hierarchy as a whole, as the relevant paradox 
>>> indicates.
>>> Nevertheless, it is the the hierarchy as a whole that interests us.
>>> I would be grateful for any feedback from readers who do get a chance
>>> to look at my notes. This draft does need more work, but I worry 
>>> about
>>> spending too much time on the wrong thing. The bibliography is
>>> absolutely bare-bones, and I am hopeful that readers will suggest
>>> references, including their own work, which they feel would be
>>> appropriate.
>>> Andre Kornell
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>>> FOM at cs.nyu.edu
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