Apparently I forgot to press "add" because I had a whole thing written. I will say this: the foundations of mathematics is still a living discipline. The titans of mathematical formalization built the foundation as a consistent one. But more and more, people are looking into paraconsistent logic, where the principle of explosion does not hold: A and ~A does not necessarily imply B.

There's a lot of literature on this topic. I rather like this primer: https://www.iep.utm.edu/para-log/

There are also attempts to develop real analysis, probability theory, and other existing bodies of mathematical work, in paraconsistent logics. It's one of my areas of interest, and specifically I want to reconstruct Bayesian inference in a paraconsistent probability theory: Article On the Brittleness of Bayesian Inference

Dear Sen Wong ,

Godel's incompleteness theorems demonstrate that all the time we can add a new axiom (P) in both (Arithmetic model ) and (Geometry model) where the new model stays consistent. One can deduce that our knowledge is limited and we always expect to invent and introduce new postulates to the existing model but all accumulative facts will stay true.

In this sense, incompleteness is natural and necessary to preserve development.

For example, different statements were added to replace the (parallel axiom) in Euclid's model for geometry which caused the creation of several non-Euclidean geometries as the hyperbolic, elliptic and parabolic geometries. The new geometries give new vision about our universe and inspired Einstein to discover the relativity and to show a better description of our surrounding.

Best regards

Thanks, guys. Well, my question is really pointing towards whether it makes 'sense' anymore to give the kind of foundations to mathematics as prescribed by the logicists / intuitionists / formalists / set theorists / category theorists ... and what have you? 'Sense' from the perspective of a mathematician (pure and applied), a philosopher, a logician, a linguist, etc.. What are the requirements for a foundation? or rather what kind? Anything that goes beyond consistency and completeness, expressive power, etc.? For example, if mathematics is not just a formal game, do we need some sort of ontological concern / requirement? ... The thing is, do mathematicians care at all besides the formal aspects of math? Or, why should they care?

Hi Peter, that's partly my point. People are doing alternative theories, like category theory vs set theory, but are they really doing foundations of mathematics?

Hi Daniel, take your time, if you can retrieve what you have lost.

Category theory isn't an alternative to set theory, Sen Wong. It's more an abstraction of it, just like abstract algebra is an abstraction of regular algebra. If we're talking about alternatives, then we're talking about reformations, such as paraconsistent set theory et al.

Dear Daniel, I'm not sure I understand what you mean by 'alternative'. But by "category theory vs set theory", I mean using category theory instead of set theory as a foundation for mathematics. When I use 'alternative', I mean it within the context of the foundations of mathematics. There is actually another sense of 'alternative' in this regard. If we buy the idea that categories are structured sets whereas sets are unstructured categories, then categories and sets have a kind of complementary relationship. There are indeed people out there thinking about category theory as a foundational framework. I'm not an expert in the theory, so it ends here. The question being posed here is simply that are there people still seriously researching on the so-called foundations of mathematics and for what reason if different from those of Frege, Hilbert, etc. The question was asked out of curiosity. Now in the case of category theory again, merely as an example, it would be 'educational'(?) to hear from category theorists, who actually intend category theory as a foundation (of mathematics), on how it can be done and why, not that I'm particularly interested in category theory per se.

Dear Issam, may I raise another question here. In order to preserve the development of a knowledge system, what kind of foundation would be preferable ?

One commonplace I run into is observation that the efforts to provide a foundation that all mathematics can be founded on does not seem to intrude over-much with the practice of mathematics/mathematicians. It seems not every mathematician is concerned about foundational issues (or philosophy of mathematics issues).

There is a long-going moderated discussion list, [FOM], [email protected] and list server (https://cs.nyu.edu/mailman/listinfo/fom) that has been discussing most of those concerns and with attention on reverse mathematics and prolific contributions by Harvey Friedman, among others.

Consideration of the various different approaches and which goes at the foundation level, and which flavor of that (e.g., which particular set theory) have been discussed at various times and searching the archive might be useful before broaching one of these anew.

Aside: AFAIK, no one has successfully established that Gödel is escaped by moving the pea under a different foundational approach.

Sen Wong " Did Gödel's incompleteness theorems really shatter the dream of looking for some foundational base for mathematics? Do we need any foundations, whether in the sense of the logicist, the formalist or the intuitionist (or what have you), for mathematics at all? "

Well, in the paper: T. J. Stępień, Ł. T. Stępień, "On the Consistency of the Arithmetic System", J. Math. Syst. Sci. 7, No.2, 43-55 (2017), http://www.davidpublisher.org/Public/uploads/Contribute/58d876b1d91a2.pdf ; (arXiv:1803.11072), a proof of the consistency of the well-known Arithmetic System, has been included. This proof was done within this Arithmetic System. Hence, one can prove that Gödel's second incompleteness theorem is invalid.

Besides, in order to formalize mathematics, you need functional calculus, but then you need first-order logic. One of the most explored problems in mathematical logic, has been the problem of entailment (entailment problem, relevance problem). In the paper: T. J. Stepien and L. T. Stepien, "Atomic Entailment and Atomic Inconsistency and Classical Entailment" , J. Math. Syst. Sci. 5, No.2, 60-71 (2015)), http://www.davidpublisher.com/index.php/Home/Article/index?id=235.html ; (arXiv:1603.06621), a new solution of the entailment probem has been given. In short, there has been presented the so-called atomic logic. Next, in the paper: T. J. Stepien and L. T. Stepien, "The Formalization of The Arithmetic System on The Ground of The Atomic Logic" , J. Math. Syst. Sci. 5, No.9, 364-368 (2015)), http://www.davidpublisher.org/index.php/Home/Article/index?id=18008.html ; (arXiv:1603.09334) it has been proved that one can formalize Arithmetic System on the ground of the atomic logic.

Dear Dennis, Thanks for the link. I definitely will check it out.

Dear Lukasz, Thanks for your papers. I'm afraid it will take me some time to go over them. In the meantime, I wonder if you are familiar with Lesniewski's tripartite hierarchy (the quantified sentential calculus - Protothetics, the calculus of names - Ontology, and the theory of parts and wholes - Mereology). His choice of Mereology (together with his ideosyncratic logical systems) as a possible foundation of math, the way I see it, is not merely an attempt to rule out potential pardoxes (that once plagued the Cantorian sets) but an ontological concern (which doesn't seem to bother many working mathematicians) that tries to rule out 'fictional' mathematical objects, for example, the null set. The reason why I mention Lesniewski's systems is because he proposed something which to me is a genuine concern. This kind of foundational study (next to securing consistency, rigorizing definitions, producing proofs) makes sense to me. But it seems that mathematicians in general do not care. Nowadays, physics seems to have become a playground for mathematicians. We have for example string theory, but do we have a physical theory? This is open for anyone to chip in.

I wonder if there are not two separate difficulties.

The first is identification of an agreeable foundation for mathematics, quite apart from philosophical logic issues that tend to be about acceptable semantics, not so much what is deducible given a proper logical formulation in a favored system.

With regard to Gödel's incompleteness, I don't think there is any expectation that there is a foundation that escapes that result (i.e., either inconsistent or incomplete). That's simply because the incompleteness result supports the thesis that it applies for any deductive system strong enough for number theory. That has arithmetization of the system immediately available, along with diagonalization and whatnot.

It would be an interesting result to establish that incompleteness is actually unimportant for the practice of mathematics, in that it just doesn't arise, except for the fact that there are known undecidable theorems and cardinality arguments demonstrate that there are always more than one can make go away.

For computation, of course, we seem to be seriously stuck with undecidability, since the halting problem can become a concern in practice. Now, the P =? NP question seems to be of greater practical interest for (deterministic) computation, although better heuristics keep showing up without resolving that and this does not seem to be a foundational issue either way.

This book just came to my attention. It is a collection of papers entitled "Foundations of Mathematics." https://andrescaicedo.wordpress.com/2019/02/01/foundations-of-mathematics/

Dear Dennis,

You are probably right. Indeed I'm thinking in that direction. The identification of an agreeable foundation for mathematics seems to be off the map in many if not most cases.

"It would be an interesting result to establish that incompleteness is actually unimportant for the practice of mathematics ..."

Now that is something worth mulling over.

I don't want to get into undecidability at this point for fear of distraction. But I would say this little bit: I find the total computable function assumed in the halting problem suspicious but I'm not able to pinpoint the problem although I have some vague idea.

Sen Wong I think the point of the halting demonstration is that such a total computable function cannot exist, by first presuming that it does and demonstrating the contradiction. A partial computable halting-determination function is problematic depending on how a program for it fails, presumably by not halting, leaving the problem of knowing whether it will halt eventually and what does that reveal about the program code on which it (seemingly) does not halt.

There are heuristics for checking whether a computable function is total (and hence halts) and for detecting non-terminating cases, and that may be good enough. Some computer program codes are easier to reason about, also.

There are computer languages that only allow total computations, but then there is the problem that they are evidently not Turing Complete and probably inadequately recursive. That is, they can't express a universal function and maybe not even their own checker.

Having said all of that, I agree that it is useful to look more closely at the Halting Problem. It interests me too, with regard to useful heuristics and constraints.

Dear Sen

Thank you for your reply and for these comments. I have never been engaged in Lesniewski's hierarchy, however maybe in the future I will deal with it. You have written among others, " We have for example string theory, but do we have a physical theory? " Well, string theory would be a physical theory, if one were be able to confirm it experimentally. Special Theory of Relativity, General Theory of Relativity and Quantum Mechanics, were confirmed experimentally, and they are physical theories.

Lukasz T. Stepien makes a good point about mathematically-expressed theories applied to nature (i.e., physics). That is not the subject matter of Foundations of Mathematics, of course. Mathematics does not require that theoretical entities be related to characteristics of nature in the manner of physical theories. Assumptions otherwise always bring me back to observations that Einstein had on the matter. http://orcmid.com/blog/2010/02/abstraction-einstein-on-theoryreality.asp

There is a vibrant foundations of mathematics community (FOM - https://cs.nyu.edu/mailman/listinfo/fom/) moderated by Martin Davis, formerly Courant Institute of Mathematical Sciences professor and an associated editorial board with significant pedigree. Martin is currently a visiting scholar at Berkeley.

Dear Lukasz, "string theory would be a physical theory, if one were able to confirm it experimentally". That is exactly the point. Peter Woit's 'Not Even Wrong' has a lot to say about this and it's not very favourable. In the case of STR, GTR and quantum mechanics, they have been challenged for many decades. A recent paper titled "The Ghost of Modality in Theoretical Physics" (by Akira Kanda, Renata Wong and Mihai Prunescu) came to my attention. It is to be published, possibly this summer, as a chapter in "Old Problems and New Horizons in World Physics". Also not a very favourable review. As I'm no physicist, I'll stay away from the controversy.

Sen Wong and Lukasz T. Stepien, string theory is still a physical theory. It is an extension of quantum mechanics. The issue is that there's no test that helps us differentiate its validity form that of QM. But we don't confirm theory anyway. We falsify it.