I wonder if I might elicit a comment from you about a previous OP of mine, which I believe might be closer to the point of your criticism of positivism:

'Scientific method relies on the ability to capture the measurable attributes of objects, in such a way as to be able to make quantitative predictions about them. This has been characteristic of science since Galileo, who distinguished those characteristics of bodies that can be made subject to rigourous quantification. These are designated the 'primary attributes' of objects, and distinguished, by both Galileo and Locke, from their 'secondary attributes', which are held to be 'in the mind of the observer'. They are also, and not coincidentally, the attributes which are specifically amenable to the treatment of mathematical physics, which lies under so many of the spectacular successes of science since Galileo.

This was part of the essential discovery of the 'scientific revolution': that insofar as you can represent an object mathematically, that you can use mathematical logic to predict its behaviour. The greater the amenability of an object to mathematical description, the more accurate the prediction can be: hence the high estimation of physics as the paradigm of an 'exact science'.

Bertrand Russell said that 'physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.' And within the domain of applied mathematics, the applicability of mathematical logic to all kinds of objects yields nearly all of the power of scientific method. But Russell makes a philosophically important point, that the power of mathematics in the physical world depends on a fundamental abstraction, a boiling down to its precisely-quantifiable attributes.

In other words, what can be expressed in quantitative terms can also be subordinated to mathematical analysis and, so, to logical prediction and control. It becomes computable, countable, and predictable by mathematical logic. That is of the essence of the so-called 'universal science' envisaged on the basis of Cartesian algebraic geometry.'

That is much nearer to what I think you have in your sights, rather than pure mathematics as such.

So I agree with your rejection of positivism, but not for your reasons. — Wayfarer

Empiricism (as embodied in the principle of testability) is just a temporary stopgap solution in science. What they really want, is the complete axiomatized theory of the physical universe. So, what they really want, is provability:

https://en.wikipedia.org/wiki/Theory_of_everything

A theory of everything (TOE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all aspects of the universe.[1]: 6  Finding a theory of everything is one of the major unsolved problems in physics.[2][3]

At this level, science and mathematics will be merged into one. They actually want to get rid of empiricism and testing and science as we know it today. However, in absence of the ToE, they simply cannot.

Hilbert was relentlessly preparing the ground for proving the completeness of the ToE, as soon as the ToE would finally be ready -- back then, "any time now".

In the positivist vision of the future, there would simply be no need for empirical testing of ToE-based mathematical statements about the physical universe.

Quite a few people still believe that this is attainable. If you tell them that it is not, they will just ignore it. In that sense, sending people to the moon with Apollo 11 was a fantastic gimmick. It fueled the masses with the hope that something like the ToE would arrive very soon now. Everybody would be able to go on holiday to the moon. Positivism is also an important political program aimed at boosting the credibility of the powers that be.

Here is a quote from Reddit that brings some clarity to the subject of "truth" in mathematics these days:

"Godel's completeness theorem, applied to group theory, says that any statement that's true for every group can be proved from the axioms of group theory. Similarly, there is more than one model of ZFC. The existence of various models of ZFC is analogous to the existence of different groups. Some statements are true in one model of ZFC and false in another. Such a statement is independent of ZFC." — jgill

(1) The completeness theorem is: If a sentence is entailed by a set of premises then the sentence is provable from that set of sentences. Or, equivalently, if a set of sentences is consistent then it has a model.

But 'a theory is complete' means that every sentence in the language for the theory is either provable in the theory or its negation is provable in the theory.

Now, I'm not sure, but I doubt that (first order) group theory is complete.

What does "true for every group" mean? Sentences are true or false in models. So does"true for every group" mean "true in every model of first order group theory"?

If yes, then, yes every sentence that is true in every model of group theory is provable in group theory, as follows:

If a sentence S is not provable from a consistent set of axioms G, then G plus ~S is consistent, as follows: By the completeness theorem, it is not the case that every model of G is a model of S. So there is a model of G that is also a model of ~S. So G plus ~S is consistent. Now suppose a sentence S is true in every model of group theory. But suppose it is not provable in group theory. So ~S is consistent with the axioms of group theory. So, by the completeness theorem, the axioms of group theory plus ~S has a model. But since S is true in every model of group theory, ~S is false in every model of group theory, which contradicts that there is a model of the axioms of group theory plus ~S.

(2) Yes, if ZFC has a model M then there are other models of ZFC that are not isomorphic with M. And, yes, there are sentences independent from ZFC. But I don't know what exact claim is made with "The existence of various models of ZFC is analogous to the existence of different groups." There's nothing notable about the fact that there are different groups. Since we have Lowenheim-Skolem, it's not even notable that there are non-isomorphic models of group theory.

'Not everything that counts can be counted, and not everything that can be counted counts' — Wayfarer

I haven't the foggiest what that is supposed to mean.

the true nature of the truth — Tarskian

Run that by me again, please.

Run that by me again, please. — jgill

Most mathematical truth is not explicable, even though we have its theory.

Most physical truth isn't explicable either, even if we had its perfect theory, which we don't. The most perfect theory of the universe would only explain a very small fraction of its truth. Hence, if the goal is to explain all of the facts in the universe, it is pointless to look for the perfect theory of the universe, because the goal is unattainable. There simply is no instrument conceivable that could do it.

'Not everything that counts can be counted, and not everything that can be counted counts'
— Wayfarer

I haven't the foggiest what that is supposed to mean. — TonesInDeepFreeze

Not everything that matters is calculable.

Empiricism (as embodied in the principle of testability) is just a temporary stopgap solution in science. What they really want, is the complete axiomatized theory of the physical universe. So, what they really want, is provability:

- - -

At this level, science and mathematics will be merged into one. They actually want to get rid of empiricism and testing and science as we know it today. However, in absence of the ToE, they simply cannot. — Tarskian

What if the positivist are indeed partly right, but they won't get the answer they would want to hear? Hasn't this been obvious starting from Hilbert? He got answer, but not those one's he wanted to hear.

What if this merging of science and mathematics can happen, yet not in the way mathematicians or especially positivists want it to happen? What if a lot of science and even something as distant as the social sciences is indeed mathematical, but in the part of math that is not provable or computable?

Just make this thought experiment: What if an area of study of reality is indeed mathematical, but firmly in the non-computable and non-provable, but perhaps in the "true and expressible" (as Yanofsky put it in the text that you referred in the OP)? How will this show itself?

In my view, one thing would be certain: those people studying that part of reality and it's phenomena aren't computing data or making functions or other mathematical models about reality. They will just smile if you ask if they could explain the phenomena they are investigating by forming a mathematical model of the phenomena.

Not everything that matters is calculable. — Joshs

I should have said that that I don't know what his comment to me is supposed to mean in relation to anything I've written.

Quite a few people still believe that this is attainable. — Tarskian

Quite a few straw people, I suspect.

:up:

What if the positivist are indeed partly right, but they won't get the answer they would want to hear? Hasn't this been obvious starting from Hilbert? He got answer, but not those one's he wanted to hear. — ssu

If a positivist hears an answer that he does not like, he will typically ignore it and just carry on. Hilbert may grudgingly have accepted proof but not everybody is Hilbert.

Positivism and scientism are ideologies. It is not possible to prove them wrong. You cannot win a debate from a Marxist either.

What do you think yourself then? (Or if you have already given a satisfying view, please refer on what page you did it.)

It should be totally evident to everybody that when discussing the foundations of mathematics, philosophy is unavoidable. You simply cannot "just stick to the math" and not take a philosophical stance in my view.

Hence this thread is totally fitting for a philosophy forum.

There's many things they don't teach in school when looking at what my children have to study. Usually the worst thing is when the writers of school books are too "ambitious" and want to bring in far more to the study than the necessities that ought to be understood. — ssu

I hear awful things about the teaching of math these days and teaching in general, but I have no personal experience. I did try to help a friend's 13 year old with her math homework once and couldn't make heads or tails of it.

I looked at this. Too bad that William Lawvere passed away last year. Actually, there's a more understandable paper of this for those who aren't well informed about category theory. And it's a paper of the same author mentioned in the OP, Noson S. Yanofsky, from 2003 called A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points. Yanofsky has tried to make the paper to be as easy to read as possible and admits that when abstaining from category theory, there might be something missing. However it's a very interesting paper. — ssu

Thanks so much for that reference and to Yanofky's YouTube channel. RIP William Lawvere.

I'm an antique. Truth for me is associated with proof. — jgill

You are not old as Godel's proof, which was published in 1931. Godel's results are therefore more antique than you. Perhaps you're a logicist at heart. They thought mathematical truth was derivable from logic.

https://en.wikipedia.org/wiki/Logicism

You are not old as Godel's proof — fishfry

Not quite. The mathematicians I knew BITD had little to no interest in discussing the distinctions between provability and truth. We were mostly in classical (complex) analysis. Mostly we are gone now. A few of us remain.

Not quite. The mathematicians I knew BITD had little to no interest in discussing the distinctions between provability and truth. We were mostly in classical (complex) analysis. Mostly we are gone now. A few of us remain. — jgill

That's as true today as it was back then, logic being a niche, ignored by most math departments. But in terms of antiquity, Godel's work precedes you.

logic being a niche, ignored by most math departments — fishfry

Depending upon the quality of the university to some extent. With the exception of a 12 month post-graduate program I took at the U of Chicago for the USAF, my entire education was in large state universities (4).

I checked at what Harvard has to offer and they have two undergraduate courses in mathematical logic (and probably foundations), but at my last Alma Mater there is nothing of that kind offered at any level.

Depending upon the quality of the university to some extent. With the exception of a 12 month post-graduate program I took at the U of Chicago for the USAF, my entire education was in large state universities (4).

I checked at what Harvard has to offer and they have two undergraduate courses in mathematical logic (and probably foundations), but at my last Alma Mater there is nothing of that kind offered at any level. — jgill

I don't think it's just quality. My grad school was high quality but no logic or foundations to speak of. The one set theorist when I was there didn't get tenure and left. I think logic is concentrated in a few places but not that widely. Seems that way anyway.

I think logic is concentrated in a few places but not that widely. Seems that way anyway. — fishfry

I just checked on this past week's papers in logic posted at ArXiv.org . Four are from American universities and 13 are from foreign countries. FWIW

They thought mathematical truth was derivable from logic. — fishfry

When you say "mathematical truth" do you also refer to axioms? Me and another user had a disagreement about the definition of logicism as it seems hard to source — no surprise. SEP presents both a "weak logicism" and "hard logicism":

The strong version of logicism maintains that all mathematical truths in the chosen branch(es) form a species of logical truth. The weak version of logicism, by contrast, maintains only that all the theorems do.

The article The Three Crises in Mathematics: Logicism, Intuitionism and Formalism says:

The formulation of the logicists' program now becomes: Show that all nine axioms of
ZF belong to logic. — https://www.jstor.org/stable/2689412?seq=1

While looking for links, I found something extremely funny:

The paper resolves the great debate of the 20th century between the three philosophies of mathematics-logicism, intuitionism and formalism—founded by Bertrand Russell and A. N. Whitehead, L. E. J. Brouwer and David Hilbert, respectively. The issue: which one provides firm foundations for mathematics? None of them won the debate. We make a critique of each, consolidate their contributions, rectify their weakness and add our own to resolve the debate. The resolution forms the new foundations of mathematics. Then we apply the new foundations to assess the status of Hilbert’s 23 problems most of which in foundations and find out which ones have been solved, which ones have flawed solutions that we rectify and which ones are open problems. Problem 6 of Hilbert’s problems—Can physics be axiomatized?—is answered yes in E. E. Escultura, Nonlinear Analysis, A-Series: 69(2008), which provides the solution, namely, the grand unified theory (GUT). We also point to the resolution of the 379-year-old Fermat’s conjecture (popularly known as Fermat’s last theorem) in E. E. Escultura, Exact Solutions of Fermat’s Equations (Definitive Resolution of Fermat’s Last Theorem), Nonlinear Studies, 5(2), (1998). Likewise, the proof of the 274-year-old Goldbach’s conjecture is in E. E. Escultura, The New Mathematics and Physics, Applied Mathematics and Computation, 138(1), 2003. — Edgar E. Escultura

Love me some crazy folks.

I just checked on this past week's papers in logic posted at ArXiv.org . Four are from American universities and 13 are from foreign countries. FWIW — jgill

Does that make Americans illogical? :-)

When you say "mathematical truth" do you also refer to axioms? Me and another user had a disagreement about the definition of logicism as it seems hard to source — no surprise. SEP presents both a "weak logicism" and "hard logicism":

The strong version of logicism maintains that all mathematical truths in the chosen branch(es) form a species of logical truth. The weak version of logicism, by contrast, maintains only that all the theorems do. — Lionino

I am definitely not authoritative on that. I know that Russell wanted to develop math from logic, and Gödel busted Russel's dreams. Beyond that I am totally ignorant.

The article The Three Crises in Mathematics: Logicism, Intuitionism and Formalism says:

The formulation of the logicists' program now becomes: Show that all nine axioms of
ZF belong to logic.
— https://www.jstor.org/stable/2689412?seq=1 — Lionino

I should read that. Will dispatch a clone.

I should read that. Will dispatch a clone. — fishfry

It is the same article as the reading for my Metaphysics of Mathematics thread. Tones didn't love it.

It is the same article as the reading for my Metaphysics of Mathematics thread. Tones didn't love it. — Lionino

So many articles, so little time.

Adderral:

I know that Russell wanted to develop math from logic, and Gödel busted Russel's dreams. Beyond that I am totally ignorant — fishfry

I agree. "Truth" is negotiable it seems. The word should be avoided in mathematical discussions.

Tarski's Undefinability Theorem says (Wiki):

Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation". It is possible to define a formula T r u e ( n ) whose extension is T ∗ , but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic. However, this formula would only be able to define a truth predicate for formulas in the original language L. To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on

There is nothing wrong with referring to truth in mathematics. (1) The everyday sense of 'truth' doesn't hurt even in mathematics. When we assert 'P' we assert 'P is true' or 'it is the case that P'. (2) There is a mathematical definition of 'true in a model'.

Just to be clear: Tarski did not disallow the notion of 'truth', but rather he sharpened it to 'true in a model'. The undefinability theorem doesn't vitiate the notion of truth, especially as formalized as 'true in a model'; rather the undefinability theorem is just that in certain interpreted languages there is no definition of a truth predicate.

I agree. "Truth" is negotiable it seems. The word should be avoided in mathematical discussions. — jgill

But of course you yourself know that's not true. I assume you think of your research as discovering truths about abstract mathematical structures that have some Platonic existence in the conceptual realm. You surely feel that the things you study are true. Do you not?

Tarski's Undefinability Theorem says (Wiki): — jgill

What made you quote that? Not sure of the relevance. It's another diagonal argument.

In any event, my sense is that most mathematicians are at heart Platonists. The things they study are real. The number 5 is prime, and there is no possible world in which it isn't. The number 5 is prime even when there are no intelligent minds in the universe to comprehend it. The fact that 5 is prime is True even before mathematics exists. There is indeed truth about the things mathematicians study.

Hasn't this been your experience?

Adderral: — Lionino

Uh-oh, insert Biden joke here.