So I agree with your rejection of positivism, but not for your reasons. — Wayfarer
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]
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
'Not everything that counts can be counted, and not everything that can be counted counts' — Wayfarer
Run that by me again, please. — jgill
'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
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:
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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
Not everything that matters is calculable. — Joshs
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
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 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
I'm an antique. Truth for me is associated with proof. — jgill
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
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. — jgill
They thought mathematical truth was derivable from logic. — fishfry
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 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
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
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
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 know that Russell wanted to develop math from logic, and Gödel busted Russel's dreams. Beyond that I am totally ignorant — fishfry
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
I agree. "Truth" is negotiable it seems. The word should be avoided in mathematical discussions. — jgill
Tarski's Undefinability Theorem says (Wiki): — jgill
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