It is terrible. I mentioned why earlier in this thread. — TonesInDeepFreeze

Sorry to hear that, I really enjoy his physics videos. He has one on why nobody has ever measured the speed of light that's most ... illuminating.

I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.) — Wayfarer

That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.)

Thank you. Have a G'day.

The philosophical implications of Godel's theorems are usually very overblown. — Pfhorrest

All of the various self-reference paradoxes have always seemed trivial to me, e.g. "This sentence is false." Who cares? Russell's paradox seems just the same, just dolled up in mathematical/logical language. Ditto with Godel's incompleteness theorem. Do these "paradoxes" really have a significant, real-time, practical impact on the effective use of mathematics and computer science in the real world? Or is it only guys who are too smart playing around with trivia as if it mattered?

What is mathematical truth is an open question in the philosophy of mathematics that has been much debated over the last 100 years, since Tarsky resurfaced it. This exchange illustrates the traditional axis of controversy between Platonists and formalists, realists and anti-realists. — SophistiCat

I seem to recall reading somewhere that Gödel was a mathematical Platonist. Are you suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language?

My disagreement with @Pfhorrest seemed to perhaps hinge on his use of the term "definitely true". He hasn't responded to say whether he would claim that "There's never a statement in any given language that is both true according to the rules of that language and also not provable in that language, because to be true according to the rules of a language just is to be provable in that language." (The statement he made leaving out the word "definite").

Would a formalist allow that there could be mathematical truths that cannot be proven? If so do formalists accept that Gödel has proven that there are such truths? If not should they reject his theorem altogether?

I seem to recall reading somewhere that Gödel was a mathematical Platonist — Janus

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason. — Godel and the nature of mathematical truth, Rebecca Goldstein

I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.)
— Wayfarer

That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.) — TonesInDeepFreeze

I skimmed it. I will go back and read it again. I found the book you mention, it seems eminently readable from the preview, I will add it to my list. Thank you.

All of the various self-reference paradoxes have always seemed trivial to me, e.g. "This sentence is false." Who cares? Russell's paradox seems just the same, just dolled up in mathematical/logical language. — T Clark

I suspect there's something you're not seeing here. When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. I think, in layman's terms, what is at stake is elucidating a set of mathematical and logical principles which are both consistent and complete 'all the way down', so to speak. As the Verisatum video mentions, David Hilbert had said 'we can know, we must know', referring to the 'formalist program', the aim of which was to produce such a complete and consistent set of principles. So, I think paradoxes of self-reference, and later, Godel's theorem, are seen to undermine forever this possibility. 'We don't know, we can't know'. So it has bearing on the limitations of knowledge, as far as I can discern.

I don't mean to be too intrusive but I do want to pick your brain regarding some interesting aspects of Godel's theorems but in a much broader context. — TheMadFool

From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.

You might find the answers in there.

When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. — Wayfarer

If I may interrupt for a second, Bertrand Russell did not approve of re-stating his paradox as the barber paradox:

You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning. — Russell

Oh, thanks. That is a good observation. I had the impression that it was a form of the paradox which Russell had sent to Godel.

Would you say the below is a fair description of what Gödel is saying?

Whilst the equation is true according to the rules of the math. The rules of the math cannot prove the equation true. To prove the equation true we need to look outside the rules of the math.

All of the various self-reference paradoxes — T Clark

"self-reference" used pejoratively in reference to Godel's theorem is a red herring. The self-reference is seen by looking outside the object language. The theorem can be proven in finitistic combinatorial arithmetic. The proof methods are no more suspect than those of proof in finitistic combinatorial arithmetic.

Do these "paradoxes" really have a significant, real-time, practical impact on the effective use of mathematics and computer science in the real world? — T Clark

The proof of the incompleteness theorem does not rely on paradox. Anyway, it's pretty rare for the various non-foundational branches of mathematics, especially applied mathematics to be concerned with the incompleteness theorem. But there are important mathematical questions that are elucidated by the incompleteness theorem, including "There is no general method for deciding whether or not a given Diophantine equation has a solution." That settled a question that even a student of high school algebra might wonder about. Basic mathematical curiosity alone leads to the question whether there is a mechanical procedure to determine whether any given Diophantine equation has a solution. And there are other answers in mathematics that incompleteness elucidates. And the methods and context of the incompleteness theorem led to the earliest developments in computability and recursion theory, as those even became branches of mathematics in light of the techniques and context of the incompleteness proof. And, for philosophy of mathematics, Godel's theorem is a central concern. Perhaps most saliently is that (put roughly) incompleteness settles that Hilbert's hope for axioms that would settle all mathematical questions cannot be achieved.

To be clear, Franzen is taking exception to the theorem being incorrectly co-opted in many of those context.

I thought that was the point I was trying to make in quoting from his book, but thanks for spelling it out to clear up any ambiguity.

Are you [SophistiCat] suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language? — Janus

Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.

(1) Sentences are not true in a language. They are true or false in a model for a language.

(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.

(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.

(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system.

Would a formalist allow that there could be mathematical truths that cannot be proven? — Janus

Any formal statement can be proven in some system or another. The incompleteness theorem, where it bears on truth, gives us that for a given system S of a certain kind, there are true arithmetical statements that are not provable in S. But those statements are provable in other systems. Even if a statement is arithmetically false, there are systems that prove the statement. Even if a statement if logically false, there are system that prove that statement (though, of course, those systems are inconsistent). So, if one is self-admittedly speaking only quite loosely to say "there are truths that cannot be proven" then we must regard that as standing for the more careful, "for a given system S of a certain kind, there are arithmetic truths that are nor provable in S".

I thought that was the point I was trying to make in quoting from his book, but thanks for spelling it out to clear up any ambiguity. — Wayfarer

Yes, your quote from the book is well taken. And it is clear in the context of the book, but some people might not realize that context, so I just wanted to underline it.

Would you say the below is a fair description of what Gödel is saying? Whilst the equation is true according to the rules of the math. The rules of the math cannot prove the equation true. To prove the equation true we need to look outside the rules of the math. — Pop

No.

(1) The Godel sentence is not an equation.

(2) "rules of math" is unclear.

(3) We don't look outside the "rules of math" even given a reasonable understanding of what 'the rules of math" might mean.

(4) A correct way to say it this:

For a given system S of a certain kind, there are statements that are arithmetically true but that are not provable in S.

"of a certain kind" can be rendered as "that is recursively axiomatizable, consistent, and and extension of Robinsion arithmetic"

"arithmetically true" can be rendered as "true in the standard model of the language of first order Peano arithmetic):

So the statement is not provable in the system. But when we look at the standard model, we see that the statement is true. Godel didn't himself refer to models, but it's the way we would formalize it now. And if we don't want to be so pedantic to formalize with models, we can say that we see that the statement is true by "outside the system" just looking at the way the statement was formulated and how it relates to ordinary arithmetic.

That's pretty close without splitting hairs technically, as we would split those hairs in a more formal treatment.

The thing is, though, that these kinds of ideas tend to filter through into popular culture, one way or another. There’s a profusion of ideas from current physics that have done so - Schrödinger’s cat, many worlds, the multiverse. Of course it’s true that to really understand those concepts requires, if not a degree in mathematical physics, at least some quite extensive reading and reflection.

But then, on the other hand, we have many popular intellectuals and scientists proselytising science as a world-view, telling us that science understands the world better than we do ourselves. I think that’s why Godel’s theorem has been seized on - rightly or wrongly - as a foil against the proselytising scientists - the Hawkings and De Grasse Tysons of the world who seem to claim ‘secret knowledge’ that none of us can access without years of study, but who also evince little insight into classical humanism and philosophy proper.

Anyway, I suppose to answer my own objection, that is the rationale behind books such as Franzen’s, which at least enable the educated layman to better consider the issue on its own terms.

I don't take exception to the Goldstein quote. But her book about incompleteness needs to be read critically. As I recall (though I can't cite specifics from memory) she gets too casual sometimes to the point of misstating certain key points.

The Franzen book is the one to read.

:up:

From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.

You might find the answers in there. — Wayfarer

Thank You Wayfarer

Relationship with the liar paradox [of Godel's Incompleteness Theorems]

Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence. — Wikipedia

Please carry out your own investigations into the issue if you're interested of course. Good luck.

Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result — Wikipedia

Yes, analogues.

I suspect there's something you're not seeing here. — Wayfarer

I think you're exactly right. That was the point of my post. It's not that I think they're wrong and I'm right. I just don't get it. I'm hoping someone will answer my questions - why do these seemingly trivial paradoxes and inconsistencies matter so much? Where do they meet the world?

When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. I think, in layman's terms, what is at stake is elucidating a set of mathematical and logical principles which are both consistent and complete 'all the way down', so to speak. — Wayfarer

But why, in any practical sense, does that matter? Does it make mathematics less useful or effective in any significant way? It seems Platonic. Forms existing without relation to physical reality.

Basic mathematical curiosity alone leads to the question whether there is a mechanical procedure to determine whether any given Diophantine equation has a solution. And there are other answers in mathematics that incompleteness elucidates. — TonesInDeepFreeze

Basic mathematical curiosity is a pretty good reason to study something. I don't have any problem with that.

I just want to make it clear - I don't doubt the results of these brilliant mathematicians work. I'm not like one of those relativity deniers who think that I can see something that mathematicians and scientists have worked on for centuries.

Where do they meet the world? — T Clark

See sections 4.4 and 4.5 here:

https://plato.stanford.edu/entries/goedel-incompleteness/

For philosophical concerns about mathemtics see:

https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm

and

section 6 here again:

https://plato.stanford.edu/entries/goedel-incompleteness/

See sections 4.4 and 4.5 here: — TonesInDeepFreeze

I'll spend some time with your referenced text. Then I'll come back later and demonstrate more of my mathematical ignorance.

Thanks.

But why, in any practical sense, does that matter? Does it make mathematics less useful or effective in any significant way? It seems Platonic. Forms existing without relation to physical reality. — T Clark

Isn't it all to do with the foundations of mathematics and logic? 'Foundations' suggests to me something real, a system of thought that can be anchored against a, or the, absolute. Both Frege and Russell were attempting that in different ways. There's an SEP article on Russell's paradox here. So it may not matter in the practical sense of you and I carrying on with our lives, but it is a philosophical issue of great significance.

Even as regards to 'Platonic forms' - if they're real, not in the sense of being only 'in someone's mind', then that's significant. Because it suggests that 'what is real' extends well beyond what is, well, materially existent. If forms, and numbers, are real, then they're real in a different sense to the objects of physics, no matter how subtle. That's what fascinates me about platonism.

Then I'll come back later and demonstrate more of my mathematical ignorance. — T Clark

Hey we're all in the same boat! In fact I bet my ignorance is bigger than yours!

There's an SEP article on Russell's paradox here. So it may not matter in the practical sense of you and I carrying on with our lives, but it is a philosophical issue of great significance. — Wayfarer

I resist philosophical labels, but I've come to the conclusion that I probably am a pragmatist. I have a strong resistance to philosophical issues that don't have practical consequences. As I said, let me spend some time reading.

Hey we're all in the same boat! In fact I bet my ignorance is bigger than yours! — Wayfarer

Oh, yeah! We'll see about that.

Then I'll come back later and demonstrate more of my mathematical ignorance. — T Clark

I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of the mathematics mentioned there.

Here's a more direct reference to Godel's incompleteness theorem vis-à-vis the Liar paradox.

Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G" — Wikipedia