I do believe this opinion places you very high on the 'confidence' axis, but not especially far along the 'competence' axis.
> The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.
Sorry I meant to quote this bit at the beginning of my comment. Parent comment which I was replying to talks about both.
that is disrespectful and very very very short sighted.
also, go read up (...on Gödel, Cantor, Turing, Tarski, etc...)
My training is as an applied physicist. We physicists have an interesting relationship with math. Obviously math is essential to the work that we do, but the physical world decides whether the math is right, not the other way around. Our mathematical models technically permit things like negative mass, time flowing backwards, or magnetic monopoles. But that doesn't mean tachyons, time machines, or fundamental magnetic particles exist--they don't, so far as we know. So I'm trained to actively disregard non-physical, not relevant mathematical implications. I'm sorry if this offends a pure mathematicians sensibilities, but pragmatically it is very useful.
Or take a different field: in computational semantics, a branch of formal linguistics, there are many models for inferring a formal logical statement from an example written sentence or spoken utterance, and then determining the validity (truth) of the statement. These models get caught up on stuff like "This sentence is false." What's the truth value for that sentence? If it is true then it must be false, and if it is false then it must be true. Error, validity of this statement can't be determined! But hey, it turns out that in practice this basically never happens unless the speaker is really confused, misspeaks, or deliberately evasive. Real sentences don't have this self-referential, circular logic structure because that's not how people think or communicate.
Now "this sentence is false" goes back to the greeks, IIRC, and Gödel's theorem is slightly different. Gödel's main work is in the formalization of proofs and proof systems, and I don't want to take away from that in any way. But the incompleteness theorem always seems to be explained through these sorts of self-referential examples and I have yet to ever see it reduced to a practical problem with real-world implications. Hence my question. Does Gödel's incompleteness theorem actually constrain a real world application of proof systems, where we tend to be interested in non-cyclical logical arguments?
Never mind then, I’m sorry I bothered.
That was the belief before the 1800s. But with the discovery of non-euclidean geometry, math has been divorced from the physical world. Math is simply a system of axioms and proofs. Math is purely abstract and logical. Whatever math that physicists use just simply happens to align with the physical world.
Also, the physical world doesn't confirm whether the math is "right". Math is deductive, not inductive. As long as the math is derivable from the axioms, it is right. The physical world/experiments determine whether the mathematical model aligns with the physical world. The physical world has no say in math. Not anymore.
> So I'm trained to actively disregard non-physical, not relevant mathematical implications.
In the past, when a mathematical model predicted something ( relativity to quantum physics to elements ), experiments were conducted to determine whether the models aligned with the physical world. If the mathematical models make predictions that physicists currently can't verify with experiments, should we disregard it? Do we need technology to advance to where we can conduct experiments. Do we need physics to become more abstract? Perhaps model the physical world in the virtual world. Would experiments in the virtual world be applicable to physics? Seems like physics is both at a dead-end and on the cusp of a revolution.
One of the things it inspired was Turing's work on uncomputable numbers. It turns out that computability and incompleteness are intertwined, which at least I find interesting. And without the "uncomputable numbers" malarkey, I wonder if the Turing Machine formalism would exist (answer, "probably, but maybe looking different and with another name"). And, well, the Halting Problem is essentially Gödel Incompleteness (imagine handwaving here).
As for proof systems, again, the answer is "probably". Knowing that there are true, unprovable, statements in a formalism is something that informs how you approach it, you need to put a limit on how far to go before you say "I don't know" and taht is in and of itself important.
The proof of Gödel's result uses the paradoxical statement "This statement is false", but that's being used to prove this very general result about all systems. So the hunt is then on to find "natural" statements that are True but Unprovable.
But the "unprovable" bit should more completely be stated as "unprovable in a specific axiomatic proof system". If we want to prove that statement S is "True but Unprovable" then we must actually prove that it's true. So if we've proved it's true, what does it mean to say it's unprovable? We just proved it! What's going on?
So let's take a specific example.
Peano Arithmetic (PA)[0] is an axiomatic proof system intended to capture Natural Numbers and their behaviour.
The "Goodstein Sequence" G(m) of a number m is a sequence of natural numbers ... you can find the definition here[1]. It's not hard, but it's longer than I want to reproduce here.
Goodstein's Theorem (GT) says that for every integer m greater than 0, G(m) is eventually zero.
It has been proven that GT cannot be proved in PA, but it can be proved in stronger systems, such as second-order arithmetic.
So the statement of GT is not self-referential, along the lines of "This Statement Is False" sort of thing. It's an actual statement about the behaviour of integers, so it's not a self-referential trick.
Your question now is: What's the point? How is this useful or relevant?
Much of modern (pure) mathematics is chasing things because the mathematicians find them interesting. The vast, vast majority will never, of themselves, be useful by (what I expect are) your standards.
But it was once thought that factoring integers was of no practical use, and only pursued or investigated by cranks. Imaginary Numbers were thought to be bizarre, useless, and dangerous. Non-Euclidean Geometry was thought to be utter nonsense, and held up as part of the "proof" that the fifth postulate was unnecessary and was deducible from the other four. All three of these now form critical components in modern technology.
Even more, to the average person on the street, anything to do with algebra is completely pointless.
For you, Gödel's theorem is completely pointless and useless and probably of no interest at all, but it helps us understand the limitations of formal systems. The techniques that have been developed in the time since it was proved have helped us understand more about what computer verification systems might or might not be able to accomplish.
Of itself, Gödel's theorem might not be of direct, immediate, and practical use, but the work it has inspired has tangentially been useful, and may yet be moreso.
But not for everyone. After all, some people don't care about the Mona Lisa, or Beethoven's Fifth Symphony, or Michaelangelo's David, or the fact that people have walked on the Moon, so why should people care about results in Pure Mathematics?
That's the thing about Pure Mathematics. Sometimes it ends up being useful in ways we never expected.
[0] https://en.wikipedia.org/wiki/Peano_axioms
[1] https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_...
It is fated to be proved as a trivial corollary to some more important mathematics; a corollary that no one would have bothered with if not for the historical importance.
The unsolved Twin Prime Conjecture, of roughly the same age, is expected to lead to much more interesting mathematics if it is proved.
But one of those two must be true, you just can't prove it. Of course you can add a new axiom that allows you to prove one or the other (or accept one of those statements as an axiom), but you will still have other statements that you can't prove.
https://en.wikipedia.org/wiki/Continuum_hypothesis#Independe...
Also:
https://en.wikipedia.org/wiki/Axiom_of_choice#Independence
And thanks to Wikipedia for this elaborate list:
https://en.wikipedia.org/wiki/List_of_statements_independent...
Three alternative accounts:
* Both the CH and ¬CH mathematical universes really exist, so we just have to choose which one we're more interested in at a given time. Like one might say there are the "reall numbers" and the "realle numbers", both valid and interesting constructions which humanity was just slow to recognize the distinctions between (because they were initially less relevant to our interests and our day-to-day lives).
* We are actually ultimately thinking about one or the other of them, or are in some sense in one or the other mathematical universe, but we don't know enough about our intuition about the reals to be able to specify or explain which one. (Maybe we need other properties whose obviousness or relevance humanity is not smart enough to notice?)
* Some finitist or ultrafinitist approach is actually right: the real numbers are a formalism that, while reasonably motivated by historical attempts to "complete" mathematics in various ways, doesn't correspond to anything Platonically real or to anything intellectually relevant to humanity. (In this account, there is potentially no answer to the question because the real numbers don't exist at all. Neither CH nor ¬CH refers to a mathematical reality, just to games about formalisms.)
It's also a provable statement (via excluded middle)
My understanding comes from Keith Devlin’s wonderful book “Mathematics: A new golden age”.
IMO the most weird thing is the following, intertwining consistency of ZFC with Diophantine equations[2]:
> One can write down a concrete polynomial p ∈ Z[x1, ..., x9] such that the statement "there are integers m1, ..., m9 with p(m1, ..., m9) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). [...] the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent.
[1] https://en.wikipedia.org/wiki/List_of_statements_independent...
[2] https://en.wikipedia.org/wiki/List_of_statements_independent...
But in general, Gödel applies to all formal systems that satisfy certain properties, it's just that the exact unprovable sentences will be different (since you may just add that sentence as an axiom) that's why the general example is very abstract. It shows that sufficiently rich theories are not only incomplete, but also incompletable.
Goodstein's theorem and the Paris-Harrington theorem are some examples of this for ZFC. There are several more, maybe a logician could chime in
The CH is undoubtedly meaningful, natural, and of huge interest to (a subset of) mathematicians.
There are huge branches of mathematics that don't find immediate practicality in physics. Often these end up being practical in cryptography or quantum mechanics, but sometimes they don't. But to dismiss the entire field that relates to the cardinality of real numbers as uninteresting if someone can't give you a practical application of it shows a simple disregard for other fields of study.