Does Gödel’s findings boil down to part of classical mathematics (as opposed to computation) is flawed?

Question

According to artificial intelligence researcher Joscha Bach, only classical mathematics is affected by Gödel’s incompleteness theorem however not computation where calculations are performed in a step-wise and thus time-dependent manner.

In the YouTube video “Donald Hoffman Λ Joscha Bach on Consciousness, Free Will, and Gödel“, Bach explains that classical (= “stateless”) mathematics is a time-independent way of performing all necessary operations at once, so that e.g. pi is considered a value. Whilst in computation which is only progressing in a step-by-step (state-to-state) fashion pi just becomes a function that allows one to know more and more digits of pi by further iterating through the algorithm – something like the last digit of pi can never be reached.

Now, Bach holds that unlike computation, part of mathematics is based on erroneous assumptions (like that one of infinity) which is why new axioms will have to be added to fix the issues Gödel discovered:

“so basically what Gödel and Turing have shown is that stateless mathematics doesn't work and the opposite is true there is no deeper notion of truth than proof … Gödel has not discovered that mathematics cannot reach truth but that truth is no more than the result of a sequence of steps that is compressing a statement to axioms losslessly”

'Donald Hoffman Λ Joscha Bach on Consciousness, Free Will, and Gödel [Theolocution]' between time-stamps https://youtu.be/bhSlYfVtgww?t=3986 and https://youtu.be/bhSlYfVtgww?t=4193.

Could you please let me know your thoughts about this? Do you agree or disagree, and why?

Thank you in advance.

Answer 1

"there is no deeper notion of truth than proof"

I think that idea, is a bit sad really. Just start doing philosophy. What does it mean? Proof? System? Symbols? If truth is all about proof, we should be able to begin only there. But you don't. You pick axioms, see where they get you, amend them. So from the very beginning of thinking about thinking you have to ask, long before proofs, how do you get good axioms? Is it by judging some kind of coherence in the logical structures that can be built with them?

I would say mathematical proof is all fine and great, once you are sure you have the correct abstractions. But in that gap, checking you have good abstractions, is all of science: with a different & more nuanced concept of truth. As discussed here: Are numbers, given just as mathematical objects, quantities in themselves? See also Cartwright's 'How The Laws Of Physics Lie'.

It's a shame, that Bach makes I think a very good point, that mathematics just elides notions of computational complexity. But undercuts that, with seemingly a naive understanding of the substance of Godel Incompleteness. You can't 'fix' it with some extra axiom. That is, humourously problematic.

I would look to Deutsch & Marletto's Universal Constructor Theory, to understand how counterfactuals can bridge physics and mathematics. We can think also about Feynman diagram vertexes, and the stubborn problem of quantum renormalisation. This is a good and important avenue to think about.

I see Hofstadter's deepest point as that, when you look at systems where an agent is able to represent itself in it's model, you get feedback, that we call identity, and that can be related to intentions. When such a strange-loop agent undertakes a process of situating itself in it's environment, generating a meaning-cosmology about how to act, you get a tangled hierarchy.

That is another way of saying, Godel Incompleteness is really about how 'mindliness' can emerge in the math, and generate new rules, yet not be fully constrained by them in the future, because of the capacity to reformulate formal systems and add axioms in intentional, goal-oriented ways. It's strikingly similar to me to the difference between thermodynamically closed systems, where physics is easy and logical, and open systems, like biology, where a lot more care must be taken

You might also consider Popper's point about hypothesis generation: that's also an area not suitable for being recursively ennumerated. There is no law or rule or proof-system for the next piece of science: finding it involves creativity, novel-systemising, of everything we have so far.