Hacker News Reader

Comment by krukah | Hacker News Reader

I'm always fascinated by how Gödel's incompleteness theorems, Cantor's diagonalization proof, Turing's halting problem, and Russel's paradox all seem to graze the boundaries of logic. There's something almost terrifying about how everything we know seems to "bottom out" and what we're left with is an embarrassingly small infinite set of truths to grapple with.

It really feels to me as if the distinctions between countable vs uncountable; rational vs irrational; discrete vs continuous; all represent the boundary between physics and mathematics – an idea I wish I could elaborate more precisely, but for me stands only on a shred of intuition.

I've been interested lately in Stephen Wolfram's and Scott Aaronson's writings on related ideas.

Aaronson on Gödel, Turing, and Friends: https://www.scottaaronson.com/democritus/lec3.html

Wolfram on computational irreducibility and equivalence: https://www.wolframscience.com/nks/chap-12--the-principle-of...

View on HN

In some sense, I find it reassuring that the practice of logic (and mathematics as well) is deeply rooted in the human condition.

No matter what you choose to believe about the metaphysics of math and logic (i.e. an opinion on platonism) we end up in practice with limitations about _how_ we can know mathematical and logical truths, and there I feel we end up with something warm and vibrant and human, and not at all the cold and precise thing that logic is sometimes presented as.

You may find Constructor Theory interesting. An attempt to express physical laws solely in terms of possible and impossible transformations.

“These include providing a theory of information underlying classical and quantum information; generalising the theory of computation to include all physical transformations; unifying formal statements of conservation laws with the stronger operational ones (such as the ruling-out of perpetual motion machines); expressing the principles of testability and of the computability of nature (currently deemed methodological and metaphysical respectively) as laws of physics; allowing exact statements of emergent laws (such as the second law of thermodynamics); and expressing certain apparently anthropocentric attributes such as knowledge in physical terms.”

https://arxiv.org/abs/1210.7439

> Gödel's incompleteness theorems, Cantor's diagonalization proof, Turing's halting problem

If you’re looking for a generalisation, these are all instances of Lawvere’s Fixed Point Theorem, although my grasp of category theory is nowhere near enough for me to claim to have any insight into this abstraction.

Physics only seems to be continuous when you “zoom out”.

I do agree that Godel’s incompleteness is effectively a statement about integers. As is our model of computation (lambda calculus and church-Turing thesis)

All boundaries are anthropomorphic in bias and nature. Humans excel at edge detection, all labels are artificial.

Well yeah, we’re finite beings so the things “like us” tend to be finite as well — what we study in physics.

A real number is what happens when you reach the end of an infinitely long light ray — the sum of such a journey. We don’t experience things like that.

Yet things like spinors seem to accurately model particle physics, hinting that there’s more to the story.

Reply on news.ycombinator.com