Godel's incompleteness theorem

Which is a mathematical model and doesn’t necessarily reflect the physical universe, similar to how a perfect sphere doesn’t necessarily reflect the shape of the Earth.

Edit: upon reflection, I think maybe you and I are interpreting "The set of natural laws" in the GGP differently. I think maybe maybe you're interpreting this as a statement about how the universe actually works, instead of a statement about the set of laws enumerated by modern science (or any logic-based successor of modern science). The GGP mentioning the axioms of the laws however, makes it clear they're talking about the set of laws discovered and discoverable via the scientific method.

In other words, there are two ways to talk about the laws of nature: (1) some set of Platonic ideal laws existing outside of human experience, that actually govern the universe and (2) the set of approximations of these ideal laws that could possibly be discovered by a logic-based science. The GGP's mentioning of axioms means they're clearly talking about (2), but your statement makes much more sense if meant about (1).

Everything below is based on my original understanding of what you wrote, which I believe is a misunderstanding on my part.

I'm not sure what you're getting at with regard to Godel's incompleteness theorem not applying to modern science.

Are you arguing that Godel's incompleteness theorem doesn't apply to the mathematical logic model(s) at the heart of the modern scientific method, because a scientific model is an approximation of reality? The match between the models and reality has no bearing on the limits of the structure of the models themselves.

The original statement was about the limits of the logical structure of the models underlying modern science, having nothing to do with how well they actually fit reality.

Are you perhaps saying something along the lines that because scientific models don't precisely match reality, it's okay to introduce axioms that make them complete (able to prove all true statements in their domain) but inconsistent (and therefore able to construct proofs that false statements are true)?

I dispute that knowledge obtained through scientific enquiry is rigorously based on mathematics, and therefore I do not believe that results about mathematical provability necessarily imply anything about the limits of scientific knowledge.

Gödel's work was concerned with formally defined abstract systems, and his results demonstrate the limits of mathematical proof within such systems.

But science never proves anything. Science often uses the language of mathematics to express and to quantify ideas, but the core of science is observation, hypothesis-forming, and experimentation. Scientists apply logic to rule out theories, but it's an informal application of logic, not a formal one, because you can never precisely define a theory the way you can precisely define a mathematical object.

Science may appear to be a rigorous discipline, but it is at best diligent, not rigorous - not in the sense that a proof of a mathematical theorem is rigorous. A proven theorem must necessarily be true - that's what the proof demonstrates. Meanwhile, a scientific theory is only conditionally true, inferred from the evidence and hypotheses about the underlying mechanisms of the universe. All scientific truth is subject to revision if contradictory evidence arises.

Science is a system of best guesses based on what we have observed. Some of those guesses have proven very useful, and very reliable at predicting the future. But none of those guesses are fundamentally based on, or necessarily limited by, formal axiomatic systems of logic.

For all we know, there is a low upper bound on the complexity of the universe, and it might be completely explainable and understandable through the lens of science, without getting anywhere near the towering near-infinities of abstract mathematical thought.

Alternatively, if the universe was a formal system about which there were unprovable truths, one could simply add that unprovable truth as an axiom, to form a larger formal system, which itself would have unprovable truths, but as long as the universe was entirely contained within that larger system, you could prove every truth relevant to it, without being limited by Gödel.

If someone is seeking unknowables in the real world, something like the uncertainty principle could be a closer match.

We were out of oranges. Then Mary left an orange in the kitchen. Then Bob left an orange in the kitchen.

Q: How many oranges are there in the kitchen? A: 2.

Not so fast. Addition is a mathematical model and doesn't necessarily reflect the physical universe.

You're not actually refuting the main point based on this story. Yes, mathematics is not necessarily a reflection of the physical universe. It can be, in cases like addition, but it doesn't necessarily need to be. Which was the parent's point.

> mathematics is not necessarily a reflection of the physical universe

Nobody knows if it is or not. It could also be that the physical universe is a reflection of mathematical truth.

> It can be, in cases like addition, but it doesn't necessarily need to be. Which was the parent's point.

Again, nobody knows if it what you claim is true or not. What we do know is that all scientific models are based on math, and if there is something wrong at the foundations of math, then all science is pseudoscience. Gödel's theorems are at the foundations of math. You can't just decide that you are skeptical about that part of math being applicable to physical models but addition is fine.

The real problem with Gödel is that it says something that a lot of people don't like to be true.

You are misunderstanding my point. Mathematics is not related to the natural world at all. 1 plus 1 will equal 2 regardless of whether there is a universe to experience it. It turns out certain types of math are useful to us, it doesn't mean that all types are. For example, infinite numbers exist, but the natural world cannot have an infinite or infinitessimal amount of things, there are upper and lower limits. Does that mean infinities don't exist? Of course not.

> Mathematics is not related to the natural world at all.

This is your personal belief, unless you can justify it. I won't hold my breath, because this discussion has been raging for centuries with no end in sight.

> For example, infinite numbers exist, but the natural world cannot have an infinite or infinitessimal amount of things, there are upper and lower limits.

Nobody knows if the natural world is finite or not. Consider for example the many worlds interpretation of quantum mechanics. We just don't know enough to assert such as thing.

You allude to a physical universe which has some properties though (laws).

“Laws” are human constructs. Mental models of how we think the universe works. Models might be useful but they are not reality.

Well things happening the same way everywhere is a sort of medium with rules.

Although, some would argue the universe would cease to exist without humans to observe it (and initiate collapse :))

We don’t know if things happen the same way everywhere. We also don’t even know what “everywhere” is. However we have mental models/laws that seems to approximate what our senses tell us in certain limited contexts. That makes the models useful.

True, we don’t agree on what everywhere means. I guess I was confining myself to the observable limits of spacetime. Outside of that, who knows :)

true, but also for some math isnt a model but a discoverable part of nature. depends on who is doing the philosophical arguing there. we dont know for sure, ironically because of godel? therefore its a true part of nature?

As with many such philosophical arguments, I think it's useful to split the word into two definitions, and the philosophical argument vanishes into a semantic argument. There are two definitions for mathematics. One definition is the rules that have existed as long as time itself, and the other definition is the subset of these rules that humans have discovered.