Downvoted because it’s an argument that Collatz etc. “will never be solved”, but it proves too much, the argument applies equally well to every other conjecture and theorem in math, including the ones that have in fact already been solved / proven long ago.
I am looking for a counter example - one that doesn't go above the arithmetic level for both system and level of proof - can you name any?
I don’t understand what “go above the arithmetic level” means.
But here’s another way that I can restate my original complaint.
Collatz Conjecture:
Steve Conjecture:
Steve Conjecture is true and easy to prove, right?
But your argument that “That's why it will never be solved” applies to the Steve Conjecture just as much as it applies to the Collatz Conjecture, because your argument does not mention any specific aspects of the Collatz Conjecture that are not also true of the Steve Conjecture. You never talk about the factor of 3, you never talk about proof by induction, you never talk about anything that would distinguish Collatz Conjecture from Steve Conjecture. Therefore, your argument is invalid, because it applies equally well to things that do in fact have proofs.
This is very closely related to a discussion in my post on simulacra levels, but I think this is importantly false.
Yes, mathematics refers to nothing outside itself, but that is not to say that things within it cannot prove other things within it. It is a highly abstract field (my simulacra level 5, which is a form of Baudrillard's 4), but that is not why it hasn't been solved.
Take the question "what is 2 + 2". This is made up of pure abstraction (calling this abstraction simply "1s" is not actually correct - "1" is a quantity defined in this abstraction), and does not directly apply to reality, except through a translation layer where the symbol "2" is equated to * *. However, the answer is still 4.
Stop. Look deeper.
What is 7? Just: 1 1 1 1 1 1 1
What is 4? Just: 1 1 1 1
What's 7+4? Just: 1 1 1 1 1 1 1 1 1 1 1
This isn't an abstraction. This is the fundamental reality beneath all numbers.
Now look at the Collatz Conjecture. Really look:
Take 7: 1 1 1 1 1 1 1
If odd: Triple and add 1
→ 22: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
If even: Halve it
→ 11: 1 1 1 1 1 1 1 1 1 1 1
→ 34: 1 1 1 1...
What are we really doing? We're using 1s to make rules about how 1s should jump around, hoping the 1s will eventually land in a certain pattern of 1s.
We're trapped inside the system we're trying to understand.
Look at other unsolved problems:
- Goldbach: Can every even number of 1s be split into two prime clusters of 1s?
- Twin Primes: Are there infinite pairs of prime clusters of 1s separated by two 1s?
- Riemann: How are the prime clusters of 1s distributed?
For centuries, they resist. Why?
Because in each case, we're:
- Using collections of 1s
- To understand collections of 1s
- Through rules made of 1s
- With proofs built from 1s
"But what about higher mathematics?" you ask. "Topology? Complex analysis? Category theory?"
Look closer. Every mathematical framework, no matter how abstract, is built on the concept of 1. On counting. On the successor function. On the fundamental act of distinguishing one thing from another.
"But what about axioms?" you ask. "What if we build different foundational rules?"
Look at what you just said. To create axioms, you need to:
You can't even express the concept of "different axioms" without first having the concept of "one versus another." The very act of trying to escape through axioms requires the trap you're trying to escape from. There is no meta-level escape. The ability to count - to distinguish one thing from another - is prerequisite for all mathematical thought. Even attempting to create a system without 1s requires 1s to do it.
There is no escape. Every attempt to step outside the system of 1s must use tools built from 1s. The circularity is complete. Absolute. Inescapable.
This isn't coincidence. It's not temporary. It's a fundamental limit of self-reference.
When a system tries to fully understand itself using only its own elements, it gets trapped. Not just trapped temporarily - trapped fundamentally, inescapably, permanently.
Next time you see Collatz, don't just see the numbers. See the 1s trying to understand themselves.
That's why it will never be solved. Not because we're not smart enough. Not because we haven't found the right approach. But because the very act of proof requires the tools that are trapped in the self-reference.