The Self-Reference Trap in Mathematics
post by Alister Munday (alister-munday) · 2025-02-03T16:12:21.392Z · LW · GW · 21 commentsContents
The Pattern The Revelation None 21 comments
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:
- Distinguish one axiom from another (1s)
- Build logical rules connecting them (more 1s)
- Create a system of proof (even more 1s)
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.
The Pattern
When a system tries to fully understand itself using only its own elements, it gets trapped. Not just trapped temporarily - trapped fundamentally, inescapably, permanently.
The Revelation
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.
21 comments
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comment by Steven Byrnes (steve2152) · 2025-02-03T16:24:20.729Z · LW(p) · GW(p)
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.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T17:26:49.791Z · LW(p) · GW(p)
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?
Replies from: steve2152↑ comment by Steven Byrnes (steve2152) · 2025-02-03T17:48:44.401Z · LW(p) · GW(p)
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:
- If your number is odd, triple it and add one.
- If your number is even, divide by two.
- …Prove or disprove: if you start at positive integer, then you’ll eventually wind up at 1.
Steve Conjecture:
- If your number is odd, add one.
- If your number is even, divide by two.
- …Prove or disprove: if you start at a positive integer, then you’ll eventually wind up at 1.
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.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T17:54:56.851Z · LW(p) · GW(p)
The Steve Conjecture still requires universal quantification - "For ALL positive integers, this process leads to 1." That's above pure arithmetic level.
In pure arithmetic we can only verify specific cases:
"1 is odd -> 2 -> 1"
"3 is odd -> 4 -> 2 -> 1"
"5 is odd -> 6 -> 3 -> 4 -> 2 -> 1"
To prove it works for ALL numbers requires stepping above arithmetic to use induction or other higher structures.
Replies from: steve2152↑ comment by Steven Byrnes (steve2152) · 2025-02-03T18:25:00.948Z · LW(p) · GW(p)
Your post purports to conclude: “That's why [the Collatz conjecture] will never be solved”.
Do you think it would also be correct to say: “That's why [the Steve conjecture] will never be solved”?
If yes, then I think you’re using the word “solved” in an extremely strange and misleading way.
If no, then you evidently messed up, because your argument does not rely on any property of the Collatz conjecture that is not equally true of the Steve conjecture.
Replies from: transhumanist_atom_understander↑ comment by transhumanist_atom_understander · 2025-02-03T18:46:15.843Z · LW(p) · GW(p)
Yeah, just went through this whole same line of evasion [LW(p) · GW(p)]. Alright, the Collatz conjecture will never be "proved" in this restrictive sense—and neither will the Steve conjecture or the irrationality of √2—do we care? It may still be proved according to the ordinary meaning.
Replies from: steve2152↑ comment by Steven Byrnes (steve2152) · 2025-02-03T19:33:17.374Z · LW(p) · GW(p)
Yeah it’s super-misleading that the post says:
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?
I think it would be much clearer to everyone if the OP said
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?
- The claim that one odd number plus another odd number is always an even number: When we squash together two odd groups of 1s, do we get an even group of 1s?
- The claim that √2 is irrational: Can 1s be divided by 1s, and squared, to get 1+1?For centuries, they resist. Why?
I request that Alister Munday please make that change. It would save readers a lot of time and confusion … because the readers would immediately know not to waste their time reading on …
comment by Alister Munday (alister-munday) · 2025-02-03T16:30:40.550Z · LW(p) · GW(p)
ALL solved conjectures have their proof or system above arithmetic level
NONE that stay in arithmetic are solved
↑ comment by mishka · 2025-02-03T17:47:31.893Z · LW(p) · GW(p)
When a solution is formalized inside a theorem prover, it is reduced to the level of arithmetic (a theorem prover is an arithmetic-level machine).
So a theory might be a very high-brow math, but a formal derivation is still arithmetic (if one just focuses on the syntax and the formal rules, and not on the presumed semantics).
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T18:10:30.317Z · LW(p) · GW(p)
The theorem prover point is interesting but misses the key distinction:
- Yes, theorem provers operate at a mechanical/syntactic level
- But they still encode and use higher mathematical concepts
- The proofs they verify still require going above pure arithmetic
- The mechanical verification is different from staying within arithmetic concepts
In pure arithmetic we can only:
- Do basic operations (+,-,×,÷)
- Check specific cases
- Compute results
Any examples of real conjectures proven while staying at this basic level?
Replies from: mishka↑ comment by mishka · 2025-02-03T18:35:03.371Z · LW(p) · GW(p)
Yes, the technique of formal proofs, in effect, involves translation of high-level proofs into arithmetic.
So self-reference is fully present (that's why we have Gödel's results and other similar results).
What this implies, in particular, is that one can reduce a "real proof" to the arithmetic; this would be ugly, and one should not do it in one's informal mathematical practice; but your post is not talking about pragmatics, you are referencing "fundamental limit of self-reference".
And, certainly, there are some interesting fundamental limits of self-reference (that's why we have algorithmically undecidable problems and such). But this is different from issues of pragmatic math techniques.
What high-level abstraction buys us is a lot of structure and intuition. The constraints related to staying within arithmetic are pragmatic, and not fundamental (without high-level abstractions one loses some very powerful ways to structure things and to guide our intuition, and things stop being comprehensible to a human mind).
↑ comment by transhumanist_atom_understander · 2025-02-03T17:19:50.424Z · LW(p) · GW(p)
Yeah, I think this is the distinction I'm struggling with. To start with proofs in Euclid, why can I prove that the square root of two is irrational? Why can I prove there are infinite prime numbers? If I saw that they are escaping this self-reference somehow, maybe I'd get the point. And without that, I don't see that I can rule out such an escape in Collatz.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T17:25:28.720Z · LW(p) · GW(p)
Both examples perfectly demonstrate my point:
- √2 irrationality - requires real numbers (beyond arithmetic)
- Infinite primes - requires infinity (beyond arithmetic)
These seem "simple" but actually step above pure arithmetic to even state them. That's exactly the pattern - we need to go above arithmetic level to prove things. Can you find any famous proofs that stay purely within natural numbers?
Replies from: transhumanist_atom_understander↑ comment by transhumanist_atom_understander · 2025-02-03T17:34:27.396Z · LW(p) · GW(p)
Yes, the proof that there's no rational solution to x²=2. It doesn't require real numbers.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T17:38:28.318Z · LW(p) · GW(p)
The proof requires:
- The concept of rational numbers (not just natural numbers)
- Proof by contradiction (logical structure above arithmetic)
- Divisibility properties beyond basic operations
We can only use counting and basic operations (+,-,×,÷) in pure arithmetic. Any examples that stay within those bounds?
Replies from: transhumanist_atom_understander↑ comment by transhumanist_atom_understander · 2025-02-03T17:43:11.727Z · LW(p) · GW(p)
We can eliminate the concept of rational numbers by framing it as the proof that there are no integer solutions to p² = 2 q²... but... no proof by contradiction? If escape from self-reference is that easy, then surely it is possible to prove the Collatz conjecture. Someone just needs to prove that the existence of any cycle beyond the familiar one implies a contradiction.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T17:44:41.397Z · LW(p) · GW(p)
Actually, your example still goes beyond arithmetic:
- "No integer solutions" is a universal statement about ALL integers
- Proof by contradiction is still needed
- Even reframed, it requires proving properties about ALL possible p and q
In pure arithmetic we can only check specific cases: "3² ≠ 2×2²", "4² ≠ 2×3²", etc. Any examples using just counting and basic operations?
Replies from: transhumanist_atom_understander↑ comment by transhumanist_atom_understander · 2025-02-03T17:47:03.361Z · LW(p) · GW(p)
Not off the top of my head, but since a proof of Collatz does not require working under these constraints, I don't think the distinction has any important implications.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T17:53:49.139Z · LW(p) · GW(p)
Actually, Collatz DOES require working under these constraints if staying in arithmetic. The conjecture itself needs universal quantification ("for ALL numbers...") to even state it. In pure arithmetic we can only verify specific cases: "4 goes to 2", "5 goes to 16 to 8 to 4 to 2", etc.
Replies from: transhumanist_atom_understander↑ comment by transhumanist_atom_understander · 2025-02-03T18:01:13.381Z · LW(p) · GW(p)
Alright, so Collatz will be proved, and the proof will not be done by "staying in arithmetic". Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by "staying in arithmetic". It doesn't matter.
Replies from: alister-munday↑ comment by Alister Munday (alister-munday) · 2025-02-03T18:09:09.520Z · LW(p) · GW(p)
This shows why it DOES matter: Both Collatz and p² = 2q² require going above arithmetic to even state them ("for ALL numbers..."). That's the key insight - we can't even formulate these interesting conjectures within pure arithmetic, let alone prove them. In pure arithmetic we can only check specific cases:
- "4 goes to 2"
- "3² ≠ 2×2²"