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...
Dear LessWrong community, Before diving into this paper, try something: Think of your favourite approach for potentially proving the Collatz conjecture - whether through dynamical systems, ergodic theory, or any other sophisticated mathematical framework. Now mentally trace that approach to its foundations. Notice how every path ultimately requires you to...
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:
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.
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.
Actually, your example still goes beyond arithmetic:
In pure arithmetic we can only check specific cases: "3² ≠ 2×2²", "4² ≠ 2×3²", etc. Any examples using just counting and basic operations?
The proof requires:
We can only use counting and basic operations (+,-,×,÷) in pure arithmetic. Any examples that stay within those bounds?
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?
Both examples perfectly demonstrate my point:
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?
ALL solved conjectures have their proof or system above arithmetic level
NONE that stay in arithmetic are solved
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... (read 372 more words →)
While you're correct that arithmetic operations can be derived recursively from succession, the paper's core insight isn't about formal derivability. Rather, it suggests some mathematical behaviors are "irreducible" - they arise directly from how properties interact rather than from deeper patterns waiting to be discovered. This may explain why certain simple-looking conjectures resist proof: we're seeking deeper explanations when the behaviour itself IS the fundamental interaction.
Dear LessWrong community,
Before diving into this paper, try something: Think of your favourite approach for potentially proving the Collatz conjecture - whether through dynamical systems, ergodic theory, or any other sophisticated mathematical framework. Now mentally trace that approach to its foundations. Notice how every path ultimately requires you to reason about basic arithmetic properties - the very things you're trying to prove things about. It's like trying to use a microscope to examine the microscope's own lens.
This paper proposes that this circularity isn't a failure of our proof techniques, but rather points to something fundamental about mathematical truth itself. It suggests there's a category of mathematical behaviours that are unprovable in a... (read 1136 more words →)
The theorem prover point is interesting but misses the key distinction:
In pure arithmetic we can only:
Any examples of real conjectures proven while staying at this basic level?