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 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.
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.
Yeah, just went through this whole same line of evasion. 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.
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 …
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:
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.
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.
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.
Yeah, just went through this whole same line of evasion. 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.
Yeah it’s super-misleading that the post says:
I think it would be much clearer to everyone if the OP said
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 …