Whether it will say anything very significant practically or only philosophically is a different question. Maybe it is something like the discovery of transcendentals.. finding out that most of the number line won't have a tidy algebraic closed-form isn't exactly a make-or-break deal for the program of mathematics itself, and it also doesn't matter much to people who are doing engineering
You're right!
What could darn possibly matter these days, in the whole entirety of the realm Mathematics, if it does now somehow have a measurable impact on backprop ?
> Just as bad as what, personally only being interested in something if it relates to the hot topic of the day?
On topics that don't inherently suggest any AI application, asking about AI applications from an AI-cynical perspective is even worse than asking from an AI-hype perspective. They both suck even if they're genuine, and the cynicism pushes it a little bit further.
At school they say: There are no stupid questions, only stupid answers. Consider what negative value you have brought to this conversation, without providing any insight whatsoever.
> doing so and then trying to turn it
Do you have me confused with someone else? I did not reply until after you said why you asked. I didn't misinterpret anything.
I'm not trying to do any sneaky redirection. I just think it's bad to bring up AI on non-AI topics!
> At school they say: There are no stupid questions, only stupid answers. Consider what negative value you have brought to this conversation, without providing any insight whatsoever.
You said you dislike the AI dominating the front page, right?
You're contributing to that when you ask questions like the above.
Questions are usually good but sometimes a question is so off-topic that it detracts from the conversation.
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└─┘ (if odd n then 3*n+1 else n) `div` 211100 == 111 == 11100000000, in terms of the next odd iteration
Even numbers don't really count in the process surely? All collatz does is essentially ignore those zeroes
Plugging in values from OEIS A006884, it looks like the maximum ratio between the maximum and starting values goes down until around 4255, then picks up again, gradually increasing from there. Eyeballing the growth rate, I suspect there's a counterexample to this interpretation somewhere before 10^1000. (Does anyone have an element of A006884 greater than 2358909599867980429759? That's 140 bits maximum to 71 bits starting.)
It would seem simple, but many simple iterative calculations get us to Turing machine territory regarding computability.
Some kind of structure there that Collatz probing is sketching
edit: however we could consider the weaker definition of "forever", and consider there are some outliers that go on "for a long time" per post title, probing structure with these loops and spokes. :D
I especially like how he highlights that Collatz conjecture shows that a simple dynamical system can have amazingly complex behavior; also 3n-1 variant has two known cycles - so "any proof of the Collatz conjecture must at some point use a property of the 3n+1 map that is not shared by the 3n-1 map." And this property can't be too general either - questions about FRACTRAN programs (of which Collatz conjecture is a special case) can encode the halting problem.
If you haven't seen it, FRACTRAN itself is amazing - https://www.cs.unc.edu/~stotts/COMP210-s23/madMath/Conway87.... and the paper is pure joy to read.
But I’m interested in hearing the counterarguments that Collatz likely is provable within PA and why this would be the case.
Goodstein's theorem by contrast is obviously provable in slightly stronger theories than PA, and it involves a fast-growing sequence which suggests it's out of weaker theories' reach. In fact it encodes ordinals up to eps_0 in a natural way, so its equivalence to CON(PA) is unsurprising. The Collatz conjecture is nothing like that. It's beguilingly simple by comparison.
throwaway81523•2mo ago