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>Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist why on earth would you e
by permo-w 3y ago
>Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist
why on earth would you ever think that
- zimpenfish 3y ago> naively I think that answers your question?
- permo-w 3y agoit's clear that the thought is naive, but that doesn't explain why it was come to
- deleted 3y ago[deleted]
- xanderlewis 3y agoWhy do you assume the sun will rise tomorrow?
- bryanrasmussen 3y agoI don't just assume, I have my bottom dollar riding on it!
- permo-w 3y agothis analogy illustrates the opposite of what I think you intend it to. if we could artificially iterate the sun rising infinite times, somewhere very high up, maybe after 10^10, it eventually would no longer rise
- xanderlewis 3y agoTrue, and that did occur to me as I was writing it. The point was more that any length of time greater than a few thousand years is as good as infinite for a human, so (for, you know, everyday purposes) we might as well assume that the sun will keep on rising forever.
- permo-w 3y agowell, true, but it seems strange to me to apply such everyday processes to something as comfortably infinite as maths
- xanderlewis 3y agoThen you have better intuition than some!
- deleted 3y ago[deleted]
- Buttons840 3y agoI think you're being purposely dense. You pretend that it's unusual that a human would view a pattern that has occured 10 billion times and then expect that it would continue forever. But, I'll answer anyway. It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to reveal all patterns between the numbers. Maybe we could somehow say "if a counter-example exists, it must be smaller than N". I don't know what kind of math would allow us to formalize the possible patterns and what N is, I don't think it's been discovered yet. Maybe such math doesn't exist, but certainly we haven't discovered all mathematical tools for solving such problems, so maybe it does exist.
- kevinventullo 3y agoI think what you’re describing is very roughly equivalent to Busy Beaver numbers: https://en.m.wikipedia.org/wiki/Busy_beaver https://en.m.wikipedia.org/wiki/Busy_beaver This kind of assumes that “A simple problem, involving basic arithmetic and small numbers, 1, 2, and 3” can be encoded into some bounded-state Turing machine.
- permo-w 3y agoif you have absolutely no grounding in maths or science, then sure, it would be unusual to think that way, but if you do, which you clearly did, then it should no longer be unusual whatsoever. it would be trivial to artificially construct a series that equals 1 until n=10^10 at which point it equals pi/785498. if these series do exist, then - without additional information - why would you ever think that one you're studying is not one of them? even from a metaepistemological standpoint, had you never come across the notion that, for example, Fermat's Last Theorem could not have been proven simply by showing that 10^10/infinity of the possible outcomes had been shown to follow it? >It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to reveal all patterns between the numbers does it? why?