I would say both “very large unknown positive number x”—“very large unknown positive number y” and inf—inf are undefined. However, whereas the value of 1st difference can in theory be determined by looking into what is generating x and y, the 2nd difference cannot be resolved even in principle.
inf—inf can sometimes be resolved under certain assumptions with richer representations of infinite outcomes, e.g. if both infinities are the result of infinite series over a common ordered index set (e.g. spacetime locations by distance from a specific location, moral patients with some order), you can rearrange the difference of series as a series of differences. This doesn’t always work, because the series of differences may not always have a limit at all.
inf—inf can sometimes be resolved under certain assumptions with richer representations of infinite outcomes, e.g. if both infinities are the result of infinite series over a common ordered index set
Right, but I would classify these cases as resolving “very large unknown positive number x”—“very large unknown positive number y”. It looks to me that infinite series are endless in the sense that we cannot point to where they end, but they do not contain infinity.
For example, the natural numbers 1, 2, … go on indefinetely, but any single one of them is still finite, so I would say they can be represented by 1, 2, …, N, where N is a very large unknown number. From the point of view of physics, I am pretty confident we could assume N = TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3) while explaining exactly the same evidence.
Hi Linch,
I would say both “very large unknown positive number x”—“very large unknown positive number y” and inf—inf are undefined. However, whereas the value of 1st difference can in theory be determined by looking into what is generating x and y, the 2nd difference cannot be resolved even in principle.
inf—inf can sometimes be resolved under certain assumptions with richer representations of infinite outcomes, e.g. if both infinities are the result of infinite series over a common ordered index set (e.g. spacetime locations by distance from a specific location, moral patients with some order), you can rearrange the difference of series as a series of differences. This doesn’t always work, because the series of differences may not always have a limit at all.
See:
https://forum.effectivealtruism.org/posts/N2veJcXPHby5ZwnE5/hayden-wilkinson-doing-good-in-an-infinite-chaotic-world
https://link.springer.com/article/10.1007/s11098-020-01516-w
Right, but I would classify these cases as resolving “very large unknown positive number x”—“very large unknown positive number y”. It looks to me that infinite series are endless in the sense that we cannot point to where they end, but they do not contain infinity.
For example, the natural numbers 1, 2, … go on indefinetely, but any single one of them is still finite, so I would say they can be represented by 1, 2, …, N, where N is a very large unknown number. From the point of view of physics, I am pretty confident we could assume N = TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3) while explaining exactly the same evidence.
Saved to watch later. Thanks for sharing!