Sorry; you are right. I had seen and respected that you had said this originally, but forgotten it. With that back in mind, I have no quibble with your posts. (What caused me to forget this immensely sensible statement, I suppose, was your stating that “Hari Seldon is wrong… There is no way that that his pile of coconuts approaches zero as time approaches 1 hour”.)
Yes, that’s what I mean.
I never felt you were quibbling with me, and I certainly have no quibble with your posts! This is what mathematicians do when discussing an interesting problem. Now we just need to drink lots of beer, submit this to a journal, and try to figure out how to secure funding to do further research in this area.
I would say that my approach is not analogous to knowing the value of a function for 0 <= t < 1, but rather that it’s analogous to knowing the derivative of a function for 0 <= t <= 0, plus a suitable initial condition (e.g. no coconuts in either cave at t=0). Certainly, in the case of actual functions f(t), you can (often) say what the value of f(1) is in the second case, where you can’t in the first. Not the One True Model, but that gives it a big advantage.
We can, though, rule out answers that can be shown to be incorrect. Look at the pipe scenario. “Is mango 1 in the pipe” and “Does section 1 have a mango” must have the same answer. To go further, every approach I’m aware of that will give no coconuts in the ape’s cave will also give no mangoes in section 1. Which means you’re left arguing that there are no mangoes in the pipe, after putting in an infinite number of them, and removing none.
That’s absolutely correct, but the description I gave excluded that. But I described only the actions of gorilla grabs 1000 coconuts, monkey retrieves one and specifically the lowest number one in the gorilla’s cache. Obviously tossing a coconut back and forth invalidates the conclusion. But that’s not what happened. Once a coconut gets into the monkey’s cave it never leaves. And every coconut eventually does.
If you don’t see it, well you don’t and no further explanation by me is likely to help so I will refrain from further comments on this thread.
I think everyone in this thread agrees that “Out of the natural number-indexed coconuts on the island, each leaves the ape’s cave and enters the monkey’s cave in less than an hour. If coconuts remain in the monkey’s cave forever upon entering it (not just remaining there throughout “interval 1”, “interval 2”, “interval 3”, …, but furthermore, we can be assured, at all later times as well (to which we might attach transfinite interval numbers)), then the state of the world after one hour (at the start of interval omega, so to speak) is such that all the natural number-indexed coconuts are in the monkey’s cave (and, thus, none are in the ape’s cave)”.
It’s the principle “If coconuts remain in the monkey’s cave forever upon entering it (not just remaining there throughout ‘interval 1’, ‘interval 2’, ‘interval 3’, …, but furthermore, we can be assured, at all later times as well (to which we might attach transfinite interval numbers))” which people take issue with considering as implicit, because exactly analogous principles must just as well fail.
But, of course, you can always just explicitly say “Once a coconut enters the monkey’s cave, it remains there forever (not just throughout the finitely-indexed intervals, but furthermore, we can be assured, at all later times)”, and then there is zero ambiguity. The only problem was that this wasn’t actually explicitly stated originally.
I doubt the monkey’s cave is big enough to store all the coconuts he’s half-inched from the ape. Perhpas he could use the Hilbert’s Hotel, but I hear that’s full-up already.
This is a really fun thread to skim, and try to pinpoint precisely where the transition from the abstract to the absurd occurs. I think it’s somewhere around mangos.
The problem I have with this is that you’re jumping from “there are no mangoes in any particular section of pipe” to “there are no mangoes in the pipe”. But the pipe is infinitely long. It seems equally true to say, “for any particular section of pipe, there are mangoes further down the pipe”. I would not say that because any finite section of pipe is empty, that the infinite pipe is necessarily empty.
Indistinguishable, I’d say that objects remaining where they are unless interacted with is assumed to be the normal situation for a physically stated problem. Note that my approach ignores whether or not the pile is infinite, and simply shows that if you assume time 1, there will always be some time in the past at which a coconut was stolen by the monkey.
No problem; once his cave is full, he just throws them back on the pile.
Analogously, you might also say the normal situation for a physically stated problem is that a previously increasing quantity does not suddenly drop to zero specifically at time t unless some special interaction occurs specifically at time t to arrest its growth. Yet you propose the ape’s coconuts suddenly drop to zero specifically at one hour from the start, despite no interaction occurring specifically at one hour from the start and the ape’s number of coconuts steadily increasing until then. Which is fine; the norm for a physically stated problem is to assume a number of continuity principles which cannot all hold here. Something has to give, one way or another.
Personally, I think that the simplest model is one where the monkey ends up with every indexed coconut, but that the ape ends up with a countably infinite pile of coconuts none of which are validly indexed.
Let f(t) be the number of coconuts the ape has at time t. A continuous function f:[0,1] -> R is bounded, since [0,1] is compact. Since the number of coconuts the ape has grows without bound as a function of time (I think/hope we all agree with that!), we are left with the inescapable conclusion that f is not continuous on [0,1]. So we must agree that either f(1) is undefined, or f(1) is defined in some way such that f is not continuous. Which is fine, we can define f however we want–there’s no problem doing so and no one has argued otherwise. It’s just that, you know, this wasn’t specified when the question was originally posed, and it doesn’t jibe with our everyday intuition. Again there’s no problem with that.
This is what I’ve been arguing for a couple of days now, where by “none of which are validly indexed” I specifically mean “indexed in such a manner as to have asymptotic density zero in the set of natural numbers.”
What does “validly indexed” mean?
Colloquially, I would have thought the scenario specifies that they are “validly indexed” when it numbers the coconuts in batches of a thousand.
The OP quote pretty convincingly makes the case that the limit as N ->infinity is indeed 1. (Sure, it’s not continually increasing for every single N, but with the right N you can get as close as you want to 1 for that N and all greater Ns, which is I believe the usual definition of limit). That’s pretty intuitive: as numbers get bigger, they get more digits, which gives more chances for a three. Very very few (relatively) numbers with a million digits don’t have a 3 somewhere in all those digits, and even (relatively) fewer numbers with two million digits have no threes.
I think it’s less about defining ‘percentage’ in a peculiar way, and more about choosing a way to translate the vague English into a precise mathematical question (English isn’t very precise when infinity gets involved). If one chooses the limit formulation [What’s the limit as N goes to infinity of the ratio of numbers less than N that have a ‘3’ in the decimal representation to all numbers less than N?] the answer is pretty clearly “one”, whereas if one chooses the infinite set approach, the answer is “there are just as many members of the set of numbers with threes as without, but it’s meaningless to speak of the ratio of the size of the two infinite sets”.
One could argue that given a mathematically vague English question, the interpretation that gives a mathematically valid answer is a better interpretation, and therefore the ‘right’ answer to the question of threes is indeed 100%.
Not as an argument, but just as a proposal for discussion, consider this reframing:
A) There’s a signpost followed by a long white road to the east. The road’s length is infinite, but every particular point on it is a finite distance from the signpost.
A monkey and an ape start off at the signpost with paintbrushes; the monkey’s paintbrush is red and the ape’s paintbrush is blue. Each runs off away from the signpost, painting the road below them; the ape runs at 1000 MPH and the monkey runs at 1 MPH.
So every hour, the number of red miles has gone up by 1 and the number of blue miles has gone up by 999 (the gorilla paints 1000 new miles blue, while the monkey reclaims 1 blue mile as red).
After a minimal eternity, how much of the road is blue? (If you like, compress this eternity into an hour, in the usual way; the first Old Style hour takes up only 30 New Style minutes, the second Old Style hour takes up only 15 New Style minutes, and so on). Well, every piece of the road has been first hit by the gorilla and then by the monkey and then left alone, so we might say, the entire road is red and none of it is blue.
B) There’s a signpost followed by a long white road. The road’s length is infinite, but every particular point on it is a finite distance from the signpost.
A monkey and an ape start off at the signpost. The monkey stands absolutely still. The gorilla runs east down the road at 999 MPH, painting it blue. The signpost runs off to the west at 1 MPH, laying down new bits of red road.
So every hour, the number of red miles has gone up by 1 and the number of blue miles has gone up by 999.
After a minimal eternity, how much of the road is blue? Well, after the first hour, the first mile of the road was painted blue and thereafter left alone. After the second hour, the second mile of the road was painted blue and thereafter left alone. And so on. So infinitely many miles of the road are blue (everything east of the immobile monkey); at the very least, that first mile surely still is.
Of course, A) and B) are the exact same situation, just from the frame of reference of the signpost and the monkey, respectively.
No, I’m not doing that at all.
I’m observing that there is a symmetry in the pipe scenario. It can be described equally as “slide the line of mangoes into the pipe, one mango per step.” and “slide the pipe over the mangoes, one section per step.” Because of that symmetry, mango 1 is in the pipe at the 1 hour mark if and only if section 1 has a mango in it at the 1 hour mark.
A thought: This requires the primates to both move infinitely fast. So suppose, additionally, that they both have some finite speed, and that the “miles” scale in the same way as the time intervals. In this case, even though at any time the gorilla always has more “miles” of road than the monkey, the “real” length of road claimed by the monkey eventually surpasses the “real” length of road claimed by the gorilla, and all the limits work out just the way you’d expect them to.
Oh, and as for “invalid labels”, the way I prefer to think of it, let’s say that our numbers are represented by dots: A single dot means “one”, a pair of dots means “two”, and so on. In the model I prefer for the coconut problem, the monkey ends up with every coconut with a finite number of dots, and the ape ends up with a pile of coconuts each one of which has an infinite number of dots. They’re labeled, but they’re not validly labeled, in the sense that their labels don’t represent integers.
A bit odd, though, that the ape ends up with coconuts which supposedly didn’t exist at the start… I’m still happy to consider that possibility, though. 