charlie56

Part of an ongoing debate the questions are surely already answered somewhere, but we haven't found where, so we turn to the esteemed members of this forum in search of illumination: Thought experiment. 'Gas in a box' (there's a Wikipedia article by that title! but the math is above us...). Closed system, no heat in or out, etc., just the molecules bouncing off each other and the walls (which magically do not erode or change their elasticity, etc. etc). My friend says that if the system of molecules in the box is allowed to evolve for long enough, there is a probability approaching certainty that at some point an earlier state will be repeated, with all variables at exactly their earlier values, so that from that point the same evolution will recur; from that moment, the system will be 'locked in' and the gas molecules will cycle through an infinite number of identical evolutions. My friend argued that the system as defined must eventually repeat itself. I argued that the number of possible states is effectively infinite, so repetition is unlikely; he replied that the time available to run the experiment is also infinite, so probability of recurrence approaches certainty. I suggested the example of two rotations in an unalterable ratio of pi as an example of a simple mechanistic system that would never repeat any earlier state (i.e. given two perfect wheels turning, the ratio of their rotations being any irrational number, two points opposite each other on a line passing through the centers of the rotations will never be there again). I argued (perhaps quite ridiculously) that if a system with such a limited number of variables (P of wheel 1, P of wheel 2) could have infinite possible states, a system with a very large number of interacting variables would probably have a large number of infinitely variable interactions--I suggested something weird like a non-linear or higher-order infinity that would trump his 'long, lo-o-ong time' infinity--i.e. that there could be an infinite number of possible system states that would not occur in his infinity of duration. (my infinity is bigger than yours...It's like a line from the Bible, 'infinity of infinities' hah. Before you laugh, review Cantor, or consider the non-countability of the real numbers.) We (an editor and a substance abuse counselor) asked for help with this, and it was pointed out that there's a question about our initial setup that affects the outcome: if space, time, energy and so motion are all continuous, then the number of possible states is infinite and (so I say) the likelihood of repetition very low. But if some underlying condition is quantized (for instance, if space is not infinitely divisible but discontinuous/textured at say Planck length, or if energy is quantized...um, yes...), then there may be only a finite number of relative positions/velocities, which would suggest that repetition may be inevitable (thanks to sr. mmbr. overtone in the math forums for help on this piece!). My argument against inevitable recurrence looked to be in the poorer position at this point... Then we really started feeling ignorant, when we realized that even in our magical thought-experiment box the molecules may not bounce like billiard balls, with fully deterministic paths, but that they probably interact with each other at a quantum level to some extent, making the molecules 'bounce funny' and adding additional degrees of randomness to the system, and perhaps greatly increasing the time needed for probability of repetition to approach one. Then we realized that, although we've magically excluded cosmic rays, wandering neutrinos and other interlopers from our box, we have enclosed a portion of 'space'--and so our molecules of gas may (?) bump into virtual particles from time to time, potentially adding another randomizing factor... Interestingly, if the system has a beginning, S1, with succeeding states S2, S3, etc., there is only a very small likelihood that the point of recurrence will be S1' with the identical succeeding states S2', S3' etc. On the first pass, if and when the system did repeat (and enter the first cycle of recurrence) it would most likely be at a point in the first evolution significantly subsequent to S1, so that the 'lead-in' would be chopped off, and the recurring cycle would be shorter than the first evolution. It would also likely be much shorter than the longest possible evolution of the system (one passing through the maximum number of possible states of the system that can be done in a single loop before recurring), leaving a number of possible states never entered at all. Then we wondered if there is a difference between the maximum number of possible states, and the maximum number that can be passed through in a single cycle and still make it back to 'go'... So we're still not sure whether the system will probably repeat some earlier state (given infinite time) or whether number of possible states is finite or not, etc... We realize that our descriptions of things are likely quite clunky, and that our questions probably have a high 'duh' factor to the initiated, but we are having a lot of fun with this because physics is cool. If anyone would be so kind as to point out errors, discuss the questions about finite vs. infinite possible states, various randomizing factors, etc., OR would point us to an accessible source that will shed light on things, we will be very appreciative--downright tickled, that's what we'll be. Thanks, Chuck

Good video 00:00 'Zero is a perfectly good number' then why when we try to use it as a number does it produce so many contradictions? 2:15 Infinity is not a number, it's a concept... I would still argue that all these weird things happen 'at zero' because it's not a number, but a concept...the concept of 'no number here'....

Um, I'm not much of a mathematician, and repling to an old thread, hah, and tired, but two thoughts occur: 1) Zero was not originally a number, but a placeholder. At that point in history, it didn't make sense to divide anything by 0 because 0 was not a number, just the sign for the absence of a number--a mark that meant 'no quantity'. Nothing to divide by. You can't put emptiness into a bottle an infinity of times, or zero times, or one time. Because it occupies the center of our coordinate graphs, however, we came to think of it as like other numbers; but it if like Sambridge we want to think about it 'outside the view of math' we'll notice that it is a little paradoxical--a quantityless quantity. It's really still NOT a number, just a sign that says 'no number exists here'. So it makes sense that you can't divide anything by 0--you can only divide numbers by other numbers. 2) But, oh my, we have redefined zero as one of the real numbers, because we want to play up and down the number line, and all across the grid, and we can’t do it safely if there’s just a hole at the center of everything, now can we? I strongly suspect that this idea (0 = some number) sits in the center of our graphs like a little liar’s paradox or Gödel sentence ("'This statement cannot be proved in theory T' cannot be proved in theory T") to remind us of the incompleteness of any mathematical logic we happen to define. Perhaps because 0 is both a ‘real number’ and a quantityless quantity, any statement n/0 = x is simply a self-contradiction, like ‘this sentence is false’—if it is false, then it’s true… oh my. So, just as the liar’s paradox has no truth value—it is neither true nor false—n/0 has no numerical value, not n, not infinity. If you must have a ‘value’ for 1/0 it would have to be 0 itself: 1/0 = 0 i.e. ‘not a number’, i.e. ‘this is not math’. Lol, trying to figure out how to turn ‘this is not math’ into a Godel sentence…lol… . ‘In my bad dreams I fear the zero-point on the number line, and when I wake I am thankful that mathematics does not map reality. In our equations, how easily we handle zero, making nothingness our toy; and then I tremble anew: how much more fearsome are the real abysses we face?’ Effron T. R. Ygäll ‘The Logic of Absurdity’. PS 1/0 can't = infinity because 0 times infinity = 0 not 1 ;;; keep on ...no, please stop ...ahahaha ... it tickles ... Charlie

Probably a duh question: If we define a mathematical object made up of two identical forever-rotating circles, everything perfectly regular (circular circles, axes central, rates of rotation unvarying, etc.) with the rates of rotation set in a non-integral ratio, such as 1 to pi, that is, the circles have a ratio of rotation that is a non-terminating, non-repeating fraction, will any particular state of that object ever recur--will two points that pass each other on a line through both axes ever be in that position again? I think that, because the ratio of rotation is non-integral, no state of the system is ever going to be repeated. Of course, I am thinking in a plane--would the answer change in non-Euclidean spaces? What if the two circles are in different parts of a saddle-shaped 'plane', for instance? Are there other factors I haven't considered that could affect the outcome? I am a non-mathematician--an editor--in a debate with another non-mathematician, who fancies himself a Nietzschean (Eternal Recurrence, bla bla). He says that, given enough time, any dynamic system with a limited number of variables must either stop (run out of steam, as it were) or repeat itself. For instance, given unlimited time and an unlimited energy source, the molecules of gas in a box will eventually return to some earlier state, and then repeat the same motions again, to infinity. I've argued that, since non-terminating, non-repeating fractions don't repeat, and since several such fractions seem to be fundamental constants in the universe (pi, e, etc.) the idea of a 'self-repeating cosmos', or of a self-repeating stochastic system the size of even a small sub-universe like ours, is implausible, and that the molecules might not ever return exactly to any previous state of energy and motion. Of course, neither of us knows enough to resolve the dispute one way or the other. My gratitude to anyone who takes the time to read this, and my profound thanks to anyone who undertakes an answer to any part of it. I realize some aspects of it are physics/thermodynamics related, and will put a related query into a physics-related forum. Chuck