Monday, December 19, 2005

Traversing the Infinite

Common sense would have us believe that you cannot travel an infinite distance in a finite amount of time. But do we actually have any good reasons to think that this is logically (rather than merely physically) impossible? On the contrary, the sort of exponential acceleration one imagines in accelerating Turing machines would seem to allow for the possibility of infinite traversals.

The idea is seen most simply in Zeno's paradox. Zeno taught us that any movement necessarily traverses an infinite number of spatial intervals (first half the distance, then half the remainder, then half of the new remainder, ad infinitum). But that's no problem, because the time it takes to traverse each interval is "accelerating" in the sense mentioned above - each interval will be traversed in half the time of the previous one. If we're travelling at a constant speed, and cross half the distance in one minute, then we will reach our destination after two minutes. The time it takes to traverse each of the intervals follows the pattern: {1, 1/2, 1/4, ... 1/(2^n) ...} which sums to 2. The paradox only gets off the ground if we make the false assumption that an infinite series cannot yield a finite sum. Contrary to Zeno, movement is possible after all. (What a relief!)

In the above example, we achieved the required 'acceleration' by travelling at a constant speed across decreasing distances. (Recall that each spatial interval was half the length of the one that went before.) But we could achieve the same effect by increasing our speed across constant distances. So let's take the infinite distance we want to traverse, and break it up into (infinitely many) 1 metre intervals. Suppose I am accelerating in such a way that it takes me 1 minute to cross the first metre, 1/2 to cross the 2nd, 1/4 for the third, and so on, in general taking 1/(2^n) minutes to travel the (n+1)th metre. After two minutes, I would be finished, having travelled an infinite distance in that time.

So contrary to common sense (and the old post at Mathetes that inspired this one), it seems logically possible to traverse the infinite. Any objections?


  1. I believe Zeno's Achilles argument is an argument against the infinite divisibility of space, not against motion. The paradox arises when the arguments against discrete space are introduced.

    The Achilles can be re-formulated in terms of a photon bouncing off an infinite sequence of decreasing mirrors. Then the photon moves to a convergence point in finite time, but has no definite direction, even though presumably it is still moving at constant velocity. The argument survives the infinite-series convergence counterargument.

  2. It is not only logically possible, but physically possible as well, if by a finite amount of time you mean "proper time", that is time measured according to the clock carried with the motion.
    There are trajectories allowed by special relativity (the velocity never surpasses the speed of light) which accelerate at increasing rate so that one can get infinitly far away (distance measured in the rest frame)in a finite amount of proper time. But as it takes an infinite amount of time as measured in the rest frame, perhaps you wouldn´t count it as "traveling an infinite distance in a finite amount of time". It isn´t if the time is measured by an external observer that stays at rest; it is if time is measured by the traveler.

  3. Some objections:

    1. Where are you at the two minute mark?

    This is especially problematic if you've been traveling in a circle rather than a straight line (which seems preferable for your example, because it would be better if you don't have to make the assumption that space is infinite). This example, like HV's example of a photon between two mirrors, is similar to the old paradox about flipping a light switch every 1/(2^n) minutes - is the light on or off after a minute (i.e. after infinitely many flips)?

    2. If we take HV's interpretation of Zeno, couldn't your version be thought of as a demonstration that time is not infinitely divisible? Many physicists think that this is true: time is quantized. The speed of light is attained by moving one quantum of space in one quantum of time.

    3. You seem to be assuming a Newtonian universe. To show logical possibility, is it sufficient to show that it would be possible in a universe that is very different from our own (even if our universe sometimes sort of looks like it to our eyes)? Basic concepts like "time", "space", "velocity", and "matter" may not be anything like what you're implicitly assuming them to be in your example, and in this case that seems important.

    4. For Alejandro: wouldn't the accelerating trajectories consistent with special relativity require an infinite amount of energy? If so, you're assuming infinite energy to prove infinite traversal in finite time (with measurements taking place in different reference frames). Although maybe you could just talk about photons instead of objects with mass - aren't they timeless from their own reference frame?

  4. Math geek that I am, I worked out the relevant equations for this scenario. We get x=-1.44ln(2-t)+1, v=1.44/(2-x), and a=1.44/(2-x)^2. Note that note of these really go to infinity, they just can be made arbitrarily close by making t arbitrarily close to 2. At t=2, their values are not infinite, but undefined. So I think what you've proved is not that it's possible to cross infinity, but that it's possible to cross an arbitrarily large distance by going arbitrarily fast, which isn't a surprising result.

  5. In a discussion of actual infinites, using Zeno's paradox to prove that we can traverse and actual infinite just begs the question; it's being assumed that the space being traversed is actually infinitely divided (not just infinitely divisible). Personally, I think the notion of an actual infinite residing within every finite amount of distance is self-contradictory. Let's assume, as Zeno's paradox does, that there is an infinite amount of intervals between every finite amount of distance. An interval covers a certain amount of distance. So if there is an infinite amount of intervals there must be an infinite amount of distance. This then says that there is an infinite amount of distance between every finite amount of distance, which is self-contradictory.

  6. I think you have in effect made a mapping onto the natural numbers (the numerated metre intervals) and in a sense proved that points in the acceleration space are countable.

    But if distance is represented by real numbers rather than natural numbers I don't think that you would get a convergent series.

    To extend your argument, one could posit an accerating acceleration which is also acclerating and so on up through an infinite number of derivatives. The conclusion from that would be travelling an infinite distance in a vanishingly small amount of time.

  7. Blar - There's a paper somewhere where someone responds to the lightswitch paradox by arguing that there isn't necessarily any end state at all. The sequence we're talking about is concerning the open interval up to (but not including) t=2. As such, it just doesn't say anything at all about what's happening at t=2. Maybe the world explodes, or the light winks out of existence, or something. (I would have thought that it must have had a final state before winking out, so I'll need to chase up what the response to that sort of objection is.)

    Don - "if there is an infinite amount of intervals there must be an infinite amount of distance".

    Why think that? It seems to ignore the fact that an infinite series can converge to a finite sum. I even gave an example where we had an infinite number of intervals which clearly covered a finite distance.

  8. Blar,
    A priori I don´t see any reason why the "hyper-accelerated" trajectory would be impossible because of the energy expense. The trajectory has an steadily increasing acceleration, which implies a steadily increasing force, but I don´t see why this should be impossible in principle (other than by obvious technological reasons). The "total" energy given to the motion may be infinite, but it is spread over an infinite amount of (coordinate, not proper) time, so the power is finite. The same happens in a motion with constant acceleration, which in ideal conditions could be produced by a constant electric field acting on a charge.

    [By acceleration here it is meant relativistic acceleration, which is second derivative of position with respect to _proper_ time. A constant or steadily increasing acceleration in the sense of second derivative with respect to coordinate time (time in the rest frame) is relativistically impossible.]

    In response to your last question: Photons are _not_ "timeless" in their rest frame, there is simply no frame of reference at all associated with a photon (in the sense there is one associated with any subluminal motion). I protested about this in my Physics 1 course, saying that "obviously" one could imagine a frame of reference attached to a photon. But I was wrong. (Drawing a couple of Minkowski diagrams may be helpful to see why -in the purported photon´s frame, the time axis and the space axis collapse together, so there is no coordinate system at all). You can´t talk at all of "how much time the photon takes to travel a distance x, measured in the photon´s rest frame". You can ask this question about any other reference frame, though, and the answer is always t=x/c.

  9. I think that "if there is an infinite amount of intervals there must be an infinite amount of distance" because, as I said in my first comment and as it seems to be somewhat obvious, "An interval covers a certain amount of distance." This is true by definition. I think it's fairly clear that if there's an infinite amount of intervals there must be an infinite amount of distance. To say an infinite series converges to a finite sum simply says that it approaches a finite sum. (Of course the infinite series, in total, might be said to equal a finite sum but that's just a way of saying the infinite series, in succession, approaches that finite sum. No one has ever actually added it all up.) Simply because we can play with numbers in math (which is theoretical) doesn't mean it can translate to the real world. One might as well suggest that a tree can have –1 apples. No one has ever added up an infinite amount of intervals (or if they have I want to know how long it took them). And no one has every traversed an infinite amount of intervals. (How the simple fact that the infinite amount of intervals decrease in length makes it possible for one to actually cross them is not clear to me. Smaller intervals shrinks the infinite?) If one is willing to say that an infinite amount of intervals can sum to a finite amount of distance in reality, then why isn't one also willing to say that an infinite amount of distance can sum to a finite amount of distance? The definition of interval seems to require that that be the case. Maybe the latter statement, as a sort of reductio ad absurdum, just reveals the impossibility of an actual infinite.

  10. What do you mean by an infinite "amount" of distance? If you mean an infinitely long distance, then the maths simply proves you wrong on that one. Each "interval" in the infinite series covers a certain amount of distance, sure, but they're decreasing so quickly that when you add them all up you will never get anything longer than 2, which certainly isn't infinite then!

  11. I think I understand what you are saying, Richard, but I also think that you are missing the point. I'm not saying that getting an infinite to "sum" to a finite isn't possible in math, that is, in theory. In fact, that's the very thing I'm assuming. (And I don't even think summing—literally summing—an infinite is possible in theory. When we use that sort of terminology it's just as a manner of speaking or for theoretical use. We're just saying that the sum approaches some finite number. Again, no one has ever added an infinite all up.) Rather, I'm showing, by means of a sort of reductio ad absurdum, the implications of saying that that is actually possible. Unless you're going to redefine the meaning of interval, it seems that if you are to say that in infinite amount of intervals can sum to a finite amount of distance then you'll also have to say that an infinite amount of distance can sum to a finite amount of distance. (As an aside, there seems to be nothing ambiguous about the terms "amount" and "interval" so I don't know why you put them in quotes in your last comment as if they could have alternative or unclear meanings.)

    In response to the reductio against an actual infinite (within every finite amount of distance) you say, "If you mean an infinitely long distance, then the maths simply proves you wrong on that one." My beef isn't with the maths. The reductio shows what one is forced to say if one asserts that the maths are true in reality (or true beyond being just a sort of theoretical use). So the reductio, in fact, assumes the maths are true in reality and shows what that ultimately means. Consequently, it's ineffective to appeal to the maths, as you have done, in response to the reductio. Say, for instance, that along the lines of the atheistic problem of evil argument one argues, as a reductio, that if Christianity is true then God exists and if God exists then evil ought not exist. One can't then respond to this reductio by saying, "Well if Christianity is true then evil does exist, just look at the devil. So you're wrong." That response is ineffective. One has to show why the reductio, as an argument in itself, fails rather than simply appealing to the very thing that the reductio assumes in order to be an effective reductio in the first place, which is what you have done here.

  12. I don't see how your argument is supposed to be a 'practical' rather than 'theoretical' one. You write:

    "Unless you're going to redefine the meaning of interval, it seems that if you are to say that in infinite amount of intervals can sum to a finite amount of distance then you'll also have to say that an infinite amount of distance can sum to a finite amount of distance."

    But that's clearly a theoretical argument, and you seem to be simply assuming that the maths is mistaken, and that an infinite number (one sense of "amount") of values must sum to a value of infinite magnitude (the other sense of "amount"). That's just a false assumption, as the p-series 1/(n^2) shows. If your argument worked, it would work against the mathematical theory too. But it doesn't.

  13. The mathematical theory, in this case, is simply a manner of speaking. No one has summed up an infinite series. The argument isn't against an actual infinity, per se; it's against the traversal of an actual infinity. The p-series just shows that certain series approach some limit as n approaches infinity. That's all. And, more importantly, that's the problem! Infinity can't ever be reached in succession, only approached, which implies that it can't be traversed. Honestly, since it can't be reached, it can't really even be approached. (1 million is just as infinitely distant from infinity as 1 is. Although, I guess one could say that it's 1 million steps closer, whatever that would mean.) I'm not assuming that the maths is mistaken, but I think you're taking the maths too literally. Again, no one has actually summed up an infinite series.

    You say, "If your argument worked, it would work against the mathematical theory too. But it doesn't." This seems to imply that my argument works against the actual world but not against the mathematical theory. I don't see how it could work against one and not the other but if that is the case (if it works against the actual world but not the theoretical world) then that is all that is needed.

    I see the distinction now in my usage of "amount." Thanks for clarifying that. However, I don't think that takes away from my argument here. In fact, if we disregard converging series for the moment we can see that if you evenly divide up a finite distance, say, every 3 units, then one will eventually reach the end. So to say that a finite distance is infinitely divided depends, interestingly, upon how one actually divides it. And this, I think, lends more support to the notion that finite distances are infinitely divisible, not infinitely divided.

    I apologize for the length of this comment but again I have to say that I just don't see how the fact that the intervals shrink makes it possible to cross an infinite number of them. It seems like we're saying you can cross an infinite number of intervals as long as you divide them up right. Why the manner in which you divide up the intervals has any affect on there being an infinite number of them is still a mystery to me. No matter how you divide them up there's still an infinite number of them. If you can't cross—cross!—an infinite number of intervals in one case, why you can cross them in another is still unclear to me. The manner in which the intervals are divided up has no affect on the number of them.

  14. Don Jr.: why do you think no one's ever summed up an infinite series? I summed up a whole bunch of them in my high school calculus class.

    Richard: you're not really talking about logical possibility, because distance, time, infinite, etc. are not logical concepts; I can travel an infinite distance in a finite amount of time is trivially logically possible; let "infinite" mean "fun" and let all the other terms mean what they mean in English, and it comes out true. So just what are we talking about, if not logical possibility? Something strong weaker than physical possibility, as you say. Metaphysical possibility? Conceptual impossibility? Something else?

  15. Jonathan, you've actually completed an infinite number of tasks? I thought it was rather clear that by saying no one has actually added up an infinite series I meant that no one has summed up each member of the series which would mean completing an infinite number of tasks which would be traversing an infinite, which is not actually possible.

  16. The only difference between a converging and a diverging infinite series is that one approaches a limit while the other shoots off to infinity. Why that has anything to do with completing an infinite amount of tasks is not clear.

    "This goes on forever. We'll never finish. How many more steps?"
    (. . . An infinite amount of time later . . .)
    "This goes on forever. We'll never finish. How many more steps?"
    (. . . Ad infinitum . . .)

    "I see it up ahead. We're so close to the limit. How many more steps?"
    (. . . An infinite amount of time later . . .)
    "I see it up ahead. We're so close to the limit. How many more steps?"
    (. . . Ad infinitum . . .)

    What's the difference? Neither task will ever get finished.

  17. Johnathan, you can traverse a fun distance in a not fun amount of time?

  18. If you are talking physics, you hit a wall at c. At which point you have used up all the energy in the universe, and having done so, may very well have used up all the space.

    A heck of a way to cross all the available distance though. It is going to take a heck of a lot of interesting engineering.

  19. My problem with this lies in that fact that an infinite distance is not truly a distance at all. Distance is defined as the traversal of the distance between two points. No points lie along an infinite expanse in so much as you never reach a new point, they are all identical. To argue such a point as traversing the infinite is contradictory as traversal is not something one is capable of on an infinite line. In short, all points on an infinite line are identical in that they none are closer to the "termination" point of the line on which they rest, being as by the definition of the infinite there is not point of termination. In short, you cannot travel along an infinite distance at all, even with infinite time it is impossible being as time is dependent on motion, which is impossible in the situation presented here. So time doesn't even exist in an infinite space... traversal is not possible in any sense as motion itself does not exist.

  20. "Distance is defined as the traversal of the distance between two points."

    Why think that? (It's a circular definition, for one thing.)

    "all points on an infinite line are identical in that they none are closer to the "termination" point"

    But distance from the termination point is not an appropriate criterion for identity! Obviously the points are not all identical because any two of them are finitely distant from each other!

  21. Then what is it that defines the identity of these points? If it is the points around it, what defines these, or the points used to define these? Ultimately, the true definition of distance and the identity of the points defining any traversal through a given region becomes subjective insomuch as the identities are dependent on perspective. Does this then not remove the notion of traversing an infinite distance? My point is simply this, the identities of these points which define the actual traversal through a given distance are dependent on termination points which are ultimately subjective in nature. Distance and time then are subjective in nature if one chooses to speak in terms of traversing an infinite span. Unless there are defined points of termination, one must assign subjective points of termination to give any notion of identity (and therefore distance/time) to the points that are crossed.

    I would also like to point out that an infinite line is composed of infinitely many lines, meaning that all the points making up that line are infinitessimal, meaning in turn that their relationships to eachother are themselves infinite.
    - All in subjective truth

  22. "demonstration that time is not infinitely divisible?"

    I think that is not just some scientists, it is the accepted orthodoxy.
    To put it more scientificaly - basicaly your zeno experiment or any other will fail in that it does not reflect reality. just like if you stick two rabits n a box and get left with two adults and ten babies it isn't exactly a miracle.

    There is a point at which a mathmatical description of a person running a a distance in a period time and it is an error to then suggest that he or any part of him or any meaningful description of his travel could talk about him traveling some fraction of that distance in some fraction of that time.

  23. Dum dee doo...

    Zeno's paradox rests on a mistake in summation of a series. If you add together 1 second plus 1/2 second plus 1/4 second plus 1/8 second, etc, you get only 2 seconds. So while you are only getting slightly closer for an infinity series of time intervals, that infinity series adds up to a finite amount of time.

    But you're misusing the word "logical". There's nothing logically inconsistent about teleporting from the start to the end of the infinity traversal, and just skipping all the points in the middle.

    But since you're playing games with accelerated turing machines, here's one for you:

    A Turing Machine that can solve the halting problem:

    Define a turing machine as a tape with markings on it that are blank or X, and a box with some states (call it 1 state) that reads the markings, moves, etc, yadda-yadda, insert standard parts of Turing Machine definition here.

    Now we need to define two more things: a function saying what the markings on the tape mean before the machine starts, and a function saying what the markings on the tape mean when the machine ends.

    1) Define the meaning of the markings on the tape before the machine starts as being the compound of two functions F1 and F2.

    F1: represent numbers in binary with an X at the start and end, with the head resting on the left-most X. So XX is 0, X X is also 0, X X is 1, XX X is 2, etc. Any starting state with an X to the left of the head, or with less than two Xs is invalid. Any starting state with Xs more than a finite distance apart is invalid.

    F2: Number the possible turing machine in (insert your favorite way here) and define this function as representing a turing machine if it represents that machines number.

    F2(F1(tape)) clearly gives a mapping from Xs and blanks to turing machines, for input.

    2) Define the meaning of markings on the tape when the tape ends as follows:

    F3(F2(F1(tape))) where F2 and F1 are defined as above and F3 is as follows:

    F3: This function is True of a Turing machine that halts, and False of a Turing machine that does not halt.

    This is the point at which you smile and say "that's cheating!".

    Well, it is. And? So? I've not violated any rules of logic. It's a perfectly valid turing machine. And with a head of only one state (0: always halt) it will calculate the Halting Function every time. So there you go: there's nothing illogical at all about solving the halting problem, with no infinite inputs, outputs or traversals required.

    My point (at the end of this long rant) is that to say something is "logically possible" is vacuous. Logical possibility does no work in the rhetoric of real arguments. It's logical impossibility that does all the work, and then it builds up impossibility off conflicts in premises.

  24. Following up on a point made in a comment by A Romantic Individualist, a standard conception of finite distance in one dimension is the distance between two points. So if a model of infinite distance could be arrived at where infinite distance in one dimension is the distance between two points, that would be a conception of infinite distance more like finite distance, and so would be more likely to actually be the correct conception of infinite distance. Such a conception can be worked out.

    Similarly, finite (natural) numbers (when order is taken into account) have first elements with successors, last elements with predecessors, and middle elements with both. So a conception of infinite number that had these properties would be more likely to actually be the infinite (natural) numbers.

    The problem is, in the original post, before grappling with -- ‘Common sense would have us believe that you cannot travel an infinite distance in a finite amount of time’ -- we must first ask: What is an infinite distance?

    Similarly to resolve paradoxes of infinite number, it is first necessary to ask: Which objects are the infinite (natural) numbers?

    However, it may take some time before the importance of these questions is realized.

  25. Say there is a device at the end of the infinite road that you wish to walk on, that adds the same number of steps that you take. Let say you take 10 steps forward, and at the same time 10 more steps were added to the end of the road. If this is the case then your progress is zero steps toward the end of the infinite road. So, even if the number of steps that you take is infinite the steps that were added is also infinite. Progress is still zero.

    Now, only if the road is finite can you any make progress on that road, since there are no additional steps added or the limit of the steps that can be added is reached at some point.

    Therefore, it is only with the idea of a limit that something that is infinite could be traversed. However, this limitation would then convert the infinite into the finite by that limiting constraint.

    In Calculus, the only way we can divide by zero is with the limit. Where by differentiation we are only dividing by a number that is infinity close to zero and not equal to zero itself.

    Therefore, it is not logically nor mathematically possible to traverse the truly infinite. In mathematics we appear to traverse the infinite only by limiting that infinite.


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