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. 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.

  2. 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?

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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!

  8. 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.

  9. 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?

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

  11. 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.

  12. "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!

  13. "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.

  14. 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.

  15. 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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