Patterico's Pontifications

7/3/2016

Zeno’s Paradox Solved?

Filed under: General — Patterico @ 11:59 pm



I am bored by politics. How about some abstract philosophical/theoretical physics musing, from a complete and total amateur. If nothing else, it’s a chance for you to teach me something.

You have probably heard of Zeno’s paradoxes, one of which is that to get anywhere, you first must travel half the distance, and then half the remaining distance, and then half the remaining distance, and so on. According to this theory, you never get there.

I have seen various solutions to this, but I am not sure anyone has proposed mine. Perhaps they have, but here it is anyway:

My proposed solution is that there is a minimum distance in the universe, which cannot be subdivided into smaller distances.

Imagine a debate between Smarty Pants #1 and Smarty Pants #2. They are debating whether there is a smallest distance in the universe.

Smarty Pants #1 says there is no such thing as a smallest distance. He says: imagine the distance equal to .001 of an inch. I can add a zero and make it .0001 of an inch. If you imagine the distance equal to .00000000001 (ten zeroes) of an inch, he says, I can add a zero and make it .000000000001 (eleven zeroes) of an inch. And so on. No matter how many zeroes you write, I can add another.

Smarty Pants #2 has two responses. His responses are not designed to show there is a smallest distance, but to show that there could be one.

RESPONSE ONE: A theoretical number does not necessarily translate to the real world. For example, light travels 186,282 miles per second. It is also generally accepted that nothing can travel faster than light. There are numbers higher than 186,282, but that does not mean that any object can travel 187,000 miles per second, even though the number 187,000 exists. Similarly, the fact that you can express an incredibly small fraction of an inch numerically, in theory, does not mean that this distance exists in the real world.

RESPONSE TWO: Remember how you said that you can always add another zero to a very small decimal representing a small fraction of an inch? Fine. Go ahead and do it. Write the smallest number you can imagine, representing the smallest fraction of an inch you can express in numbers. I’ll wait right here while you do that. Are you done? Great! Turns out the smallest distance is smaller than that. What’s that you say? You can make it smaller? Fine, go ahead. Ready? Yeah, the smallest distance is still smaller than that. You say you can add zeroes all day? I’m here all day, too — and no matter what number you write, I can claim that the real smallest distance is smaller still.

See, Smarty Pants #1 thinks he wins the theoretical argument by positing that he can always add another zero. But Smarty Pants #2 trumps him by simply claiming that the smallest distance in the world could be smaller still. Any number that #1 comes up with, #2 just says the actual smallest distance is smaller.

And if that minimum distance exists? Zeno’s paradox is solved!

Some day, someone will prove this is true, in almost the same way I have described. I want the credit. And I want it now. Someone send the link to Stephen Hawking pronto.

P.S. I think all this may have something to do with the “Planck length” but I’m not sure. Let those who know more than I do instruct me.

100 Responses to “Zeno’s Paradox Solved?”

  1. If any of this makes sense to you, then I applaud you. With a slow, loud clap.

    Patterico (86c8ed)

  2. It makes sense to me and I understand the paradox. I do not think I grasp what you mean by having provided a solution though. The idea that there is a minimum distance has probably been given before. That was the “solution” that gave us the concept of atoms, they were the smallest matter that could exist until we split them.

    I’d say that the solution to the paradox is simply that the paradox creates itself by the question it asks. The idea of a journey by halves is what creates the paradox, but the idea that you can go smaller and smaller is really the question. That entirely depends on the dimension. For Width, Height, Depth, and Time it makes sense to ask that question, but I don’t think there is any “answer” to the paradox. You can always halve the time, it’s just a question of when is it meaningful?

    If you take time for an example, you can continually slow it down to smaller increments, but at some point it’ll still seem to stand still.

    Dejectedhead (65d226)

  3. Oy vey! Applying division to calculate distance traveled is like using an airhose on your car’s tires to fill your gas tank with gasoline. Hello! That’s not the way you do it! You are using the wrong method! What you need to apply is subtraction. S-u-b-t-r-a-c-t-i-o-n.

    And unlike Zeno’s time, we now have the concept not only of zero but also of negative numbers. You can have that arrow sticking halfway through the other side of your enemy and Achilles lapping the turtle three times. With subtraction.

    Come the Revolution, philosophers will be the ones spreading the “organic fertilizer” at the agricultural re-education communities. With their bare hands, because they won’t be able to figure out which end of the hoe to hold.

    nk (dbc370)

  4. Zeno’s requires a massive paradigm shift:

    “Almost been there. Almost done that.”

    navyvet (c33501)

  5. Thank goodness you stayed in law and not science. There is a limit, after all, even to legal thinking. A thought, one supposes, that Newton, Riemann and all those that have made your iPhone possible along with the modern world recognized long ago.
    The point is that if you accumulate all the legal minds in the universe ans measured them, you would still find it is less than the proposed minimum distance.

    cedarhill (a329d3)

  6. When something is obviously false–we do get there–it’s wrong, grinning cleverness of the
    promoter notwithstanding.
    So what’s the answer? Tell Zeno, “no more for you buddy, and lets have your keys.”
    Okay, “butbutbut, if we only go half way….and then half, HOW do we get there?”

    Planted axioms have wasted more cumulative human time than Hugh Hefner’s work.

    Richard Aubrey (472a6f)

  7. In undergrad and grad courses we always called physics the inexact science. Basic physics is always about measurement. Your measurement is only as exact as the tool you use. (Try measuring an atom with a yard stick.) Einstein said that if you travel far enough in the universe, you will return to the starting point. Which tool would be appropriate to prove Einstein right or wrong? (Hint: alcohol consumption helps the thought pattern, and at my age, it is hard to remember if LSD helped or not.) It was easier teaching college level physics, to keep to the practical and leave the theoretical physics for after class.

    Happy 4th of July.

    Semper Fi

    EldonH (e0559f)

  8. Just a shot,
    As one halves the distance, one is also halving the time it takes to do it.

    I always marveled about “pi”, you have a piece of string the length of the circumference, and one the length of the diameter, but the ratio of the lengths appears to be a number that cannot be specifically determined in the manner of a number with decimals.

    John Lennox, mathematician, talks about similar things in his discussions about faith and science. Science knows much less than it thinks it does when it comes to the very fundamentals of existence.

    MD in Philly (f9371b)

  9. A kettlebell will teach you all the philosophy you need.

    Steve57 (ecac13)

  10. But then, this thread would not be so interesting.

    Steve57 (ecac13)

  11. Just a shot,
    As one halves the distance, one is also halving the time it takes to do it.

    That’s pretty much exactly the resolution to the paradox.

    Zeno’s paradox has been resolved by Calculus & Real Analysis for quite a while now. I urge everyone to study these fascinating disciplines.

    CayleyGraph (353727)

  12. yes yes this is why i always ride in the front train car

    that way you overshoot where you need to be and the paradox is all like dude that’s no fair

    and i’m like deal with it pooper

    happyfeet (28a91b)

  13. The reason that Achilles can catch the tortoise is because the sequence 1, 1/2, 1/4, … converges. It has a finite sum; 2.

    If the problem was stated differently, then he might not ever catch it.
    Suppose that the tortoise’s progress was 1/2, then 1/3, then 1/4, then 1/5, etc.
    Then Achilles would never catch him.
    (because that series diverges; it has no finite sum).

    Anonymous Coward (4451ce)

  14. Ah, my college calc comes through…
    You would probably enjoy my math joke.

    A question, Cayley, if you feel qualified,
    I once heard a college math or engineering instructor say that hs AP calc was. “too simplified”, and students were better off not taking it, unless they can take “true college level” while still in hs. Does that make sense to you?

    MD in Philly (f9371b)

  15. Planck length

    In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; it is often guessed that spacetime might have a discrete or foamy structure at a Planck length scale.

    Anachronda (991b9f)

  16. There’ two problems with exceeding the light barrier one is mass and the other is time, as one approaches the mass becomes and one year pasdrd at the point one perceives one hour, commitant with that are massive gravitational forces that would crush anything inside. In star trejthey created inertial dampeners but that just won’t do.

    narciso (732bc0)

  17. Zeno was neither ignorant nor stupid. He knew his paradox was bulls**t. He used it as a test to weed out the inferior intellects who did not see it as bulls**t and wasted their time with sophomoric “solutions”.

    nk (dbc370)

  18. MD@14
    That matches my experience.
    I took AP Calc. in high school, got all As, and was woefully unprepared for college Calc. I basically started from scratch my first year of college.

    And that was forty years ago. I shudder to think of how matters now stand.

    kishnevi (b1c03d)

  19. And I’ve marked all you guys down for an arugula farm in Iowa, BTW.

    nk (dbc370)

  20. Anachronda

    not a plank owner.

    Am a shellback, though.

    Steve57 (ecac13)

  21. Capitalization is not me strong point.

    Steve57 (ecac13)

  22. John Lennox, mathematician, talks about similar things in his discussions about faith and science. Science knows much less than it thinks it does when it comes to the very fundamentals of existence

    Good scientists know that.
    But the very fact that we can describe the universe scientifically: that it has a regularity reduceable to mathematical formulation: has been seen as proof of God’s existence since medieval times (although of course they expressed the idea in terms of medieval physics).

    kishnevi (b1c03d)

  23. Steve, don’t tell me you never walked the Planck.

    kishnevi (b1c03d)

  24. And I’ve marked all you guys down for an arugula farm in Iowa, BTW.
    nk (dbc370) — 7/4/2016 @ 6:53 am

    \

    do I get to bring my longhorn steer?

    https://www.youtube.com/watch?v=73WjD1sPwwU

    Steve57 (ecac13)

  25. The Plank’s Length is whatever I cut it….

    oldirishpig (847652)

  26. he/she/it can pull a plow.

    Steve57 (ecac13)

  27. Math is hard.
    But in college, I could always figure out how much money the other guys owed me for their share of the beer keg!

    Cruz Supporter (102c9a)

  28. @3:53, got to smell the traffic cones.

    Well, who’s walk is it?

    Steve57 (ecac13)

  29. Sure, you can, Steve. We will need the steer, and others, to provide the organic fertilizer the philosophers will be spreading. However, it will be the philosophers who are undergoing reeducation who will be pulling the plow, so they can see for themselves whether it is possible to reach the end of the furrows.

    nk (dbc370)

  30. I’ll bring the herd.

    Steve57 (ecac13)

  31. If the radius of an electron is 9.11 to the negative 28th power, tha

    narciso (732bc0)

  32. Quarks. Would to be to the negative 34th power but why quibble.

    narciso (732bc0)

  33. A question, Cayley, if you feel qualified,
    I once heard a college math or engineering instructor say that hs AP calc was. “too simplified”, and students were better off not taking it, unless they can take “true college level” while still in hs. Does that make sense to you?

    My AP Calculus class seemed alright to me, although I may be nostalgically glossing over some problems.
    My guess is that he’s upset that his class didn’t define limits rigorously enough for his tastes, which can lead to confusion about the nature of differentials. I’d love it if more people knew how to deal with open neighborhoods & distance metrics, but I have to grudgingly admit that it’s useful for engineers to be able to deal with differential equations even if they don’t have a rigorous understanding of analysis.

    CayleyGraph (353727)

  34. The definitive answer is Grandma’s Christmas cookies, arrived at after many years of direct and cross examination of the driver from the back seat.

    Are we there yet?

    papertiger (c2d6da)

  35. “No we still have half way to go” incorrect.

    “Almost there.” better.

    “Here. Have a cookie.” definitive.

    papertiger (c2d6da)

  36. Whys that Interesting, the known universe is give or take, 4.6 timrsten to the negative 24th power.

    narciso (b863a7)

  37. There is a classic (i.e., my older brother told it to me 50 years ago) risque joke about an engineer, a mathematician, a beautiful woman, and Zeno’s paradox. Punchline: “But the engineer knew he could get close enough for all practical purposes.” The rest of the joke is left as an exercise for the reader.

    bfwebster (a3bc56)

  38. papertiger,
    What if Mom tells you to share the cookie with your brother and sister? Will you break it into thirds? (LOL)

    Cruz Supporter (102c9a)

  39. Quantum theory teaches us there exists a none zero probability that the driver will push on through to Aunt Lea’s house, where there is no cookie.

    papertiger (c2d6da)

  40. I refuse to accept your government imposed cookie shortage.

    papertiger (c2d6da)

  41. Grandma, being an advocate of free enterprise, recognized a niche market in need, and sent a batch of cookies ahead to Aunt Lea’s house.

    Zeno’s paradox solved!

    papertiger (c2d6da)

  42. This is the ultimate “need for closure” test. The more time you spend contemplating this paradox, the higher you score on the psychosis scale. Don’t believe me? Google “need for closure”.

    Jcurtis (65b39a)

  43. No its lesson i. The difference between artificial and real limits

    narciso (b863a7)

  44. If you deal with an effective field theory, are all dimensionless couplings inevitably comparable to one? The answer is clearly “No”. Why do we expect that it should “normally” be so? Well, it’s because we describe our ignorance by a uniform prior. With a uniform probability distribution for a certain dimensionless parameter, it is unlikely that the parameter will be extremely tiny.

    But the uniform probabilistic distribution is not a God-given law of the Cosmos. It is just a convenient trick to make balanced expectations – expectations that often turn out to be wrong anyway as soon as we figure out how the system works in more detail, as soon as we discover new reasons that make some special expectations more meaningful. Naturalness is thus not an unbreakable law of physics either. Even if you are a huge optimist, it is just a useful tool to quantify how unexpected the values of numerical parameters within a certain framework are.

    – Lubos Motl (the only physicist I have bookmarked) this is an excerpt of a page I found searching “Zemo”. http://motls.blogspot.com/2007/05/doomsday-arguments.html

    papertiger (c2d6da)

  45. But, can the “Planck Length” actually be determined since no absolute vacuum exists?

    askeptic (bfa00e)

  46. Zeno of course. Zemo is the movie character.

    papertiger (c2d6da)

  47. …and, how many steps are required to “Walk the Planck”?

    askeptic (bfa00e)

  48. I remain, as always, your imperfect servant.

    Steve57 (ecac13)

  49. You guys must have very tiny feet. I have big feet.

    Garity (e5ec9a)

  50. 42

    NJRob (a07d2e)

  51. You can change “half” the distance for each subsequent step to 99.9% the distance of each subsequent step and you ll still never reach the end. The important question is: “are you alright with that?”. If not, see if you can make yourself alright with it through self healing. “Let it go”, as they say.

    Jcurtis (65b39a)

  52. Thanks.
    Good scientists who maintain humility (is that the only kind) know that,
    others tell the public that science has disproven the existence of God, without realizing that their understanding is severely self-truncated.

    Back in my day we didn’t have AP courses, if you wanted to take calc you drove over to the local college and took what the freshman math, engineering, and science students took.
    Trying to know how to advise the youngest, still in hs.

    MD in Philly (f9371b)

  53. What’s that one about lawyers and splitting hairs…?

    bud (791f49)

  54. Actually, the solution to Zeno’s paradox is very simple: all you have to do is realize that the sum of an infinite series can be a finite number.

    That’s all there is to it.

    Steven Den Beste (99cfa1)

  55. Actually, the solution to Zeno’s paradox is very simple: all you have to do is realize that the sum of an infinite series can be a finite number.

    That’s all there is to it.

    That’s difficult to conceptualize. Is t hat the solution offered by calculus that CayleyGraph mentioned above? (I took an advanced calculus class in high school, but a) I don’t remember any of it, and b) I probably didn’t belong in it, narrowly avoiding a C in the class.)

    Patterico (86c8ed)

  56. I think this is how you conceptualize it. It’s a bonus that the explanation is courtesy of Cornell.

    DRJ (15874d)

  57. The concept is the “limit.” An infinite series analyzed as its limiting value at infinity is a fundamental idea behind the calculus. It would be fair to say that Newton and Leibnitz solved Xeno’s paradox.

    Assuming that a series converges, there is a definite (and fairly obvious) limiting value. In Xeno’s case the incremental distance is X * (1/2)^n, where X is the original distance. Whatever X is, (1/2)^n approaches zero in the limit, and can be analyzed AS zero when n = ∞.

    Kevin M (25bbee)

  58. Actually, the solution to Zeno’s paradox is very simple: all you have to do is realize that the sum of an infinite series can be a finite number.

    That’s all there is to it.

    It requires a bit more than that; specifically, it requires knowing that there’s two infinite series in the problem.

    If you move half of the remaining distance over the course of one second, then move half the remaining distance over the course of another second, then half the remaining distance in a third second… then no matter how many seconds pass, you’ll never move the complete distance, no matter how many seconds you wait.

    Of course, Zeno didn’t specify how much time each step takes. My understanding is that the paradox was supposed to describe the movement of an actual, physical person covering the distance in question. So, the time each step takes would decrease commensurate with the decrease in distance (if the person’s speed was constant, the decrease would be proportional).

    Is t hat the solution offered by calculus that CayleyGraph mentioned above? (I took an advanced calculus class in high school, but a) I don’t remember any of it, and b) I probably didn’t belong in it, narrowly avoiding a C in the class.)

    The solution I mentioned is noticing that “distance moved per step” and “time taken per step” are both infinite series that nonetheless converge. Whether or not a high school calculus class would cover it depends on how precisely they defined limits, how much they covered series, and how many examples they took from kinematics. Most high school classes do the third, but less on the former two.

    CayleyGraph (353727)

  59. Assuming that a series converges, there is a definite (and fairly obvious) limiting value. In Xeno’s case the incremental distance is X * (1/2)^n, where X is the original distance. Whatever X is, (1/2)^n approaches zero in the limit, and can be analyzed AS zero when n = ∞.

    Be careful, though, ’cause if the series had been X / n, the individual terms would still approach zero, but the sum would increase without an upper bound.

    This is probably what MD in Philly’s instructor was talking about. It’s a bit easier to see through the problem with precise language, but the imprecise language is so much easier to teach.

    CayleyGraph (353727)

  60. You’ll never get 1/100th of the way there. Or 1/1,000,000,000th. Or any fraction of the distance at all. With your way of thinking. What makes the first 1/1,000,000,000th of the distance any less subject to the divide by 2 nonsense than the last 1/1,000,000,000th or any other 1/1,000,000,000th in-between? You’re frozen solidly in place for all intents and purposes, my friends.

    nk (dbc370)

  61. The paradox is a paradox only for those who think that the concept infinity is metaphysical, rather than methodological or epistemological. If infinity is real, than the universe may be infinite in at least 2 directions: infinitely big, yet made up of stuff that is infinitely tiny (the tiny stuff of Zeno’s Paradox).

    But infinity as metaphysical or as real never made such sense. So paradox solved.

    Brian (8569a3)

  62. Regarding the idea that there is a smallest distance which cannot be further divided. Simply extend an asymptotic curve outward. As it approaches infinity it becomes zero in infinitely smaller steps. https://en.wikipedia.org/wiki/Asymptote.

    Zeno’s Paradox is a trick, not a paradox. Imagine a guillotine traversing its run. Does it stop halfway to judge its position? The trick is this idea that you must evaluate an infinite series of halfway points, when in fact there is no pause at halfway points, therefore duration of pauses at halfway points = 0! Multiply zero times an infinite number of halfway points and the result is zero. I.E. 0*∞ = 0

    Jack (ff1ca8)

  63. I.E. 0*∞ = 0?

    It is actually whatever you want it to be. It might be 3, it might be 4, it might be 5. It might be all of them at once.

    Kevin M (25bbee)

  64. I think all this may have something to do with the “Planck length”

    Tee – fuckin’ – hee

    JP (bd5dd9)

  65. I think all this may have something to do with the “Planck length”

    Or, as we call it now, the “Trump length.”

    Kevin M (25bbee)

  66. This also says there there can always be yet another missing Hillary email somewhere out on the infinitely expanding universe of Clinton lies.

    Pts (ce7fc3)

  67. There’s a left wing Barack/Hillary axiom which states that increasing the size of the federal government actually reduces the size of the national debt.

    In other words, 2 + 2 = – 6

    Hopefully, not too many conservatives will sign on to that equation.

    Cruz Supporter (102c9a)

  68. there is a minimum distance in the universe, which cannot be subdivided into smaller distances.

    That explanation would be something close to classical physics (although classical physics didn’t have that)

    A later explanation could be that notions of time and distance break down when you get to very small quantities.

    The best explanation actually is that Xeno was right.

    continual observations will prevent motion …

    — Alan Turing as quoted by A. Hodges in Alan Turing: Life and Legacy of a Great Thinker p. 54

    Sammy Finkelman (09e4a9)

  69. Xeno was right. In principle, no motion is possible.

    The only reason there appears to be motion is the uncertainty principle. At very small units of distance and time, we usually cannot know where a particle is, and it is maybe a bit here and a bit there.

    If, on the contrary, because of temperatures near absolute zero, and careful probing, we know precisely where a particle is, and do not give it a decent interval of time where we do not look, all motion, and all change stops

    This is known as the Quantum-Xeno effect and explains superconductivity, and can also stop a radioactive nucleus from decaying.

    References:

    http://www.news.cornell.edu/stories/2015/10/zeno-effect-verified-atoms-wont-move-while-you-watch

    One of the oddest predictions of quantum theory – that a system can’t change while you’re watching it – has been confirmed in an experiment by Cornell physicists.

    https://en.wikipedia.org/wiki/Quantum_Zeno_effect

    The quantum Zeno effect (also known as the Turing paradox) is a situation in which an unstable particle, if observed continuously, will never decay.[1] One can “freeze” the evolution of the system by measuring it frequently enough in its known initial state….

    Sammy Finkelman (09e4a9)

  70. 63, you’re right that it’s not a paradox. It’s not actually a trick either because you can do division of lengths on a line on a plane without really having to add worry about any element of the laws of physics. And even in your example, you don’t have to account for gravity if you don’t want to. But a different example that would leave the gravity out and amount to the same thing as this Zeno thing: “a moving object ( moving at the speed of light, or snails pace, doesn’t matter ) is losing half of its rate of speed every minute ( or hours or seconds if you prefer ), how many minutes before it comes to a complete stop?”

    Of course, it never comes to a stop.

    It’s really just a useless and uninteresting remnant of human invented math.

    jcurtis (65b39a)

  71. jcurtis (65b39a) — 7/4/2016 @ 2:14 pm

    how many minutes before it comes to a complete stop?”

    Of course, it never comes to a stop.

    Yes, it does, if you set up a system to detect how fast it is moving. Speeds below a certain limit cannot be observed. This hits before you get to any kind of minimum possible distance.

    Sammy Finkelman (09e4a9)

  72. Zeno’s paradox is like socialism/communism; it seems to make great sense on paper and in the halls of academia, but it simply doesn’t jive with reality. People and objects do in fact make it from point A to point B.

    PaddyO' (a8e631)

  73. 72, what your puny little eyeballs can “observe” is irrelevant here.

    jcurtis (35d9f0)

  74. As a Philosophy major I dealt with Mr. Zeno by simply standing up and walking across the room and then borrowed a razor from Mr. Occam.

    Anchovy (4a1728)

  75. 73. PaddyO’ (a8e631) — 7/4/2016 @ 2:50 pm

    Zeno’s paradox…simply doesn’t jive with reality. People and objects do in fact make it from point A to point B.

    We know it works in reality, but Xeno raises the question:

    How does it work in theory?

    Sammy Finkelman (09e4a9)

  76. sorry I was spitballing earlier, it’s 4.72 to the positive 24th, so the negative distance, is greater than the smallest negative distance,

    narciso (732bc0)

  77. Even though I majored in philosophy, I never did get a completely satisfying answer to this. Brian, at #62 is the closest answer to me, which is basically saying that even if we can imagine numbers that small, it doesn’t imply that they exist in the world. In general, it implies that between any two points in space, there are an infinite number of units of distance between them.

    There is another, similar, paradox by Zeno that got me interested in philosophy in the first place. I knew from its conclusion that it was false, but I didn’t know exactly why its premises were false:

    If a thing moves, then it must move either in the place where it is or in a place where it is not. But it cannot move where it is, nor can a thing move where it is not; therefore it cannot move. – Zeno

    Notice that time can also be split up this way – there are an infinite number of smaller units of time between each second, or even millisecond, etc.

    Tillman (a95660)

  78. If it can’t be measured, then there is no minimum distance.

    RokShox (8d0c14)

  79. Was anybody here rash enough to listen to the “sound” of two black holes merging? Two different ones have appeared, one for each of the gravitational sensor array discoveries. The “sound” starts very low and slow. As they spiral into each other the pitch becomes higher because these two immense masses are spinning faster and faster. So you hear a rising pitch whoooop up to the moment their event horizons merge. (I’m not sure they can accurately pinpoint that event. Nobody knows what goes on inside the event horizons. They may continue to spin but the disturbance does not escape.)

    Zeno is not unlike this merger. If you created a click with the pitch doubling each time the distance remaining to be traveled divides by two. The sound would whoop through human hearing as a low buzz to something out of human hearing range in about 10 steps. So amend the “paradox” a little. Travel a tenth the distance, then a tenth the remainder, and so on. Then you’d hear something meaningful.

    But, I digress. Note that nothing in the paradox talks about time. Once we consider time and have the traveler move at a constant velocity we can sneak up on calculus from the algebraic world. Each time the traveler cuts the distance remaining in half it takes him half the time. So you can add all these times together and figure out when he becomes “close enough” or when he jumps out of the problem, your choice. 1/2 + 1/4 + 1/8 + 1/16 ….. carried out as far as you want approaches as close as you want to 1. So the last infinitesimal step takes an infinitesimal period of time. Our real world traveler doesn’t care. He just keeps going a little longer than our normalized travel time above and jumps out of the paradox, in a teeny tiny itsy bitsy little polk… er amount of time. (I gave by bikini up when I gained weight.)

    Anyway, that’s the simple minded resolution to the paradox. And it raises another hair raising question. If space is quantized, is time also quantized? If so how might we distinguish the effects this would produce from a simulation on some cosmic sized computer?

    {^.-} Nitey nite, boys.

    JDow (babe94)

  80. unless we get around this wicket,

    http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/ltrans.html#c2

    narciso (732bc0)

  81. well. Shit

    Steve57 (ecac13)

  82. If we are dealing with a finite distance then it makes sense the solution is a minimum measurable distance that cannot be subdivided.

    But if we are dealing with infinity, then can’t it be divided infinitely?

    DRJ (15874d)

  83. Infinity is a human concept meaning, “Oog not know what over hill. Oog not go there.” A rationalization for the limits of human knowledge and the limitations of human intellect.

    But we do know what happens when Achilles chases a turtle. Turtle soup.

    nk (dbc370)

  84. To infinity and beyond!

    DRJ, you are right.
    Between any two points there is another point. Between any two numbers there is another number. Geometrically speaking the dash symbol, merely a very short segment, – , contains an infinity of points.

    Which means that given any distance, you can bisect it to produce two shorter distances. And then bisect those, and so on.

    That’s why calculus with the concept of limits was needed to solve Zeno.

    It’s only with actual measurements and measuring devices that you can talk about “shortest” distance, and mean the shortest distance human senses and technology allow us to measure.

    kishnevi (857e3e)

  85. kishnevi (857e3e) — 7/5/2016 @ 8:48 am

    Between any two points there is another point.

    Not actually true, if you understand quantum physics.

    Between any two numbers there is another number.

    We can make fractions, but in reality, we can’t divide infinitely. In reality there are only whole numbers.

    Geometrically speaking the dash symbol, merely a very short segment, – , contains an infinity of points.

    No, it doesn’t. Take any proof of that you want to make – modern physics makes nonsense of that.

    Which means that given any distance, you can bisect it to produce two shorter distances. And then bisect those, and so on.

    That was proven wrong 100 years ago.

    That’s why calculus with the concept of limits was needed to solve Zeno.

    Calculus does not reflect reality. And it doesn’t help Xeno either.

    We need to completely re-do mathematics.

    There is no such thing as the square root of two. [√2] It is not an irrational number – there is no such number.

    There is no such number as Pi [π] . There is no such thing as a circle, either, or a infinitely straight line. Masses may attempt to fall into a circle, but they can never reach that state.

    Sammy Finkelman (09e4a9)

  86. DRJ (15874d) — 7/5/2016 @ 8:32 am

    But if we are dealing with infinity, then can’t it be divided infinitely?

    The whole concept of infinity is wrong, and nonsensical. We can have an unlimited number of numbers, but to perform calculations dealing with infinity is like dividing by zero. It should be prohibited. Not by law, but by custom.

    Sammy Finkelman (09e4a9)

  87. Isn’t the correct spelling (but not the pronunciation) Xeno? How is it/was it pronounced in Greek? And is that ancient Greek, or current day Greek?

    Sammy Finkelman (09e4a9)

  88. Pat,

    Zeno’s Paradox is a mathematical concept, thus it must remain in the realm of mathematics, not physics.

    In pure math, there are things called “infinite series,” comprising an infinite number of finite values added together. Many infinite series add up to infinity itself… such as the infinite series of the counting numbers: 1+2+3+4… continuing forever. That leads to infinitity.

    HOWEVER, some infinite series actually add up to a finite number! The easiest example is this:

    1/2 + 1/4 + 1/8 + 1/16… and so forth forever. That infinite series adds up to 1. In fact, it is the mathematical representation of Zeno’s Paradox. An infinite series that adds up to a finite number is called a convergent series; an infinite series that adds up to infinity is called (you guessed it!) a divergent series.

    There are infinitely many divergent series and an infinite number of convergent series. This URL is very easy and will ease you into the concept of infinite numbers adding up to a finite sum:

    http://www.mathsisfun.com/algebra/infinite-series.html

    Simply put, Zeno’s Paradox is not a paradox at all; it’s a mathematical abstract. You can think of it thus: While you have an infinite amount of “moves” (i.e., from 1/2 to 1/4 or from 1/16 to 1/32), each move takes half the time to traverse. Since that kind of infinite series is convergent, it will take a finite time to traverse overall (one unit, whatever that unit is).

    Now if we’re talking about real physics, I’m way out of my league; my degrees are in math, not physics — physics requires familiarity with the real world, and I’m not qualified!

    But I vaguely recollect that there is a “smallest length” in the universe; that is, it’s granular. So you might think you wouldn’t actually be able to move, due to that darned Zeno.

    However, since we are in the real world now, and since we seem to be able to move, obviously applying Zenoism to a physical distance is incorrect.

    Stepping waaaaaaaaaaaay out of my field, I have the feeling that at quantum levels of the universe, things like electrons don’t actually “move” through space… there is a probability sphere (perhaps of the size of the universe itself); and all we can say is that the probability of the electron being within some particular smaller sphere (like very close to the associated atom) is computable.

    That is, we don’t know how electrons “move,” or if they move at all, rather than miraculously appear here, there, or anywhere. But it’s probable that the electron is within some certain distance from with which the atom is associated.

    But the probability that I am completely bonkers on this point asymptotically (but rapidly!) approaches 100%. So don’t take my word for it, anent the physics part.

    Dafydd (d4fbf5)

  89. Isn’t the correct spelling (but not the pronunciation) Xeno? How is it/was it pronounced in Greek? And is that ancient Greek, or current day Greek?

    No, the correct spelling is Zeno. Use a Zeta (Z), not a Chi (X).

    Chuck Bartowski (8489f0)

  90. Dafydd (d4fbf5) — 7/5/2016 @ 2:39 pm

    So you might think you wouldn’t actually be able to move, due to that darned Zeno.

    However, since we are in the real world now, and since we seem to be able to move, obviously applying Zenoism to a physical distance is incorrect.

    As a matter of fact, Xeno is correct. His argument is correct – we are indeed unable to move, and we are only saved by the uncertainty principle.

    http://www.news.cornell.edu/stories/2015/10/zeno-effect-verified-atoms-wont-move-while-you-watch

    In the quantum world, the folk wisdom really is true: “A watched pot never boils.”

    http://io9.gizmodo.com/the-quantum-zeno-effect-actually-does-stop-the-world-977909459

    Zeno was correct. It is possible to freeze the world, if you manage to measure it just right. It’s just also possible to hurry it along.

    http://arxiv.org/abs/quant-ph/0104035v1#

    https://en.wikipedia.org/wiki/Quantum_Zeno_effect

    The Quantum Zeno Effect is used in commercial atomic magnetometers and naturally by birds’ magnetic compass sensory mechanism (magnetoreception)

    Stepping waaaaaaaaaaaay out of my field, I have the feeling that at quantum levels of the universe, things like electrons don’t actually “move” through space… there is a probability sphere (perhaps of the size of the universe itself); and all we can say is that the probability of the electron being within some particular smaller sphere (like very close to the associated atom) is computable.

    That is, we don’t know how electrons “move,” or if they move at all, rather than miraculously appear here, there, or anywhere. But it’s probable that the electron is within some certain distance from with which the atom is associated.

    But the probability that I am completely bonkers on this point asymptotically (but rapidly!) approaches 100%. So don’t take my word for it, anent the physics part.

    Sammy Finkelman (88f52d)

  91. Stepping waaaaaaaaaaaay out of my field, I have the feeling that at quantum levels of the universe, things like electrons don’t actually “move” through space… there is a probability sphere (perhaps of the size of the universe itself); and all we can say is that the probability of the electron being within some particular smaller sphere (like very close to the associated atom) is computable.

    That is, we don’t know how electrons “move,” or if they move at all, rather than miraculously appear here, there, or anywhere. But it’s probable that the electron is within some certain distance from with which the atom is associated.

    That’s right, I think.

    Sammy Finkelman (88f52d)

  92. I think Aristotle essentially resolved Zeno’s paradoxes. There is more to be said of course: developments in 19th century mathematics, not least set theory, were very important in further resolving them. But what Aristotle did was distinguish between a potential infinite and an actual infinite. A potential infinite is an amount increasing towards infinity as a limit but never actually reaching it. At any point, a potential infinite is a finite amount that is increasing indefinitely. An actual infinite would be when such a process is complete and is no longer finite. The problem is an actual infinite amount seems to be a metaphysical impossibility (google Hilbert’s Hotel). Zeno’s paradoxes trade on switching between potential and actual infinites — which isn’t his fault since the distinction hadn’t been made yet when he came up with his paradoxes. So, contra Patterico, any finite amount can be potentially divided without limit: that’s a potential infinite. But you’ll never finish and reach an actual “infinitieth” of a distance, since that would represent an actual infinite.

    Jim S. (3ab561)

  93. I”m going to call it ‘Patterico’s Paradox’.

    The more I read your blog, the smarter I am.

    The more I read your blog, the more I realize I don’t know and therefore the dumber I am.

    Johnny Mustard (b996d4)

  94. Even the dumbest sprinter at the Olympics will run through, rather than to, the end of the race.
    Showing that athletic intelligence wins again.

    On that same note, in the above joke I think that by the time when the engineer and mathematician finished ruminating, the beautiful woman had already left with the Italian soccer player

    steveg (fed1c9)

  95. The simplest refutation of Zeno is this:

    “Okay, Zeno, I have this arrow aimed at your heart. What do you want to bet that the arrow will never reach your heart?”

    Chuck Bartowski (8489f0)

  96. “The Way that can be spoken of in words, is not the real Way.”

    — Tao Te Ching

    Good morning.

    hunson abedeer (80144e)

  97. Dafydd is shocked!

    atoms-wont-move-while-you-watch

    Does that mean that… atoms are made of Doctor Who weeping angels? Egad!

    Dafydd (d4fbf5)

  98. I watch atoms move by the trillions or more all of the time….

    MD in Philly (f9371b)

  99. So, you cool atoms to near absolute zero, and make a big deal about not seeing them move.
    I must be missing something.
    How about looking at an atom at room temperature?
    What, you can’t find one because they are moving too fast?

    MD in Philly (f9371b)


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