Zeno's Paradox

Discussion in 'Physics & Math' started by Xenu, Jun 22, 2002.

  1. Xenu BBS Whore Registered Senior Member

    Are you guys familiar with this philosophical riddle of Zeno's paradox?

    Here's a snippet:

    Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

    Now I've heard solutions to this over and over, but am never satisfied with them. Could someone explain this to me? In idiot terms?

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  3. overdoze human Registered Senior Member

    You cover half a distance, in half the time. Another quarter of distance, in a quarter of the time. Another eighth of the distance in an eighth of the time, and so on. Your total time for crossing the room is the sum of all those times, and your total distance covered is the sum of all those distances.

    Is that idiotic enough?

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  5. Asguard Kiss my dark side Valued Senior Member

    You keep halving so the number you are taking away gets smaller but is never 0



    it will never be 0

    this is the half life rule for radioactivity
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  7. Merlijn curious cat Registered Senior Member

    The solution is setting the end-point of the journey infinitely small piece further than where you want to go.
    in that case you'll get there with no problem at all...
  8. Xenu BBS Whore Registered Senior Member

    Overdoze and Asguard,

    I'm sorry if I wasn't clear. I know how the problem works. What I wanted to know was how the problem can be explained to be wrong mathematically. In real life it's obvious that the arrow get there, but logically when it's worked out the way it is, it doesn't work.


    I think your solution has a logical hole. When you change your endpoint you have changed your endpoint, you are now shooting for a new goal, which you can still no longer reach. Whether or not this passes through the old goal is arbitrary.

    For example, if I wanted to walk to a point right in front of a wall, I don't adjust my aim to walk into the wall.

    If you can explain your solution mathematically, maybe it'll make more sense.
  9. Adam §Þ@ç€ MØnk€¥ Registered Senior Member

    It doesn't work because the entire analogy is false to begin with. It uses division by two as an anology of motion in a way which simply never happens. Yes, if in maths you continually divide by two you get half the previous number. Big deal. It has nothing whatsoever to do with actual travelling/motion. Why? Because you will not halve the length of your strides as you move. Your steps will stay around one yard, for example. When the remaining distance is less than one yard/step, the next step takes you to the end. Basically the entire analogy from the start is false. It compares division by two with regular decrements.

    EDIT: Or, mathematically, it is comparing:

    Start at 10, and divide by 2 at each step: 5, 2.5...

    Start at 10, and use regular decrements representing steps: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0!

    The paradox is trying to say that 2 can not happen because of 1. Which is totally ridiculous. It is a comparison of two entirely difefrent things.
  10. Merlijn curious cat Registered Senior Member

    Okay, let's have it again...
    Since the aiming point is never overt, we can only percieve the end-point of the motion. What was the aiming point need not necessarily be the aiming point!
    So I want to go from A to B, but my cerebellum calculates thet I have to aim for C.

    In fact this is not the true solution, I know. But i always found that is the most insightful.
    I think the actual solution has to do with a second-order differential equation. But it's rather stricky.

    Adam, the length of strides do not matter at al, because originally the story was about an arrow flying to a mark.
  11. Adam §Þ@ç€ MØnk€¥ Registered Senior Member

    The stride wa san example. The same applies for arrows in flight. In any given unit of time during the flight it will cover a certain distance. That distance per time unit does not halve with each successsive time unit.
  12. Merlijn curious cat Registered Senior Member

    Oh I forgot...
    In my A...................B..C, the distance between B and C is zero, still it will in fact solve Zeno's paradox.

    Adam, I know... I was annoying you. sorry. So you think the paradox is there because of our choice of reference. You're right. However, that does not solve the paradox. Also it's a fun exercise, don't you agree?
  13. Asguard Kiss my dark side Valued Senior Member

    what i learned in science is you touch at infinity

    adam: this DOSE have a practical purpose, its used for half lives of radiactive isotopes
  14. Adam §Þ@ç€ MØnk€¥ Registered Senior Member

    I know division by two serves a purpose. But the fact remains it is still only division by two. It's really not that interesting. Zeno got his name remembered for using a false preposition which basically doesn't work.

    EDIT: Fixed a BOLD marker.
    Last edited: Jun 23, 2002
  15. Xenu BBS Whore Registered Senior Member

    Alright, let me restructure the problem so that your guys' previous solutions and answers wouldn't work.

    ****"Xenu's Paradox"****

    The first concept to understand should be simple.

    The whole is equal to the sum of its parts

    This isn't true in things such as say... Gestalt Psychology, but for mathematics it seems to be a logical truth.

    For example, I live in city A and am driving to city C, in between is city B. The distance between A and C is equal to the sum of distances of A and B plus B and C. This is a no brainer, I know.

    Now, let's take a line segment. It's 2 cm (feet, inches, or whatever) long.

    Represented on a number line:


    Mathematically this line segment can be divided into an infinite number of line segments. How can this be so? Think about it. For instance, I could have a start of a segment at the real number 1.99998 and have an end point at 1.99999, making the segment .00001 cm long. But between these two point are real numbers such as 1.99999999999999...repeating off into infinity, or 0.44444444444 repeating, or what have you - logically making it possible to divide the line segment into an infinite number of parts.

    Now the major question is, how big are each of these line segments? I'd have to guess that each line segment would have to be infinitely small, represented by:


    So given the rule above, The whole is equal to the sum of its parts, the line segments mathematic length is equal to:

    Infinity * 1/Infinity or simplified:

    Infinity/Infinity which is equal to:

    1? - seems initially intuitive, 0? - infinity divided by anything = 0, but mathematically I think it is equal to: Undefined (or that big fat E you get on your calculator at times).

    Even if you didn't say that each segment was infinitely small, anything divided by infinity equals 0.

    It seems that no matter how you work it, you don't get 2 cm.


    Alright guys, please show me the answer to this, and definitely show me any logical flaws. This should be fun. I'm a little befuddled myself.

  16. Merlijn curious cat Registered Senior Member

    Your post reminds me of an early book by Paul Davies. Beyond the Edge of Infinity or something it was called.
    When you seperate all the infintate points of a line segment and paste them together, you can have a new line segment of any length (e.g. l=0).

    Gestalt psychology is ancient. Hahaha. Better talk about Dynamical Systems Approach of psychology. This also has some Gestalt like elements, but it is based upon the mathematical study of complex systems.. which do have emergent properties!
    So, in mathematics it is sometimes true that the whole is more than the sum of the parts!

    But the logical flaw is that infintity cannot be treated as a number. So you cannot do mathematical operations on them.
  17. Xenu BBS Whore Registered Senior Member

    So is Freud, and they still don't stop talking about that bastard

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    Heard of it, but not much about it. Could you post some info on this, sounds interesting.

    I guess that this is kind of my point too. However, calculus deals with infinities and limits (approaching infininity). I guess I don't know enough about calculus to make any other judgements about there systems. Maybe someone else could enlighten me?
  18. overdoze human Registered Senior Member


    The problem is far easier than you make it sound. Basically, assuming you travel with constant speed, if you divide distance travelled into smaller chunks you will cover each chunk in appropriately less time. The "paradox" says you will never cover that distance, but that's not true. Since each chunk of the distance corresponds to an appropriately small piece of time, adding all the chunks of distance and all the time gives you a finite time for covering the entire distance.

    With respect to infinities, sure you can do math with them. But you have to be careful not to fudge the math. There are different "degrees" of infinity. For example, there are infinitely many distinct integer numbers, and there are infinitely many distinct real numbers, but there are more real numbers than integers. So in effect, you have one infinity greater than another infinity (which is bad language, but that's what you have.) If you want to be more PC, you'd say that the cardinality of the set of real numbers is greater than the cardinality of the set of integers even though both sets are infinite in size. So you can indeed compare infinities based on their cardinality.

    If you divide a distance into infinitely many chunks, this set of chunks will have a certain cardinality. In one-to-one correspondence to it will be the infinity of time intervals, each corresponding to its own chunk of distance. So as long as you keep track of that one-to-one correspondence you will be fine. You can split the distance into an infinity of segments, and you can put it back together again to get the same distance. For a slightly different distance the set of segments of the same length will have a slightly different cardinality, so adding it back together will give you back that slightly different distance.
  19. Zero Banned Banned

    True. In Zeno's day they didn't have the concept of limits. If you calculate the limit of that paradox, you get a finite answer. Also, there is a humbling, practical answer to that. You can walk across a room, duh.

    Poor Zeno, he wasted a lifetime on some useless thing that a high schooler could calculate in five seconds today. Case closed.
  20. Xenu BBS Whore Registered Senior Member

    reply to zero's post

    Could you (or anyone else) show me how to calculate either/both Zeno's paradox or Xenu's paradox?

    Well of course you can, and Zeno wouldn't deny it.

    You consider it a waste of time because you don't see the importance of it, the way he did. Also, the case is not closed for me.

  21. Xenu BBS Whore Registered Senior Member

    response to overdoze

    I'm not satisfied with you answer to Zeno's paradox. I'll break apart your argument here.

    You have changed the nature of the problem. The way it's set up, the problem has nothing to do with time. The problem that I posted here only has to do with distance.

    This sentence is confusing. I am only going to discuss distance here. What I think you are getting at is this: You have a distance. All of the segments, that make up that distance add up to that distance, no matter how many of them there are. Duh, simple.

    Yes it's easy to see a solution when you view it post hoc, after the fact. But you have changed the nature of the problem again. What your reasoning relies on is that the distance has already been traveled, and then going back and showing how. The actual Zeno's paradox is showing the movement in process. The goal is never reached in the first place, so you can't start your math from there. Does this make sense?


    Now I want to discuss what you have to say about infinities and degrees in respects to Xenu's paradox.

    Again, this can only be done post hoc (after the fact). In order for this to work you have to start out saying that the segment is two cm long, chop it into the infinite segments and then attach a degree to it. What if I started with just the infinite segments. Add up all their distances and what do you get? You won't get 2 cm.

    "Wait a minute" you might say. When I give you the infinite segments, I also have to give you a degree with it. For instance a degree of "2 cm". This doesn't make sense either. This implies the answer is the problem. It's like asking what is the color of Chester's white horse? It's a cop out.

  22. daktaklakpak God is irrelevant! Registered Senior Member

    The problem lies on the difference between Zeno's desire and his action. From his mind, he wants to across a room, but in reality, his action dosen't. He just tries to get as close as possible to the other side of the room, but never reach it. Duh!

    It's obvious that this Zeno guy is not doing what he wanted to do. How do you expect he gets what he wanted?

    Try to look at it in another way. If Zeno is only one micron away from his goal position, can you tell from just standing next to him and say he has not cross the room yet?

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    Last edited: Jun 24, 2002
  23. Zero Banned Banned

    It is pointless. So what if there are an infinite number of half-spaces? You don't even need limits.

    If you walk at constant speed, then the time it takes for you to cross the half distance is half that of the previously covered distance. And half of that is again half that time. And so on. It is utter stupidity to think of it that way. In constant speed problems...time required is total distance divided by speed.

    If you don't walk at constant speed and you keep getting slower, then you might not reach it. Duh. If you stop midway and you stay there forever, then you don't reach it.

    Zeno wasted his time. He doesn't even bother to make any real assumptions, such as how the walker's speed changes with respect to time, and he just twists words so that people will get confused. It apparently worked, because he confused himself. It is such a waste of time. Case closed for me, it is idiocy to discuss this further. Everyone I advise you to leave and go spend your time on something constructive.

    Last bit of prooof that Zeno's ideas are dead. No one ever joins his sect of philosophy. It is dead and gone.

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