• Jeremiah
    1.5k
    This one is probably more well known than the other one, so if it turns out to be too easy I'll move on to the next one.

    This is one of Zeno’s paradoxes, I'll paraphrase and simplify it for those that like the short and sweet:

    Say I want to cross the room, then I must first cover half the distance. Then I must cover half the remaining distance. Then I must cover half the remaining distance and so on to the end of time. So I can never get to the other side of the room. This would actually make all motion impossible, as to get anywhere I must first cover an infinite number of small intermediate distances.


    For those that want the longer version here is the original:


    The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.

    “How big a head start do you need?” he asked the Tortoise with a smile.

    “Ten meters,” the latter replied.

    Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”

    “On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”

    “Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.

    “Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”

    “Very quickly,” Achilles affirmed.

    “And in that time, how far should I have gone, do you think?”

    “Perhaps a meter—no more,” said Achilles after a moment’s thought.

    “Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”

    “Very quickly indeed!”

    “And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”


    “Ye-es,” said Achilles slowly.

    “And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.

    Achilles said nothing.

    “And so you see, in each moment you must be catching up the distance between us, and yet I—at the same time—will be adding a new distance, however small, for you to catch up again.”

    “Indeed, it must be so,” said Achilles wearily.

    “And so you can never catch up,” the Tortoise concluded sympathetically.

    “You are right, as always,” said Achilles sadly—and conceded the race.
  • Michael
    15.8k
    I don't see what this has to do with mathematics. If anything it calls into question the notion that movement is continuous. If movement is actually discrete then there's no paradox.
  • Jeremiah
    1.5k


    It is a very important paradox concerning series and calculus. Ask yourself: How is that you can have an infinite number of distances to cover, but yet still be able to cross the room?
  • Baden
    16.4k


    He just said movement could be discrete. There would therefore not be an infinite number of distances to cover. Think Planck length.
  • Jeremiah
    1.5k


    In theory length is not discrete, only our ability to measure it is.
  • Jeremiah
    1.5k
    Try to think of it this way:

    I want to cross 10 meters but to do so first I have to get to the half way point.

    10/2 = 5

    Then I have to get to the half way point of 5 meters.

    5/2 = 2.5

    And so on.

    2.5/2 = 1.25

    1.25/2 = 0.625

    0.625/2 = 0.3125

    0.3125/2 = 0.15625

    0.15625/2 = 0.078125

    etc....

    That will go on forever, it is called a series.
  • Baden
    16.4k


    We have a discipline called physics which provides a potential solution to the problem as outlined. Have a read of the papers.
  • Jeremiah
    1.5k


    This is actually sequences and series that leads to an important mathematical concept that I studied in the college classroom and was tested on. Did you take those classes? Did you read the book? Here the answer is in this book. . .

    https://www.amazon.com/Calculus-Early-Transcendentals-MultiVariable-Rogawski/dp/1464171750/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=V13FMHA65YWJKE428B0F

    That is essentially what you are doing, it is something people do when they don't want to have an actual discussion but they want to claim someone is wrong. If it helps focus on the fractions and forget we are talking about length.
  • Baden
    16.4k


    It might be better if you have this conversation with yourself. That way you can get the answer you want.
  • Jeremiah
    1.5k


    Yes, I know you are eager to put me in place because you don't like me.
  • Michael
    15.8k
    Summing an infinite series in mathematics isn't anything like actually completing a supertask, which is what Zeno proposes we must do to move. This is why these mathematical solutions to the paradox miss the point.

    The explanation of Zeno's paradox in that article is useful here:

    • Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
    • Supertasks are impossible
    • Therefore, motion is impossible

    Given that motion is possible, either motion isn't a supertask (so it's discrete, not continuous) or supertasks are possible. Thomson's lamp is an attempt to show that supertasks aren't possible, and (apparently) supertasks being possible would contradict the Church-Turing thesis (although admittedly I don't know how sure the Church-Turing thesis is).
  • T Clark
    13.9k
    This is actually sequences and series that leads to an important mathematical concept that I studied in the college classroom and was tested on. Did you take those classes? Did you read the book? Here the answer is in this book. . .Jeremiah

    I don't understand, if you know the answer, why ask the question. To summarize - the sum of an infinite series can have a finite value. The proof that Zeno's particular series does add to a finite value is that we can walk through the door. See this video prepared by the University of Helsinki Institute for Finding Out Things, which I've used before....



    ...and likely will use again. Those Finns are great practical philosophers.
  • Srap Tasmaner
    5k

    When I was a kid, I was taught, like Jeremiah here, that limits and convergent series and calculus "solve" Zeno's paradox. Greeks just didn't have as much as math as we do. Of course they didn't teach me about computability when I was 17.
  • Jeremiah
    1.5k


    Zeno's paradox in mathematics lends to the idea of infinite partial sums, which is the sum of infinite parts. While it can be divided an infinite number of times, it still sums up to a whole, hence we can cross the room. I am not sure how that fits in with supertask, but even if we decide Zeno's paradox is wrong, as clearly we can cross the room, it still highlights important mathematical concepts.
  • Jeremiah
    1.5k
    Sometimes it is good to revisit the classics.
  • Michael
    15.8k
    When I was a kid, I was taught, like Jeremiah here, that limits and convergent series and calculus "solve" Zeno's paradox.Srap Tasmaner

    And I think that's wrong. That the sum of an infinite series can be finite doesn't explain how a supertask can actually be completed (or in this case even started; which point is the first point to move to)?
  • Srap Tasmaner
    5k

    And I agree. (Should have made that clear. The computability approach actually makes more sense.)

    What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy!
  • T Clark
    13.9k
    What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy!Srap Tasmaner

    From the web - The Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. This is the quantum of length, the smallest measurement of length with any meaning. And roughly equal to 1.6 x 10-35 m or about 10-20 times the size of a proton.

    Obviously, mathematicians can keep on dividing smaller than that.
  • T Clark
    13.9k
    What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy!Srap Tasmaner

    Using Excel I see that we get to that length after only about 120 steps in our series. As I said, Excel allows us to continue on smaller and smaller than that.
  • Michael
    15.8k
    While it can be divided an infinite number of times, it still sums up to a whole, hence we can cross the roomJeremiah

    This doesn't follow. A good analogy to Zeno's paradox is that of counting the real numbers from 0 to 1. Before we can count to 1 we have to count to 0.5. Before we count to 0.5 we have to count to 0.25. And so on. But that we can sum this series isn't that we can actually count it.

    The issue has nothing to do with the suggestion that it would take an infinite amount of time but with the fact that there isn't a first number to count to after 0. We're stuck at 0 with nowhere to proceed. It's the same logic with Zeno's paradox; there's no first position after the start point. We're stuck standing still. Calculus can't solve this problem.
  • Akanthinos
    1k
    Ask yourself: How is that you can have an infinite number of distances to cover, but yet still be able to cross the room?Jeremiah

    The answer is sequence. If you ask me to get between 0.1 and 1 and give me a path that goes between 0.1 and 0.999999999999....n, of course I won't ever reach 1.

    Under natural circumstances, movement is not measurement. The action of moving from one position in the sequence to the other does not alter the sequence itself, and does not provide a step for subdivision of the furthest point. That's why when I want to reach 1 from 0.1, I get to reach 1 after I've gone through a certain sequence in the serie.

    And since infinite series are also sequential, they are no more an obstacle to reaching a specific endpoint.
  • T Clark
    13.9k


    I was referring to the Planck length in my previous response.
  • Hanover
    13k
    Your counting analogy seems to establish that the Planck discussion was irrelevant. As you're indicating, you need to accept some arbitrary discrete amount in order to move, which is a Planck in physical space, but not established in mathematical "space." In other words, if we hypothosized that space were infinitely subdividable (i.e. not discrete), Zeno still makes it to the finish line, right?
  • Michael
    15.8k
    Your comment confuses me. These two seem contradictory:

    "As you're indicating, you need to accept some arbitrary discrete amount in order to move"

    "In other words, if we hypothosized that space were infinitely subdividable (i.e. not discrete), Zeno still makes it to the finish line, right?"
  • Jeremiah
    1.5k


    The track of land is basically an arc length and we are looking for the net change from point a to point b. That's calculus.
  • Michael
    15.8k
    The track of land is basically an arc length and we are looking for the net change from point a to point b. That's calculus.Jeremiah

    But that doesn't address Zeno's paradox at all.
  • Jeremiah
    1.5k


    Sure it does, Zeno's paradox is about getting from point a (the start of the race) to point b (the turtle) over infinite small intermediate distances. This is exactly what calculus is.
  • Michael
    15.8k
    Sure it does, Zeno's paradox is about getting from point a (the start of the race) to point b (the turtle) over infinite small intermediate distances. This is exactly what calculus is.Jeremiah

    Again consider the analogy of counting the real numbers between 0 and 1. Calculus can't show that it's possible to count each of the infinite reals between them. It doesn't matter than we can sum a geometric series.
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