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posted by janrinok on Sunday February 23 2020, @08:09AM   Printer-friendly
from the brain-teasers dept.

Quartz

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Almost 2,500 years ago, the philosopher Zeno of Elea set out to challenge the way we understand the physical world through a set of brain teasers that have stuck with us for millennia. The most powerful of Zeno's paradoxes grapple with the concept of infinity while pitting observable reality against the scientific language we use to describe that reality, suggesting that elements of the everyday, like motion and speed, are actually illusory.

Example paradoxes are:

The millet paradox, which states that one falling grain of millet makes no sound but a ton of falling millet makes a big one, is more of a stoner observation than a profound question about the physical world. His paradoxes of motion and space, on the other hand, are legendary. Four of the more than 40 thought experiments he is said to have devised are most often employed as vivid introductions to the intersection of math and philosophy, where something readily apparent is a challenge to definitively prove.

Dichotomy paradox: If you want to walk across the room, you have to first walk half that distance, then half the remaining distance, ad infinitum, so how do you ever get there?

Achilles paradox: If a turtle gets a head start in a race against Achilles, Achilles has to cover half the distance between himself and the turtle in order to catch up. Then half that. And half again. And again. In an upset, the turtle wins!

Arrow paradox: At any given instant, an arrow in flight occupies a certain space, no more and no less. At the next instant, it occupies a different space. If you assume an instant is indivisible, the arrow is not in motion. So how does it move? "It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever," as Bertrand Russell put it.

Stadium paradox: Imagine three sets of three bodies in stadium rows: three As, three Bs, three Cs. The As are stationary; the Bs are moving right; the Cs are moving left at the same speed. In the same timeframe, the Cs will pass just one of the As, but two of the Bs. Crazy, right? (It doesn't seem like it, but if you think of space and time atomistically, they pass without passing.)

It took more than 2,000 years to break the dichotomy and Achilles paradoxes, and the people to do it were the French mathematical prodigy Augustin-Louis Cauchy and the German Karl Weierstrass. The mathematical answer can be summed up by the intuitive answer: Eventually, you get there.

In mathematical terms, one way of putting it is "the limit of an infinite sequence of ever-improving approximations is the precise value" (pdf). By going from one side of the room to another, you go 100% of the way across. You can chop that 100% up into infinite pieces, but those pieces converge on a limit of 100, and the sum of those pieces is the value—the infinite number of increasingly small pieces adds up to a finite number. ½ + ½ = 1, of course. ½ + ¼ + ¼ also equals 1. And so forth: the numbers you add up to get to 1 can expand to infinity, but it's not changing the end result. Not all infinite geometric series converge to a limit, but some do (pdf), predictably: "All those (and only those) in which the ratio of consecutive terms is greater than –1 and less than +1, so that the absolute values of the terms get progressively smaller."


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  • (Score: 3, Funny) by Nuke on Sunday February 23 2020, @11:16AM (11 children)

    by Nuke (3162) on Sunday February 23 2020, @11:16AM (#961365)

    2500 year old news in fact. Is SN getting desparate? Next up, the Universe is created.

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  • (Score: 2) by Gaaark on Sunday February 23 2020, @11:23AM (2 children)

    by Gaaark (41) on Sunday February 23 2020, @11:23AM (#961367) Journal

    It was pushed through, after a MAJOR rewrite, to make aristarchus happy.

    --
    --- Please remind me if I haven't been civil to you: I'm channeling MDC. ---Gaaark 2.0 ---
    • (Score: 4, Funny) by Anonymous Coward on Sunday February 23 2020, @12:27PM (1 child)

      by Anonymous Coward on Sunday February 23 2020, @12:27PM (#961374)

      Isn't any attempt to make aristarchus happy a paradox in and of itself?

      • (Score: 2, Insightful) by aristarchus on Sunday February 23 2020, @08:33PM

        by aristarchus (2645) on Sunday February 23 2020, @08:33PM (#961540) Journal

        aristarchus is not happy. He is giddy, ecstatic, consummated with bliss. Now if only janrinok understood what a slam to STEM the paradoxes are.

  • (Score: 2) by MostCynical on Sunday February 23 2020, @11:50AM

    by MostCynical (2589) on Sunday February 23 2020, @11:50AM (#961368) Journal

    Only if the git request went to the correct workflow.

    Likely, we're alpha/pre-release, and the updates are still waiting in a request queue..

    --
    "I guess once you start doubting, there's no end to it." -Batou, Ghost in the Shell: Stand Alone Complex
  • (Score: 2) by janrinok on Sunday February 23 2020, @12:12PM (6 children)

    by janrinok (52) Subscriber Badge on Sunday February 23 2020, @12:12PM (#961372) Journal

    STEM includes mathematics. Many of our community like brain teasers such as this, and they will have tried to think the paradoxes through for themselves before reading how and when they were solved mathematically.

    They are as relevant today as they were 2500 years ago, IMO.

    • (Score: 2) by Azuma Hazuki on Sunday February 23 2020, @05:57PM (5 children)

      by Azuma Hazuki (5086) on Sunday February 23 2020, @05:57PM (#961470) Journal

      I don't really get the mathematics behind most of this, or anything honestly, but in the specific case of the arrow and the tortoise, isn't it just a converging series? 1/1 + 1/10 + 1/100 + 1/1000...etc, should sum to a number much less than infinity.

      Is this a classical proof that spacetime is quantized, actually? That's what I took from this, that at some point there has to be a leas unit of spacetime...

      --
      I am "that girl" your mother warned you about...
      • (Score: 0) by Anonymous Coward on Sunday February 23 2020, @09:03PM

        by Anonymous Coward on Sunday February 23 2020, @09:03PM (#961551)

        The dichotomy paradox is a series of halfs, not how they gave it. They gave the limit version, which is not the original one given. Basically, the repeated is that you have to halfway to get somewhere, which repeats for that half, etc. Which means that your first step is to cross the smallest half. This is what is paradoxical, obviously we move but then how can one do an infinite number of actions in a finite period of time? Even a discrete universe seems to suffer from that problem, as to do otherwise would seem to make geometry and other mathematics contradictory at such a scale.

      • (Score: 0) by Anonymous Coward on Sunday February 23 2020, @09:20PM (3 children)

        by Anonymous Coward on Sunday February 23 2020, @09:20PM (#961560)

        Just realized you were talking about the other one, but the underlying idea is the same. You'd have to do an infinite number of fractions in a finite period of time.

        • (Score: 2) by pgc on Monday February 24 2020, @11:57AM (2 children)

          by pgc (1600) on Monday February 24 2020, @11:57AM (#961786)

          It is not a paradox, since the time needed for each action will eventually approach 0, therefore the total time taken will fit in a limited amount of time.

          • (Score: 0) by Anonymous Coward on Monday February 24 2020, @11:41PM (1 child)

            by Anonymous Coward on Monday February 24 2020, @11:41PM (#962077)

            But you don't catch up to the turtle until it does reach zero. The fact that it approaches zero doesn't change the fact that it will never be zero.

            • (Score: 2) by Azuma Hazuki on Tuesday February 25 2020, @03:36AM

              by Azuma Hazuki (5086) on Tuesday February 25 2020, @03:36AM (#962195) Journal

              This is what introduced me to the idea of quantized spacetime as a child, though I didn't have any words for it other than "well we KNOW we can get past any distance we want if we take long enough, so maybe that means somewhere there's a smallest piece of distance?" I was 9 or 10 at the time, IIRC.

              --
              I am "that girl" your mother warned you about...