THE DICHOTOMY PARADOX

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Imagine that you’re about to set off walking down a street. To reach the other end, you’d first have to walk half way there. And to walk half way there, you’d first have to walk a quarter of the way there. And to walk a quarter of the way there, you’d first have to walk an eighth of the way there. And before that a sixteenth of the way there, and then a thirty-second of the way there, a sixty-fourth of the way there, and so on.

Ultimately, in order to perform even the simplest of tasks like walking down a street, you’d have to perform an infinite number of smaller tasks—something that, by definition, is utterly impossible. Not only that, but no matter how small the first part of the journey is said to be, it can always be halved to create another task; the only way in which it cannot be halved would be to consider the first part of the journey to be of absolutely no distance whatsoever, and in order to complete the task of moving no distance whatsoever, you can’t even start your journey in the first place.

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    This is one of the 2400-year-old formulations of Zeno’s paradox, for which there have been a number of explanations over the centuries.

    From a mathematical perspective, the paradox is resolved using a convergent infinite series.  This is a series (such as 1/2 + 1/4 + 1/8 + …) which has an infinite number of terms, the sum of which converges on a finite value (1 in this case).  When you consider that the time it takes to cover each segment converges towards zero along with the distance, you can illustrate that a finite time is required to cover the finite distance, regardless of how finely the distance and time are divided.

    Zeno’s paradox has some interesting consequences when viewed in the context of quantum mechanics.  The entire notion that reality is quantized into discrete increments means that time and distance cannot be infinitely divided into ever smaller chunks.  There is a limit, a smallest distance (Planck length = 1.617 x 10^-35 meters) and a smallest unit of time (Planck time = 5.391 x 10^-44 seconds).  These two units are related: a Planck time is how long it takes a photon moving at the speed of light to travel 1 Planck length.  If time is quantized, then there is no point in time that exists between T and T + 1 Planck time.  Likewise, the photon does not exist at any intermediate position along its path of travel between D and D + 1 Planck length.  At time T it is at point D and at time T + 1 Planck time it is at point D + 1 Planck length.

    In the context of quantum mechanics, Zeno’s paradox is resolved.  The photon does not “move” from D to D + 1 Planck length.  At time T it exists at D and at T + 1 Planck time it exists at D + 1 Planck length.  A distance of 100 m can be repeatedly divided in half about 122 times before it reaches the Planck length and can be divided no more.  The journey starts with a movement of a Planck length rather than the postulated “no distance whatsoever”.

    dougbell Genius Answered on 22nd October 2015.
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