Reductio ad absurdum (Part 1)

I first heard of Zeno’s paradox a little over a decade ago and it has always fascinated me.

The scene is Achilles racing a tortoise, who has a head start. In order for Achilles to reach the tortoise he must first run to where the tortoise started, but by that time the tortoise has moved a bit further ahead, and when Achilles reaches that point the tortoise has moved a bit further still. Even though the distances between them keep getting shorter there is an infinite amount of halfway points that must be overtaken before Achilles reaches the tortoise, and crossing an infinite amount of points, however small, would take an infinite amount of time, therefore motion is impossible and Zeno concludes it to be an illusion.

That’s the story anyway*

Pondering this during sleepless nights, my amateur philosopher solution had always been that while trying to divide space into infinite and increasingly smaller units, one must eventually hit a limit. An indivisible quantum of space^‡. This would limit the halfway points to a finite number (the same would go for time at the planck scale^♣) and make it possible to cross the threshold (however, if you want to make the infinite steps truly work we could incorporate the theory of relativity to the paradox thusly). This then played into my other sleepless-night-question concerning whether time and space were continuous or granular (if time is granular and builds upon itself it is at least conceivable that the future does not exist. Not until we actually get there anyway).

But just as I was satisfied with my granular space/time idea I remembered Zeno’s Arrow Paradox, which gives the example of an arrow in motion. If we freeze-frame time and look at the snapshots; the individual grains of time, we find that every snapshot contains the arrow occupying a certain space. And since it is where it is and cannot occupy a space where it is not, motion is again impossible at any given moment.

This then leads us back to the nature of time. If time is granular and at any given moment motion is impossible, then time might just be a handy tool used to differentiate between one arrangement of matter and another arrangement of the same matter, meaning that it’s an adaptation and not a fundamental force of nature, which then means… yep, you guessed it, time is an illusion too!

Are these just word games or is motion really an illusion? Heck, is reality an illusion?”

The short answer is, yes. The longer answer is, how the hell should I know? But let’s explore that in Part 2. Stay tuned!

*One of the solutions offered to Zeno’s paradoxes comes from the idea of a convergent infinite series, meaning that it is possible to calculate that an infinite amount of fractions that get progressively smaller total a finite number, not an infinite one.

^‡Planck length of space is 10^-35m and may be relevant here (though this may be different from an indivisible unit).

^♣Planck time is the amount of time it takes a photon to travel one planck length, 10^-43 seconds.

This post was brought to you by Protactinum (Pa).