Patterico's Pontifications


Zeno’s Paradox Solved?

Filed under: General — Patterico @ 11:59 pm

I am bored by politics. How about some abstract philosophical/theoretical physics musing, from a complete and total amateur. If nothing else, it’s a chance for you to teach me something.

You have probably heard of Zeno’s paradoxes, one of which is that to get anywhere, you first must travel half the distance, and then half the remaining distance, and then half the remaining distance, and so on. According to this theory, you never get there.

I have seen various solutions to this, but I am not sure anyone has proposed mine. Perhaps they have, but here it is anyway:

My proposed solution is that there is a minimum distance in the universe, which cannot be subdivided into smaller distances.

Imagine a debate between Smarty Pants #1 and Smarty Pants #2. They are debating whether there is a smallest distance in the universe.

Smarty Pants #1 says there is no such thing as a smallest distance. He says: imagine the distance equal to .001 of an inch. I can add a zero and make it .0001 of an inch. If you imagine the distance equal to .00000000001 (ten zeroes) of an inch, he says, I can add a zero and make it .000000000001 (eleven zeroes) of an inch. And so on. No matter how many zeroes you write, I can add another.

Smarty Pants #2 has two responses. His responses are not designed to show there is a smallest distance, but to show that there could be one.

RESPONSE ONE: A theoretical number does not necessarily translate to the real world. For example, light travels 186,282 miles per second. It is also generally accepted that nothing can travel faster than light. There are numbers higher than 186,282, but that does not mean that any object can travel 187,000 miles per second, even though the number 187,000 exists. Similarly, the fact that you can express an incredibly small fraction of an inch numerically, in theory, does not mean that this distance exists in the real world.

RESPONSE TWO: Remember how you said that you can always add another zero to a very small decimal representing a small fraction of an inch? Fine. Go ahead and do it. Write the smallest number you can imagine, representing the smallest fraction of an inch you can express in numbers. I’ll wait right here while you do that. Are you done? Great! Turns out the smallest distance is smaller than that. What’s that you say? You can make it smaller? Fine, go ahead. Ready? Yeah, the smallest distance is still smaller than that. You say you can add zeroes all day? I’m here all day, too — and no matter what number you write, I can claim that the real smallest distance is smaller still.

See, Smarty Pants #1 thinks he wins the theoretical argument by positing that he can always add another zero. But Smarty Pants #2 trumps him by simply claiming that the smallest distance in the world could be smaller still. Any number that #1 comes up with, #2 just says the actual smallest distance is smaller.

And if that minimum distance exists? Zeno’s paradox is solved!

Some day, someone will prove this is true, in almost the same way I have described. I want the credit. And I want it now. Someone send the link to Stephen Hawking pronto.

P.S. I think all this may have something to do with the “Planck length” but I’m not sure. Let those who know more than I do instruct me.

Powered by WordPress.

Page loaded in: 0.0556 secs.