A Paradox or Two

"Do you want to hear a paradox?" Juan the Gardener asked me as I opened my eyes yesterday morning. It was late for us, near 7am, and I was still luxuriating in warm, preconscious bliss beneath the quilts.


"Sure," I said, even though I hate a paradox first thing in the morning, especially before coffee.


"Let's say you have an infinite string of numbers 1, 2, 3 ,4, 5, etc., paired off with another infinite string of even numbers 2, 4, 6, 8, 10."

"I know what even numbers are," I said. For me, 1 discussion about numbers + severe, incapacitating life-long math anxiety - no coffee = extreme discomfort, if not rage.

"The question is," he continued, "Is the even numbered string of numbers as long as the first string.?"

"Yes," I said, "because they're both infinite."


"So, because they're both infinite, they're equally long? But, how could they be if one string only contains half as many numbers?"

"I don't know," I said. "I imagine that's what makes it a paradox. Tell me another." Even though the very notion of mathematics produces instantaneous fog in my mind, and leaves me bleary and vengeful, I do love a puzzle.

The next one involved an arrow traveling toward a target, but, before it got to point 1 (the target) it had to get half way there, or point 1/2, then it had to get half way between point 1/2 and point 1, and so on, ad infinitum. It had to do with a paradox concerning motion, called the Dichotomy, dating from an Ancient Greek philosopher named Zeno.

"So, basically the original argument was because you can't do an infinite number of things in finite time, it never actually gets there. You see?" Juan said.

I groaned.

"Not really, because I've seen an arrow hit a target. I've shot an arrow myself and watched it sink into a straw bulls eye, heard it make a thunk, so I have empirical proof that it does, in fact, get there."

"Exactly. It's obvious. You can do an infinite number of things in finite time, you just have to do them faster. There's an article about it in the NY Times by a philosopher named Graham Priest."

"I have a paradox for you," I said.

"Yes?"

"Say two people are lying in bed."

"Yes?"

"One of them desperately wants coffee..."

"Yes?"

"Which of them gets up and makes coffee? The one who voices the desire, or the one who wisely chooses to let the other one stay in bed and volunteers to make it himself?"

"I'll make the coffee," Juan said.