Fun with math

This morning, my daughter challenged me with the Dichotomy Paradox at an ungodly hour of the morning.

She had snuck into our bed for a morning cuddle, you see, and thought that it was immensely unfair of me to want to kick her out so I could get out of the aforementioned bed and go and have a shower. (I was trapped between the child and The Man We Call Dad, who was likewise trapped between the other child and me.)

“Up,” I told her. “I need to have my shower.”

She responded with a drawn out moan of a word followed by a mad clutching of my arm and burrowing under blankets that I interpreted to mean “Oh, hell no!”

Or maybe “Can’t we cuddle for just a little longer, Mama?”

Either way, I now had a child glued to my side like a baby monkey grasping its mother and The Man We Call Dad as a solid, warm presence at my back, and absolutely no hope of being able to extricate myself without great effort, great complaining, and possibly some heavy machinery.

“I’ll give you ten more seconds, then I’m having my shower,” I informed her (as if I had any real possibility of finding my way to freedom in the next 10 seconds).

“I’ll keep count,” she told me oh-so-helpfully.

Right before she closed her eyes again and pretended to be sleeping.

After a while, I pointed out that I was pretty sure that 10 seconds had passed. In fact, I was almost positive that it had been more like 10 minutes.

“It’s been 8 seconds. There are 2 more seconds left.” I was told. A few minutes later, we were at 9 seconds. And then 9-1/2 seconds. And that’s where the trouble began.

“Well, before we get to 10 seconds, we have to get halfway between 9-1/2 and 10, which would be 9-3/4. And then we have to get to half of the quarter second that’s left, which is 1/8 of a second, which still leaves us 1/8 of a second to go. And then we split that in half and still have 1/16th, and then 1/32nd…” and on and on she went, ending with “…so we’ll never, ever get to 10 seconds and you’ll just have to cuddle with me forever.”

She was entirely too gleeful about this idea.

Zeno’s dichotomy paradox,” I may have moaned. There may have been some eye rolling too.

She had no idea who Zeno was, or what dichotomy meant, but she was absolutely certain that my poking her in the middle until she vacated the premises and freed me from my warm haven prison was mathematically impossible and I just couldn’t leave.

She looked so adorable in purple flannel with her hair all wild from sleep and her hands on her hips and her eyes sparkling with mischief that I almost agreed with her.

But then I pointed out that the Dichotomy paradox also meant that a child who wanted a cupcake for dessert after breakfast (don’t you have dessert after breakfast? You totally should.) would have to cross half the distance from the bed to the cupcake, and then half of that distance, and so on and so on and in the end, would never ever ever be able to actually reach the cupcake, which would mean that cupcake eating was, in fact, a mathematical impossibility.

Like Aristotle did so many years before, she instantly scoffed at the possibility of the distance between her mouth and the chocolate cupcakes on the kitchen counter being infinitely divisible and announced that you could stop at eighths, after all, and therefore eat as many cupcakes as you like for after-breakfast dessert.

I don’t know about you, but I like the way she thinks, even if she does think about math far too early in the morning.

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