The Philosopher, the Cave, and the Mathematician

I have a funny story from when I was younger, and reflecting on it just now, I realized that it is helpful for understanding one aspect of Plato’s story about the cave. Well, it’s a funny story insofar as a story about a math problem can be funny I guess.

I have a friend who’s very good at math. He has a bachelor’s degree in the subject. One day, he and I heard an intriguing word problem. I can’t exactly remember the details anymore — something about twenty people at a table who can give invitations to one another, but only to the person across the table or to the east of them, and what’s the greatest number of invitations that can be passed, except that I feel like it was somehow even more complicated than that.

Anyhow, he and I both set our minds to working on the problem. I wasn’t able to find an answer, but he was. The answer he gave, though, seemed, based on the thinking I had done, like it must be wrong. The number was too small.

I told him so, and he explained how he got to the answer. It was pretty simple actually, he said, once he realized that such-and-such a well known formula was all that needed to be applied to the initial figures. I was personally unfamiliar with that formula, but it seemed to me that it couldn’t be the right one to use, and I tried to express why, in my fumbling and less sophisticated way.

He wouldn’t be convinced. The next day he told me he had called up a friend with a degree in mathematics, explained the problem, set out his solution, and the friend had agreed with him entirely. But I was still uncertain. We kept discussing it until I found the simplest way to explain my objection.

“But the two people across the table from each other can each give an invitation to the other. Isn’t your formula assuming that an invitation only crosses the table once, in one direction? And isn’t that going to mess up the final number in a pretty big way?” It was something like that.

His eyes rolled back in his head a little bit, and his brow furrowed, and he began speaking to himself in little whispers. Finally he nodded. “Yes, you’re right. My answer can’t be correct. But what you need to know, John, is that I studied pure mathematics, and this is applied math, which is not the same thing. That’s why I missed a step.”

We never bothered finding the actual answer to the problem. I look back on that conversation with some pride, still.

But there was one particular moment in the conversation that is frozen vividly in my mind, that I will never forget. My friend had a frown. The pads of the fingers of both open hands were covering his eyes and gently massaging, trying to keep his frustration at bay. “I just don’t know how to explain this to you, John. I don’t know how to be any more clear than I’m already being. I’m not sure what it will take to show you that you’re wrong.”

He assumed that our disagreement was a consequence of his wisdom and my error. And it very well could have been! But that’s a dangerous assumption to begin with. Perhaps the most fatal assumption of all.

It’s the assumption that the philosopher first had to give up in order to become a philosopher. It’s the assumption the philosopher will encounter again and again, endlessly, upon returning to the cave.

I wasn’t the philosopher in the conversation with my friend. I just had the good luck to be less wrong than my better educated friend. But that experience gave me an insight into the life of the philosopher in the cave.

We in the cave are so certain that we know what we’re talking about, and that anyone who disagrees or doubts just knows less than we do. We have reasons, and we have confidence, but that doesn’t mean we know what we’re talking about. And insisting that it does mean that, can only leave us looking foolish when we’re eventually shown to be utterly wrong.

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