Descartes's cracked ceiling

I’ve always found the story of how Cartesian coordinates were discovered deeply dissatisfying.

The tradition is that Descartes was sick in bed and looking at the ceiling when he saw some cracks in it. He wondered ‘how would I tell somebody the exact position of any one of those cracks?’ It came to him a flash of inspiration: ‘I know: I’d tell them the distance to the crack from one wall and the distance to the crack from another wall perpendicular to the first. That will determine the position uniquely.’ And so Cartesian coordinates were invented.

All very well and fine, you’d think. It’s another Newton-and-the-apple moment. Great as far as it goes.

But look at all the things it leaves untold.

Why was he looking at the ceiling instead of reading a book like anyone sensible would do? Surely a book would have been more fun. He’d have enjoyed himself and centuries of school kids wouldn’t have been put through the agony of studying his discovery ever since.

And anyway why would he want to identify the position of a crack exactly to someone else? Couldn’t they just look and see it for themselves? In any case, the issue isn’t to determine its position, but to repair it or at least paint over it.

And then there’s the most worrying aspect of all. How did the cracks get to be there in the first place? Subsidence? Seismic activity? Did he check whether the house was still structurally sound? The cracks might have been symptoms of serious risk to members of his household. What steps did he take to minimise it?

But that’s the problem. The story’s been told and retold by mathematicians. These are people who spend for ever establishing things like with the number 1 and a successor function you can derive the whole of arithmetic, as if not knowing that ever stopped anyone counting. Really important things just pass them by.