“It is the reinstatement of the vague and inarticulate to its proper place in our mental life which I am so anxious to press on the attention.” – William James

Vagueness is hard to defend. To be vague is to be imprecise, unclear, ambiguous. In an age that worships precise information, vagueness feels like willfull laziness.

And yet, as William James pointed out, vagueness is not without virtues. Sometimes, precision is dangerous, a closed door keeping us from imagining new possibilities. Vagueness is that door flung wide open, a reminder that we don’t yet know the answer, that we might still g...

Here is a clever water fountain in Japan.

What did I notice at first? Look at the spaces that make the letters as they fall. They get bigger. Why?

How about we start with a slightly simpler case. Suppose I build a water fountain that just releases two drops of water one after the other. Maybe the the second drop is released from the same point, but 0.2 seconds later. It seems to make sense that the two drops would stay 0.2 seconds apart. And they do.

Ok, to illustrate what is going on I have created a quick vpython simulation. Here you can see what that would look like.

This does appear...