Poetry and math: pretty even if you don't quite get it

Poetry and math seem to me to share an interesting feature: it's possible to perceive a poem or a piece of mathematical reasoning as "beautiful" without fully understanding it.

When it comes to poetry, this quality is perhaps experienced most sharply when you read it in a foreign language that you know very well on a basic level, but not quite proficiently. You might read a poem in that language, and realize that you don't quite understand it. Of course, it's perfectly possible for you to not understand a poem written in your native language as well; but in the latter case, it's most likely a different kind of not understanding. With a poem written in your native language, you might not understand its semantics, even though the syntax of its building blocks will be clear to you. After all, it is your native language. When you're learning a foreign language, however, you notice that it's possible to somehow grasp the semantics even though you're not quite sure about the syntax. And so lines we don't quite get can nevertheless seem beautiful.

A similar thing can happen when you're trying to follow a difficult proof which uses techniques that you know "sort of," but not really. You might understand, and appreciate, what is meant to be done without fully grasping all the details of how it's being done.

So, for example, it's possible to know what Dylan Thomas meant when he wrote,

The lips of time leach to the fountainhead;

Love drips and gathers, but the fallen blood

Shall calm her sores.

And I am dumb to tell a weather's wind

How time has ticked a heaven round the stars.

even if you have no idea what it means to tick something round something else, and it's possible to perceive this verse as beautiful, despite the lack of such knowledge. Similarly, it's possible to appreciate the beauty of Gödel's proof of his incompleteness theorems even if you don't understand all the details of how Gödel numbering procedure works.

(Of course, you have to know at least the basics of the syntax of whatever poetical or mathematical language you happen to be reading. For example, my lack of understanding of this or this is so fundamental that no level of appreciation is possible.)

A concept is either coherent or it's not. If we can define it, then we know for sure that it is; however, it's possible for a concept to be coherent even if we can't provide a definition. I think this is why we can appreciate a poem or a proof without understanding it fully: it's because, even though we can't define the thought that's being described, we can somehow feel that it makes sense, that it's a coherent thought. In this interview, logician Rohit Parikh talks about his reaction after reading one of Wittgenstein's works: "I did not really follow what he was saying, but I realized that he was a genius and that someday I must understand him." Exactly.