See also the music video I made: https://holdenlee.wordpress.com/2014/02/08/time-bending/
Godel Incompleteness Theorem (informal version): There are true statements in mathematics that cannot be proved.
One of my students told me he was disturbed when he learned this theorem: he had always seen math as a way to get at all truths; if this was false, what was there left to have faith in?
A week later, listening to music, I came upon Loquat’s song “Time bending.”
There’s so many of them, and they’re so far away. We can’t bend time to get there, and can barely even see. The pictures they take give us clues, yet each one is from the past. I don’t know my purpose in this, but it’s obvious it’s minuscule at best.
We can never visit every star in the universe because of a built-in limitation (the speed of light), and we can never find all truths in mathematics because of a built-in limitation. But the song isn’t gloomy, it’s beautiful. Why?
It’s a common theme, that there exist things we will never know. An existentialist might say: if you can never finish, what’s the point of even taking the first step? The number of stories that can be written seems exponentially many; you can never exhaust the space of possibilities; so what’s the point of writing one book, or even fifty?
But Murakami said, “Perfection is the limitless accumulation of the imperfect.” (One way people deal with this is with religion or spirituality. Define something that encompasses infinity, and call it God. But taking a more humanistic viewpoint,) Maybe because we can’t discover everything, the things that we can discover are beautiful. Before Godel, mathematicians thought math was a fortress immune to the concept of unreachable infinity. (Maybe there are proofs that are too long to write down by a human so could practically never be discovered, but that was besides the point. What matters is whether it’s possible in theory. Math problems of practical importance are pretty much all decidable. But when we talk about in theory, we also mean we care about everything.) We find out it isn’t quite the fortress we thought it to be. But math is still special. Firstly, it’s a language powerful enough to express its own limitations – the statement of Godel Incompleteness is itself a mathematical theorem. Since his time, many math problems have been shown to be “undecidable,” and even that is a mathematical statement. Second of all, the things that we do prove with math rest on absolute truth – at least, a truth more absolute that what we express with other domains. People have discovered a vast amount of beautiful ideas in mathematics, and Godel doesn’t make them any less beautiful.
Or you can take this as a reminder that we shouldn’t look to mathematics for the source of all truth. Because it’s just one lens of looking at the world – writing, for instance, gives a window into more humanistic truths. (Even though it won’t, of course, tell you about the math statements that are true but can never be proved.)
We know some things can’t be done, but it doesn’t stop us from trying – I think that’s beautiful.