Mathematics as Beauty

Beauty - what is it? We can all point to things we find beautiful; we can even (sometimes) explain why we find them beautiful. But what is that quality called "beauty" in them that makes them so attractive, especially when "beauty is in the eye of the beholder"? Yes, it's a cliche, but it's true: so many people find so many different things beautiful.

Take mathematics, for instance. As a student of mathematics, I am sometimes asked if I studied it because I liked it or because I thought it would be useful. There is no question: I absolutely loved it. But then I'm asked, "Why?" Here, I crumble. Yes, I like "right" answers, and yes, I like logic. But is that all? Am I simply attracted to sterile and rigid ways of thinking? While I agree that there is, in fact, something comforting about knowing that "right" answers exist and that they can be found by the proper use of methods and reasoning, I've always cringed at "right answers" as my first answer; besides, meeting someone who likes getting 7 (every time!) from 4 and 3 doesn't usually make for engaging conversation.

So I'm left wondering why I like mathematics. I think it has something to do with the order and relationship of things. Think of music. There's a sense in which one must play the "right" notes in order to play the song in a way that makes it recognizable. On the other hand, there's no "right" way to play a song. One song may have dozens, or even hundreds, of arrangements, and all of them sound "good" in different ways and to different ears. But the point is this: every distinct arrangement will have enough of the "real" song in it to be recognized as an arrangement of "that" song.

Mathematics is as much about "hitting the right notes" as it is about "playing the arrangement" correctly. The difference is that instead of notes, mathematicians use particular facts, theorems, arithmetic rules, identities, etc., and their arrangement is in using various theorms or facts at various stages in the mathematical argument to arrive at particular conclusions; the song they play is what you see on the board. And I think this is where the beauty of mathematics comes in. It's not only getting the right answer that I enjoy, but in watching it unfold. I am not greatly moved by hearing middle C being played by itself, but I am greatly moved when a musician plays it (and all the other notes) at the right moment with the right strength and tone in the middle of a larger piece of music. Likewise in mathematics, knowing the Binomial Theorem itself is no big deal, but seeing how it can be used in statistical analysis to predict specific outcomes of a Binomial Random variable can be exhilirating! And all the more so when you've tried to predict such outcomes using only algebra and arithmetic.

The right order of things in relationship to each other is crucial in making awesome impressions on us. Movies and literature also illustrate this idea well. If you watched any one scene of a movie or read any one chapter from a book, its impact on you will likely be far less impressive than if you watched every scene of the movie or read every chapter of the book (this is why movie trailers entice people rather than satisfy). But even here, you could watch every scene of the movie out of the director's intended order (by using "scene selections" on your DVD menu) or read the chapters of the book out of order. By doing so, your understanding of the whole story would likely be better than if you had only watched one scene or read one chapter, but the emotive impact would likely be less than if you had viewed or read them in their "right" order; it's more enjoyable watching or reading from beginning to end.

If we only see the scene of Gandalf's fall in the Mines of Moria, its import is diminished if all we think is that an old man has fallen to his death. Frodo's tears will be misunderstood as only lamenting the death of a friend and not also the loss of his protective guide through lands unknown against foes unimagined. And the only way to "get it" is to watch from the beginning and learn who the old man with the beard is; then his loss will mean what it was intended to mean by both the movie's director and the book's author.

Not everyone likes movies, music, or mathematics - and that's OK; there's a lot of things whose beauty I don't appreciate, but that doesn't mean they're not there - it just means that I don't have the right "eyes to see." My point is that there does exist a beauty in each of them, and that it has at least as much (if not more) to do with the arrangement of the individual parts as with the parts themselves.