In my last column, I tackled the topic of beauty. I tried to identify whether beauty lies in the eyes of the beholder (subjective beauty) or whether it is universal (objective beauty). And like a good academic, I found a middle ground by claiming that beauty arises from the relationship between the object and the subject and is ultimately an expression of our longing for unity among humans, with the universe and, if one believes, with God.
How do we then identify the characteristics necessary to stimulate such longing? I want to take a rather unusual example, namely mathematical beauty, to attempt an explanation. The experience of mathematicians is very similar to the experience of every child you have ever known. At first a child will play with a simple game; maybe she will color a picture with her crayons, and she will find pleasure in her ability to choose nice colors. As the child grows, colors won’t give her enough stimulation, and she will gain pleasure from showing to others and to herself that she can “stay within the lines.” As time goes by, even this game will lose appeal, and the youngster will want to draw her own picture before coloring it.
So we see that pleasure and appreciation go hand in hand with being able to deal with increasingly difficult tasks. Similarly, in mathematics, part of the “beauty” of a problem has to do with the difficulty of the problem. If the problem is too easy, our young painter, now grown into a young mathematician, will not find herself sufficiently stimulated.
But difficulty alone is not enough to provide the stimulus; the problem also must have some sort of mysterious quality that the child-mathematician finds of interest. Mystery, thus, is an important element for beauty. This is something that, for example, painters such as Dali, Magritte, de Chirico and Ernst have cleverly exploited. In mathematics, mystery abounds and children can find it in the simplest properties of numbers and of geometric figures, if only they have a teacher willing to spend some time with them.
But mystery is not enough; there must also be a way for the mathematician to see behind that mystery and to discover an answer that must be surprising. The element of surprise is fundamental in mathematical beauty.
Let us then consider a specific example that I used on my young children with some success. Take a three-digit number, for example 375, flip its digits to 573 and subtract the smallest from the largest. In this case you will get 198. Now flip those digits to 891 and add the two numbers together. You will get 1089. Now do this yourself with several three-digit numbers, such as 523, 438, 679… What is going on? Now that you are surprised, can you try to understand why this is happening? Can you find a general formulation to describe what is happening and why? Can you play a similar trick if you take four-digit numbers?
This simple example explains both the notion of mystery (why is this happening) and surprise (at least the first time you see this problem). The answer to this question is not difficult, but if you discover it, you will feel a pleasant rush of accomplishment, and you will identify a sort of “elegance” in the answer. Elegance – intended as the opposite of clumsiness and ostentation. An elegant dress is not baggy or flashy; its prototype is the famous “little black dress” that every woman has in her wardrobe. That’s what a mathematical proof needs to showcase: restraint, simplicity, power.
As you will see with the example above (you can write to me at struppa@chapman.edu if you want to discuss the answer), the explanation for what happens displays exactly these three traits: it is simple, not using more than what is necessary, and it is powerful, because the explanation will actually bring about a deeper understanding of all numbers, not just this simple game.
How about the argument regarding unity? For one thing, you will see that the answer you have obtained shows you a general way of thinking about numbers, a way that gives you a unified way of looking at them, and that has important applications. The idea behind the proof is at the basis of the error-detecting codes that we see in barcodes every day. In other words, the answer you have obtained shows you that numbers obey some kind of general rule that applies to all of them.
More than that, I think you will see how, after figuring out the answer, you will want to discuss it with your kids, your spouse, maybe even your co-workers. A really satisfactory understanding calls for sharing, and that brings you closer to your friends. Time for a mathematical group-hug!
