Mathematics and Beauty

Why are numbers beautiful? It’s like asking why Beethoven’s Ninth Symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.

—Paul Erdös

I think that most people can agree that beauty can be found in music. People may have different taste, and find different pieces of music to be beautiful for different reasons, but I doubt that there are many people that could not name a piece of music that they think is beautiful.

The same could be said for other forms of artistic expression. There are beautiful paintings; beautiful sculptures; beautiful novels, poems, and stories; even beautiful buildings and foods. People are used to, and comfortable with, the idea that art and artistic expression can be beautiful.

On the other hand, for many people mathematics is a nothing more than a tool. They learn math in school, and are told that it is useful; that it is a tool for finding answers. They may have heard at some point in their lives that math is beautiful, but they probably dismissed that idea—how can a bunch of black and white scribbles on a page be compared to beautiful things like a flower or symphony?

There are brilliant men and women who see beauty in mathematics. Paul Erdös, quoted above, was one of the most productive mathematicians of the 20th century. The quote above amply demonstrates that he saw beauty in mathematics. Einstein once stated that ugly mathematics had no place in this world; that mathematics should be beautiful. Lewis Carroll, the author of Alice in Wonderland had a love for mathematics, and hid little mathematical references throughout his work. So there are people who can find beauty in mathematical expression.

Unfortunately for the rest of us, these men were all intellectual giants. They saw the universe from a perspective that is beyond the ken of we mere mortals. Clearly, they were different from us, and could appreciate a world that most of us will never see. It is, no doubt, their genius that allowed them to appreciate the beauty of mathematics.

Or is it? Is an appreciation truly reserved only for the greatest among us?

To this question I must respond with an emphatic “No!” The ability to see the beauty in mathematics is not dependent upon deep intellectual insight or an inborn genius. It only requires an open mind and the proper training. Like all other forms of art, mathematics is a human creation. The ability to see beauty in mathematics is an effect of human culture, rather than some inborn, genetically determined trait. That is, appreciation of the beauty of mathematics is an artifact of nurture, rather than nature.

It is my belief that most people in the United States are never taught to love mathematics, nor to appreciate the beauty or elegance of mathematics. As students, they are taught arithmetic and other mathematical tools, and told that these tools are complicated and hard to use, but that they will be necessary to them at some distant point in the future. They internalize these statements as fact, and react with apprehension whenever they have to “do math” in the future. Mathematics is reserved for mundane and tedious tasks, such as balancing a checkbook or calculating fuel milage. These are tasks that are better suited to calculators and computers.

When students are taught that these kind of manipulations constitute mathematics, is it any wonder that they cannot see any beauty in math?

The important thing to remember is that mathematics is much deeper and more interesting than we have been led to believe. In fact, if you were to quiz a mathematician, she would almost certainly claim that the boring formulae and manipulations taught in primary and secondary school are not really mathematics. Rather, they are the results of mathematical inquiry, devoid of context or real meaning. The arithmetic, algebra, and geometry of most curricula are but mere shadows of mathematics.

In reality, mathematics is less about manipulating symbols than it is about finding a certain kind of truth. Math is about asking questions, and finding simple and elegant solutions to those problems—solutions that are provably true. Many artists attempt to depict truth in their photographs or concerti, but how many can claim to produce works that are objectively, provably, unimpeachably true? In many senses, this is the unique domain of mathematics.

In the pit of my stomach, I know that there is beauty and clarity in mathematics. Mathematicians have built amazing little worlds in their minds—worlds that follow simple, regular rules, but which can produce stunning complexity and euphony. I know that this beauty is there, and it saddens me every time I meet someone who cannot see it. Being told that the Pythagorean Theorem is boring or stupid is like being told that Shakespeare’s MacBeth is dull and pointless.

As a mathematics educator, part of my job is to go beyond the requirements of government-devised standards and committee-created curriculum. Certainly, my students need to understand the basic manipulations of arithmetic and algebra, but if they leave my classroom without any deeper appreciation of mathematics, then I have not really done my job. To be successful, I need to show my students the beauty of mathematics. I need to give them context and reason. I need to open their eyes to the aesthetic of rational thought and logical reasoning.

In a sense, the purpose of this wp is to extend that mission beyond the classroom. I know that anyone can find beauty in mathematics, if only a way can be found to open their minds and hearts to it. In the future, I hope to examine some problems in mathematics that are accessible to anyone with an elementary education in mathematics, but which exemplify some of the most beautiful concepts and ideas that human thinking has produced.

If one can learn to love the music of Mozart, or the paintings of Kandinsky, or the poetry of Bob Dylan, then surely one can learn to love the elegance of Euclid’s geometry, or the simplicity of Cantor’s infinity. My hope is to act as a guide on that journey.