Beauty in Patterns

A Whiteheadian Approach to Mathematics

Beauty in Patterns

A Whiteheadian Approach to Mathematics

Elisha and I were having coffee one afternoon. She teaches mathematics at a local university, and she was telling me about the beauty she found in patterns while working on a complex mathematical problem. She hoped to inspire that passion for beauty in her students. To be honest, I didn’t quite understand the technical details, so I asked if she could explain it in a way I might relate to. She knew I was a musician, so she paused for a moment, then said, "You know how a song has a melody? That melody repeats, changes slightly, and intertwines with the harmonies around it. There’s a structure there, a pattern, but it’s not rigid—it evolves. Math is like that. It’s the recognition of patterns, sometimes obvious and sometimes hidden, that makes it beautiful."

That was a turning point for me. I began to realize that the patterns mathematicians work with aren’t so different from the patterns I experience in everyday life—in music, the changing seasons, and the stories of people’s lives. This was when I stopped seeing math as just counting and calculations. Instead, I saw it as an exploration of patterns that could be found in the world, abstracted from it, and even imagined beyond it.

Growing up, like many people, I thought mathematics was primarily about counting and calculations. However, the more I got to know mathematicians like Elisha, the more I realized that, for them, mathematical thinking is as much an art as it is a science. It’s about recognizing patterns—patterns found in the world, abstracted from it, and patterns that can be conceptually entertained and explored, even if they aren’t embodied in the world.

Influenced by the philosophy of Alfred North Whitehead, a mathematician as well as a philosopher, I began to think of these patterns as, in his words, “pure potentialities”—possibilities that may or may not be actualized in the world but are nonetheless real as potentials. The reality of these pure potentialities is not something we can perceive visually, although the signs and symbols people use to designate them are objects of visual perception. For example, the equals sign (=) is a simple visual representation of balance, but it expresses much more than just a mark on paper. It represents a relationship of equivalence between two expressions, an abstract concept of equality that goes far beyond the visual form of the sign itself. Similarly, the infinity symbol (∞) is a simple visual sign, but it represents an abstract concept far beyond its visual form. It expresses the idea of something that has no limit or end—a reality we can think about and work with in mathematics but which cannot be fully visualized or embodied.

Whitehead had another idea that helped me think about mathematics. In his book

*Process and Reality,*he distinguished between at least two kinds of pure potentialities: (1) potentialities dealing with “extensive relations” between objects in the world, such as whole-part relations or symmetry, and (2) potentialities for feeling and emotion, such as anxiety, compassion, repulsion, and hope. These are the kinds of potentials we experience in interpersonal relations and when we reflect upon the feelings and emotions of characters in plays and films. It seemed to me, as it did to Whitehead, that mathematics deals with the first kind of potential, and the humanities with the second, although, of course, they interweave. One place they come together is in the theater arts, where the performance of a play depends on set design (potentials of the objective kind) and the emotions of a character (potentials of the subjective kind). What makes a performance effective is its integration of both kinds of potential. A performance of Shakespeare's

*Hamlet,*for example, can be powerful by virtue of its set design and how the character of Hamlet comes across emotionally.

Whitehead also introduced the idea that humans (and perhaps other beings) derive satisfaction from creating and feeling “contrasts” between things. A contrast is a kind of pattern, too, where difference and distinction are organized meaningfully. These contrasts can be intellectual or spiritual, often leading to a sense of harmony, intensity, or beauty. Contrasts might exist between patterns themselves, between emotions, or between ideas. It seemed to me that mathematicians I know, like Elisha, derive pleasure from contrasts within and between the patterns they explore. In simple terms, they enjoy discovering how different elements of a pattern can coexist, complement, or even oppose one another in meaningful ways. These contrasts create a sense of coherence, connection, and harmony, much like the interplay of melodies and harmonies in music. For them, the beauty lies not only in the individual patterns but in the dynamic tension between them, where complexity and simplicity, variation and repetition, all contribute to the whole.

As I said, mathematics primarily deals with these kinds of objective patterns, and the joy mathematicians feel often stems from the meaningful and coherent contrasts they uncover. In this sense, mathematical thinking becomes a way of exploring both order and variation, seeking out the beauty in how these elements interrelate.

It’s obvious to us all that mathematics has countless practical applications. Mathematicians themselves often distinguish between pure mathematics and applied mathematics. Pure mathematics includes areas such as number theory, abstract algebra, and topology, where the focus is on developing theories and exploring mathematical structures for their own sake, without immediate concern for real-world applications. In contrast, applied mathematics involves using mathematical methods to solve practical problems in fields such as physics, engineering, economics, and biology. Examples include calculus in physics, statistics in the social sciences, and mathematical modeling in climate science. Yet, in both pure and applied mathematics, the joy of mathematical thinking is deeply felt. Whether exploring abstract concepts or solving real-world problems, mathematicians experience a sense of discovery, coherence, and beauty that comes from engaging with patterns and relationships.

I think Whitehead knew this beauty. It is aesthetic, to be sure, but also cognitive. The enjoyment of mathematical beauty includes within it a sense of understanding. It so happens that when we discern patterns in the world—and, for that matter, beyond the world—we feel that we have "understood" something. What do we understand? People will differ. Some students of Whitehead believe that the most abstract potentialities are part of the very mind of God. With Whitehead, they call them "eternal objects." This means that some forms of mathematics are, as it were, exploring the mind of God. But even if not understood in this way, mathematics is the act of understanding something about the world itself, about how it unfolds in patterns and contrasts. The world is not sameness. The world—the universe—is a multiplicity of contrasts and of patterned contrasts. Mathematics is the art of discovering them.

- Jay McDaniel