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A Whiteheadian Approach

Jay McDaniel

Mathematics is the intellectual discernment and exploration of forms of patterned beauty that can be quantified and expressed symbolically, as in the equals sign ("=") or the infinity symbol (∞). The act of discerning and exploring is a verb; the forms of patterned beauty are, in their way, the nouns, although vibrant with power of their own.

The symbols used in mathematics do not encapsulate the full essence of the patterns they represent; they merely point toward forms of beauty that transcend their visual representation. The symbols act as shorthand for mathematical and logical entities, relationships, and contrasts that are sought and discerned. The mathematical mind can think without relying on them.

The beauty of mathematical patterns lies in the harmony and contrast of abstract, non-visible forms, which are entertained conceptually. Although these forms cannot be seen with the eyes, their*embodiments* can be observed. For instance, the Fibonacci sequence can be observed in objects that display its pattern, yet the sequence itself transcends any visible manifestation. Some patterns materialize in the physical world and can be perceived, while others remain abstract, existing as potentials for embodiment.

The ontological status of a mathematical form is thus one of potentiality, not actuality—it is something that*can* be actualized, even if it is not actual. Whitehead describes some of these patterns as timeless, referring to them as "eternal objects" or "pure potentials," which reflects the Platonic side of his thought. Here, "eternal" refers to non-temporal rather than everlasting. Other patterns, which Whitehead calls "impure potentials," are more closely tied to the world, felt as relevant but not yet realized. He calls them "propositions" or ideas.

The joy of mathematics lies in appreciating the beauty of these abstractions, whether pure or impure, highly abstract or directly relevant to the world. Although mathematical objects are not agents and cannot actualize themselves, they possess inherent forms of order, which contribute to their beauty. The apprehension of these objects is an intellectual activity in its own right, a form of "prehending" or feeling their presence—specifically a "conceptual prehension"—which carries an emotional tone or "subjective form" in Whitehead's terms. This tone evokes amazement, appreciation, wonder, and even humility in the presence of potentialities that are beautiful in themselves.

The more a person feels this sense of wonder in the presence of patterned contrasts of the mathematical kind, the more they partake in the mind of a mathematician, even if they are not one by profession. The mathematical mind is not fundamentally different from the artistic mind. At its core is a shared sense of beauty, intellectually apprehended. Both disciplines—mathematics and art—are united by an appreciation for harmony, contrast, and order, revealing an intrinsic connection between the aesthetic and the intellectual.

The symbols used in mathematics do not encapsulate the full essence of the patterns they represent; they merely point toward forms of beauty that transcend their visual representation. The symbols act as shorthand for mathematical and logical entities, relationships, and contrasts that are sought and discerned. The mathematical mind can think without relying on them.

The beauty of mathematical patterns lies in the harmony and contrast of abstract, non-visible forms, which are entertained conceptually. Although these forms cannot be seen with the eyes, their

The ontological status of a mathematical form is thus one of potentiality, not actuality—it is something that

The joy of mathematics lies in appreciating the beauty of these abstractions, whether pure or impure, highly abstract or directly relevant to the world. Although mathematical objects are not agents and cannot actualize themselves, they possess inherent forms of order, which contribute to their beauty. The apprehension of these objects is an intellectual activity in its own right, a form of "prehending" or feeling their presence—specifically a "conceptual prehension"—which carries an emotional tone or "subjective form" in Whitehead's terms. This tone evokes amazement, appreciation, wonder, and even humility in the presence of potentialities that are beautiful in themselves.

The more a person feels this sense of wonder in the presence of patterned contrasts of the mathematical kind, the more they partake in the mind of a mathematician, even if they are not one by profession. The mathematical mind is not fundamentally different from the artistic mind. At its core is a shared sense of beauty, intellectually apprehended. Both disciplines—mathematics and art—are united by an appreciation for harmony, contrast, and order, revealing an intrinsic connection between the aesthetic and the intellectual.

Two Kinds of Potentialities

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 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

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

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