THE ESSENCE OF UNDERSTANDING: A JOURNEY THROUGH PERSPECTIVE AND PATTERNS

 Understanding is one of the most profound goals of human life. Across disciplines, endeavors, and conversations, we aim to understand—whether it’s the natural world, the people around us, or abstract concepts. But what does it truly mean to understand something? At its heart, understanding requires the ability to shift perspectives. This claim not only reshapes how we approach knowledge but also ties deeply to the essence of mathematics, a field that often appears rigid but is, in reality, a profound exploration of patterns and perspectives.

Mathematics: Beyond Numbers

For many, mathematics is synonymous with numbers and operations—addition, subtraction, multiplication, division, algebra, and geometry. While these elements are foundational, they do not capture the true essence of mathematics. Mathematics is, at its core, about discovering patterns, representing them with a language, and exploring what emerges when we play with assumptions and representations.

Consider a pattern formed by simply drawing straight lines. At first glance, the lines appear unrelated, but as we extend and rotate them, a deeper structure—like a parabola—emerges. By changing our perspective, we uncover the hidden beauty of the pattern. Mathematics is filled with such revelations, from tie knots and shoelaces to the symmetry of platonic solids and the rhythms of sound.

Representation: The Key to Understanding

Representation lies at the heart of mathematics and understanding. It allows us to give form to abstract ideas. Whether it’s Leibniz’s notation for calculus, a visual curve, or even musical notes, representation transforms intangible patterns into something we can see, hear, or feel.

For instance, the number four-thirds can be expressed in various forms: as a fraction, a decimal, a ratio, or even as a rhythmic pattern in music. Each representation offers a new way of understanding the same concept, revealing its multifaceted nature. This ability to shift between representations is not just a mathematical exercise but a pathway to deeper understanding.

Perspective: The Path to Insight

The act of changing perspective is what enables understanding. Every mathematical equation, such as  x + 𝑥 = 2⋅𝑥

x+x=2⋅x, is essentially a metaphor, an analogy between two perspectives. One side views the equation as addition, the other as multiplication. By holding both perspectives simultaneously, we gain insight into the relationship between them.

This concept extends beyond mathematics. Imagine viewing the Earth. From one perspective, it’s a planet in the vastness of space. From another, it’s an ocean with waves. From yet another, it’s a shore that we can feel, smell, and hear. Each perspective adds depth to our understanding. Similarly, in mathematics, when we view a structure—like an octahedron—from multiple angles, we learn more about its properties and relationships.

Mathematics as an Act of Imagination

Understanding requires imagination. In mathematics and computer science, imagination is essential for exploring patterns and structures. For example, envisioning the rhythms of four-thirds, tracing their visual representation, and connecting them to the geometry of a sphere are acts of creative exploration. These disciplines, often perceived as rigid, are among the most imaginative art forms, demanding a fluid and flexible mind.

Imagination also connects mathematics to empathy. When we imagine the world from someone else’s perspective, we practice empathy. This requires the same mental flexibility as shifting perspectives in mathematics. Both acts deepen our understanding—of others and of the world around us.

Lessons for Life

To truly understand something, whether it’s a mathematical concept or another person’s feelings, we must embrace the art of changing perspectives. This practice fosters flexibility, openness, and creativity. By training our minds to see patterns, represent ideas, and view them from multiple angles, we become better equipped to understand not only abstract concepts but also the complexities of life.

Mathematics offers a unique avenue for developing this skill. It invites us to play, experiment, and explore, transforming our minds in the process. As the saying goes, “Have a mind like water”—adaptable, reflective, and always ready to flow into new perspectives.

Understanding, then, is not a destination but a journey—a continuous effort to see the world through different lenses and uncover the beauty of its pattern.