Thoughts on Mathematics

I’ve been working in math for a while, and one thing fascinates me deeply: how definitions and naming things actually help build intuition. It’s amazing how much the choice of words shapes our understanding. For instance, consider population growth. If I say “growth is moving up this year,” people might conclude that next year the population will definitely be bigger. But if I instead say “growth rate is higher this year,” it doesn’t guarantee the population is bigger—it just says the rate of change is higher than before. Subtle differences in naming change how we reason about the phenomenon. This is not a perfect example, but it illustrates how definitions guide thought.

From my experience, it’s rare to find other fields with this level of rigor in defining concepts. Psychology, law, or even social sciences grapple with definitions, but often they are slippery and open to interpretation. In math, definitions are precise; once a concept is defined, everything else follows logically. This is why someone once said, “math is the study of science with cheap experiments.” You can explore consequences of definitions safely, without spending billions or waiting for nature to cooperate. Imagine physicists naming a particle a “massless proton” or “light particle” and then designing experiments ignoring its tiny mass. Years and billions of dollars later, they realize the particle does have mass, and theory must be revised. In math, by contrast, the cost of a “wrong definition” is mostly cognitive, not financial.

This property of math comes from its abstractness and detachment from the real world. We can study a theory that seemingly has no immediate application, yet often, as Wigner pointed out in The Unreasonable Effectiveness of Mathematics, these theories end up being useful. Studying families of sines and cosines may seem purely abstract, but they later become essential for understanding sound, radio signals, or even plant growth patterns. Nature seems to repeat structures and patterns, perhaps because it is computationally efficient—or maybe God just doesn’t mind efficiency. Either way, our abstract study of patterns often becomes a powerful tool for interpreting reality.

The question of whether mathematics is invented or discovered is tricky. I tend to lean toward invented. Nature may repeat patterns, making it feel like discovery, but we are the ones who express and explore these patterns. Take the Pythagorean theorem. Ancient people may have thought it was discovered because it describes a fact about the 2D world, such as measuring a triangle. Modern perspectives suggest invention, because once the idea exists, we can build formal axioms and theorems around it. Perhaps discovery and invention form a cycle: we invent abstract patterns, then discover their reflection in reality. In other words, everyone using the abstract tool would end up with the same results, but the study of these patterns—choosing what to define and explore—is an act of invention.

Consider studying n-dimensional Euclidean space. This is highly abstract. Yet, it turns out that nature often seems to extend dimensions to explain things efficiently. For example, Euclidean distance in n dimensions can capture some property of an object or phenomenon, even if the “distance” itself may not be physically useful. Imagine measuring the straight-line distance through the Earth between Mount Everest and the Pyramid of Giza. In practice, this is meaningless for travel. Walking along the surface, one path might be shorter, and a traveler could “win the race,” even though Euclidean distance assigns the same value to both paths. Abstract math doesn’t care about practical constraints—it provides an objective, consistent measure. Its relevance depends on the context, but it gives a framework to reason rigorously.

I think this is what makes mathematics unique and powerful: it allows us to invent abstract structures and explore them safely, while simultaneously preparing us to discover patterns when they appear in the real world. Its abstraction frees us from immediate reality, yet its structures often anticipate reality in surprising ways. Studying patterns for the sake of understanding them—whether or not they are immediately useful—becomes an investment in the future. When reality presents a problem that fits a pattern, we are ready.