Mathematics: A Blend of Invention and Discovery

Imagine you want to build a bridge. It wouldn’t make sense to place it in the middle of a flat field. You need a river to build the bridge over, where it’s situated naturally. Similarly, in mathematics, we often find ourselves at a crossroads between invention and discovery, exploring concepts that are both naturally occurring and human creations.

The Nature of Mathematical Concepts

When we study mathematics, we often find that it’s both discovered and invented. For instance, when you construct a bridge, the location for it is found based on the natural flow of the river. It’s similar in mathematics, where theorems are discovered, but the methods and proofs themselves are inventions. While we discover the inherent truths that underpin mathematical concepts, we invent the structures, names, and proofs that we use to understand and explain these truths.

Discoveries in Mathematics: The Cases of Pi and e

Consider the mathematical constants, pi; and e. These values are discoveries because they inherently exist in the fabric of the universe. For example, pi; represents the ratio of a circle's circumference to its diameter and appears in countless natural phenomena. On the other hand, the names we give to these constants—pi; and e, and the methods we use to prove their properties—are inventions. It’s interesting to note that the value of 2pi; (theta) has been debated in the context of the Tau Manifesto, which suggests that tau; might be a more natural representation. Similarly, some argue that pi;/2 (eta) is a more meaningful value. Nevertheless, the ratio of a circle's area to the area of its inscribed square, or the ratio of a circular inch to a square inch, all point to the cultural and historical aspect of pi; as an invention.

e, on the other hand, emerges from natural logarithms and is a constant that governs the growth rate in nature. The value e 2.718 arises in experiments involving logarithms, and it’s noticeable when examining the points where logarithmic values align with natural growth rates.

Geometries: A Mix of Discovery and Invention

Another area where the blend of discovery and invention is evident is in the study of geometries. Euclidean geometry, for instance, is a well-established system with a set of axioms. However, the existence of non-Euclidean geometries, such as hyperbolic geometry, suggests that there are other natural geometrical systems beyond Euclidean space. These systems can be conceived as extensions of Euclidean geometry with different curvature properties.

Consider hyperbolic geometry, which is a geometry with negative curvature. In this geometry, the concept of a 'straight line' is redefined as a line of the same curvature as the space it’s in. This leads to a fascinating transformation where shapes and lines behave differently from their counterparts in Euclidean geometry. For example, in a hyperbolic plane, a circle can be considered a straight line if it’s viewed from the perspective of the space itself. This is similar to how a Mobius geometry can be seen as a complete Euclidean plane where all circles are straight lines, due to the property 0/0 being undefined.

Infinities: Conceptions and Realities

Mathematics also grapples with the concept of infinity, which is largely an invention. While real infinities don’t truly exist in nature, the mathematical concept of infinity allows us to explore and understand the limits of systems and processes. Small infinities, or limits, are often the focus of mathematical analysis, providing a nuanced and precise understanding of mathematical behavior at extreme scales.

Conclusion

Mathematics, in its essence, is a harmonious blend of discovery and invention. From the inherent truths that emerge naturally in the world to the structures and proofs we create, mathematics is a fascinating domain that constantly pushes the boundaries of our understanding. Whether we are discovering mathematical truths or inventing new methods to understand them, the journey is as enriching as it is enlightening.