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by: Brian Vuyk, student at Redeemer University College

Mathematics, as defined by the Encyclopedia Britannica is the ‘Science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects.’ For thousands of years many scholars have struggled to understand the nature of the world around us, and the reality we are presented with. Because the nature of this study is to understand and define what we understand as reality, there is a certain measure of philosophical implications present, concerning not only the nature of our world, but also the nature of mathematics as a system of study and reality.

During the course of study of geometry, we are introduced to two main systems of geometry: Euclidean and hyperbolic geometry. Both of these systems are equally consistent, relying on each other for proof of the consistency of the other.1 Only one of these geometries may hold in any given system; there are theorems in each that would contradict if both geometries were to hold with equal authority2. This leads, however, to the question of which of these geometries holds true in the world around us?

Euclidean geometry seems to be the natural fit for the world around us, although this may very well be due to the fact that most elementary classes in geometry teach a strictly Euclidean model for geometry, emphasizing such theorems as the existence and uniqueness of parallel lines. And it appears, at least to our eyes, that we can see such shapes such as rectangles in the world around us. Engineers and architects handily use trigonometry and other mathematics in their work, resulting in robust designs. Therefore, it seams quite intuitive to assume that Euclidean geometry is the model for nature.

This may not be the case, however. Humanity as a whole plays a minuscule role in the organization of the universe; that which we may observe can hardly be any fair representation of the universe as a whole. What we interpret as Euclidean results may only appear so due to the trivial size of the line we are considering. Indeed, we have no way of observing any large3 geometric phenomena with relative abstraction from the triviality of our natural observations.