Essays on Mathematics

"There's no sense in being precise when you don't even know what you're talking about." (John von Neumann)

Prologue

For the typical primary school or high school student, the following definition of mathematics would suffice:

(Naive definition) Starting from obviously true axioms, use obviously correct inference rules to derive additional truths.

In this sense, mathematics is all about discovering indisputable truths. For example, the theorems proved in geometry would be literally true statements about the physical space. Someone may argue that there is no such physical object as a geometrical point, or a geometrical line, but this is no issue because we can reply that geometrical objects are nothing more than locations in the physical space, and thus they can happily exist even if nobody can see them materialized. As for the exotic topic of complex numbers, they can be viewed as a man-made tool that sometimes comes in handy for mathematicians in describing reality.Continue reading

“It is by logic that we prove, but by intuition that we discover.” (Henri Poincaré)

Introduction

It is our day-to-day experience that rigid objects, e.g. two books, can be placed right next to each other, in a way that they touch but don’t overlap. However, if we think of space as composed of points, meaning that space is nothing more than a set of points, the concept of touching becomes different: namely, if two geometrical objects touch each other, they always overlap. For example, when a cube touches a sphere, they will have one point in common where they overlap. This is inconsistent with our intuition that a given piece of space, even a single point for that matter, cannot be occupied by multiple rigid objects at the same time.Continue reading

“In mathematics you don’t understand things. You just get used to them.” (John von Neumann)

Introduction

Although special relativity is a theory of physics, the chief ingredient in deriving its astonishing results about space and time is mere logical thinking. Besides that, only surprisingly few initial experimental facts are needed to develop the theory.Continue reading