Mathematics as a language of nature

This book is about the fundamental equations of the physical sciences as inspiring fruits of the human quest for understanding the universe. These equations are compact statements about the way nature works, expressed in the language of mathematics. As such, the equations we are going to discuss cannot be derived based on logical reasoning alone: they have resulted from a critical dialogue between the observation of nature and the intuition and creative thinking of some great minds.

Yet, this book does not aim at teaching either mathematics or physics in exact terms. We present the equations as plainly as possible, without much comment on the empirical data justifying them – despite the fact that their existence was revealed to us primarily through the careful and critical observation of patterns and motions in nature. This book is an attempt to convey the excitement and beauty of what these equations tell us.

Physicists use mathematics as a language of nature, a language which has had to be extended regularly when new layers of physical reality were uncovered. Indeed, the natural scientist tends to use mathematics as a language, whereas the mathematician studies it for its own sake.

Expressing relations by relating expressions

The word equation comes from the Latin aequare, which in turn is derived from the word aequus, meaning equal or level. Equations may only assign a value to some variable, but in our general context they express relations between physical variables that characterize the physical system we want to consider and determine their allowed changes in space and time. The relations are mathematically expressed using relational symbols such as the equal sign ‘=’, the greater than ‘>’ and less than ‘<’ symbols. Consequently, one speaks of equations or inequalities, depending on the relational symbol that appears.

There are many equations in science, which have a varying degree of importance. The fundamental equations we will focus on mark radical turning points in our understanding. They are the fundamental nodes in a complex network of ideas embedded in a space of observable facts.

Simple equations and complex solutions

To study a system or structure, generally we encode it in a number of variables like position, population density, temperature, and so on.