What is the essence of mathematics?

Mathematics is a tool for understanding the world, and indeed, the most scientific tool. I recommend a popular science book on mathematics published in 1941, What Is Mathematics? An Elementary Approach to Ideas and Methods, often hailed as one of the greatest guides to mathematical thought. Unlike textbooks that simply teach problem-solving, this book elucidates the essence of mathematics from a conceptual perspective.

The first edition of this book had two authors: Richard Courant and Herbert Robbins. Richard Courant was an expert in partial differential equations, while Herbert Robbins was an expert in probability theory. Both authors agreed that in mathematics, geometric intuition is more important than rigorous methods, and that the purpose of mathematics is to provide humanity with the tools to fully understand the world. Complex formulas are merely the external manifestations of mathematics, not its essence. The second edition of this book was published in 1996. British mathematician Ian Stewart strongly agreed with the views of Richard Courant and Herbert Robbins, and added introductions to the Four Color Theorem and Fermat’s Last Theorem, supplementing the original text with important concepts from the development of modern mathematics since the first edition.

For a considerable period in history, mathematics and philosophy were considered separate subjects. This book also includes some content on the philosophy of mathematics, using examples to explain the deeper background of mathematical concepts and guide readers’ thinking. Being able to solve problems and apply formulas does not equate to understanding mathematics. To truly learn mathematics, it is best to understand the origins of these mathematical concepts and how these great mathematical ideas came about.