What is Mathematics? By Richard Courant

What is Mathematics?

By Richard Courant

This book was recommended to me by my maths teacher. It sets out to teach the reader the basics of mathematics much more advanced than what is taught in secondary school, but without assuming anything of the reader other than a bit of mathematical curiosity. It begins with a discussion of the fundamental concept of number, before moving on to set theory, geometry, topology and an interesting exposition of the calculus, where the Fundamental Theorem of the Calculus is shown to be intuitive and obvious before being proven.

It is relatively unique in that it presents an interesting, stimulating overview of the areas of modern mathematics in a rigorous way, yet avoids unnecessarily technical language. The author, Courant, keeps in mind at all times that he is not writing a textbook, but a “taster” of what advanced mathematics is. Exercises are dispersed throughout the book, and intended to be for the reader’s enjoyment rather than simply for “practise”. It is not essential that the reader complete them, and may in fact be well advised to attempt them only upon a second reading of the book as some are rather difficult.

Importantly, although the book is aimed mainly at non-mathematicians, it does not compromise on the abstract beauty of pure mathematics by emphasising application.

The author’s extensive coverage of Euclidean geometry, an area often forgotten in undergraduate courses, as well the enthusiasm that emanates from his effervescent use of language will make this a refreshing read for someone who already has a strong background in mathematics.