Hardy's "A course of Pure Mathematics" has been highly regarded since it was first published in 1908 because it was an innovative text: rigorous, modern, well-written. Its intended readership was always first year "honours" mathematics students. This book inspired innovation in subsequent generations of textbook writers.<p>However, in the 21st century, this book really can no longer be recommended for its original teaching purpose. As a textbook it is outdated (a term I hate, but it is true). It is now an historical curiosity - although one which I am pleased to own, and the exercises in the book are still worth a look.<p>Calculus teaching has progressed considerably since 1908. The construction of the real number system in Hardy's book, using the Dedekind Cut method is overly complicated - the use of the of Least Upper Bound is much simpler and clearer. Hardy defines the concept of integral solely as the anti-derivative; there is no discussion of Riemann sums, or Darboux sums, etc. I am sure I would not want to take Hardy's approach today.<p>I think we are better off recommending books are more modern.<p>I will start by recommending "Calculus" by Michael Spivak.
I read this book as a first-year undergrad. His style inspired me to go after rigour and proof and was a good start to serious mathematics. I always loved Hardy's work and Hardy and Wright's number theory text was also very nice through my PhD in algebra/number theory. I found Hardy's book much nicer than the contemporary calculus texts with irrelevant pictures and modern-day examples. Just straight math! Not for everyone, but it has classical, austere appeal for those who enjoy such things.
The preface left a deep impression on me because Hardy says only exceptionally talented and intelligent students will be able to finish and understand this book. It was a revelation because I grew up in schools that claimed anyone can learn maths to any level so long as they had enough interest and support. It was healthy as a young man to know I was beginning to approach my limits as a mathematician, and this book ultimately led me to focus more on computer science and programming.
It would be more useful to link to the Gutenberg page that shows links to the various formats:<p><a href="https://www.gutenberg.org/ebooks/38769" rel="nofollow">https://www.gutenberg.org/ebooks/38769</a>
In A Mathematician's Apology (1940), Hardy has lots of fun musings on math.<p>I don't have the quote handy, but he argues that pure math is closer to reality than applied math since it deals with actual mathematical objects rather than mathematical models of physical objects.
Nice book. For another old but excellent math book I recommend Geometry and the Imagination by David Hilbert. No gutenberg remake I'm afraid, maybe because of the numerous (and incredibly high quality) illustrations.
One of my favourite texts. One of those that I found influential early in academics as well as when re-reading later in my career. Even for younger students I think it can be great introduction to more formal approaches, as well as a taste for the austere.
Ive been nerd sniped by point 158 on page 353. I cant believe I slipped through so many calc classes without understanding Leibniz's rule for taking the derivative of a definite integral. I didn't actually follow through with the calculation in Hardy's book but I bet it haunts me until I do :(.
I remember having a hard time understanding limsup/liminf and only actually understood the concept after reading this book. I am sure it is not for everyone due to its age, but I think this book is a much better introduction than baby Rudin.
A bit of background on Hardy, and his choice of title:<p><a href="https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics" rel="nofollow">https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics</a>
From the title page. No Ph.D. necessary to be a legend in them days. ;-)<p><pre><code> G. H. HARDY, M.A., F.R.S.
FELLOW OF NEW COLLEGE
SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE</code></pre>
I think in the book, Hardy mentions that Mathematics is one big tautology.<p>Intial axioms setup a graph structure of theorems and the task of mathematicians is to find shortcuts in that graph.
Alternative link:
<a href="https://archive.org/download/in.ernet.dli.2015.239784/2015.239784.A-Course.pdf" rel="nofollow">https://archive.org/download/in.ernet.dli.2015.239784/2015.2...</a>