matrimc wrote:EricD: I have not read Hardy's apology but what I read about it is that it is more of an explanation piece rather than the book length exposition which is What is Mathematics?. I will have to read it one of these days. Also I have downloaded the short article "the unreasonable effectiveness of mathematics" and syet to read it
Yes, sir. Courant and Robbins is far more mathematical compared to Hardy's meta-mathematical "Apology". The positive side is that Hardy is an easy read, all the more so if you skip C. P. Snow's preface. You can read it in one evening in one long sitting.
Courant and Robbins is an awesome book. No question about that. His choice of topics are also excellent, for the most part, except when he speaks of the stuff like isoperimetric problem or Steiner's problem or minimal surfaces(made using soap bubbles), maybe the sort of stuff that fetched Jesse Douglas a Fields Medal. These are important, no doubt, but this is quite an old and unfashionable topic now. This is the general problem with that book; it is too old and has older approaches and an "applied" taste all over.
The first two chapters are ok. As regards Chap. 3, Courant and Robbins made an excellent choice of topic; but his methods are archaic and currently the approach using Galois theory is preferred for proving impossibility of the geometric constructions. As regards chap. 4, he could have connected projective geometry with the rudiments of algebraic geometry and non-Euclidean geometry with modern hyperbolic geometry. Chap. 5 also has a good selection of topics; but he is far too outdated without mentioning homology, de Rham theorem, etc.. Chap 6. is ok except for the zeal in throwing out infinitesimal quantities, and he could have gone further into founding mathematical analysis properly.
The next chapter onwards, Courant focuses too much on the "analytic" approach, throwing out the geometrical or topological approaches. Also he does not go very deep into number theory in the first chapter; e. g., he skips quadratic reciprocity, "algebraic" number theory, etc..