Resources for exploring math without a teacher

The ability to understand the beauty of math requires rigorous study. However, most people do not have access to the kind of training pure math requires.

Many of my friends easily get interested in the subject when demonstrated beautiful proofs and constructs, but I am at loss when they ask for resources to learn more, since we had never used books in our studies, instead, the teacher would show the art of math by proving theorems and building constructs together with the students using only the whiteboard.

I am looking for books or online courses on that are suitable for beginners (more on this below) on the following subjects:

• Intro to sets, mappings, boolean logic, predicate theory
• Number theory
• Rings, fields, groups, vector spaces, Galois theory, etc
• Set theory, measure theory, functional analysis
• Combinatorics, graph theory
• Language theory, lambda calculus
• Other subjects that are not in this list

Books rigorously building the field entirely from ground up (axioms) with detailed looks at all the important proofs with multiple versions and highlighting relations to other fields would be best. It is good if the book has exercises in the form of main story lemmas and side story / related proofs.

Please share your favorite pure math works below.

• Without being cliche, isn't beauty in the eye of the beholder? I mean, a book could just provide an excellent exposition without necessarily emphasizing any beauty, from which the beauty is inherent to anybody who properly understands the argument. I think a good example of this is Milnor's Morse Theory: He does not expound any beauty, but the results are beautiful and that beauty is evident by his presentation. – Tyler Holden Oct 21 '13 at 20:44
• I agree, post edited, thanks for mentioning Milnor's Morse Theory. – glebm Oct 21 '13 at 20:50

1 Answer

Any mathematician could give an exhaustive list of excellent math books, each of which presents the beauty in the subject through the clarity of the prose and insight of the author. Such books likely to be listed include any of John Lee's books, Bott and Tu's book on Differential Forms in Algebraic Topology, Milnor's book on Morse Theory, etc.

However, I don't think any of these truly answer your question, because while they are incredible books, the "beauty" is not openly stated but rather inferred. In my readings, the book which I feel comes closest to your description is

Aluffi, Paolo. Algebra: Chapter 0

It contains all of the features of an excellent book: Clear writing, excellent insight, and a sense of humour. The exercises are excellent, and no rigour is omitted. In fact, the entire book is presented with category theory and homological algebra subtly (and often not so subtly) hiding in the background. This makes it excellent from a `big picture' point of view. Depending on your exposure to algebra though, category theory might be too much abstract non-sense in too short an exposure. Also, if I had to give it any criticism whatsoever, it would be that it is riddled with typos (it's a first edition) and has only a cursory exploration of classical algebraic geometry.

• Thanks, this is the kind I am looking for! It would be great if more mathematicians commented on such books in various fields. – glebm Oct 21 '13 at 21:02