Popular math books with depth

Solution 1:

Weeks, The Shape of Space

Penrose, The Road to Reality

Gowers (ed.), The Princeton Companion to Mathematics

Poston and Stewart, Catastrophe Theory and Its Applications

Courant and Robbins, What is Mathematics?

Lawvere and Schanuel, Conceptual Mathematics

Shafarevich, Basic Notions of Algebra

Alexandroff, Elementary Concepts of Topology

Calculus , Calculus made easy by Silvanus P.Thompson

Another Fantastic Article that gives a good intuitive start for Algebraic-Geometry is : Algebraic Geometry by Andreas Gathmann

Colin Adams, The Knot Book

Solution 2:

For me, I gain a lot of intuition from a book with many well-drawn and colorful figures and then trying to draw my own for the situations they do not present. Two particular books that stand out for me in this regard are the following,

• Visual Complex Analysis, by Tristan Needham [Amazon Link]

• Discrete And Computational Geometry, by Satyan Devadoss and Joseph O'Rourke [Amazon Link]

Both books have a decent amount of motivation, and in the later book the style is very good for drawing in the reader with pictures and motivation as to why definitions are made and why certain questioned are posed and answered.

If you're looking for a book that gives the motivational explanation behind proofs and specific technical techniques then I would suggest the following,

• Mathematical Proofs: A Transition to Advanced Mathematics, by Zhang, Chartrand, Polimeni [Amazon Link]

Hopefully those are of the style that you are looking for. Note that these are actual mathematics textbooks, not expository writing.

Solution 3:

For algebraic geometry, I enthusiastically recommend Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by Cox, Little and O'Shea (Springer UTM series, ISBN-13: 978-0387946801).

The style is that of a textbook, with theorems, proofs, corollaries and examples, and not that of a popular math book, but it's very well written (in a just-casual-enough style that feels right) and formal prerequisites are minimal. The focus on computational aspects of algebraic geometry is what made it particularly interesting and lively.

Solution 4:

It is hard to ignore (the first half of) Hofstadter's "Gödel, Escher, Bach". Although there are some problems with the mathematics, it presents the basic ideas behind Gödel's Incompleteness Theorem and outlines the proof. (The second half of the book is quite good as well, but it focuses on Artificial Intelligence, and the book as a whole can be thought of as a refutation of John Lucas' stance that Gödel's Theorem precludes the possibility of artificial (mechanical) intelligence.)