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The most wonderful book I have ever read in my life was Fearless Symmetry by Avner Ash and Robert Gross, which is a good book that gives an intuition , and reasons behind the introducing fields, need for Galois theory etc.

I am interested in those books which possess the following characteristics ( as possessed by Fearless Symmetry :

  1. A good introduction to the concept, giving the reasons behind introducing theory X or some jargon Y in the arbitrary field chosen.
  2. Requires a little mathematical background behind understanding that book, must be naive-user-friendly.
  3. And must be able to convey the things in a perfect manner.

Any mathematical area is fine with me

To frame in another manner, are there any Analogues of Fearless-Symmetry ? ( in other fields like Algebraic Geometry , Topology....etc)

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@Iyengar Since you're asking for books similar to Fearless Symmetry, you may be interested in Elliptic Tales: Curves, Counting, and Number Theory by the same authors of Fearless Symmetry. It attempts to explain the Birch and Swinnerton-Dyer conjecture apparently in a similar style to the one they used in Fearless Symmetry. – Adrián Barquero Feb 3 '12 at 16:41
To all: I have cast the final vote (out of five) to re-open, as this current version is much better phrased. I'm also cleaning up some of the less relevant comments. For further "meta" discussions, please bring it to Meta or to this thread‌​. – Willie Wong Feb 6 '12 at 8:34
@Adrian: since this question has been re-opened, would you care to re-post your comment above as an answer? – Willie Wong Feb 6 '12 at 8:36
@WillieWong : Thanks a lot sir, for re-opening it again. – Iyengar Feb 6 '12 at 16:19

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

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This was a splendid answer, I touch your feet for a mighty answer, let me see how many books add to this list. Thanks a ton. – Iyengar Feb 1 '12 at 17:30
Sir I added Calculus by Thompson, Which would be a good book in providing an intuition , its my opinion, but if you differ from it, you are at your liberty to re-edit and delete my words. Thank you. – Iyengar Feb 1 '12 at 17:44
Added book on Algebraic Geometry. – Iyengar Feb 2 '12 at 18:00
Just going to point out that I would not recommend Alexandroff's topology book. Maybe if you are going to graduate school in topology and are looking for some of the starting ideas in the field, but for me it was unreadable. – Samuel Reid Feb 2 '12 at 18:49
@Iyengar If you loved Fearless Symmetry, then you should read Symmetry and the Monster by Mark Ronan… – Moderat Aug 20 '12 at 12:25

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.

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Really a good collection. – Iyengar Feb 3 '12 at 16:23

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.

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Thank you FYI . – Iyengar Aug 20 '12 at 16:43

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.)

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Thank you FYI . – Iyengar Aug 20 '12 at 16:46

Prime Obsession by John Derbyshire was a fantastic book about the Riemann Hypothesis and the attempts that have been made to solve it. It outlines the history behind the Hypothesis, as well as explaining the incisive reasoning behind the following formula using $n \in \Bbb N$ and prime $p$: $$ \sum_n \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} $$

The book also brings in the physical connections between the distribution of primes hypothesized by Riemann and the particle physics involved in describing atomic nuclei . . . A fascinating book that isn't afraid to get technical.

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Maybe Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen.

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I'd add The Nature of Computation by Moore & Mertens.

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Nice one, Thank you for your kind help. Can you edit the above answer and add it there, so that it would be good to see. I mean being at a same place rather than being discrete, But anyway its left to you. – Iyengar Feb 1 '12 at 17:51

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