A Math book with an inspiring ethos?

I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of elementary(no measures) probability. The book is rigorous "hard" math but doesn't escape from giving a solid intuitive feeling. The author discusses a topic, mentions an example, proposes different scenarios that gives back more math. His first chapter on "nature of probability" is essential. It gives a good feeling for what statistical probability means, and why/how it was defined as it is.

Question: I'm looking for other math books on fundamental mathematics(algebra, real analysis, etc...)- essential mathematics that is not very advanced(algebraic geometry for example) - of high qualities like Feller's probability text. Feller might not be used anymore, but its full of exercises that would make it a working textbook written by a master.

To be specific and not too general. I'm looking exactly for inspiring Feller style books in real analysis and abstract algebra. Rudin is good, but its not a master book. I don't know much about abstract algebra available textbooks/master expositions.

Solution 1:

Goldrei's Classic Set Theory For Guided Independent Study. I don't necessarily think he's the greatest expositor, but his educational philosophy is spot on. For instance, he starts with the real number system and asks: how do we know this system exists? One possible answer is: because the set of all Dedekind cuts of rational numbers can be made into a real number system in a natural way. Okay but how do we know a rational number system exists? Easy: we can build rational numbers as equivalence classes of integers. But wait! Perhaps there is no integer number system. But that can't be, because we can build integers out of naturals. Okay, but maybe there doesn't exist a natural number system.

At this point, the reader has an epiphany. The existence of all the major number systems can be demonstrated using set theory alone - that is, if we can build a natural number system. So if we can build such a system, then WOW! Set theory is POWERFUL. Its only at this late stage in the game that Goldrei actually starts talking about the ZFC axioms. And it works great!

Too many math books start with axioms, or esoteric definitions, without giving the reader any intuition about why they should care. Goldrei's book is a breath of fresh air in this regard. Truly, a remarkable book.

Solution 2:

Here is a link to a free down-load of virtually verbatim lecture notes for a real analysis course taught by Fields Medal winner Vaughan Jones. They were my first introduction to real math - beautiful presentation, lots of motivation:

https://sites.google.com/site/math104sp2011/lecture-notes

Another nice book on real analysis is Pugh's "Real Math. Analysis." An unsung hero, again lots of motivation, excellent pictures for a real feel, and plenty of examples.

In addition to Paul Garrett's excellent notes, here is a link to great material by Keith Conrad:

http://www.math.uconn.edu/~kconrad/blurbs/

What I especially like here aside from the great presentation is the constant pointing out of anticipated misconceptions and many, many examples looking at the topic from many sides. They are relatively short and cover a wide range of primarily algebraic topics at many levels.

Solution 3:

I don't know if this is exactly what you mean but the book visual group theory is a great way to develop intuition in abstract algebra.

Another great book is adventures in group theory where they use mathematical toys to give an insight of group theory

Finally, for a serious text I would recommend Paolo Aluffi