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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.

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    $\begingroup$ Rudin is not a master book? I beg to differ. $\endgroup$
    – Potato
    Mar 27, 2013 at 22:14
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    $\begingroup$ I humbly suggest my algebra notes, freely available at math.umn.edu/~garrett/m/algebra ... in their favor, I note that I did not feel bound to adhere to all the iconic stuff, did try to give representative worked-out-completely examples (rather than have cryptic exercises for which no example existed in the "chapter"), and honestly addressed simple, tangible cases, rather than "standard abstractions". Used category-theory ideas without the burden of formalization. Stuff like that. I only included things that have mattered in my mathematical life... $\endgroup$
    – paul garrett
    Mar 27, 2013 at 22:15
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    $\begingroup$ Perhaps you would add more specificity than the quality of exercises as to what your particular criteria are. Keep in mind that most widely used texts, Rudin included, are not in that position by accident, and you can be comfortable that most are written by masters. $\endgroup$
    – user12802
    Mar 27, 2013 at 22:17
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    $\begingroup$ @Potato I love Rudin, and it is a marvel of compactness in the "definition/theorem/proof" form, but it is very short on motivations. I tend to love that sort of book, but I understand the need of others to have a more intuitive approach alongside the hard math. Still, it's a bit silly for OP to deny it "master work" status just because it is not to his taste. $\endgroup$
    – Thomas Andrews
    Mar 27, 2013 at 22:39
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    $\begingroup$ I strongly recommend Paul Garrett's notes. Apart from their high mathematical quality, their format as small, self-contained, easily digestible units make for very enriching and yet pleasant reading. $\endgroup$
    – Georges Elencwajg
    Mar 27, 2013 at 22:56

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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.

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    $\begingroup$ This sounds really nice; this is very similar to Feller's approach, a very good expositor with fresh insights. A very simple example: " Most probability situations are just equivalent to placing r balls in n boxes" This might sound intuitive. We'd all mentally use this approach in solving problems without noticing, but stating it clearly left an impression on me. $\endgroup$
    – kmhrm
    Apr 9, 2013 at 23:14
  • $\begingroup$ Sounds great! I've an interest in reading feller now... $\endgroup$
    – goblin GONE
    Apr 10, 2013 at 11:22
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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.

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  • $\begingroup$ This looks very promising. And if I could, I'd specially double upvote the "Keith Conrad" suggestion. $\endgroup$
    – kmhrm
    Mar 28, 2013 at 19:49
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    $\begingroup$ @kmhrm To the extent you study "Artin," an excellent choice, I would also recommend this set of video lectures by Benedict Gross ( real master) for the Harvard course that closely follows it. At that time it was the 1st edition. While I would highly recommend studying from the 2nd edition, these lectures work in either case and are superb: extension.harvard.edu/open-learning-initiative/abstract-algebra $\endgroup$
    – user12802
    Mar 28, 2013 at 20:24
  • $\begingroup$ @kmhrm It just occurred to me last night to mention these notes on this link: math.uconn.edu/~salisbury which include courses given by Conrad. I have only used the top one on ANT which has some excellent material. I do prefer the "blurbs," for the specific material covered in them, but these can be good. $\endgroup$
    – user12802
    Mar 29, 2013 at 11:19
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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

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  • $\begingroup$ This is definitely in line. thanks $\endgroup$
    – kmhrm
    Mar 27, 2013 at 23:50
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I highly recommend Artin's Algebra. In my opinion, it fits your criteria nicely; topic introduction, concrete example, and then thought provoking discussions that pique your interest. The exercises are fantastic.

If Artin is not advanced enough, then I second Jorge's recommendation for Paolo Aluffi.

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    $\begingroup$ I heard about it, and it has the reputation of a challenging first introduction to Algebra; a look is a must. $\endgroup$
    – kmhrm
    Mar 28, 2013 at 19:40
  • $\begingroup$ @user18921: I can see where you are coming from. However, I feel that Artin's prerequisite is not so much intelligence as it is maturity. I feel that he really bends over backwards to explain the material to well prepared students. The text is meant for strong students, for sure, but the preparation is mostly with regards to maturity. The desire to work really hard is most certainly a prerequisite. $\endgroup$
    – user59083
    Apr 9, 2013 at 1:51

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