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I've bought some books and I guess they're written in a way that presumes a previous background or assignment in some mathematics course - these books just spit the content directly in your face. This kind of book is not suited for me (for a first reading in the subject): I'm not in a maths course, and they wouldn't be even if I were in a maths course because I like to learn the content, It's history and why study it. I've found one book on this class and I'd like to mention it as an example:

  • A book of abstract algebra, Charles Pinter;

The chapter titled Why abstract algebra? is very useful, it contains a light historical background and also some motivation for it. After this chapter the book starts to provide a normal introductory abstract algebra course.

I am looking specifically for books with this spirit, there are a lot of history books about specific fields of mathematics, for example:

  • A history of abstract algebra, Kleiner;
  • Number theory and its history, Oystein Ore.

They are nice books and they can provide some motivation for study, but my impression is that they are more a history book than a textbook that could be used in a undergraduate course of mathematics. You can suggest books in any mathematical subject you wish, but they have to attain to that criteria.

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While studying analysis/calculus, I've found the following book to be quite interesting/motivating, even though I'm not so sure you can master the subject just by reading it alone:

Analysis by Its History, by Ernst Hairer and Gerhard Wanner.

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My suggestion:

Journey through Genius: The Great Theorems of Mathematics by William Dunham.

This book contains within its pages the reason I became a Pure Mathematician. There is one chapter dedicated entirely to $\sum \frac{1}{k^2}=\frac{\pi^2}{6}$ and the ingenuity it took to figure that out. In high school our teacher tolds us the sum converged, but said it wasn't known to what. I found this book in my search for the answer and there it was, an entire chapter just on this one problem.

Every chapter is a gem of mathematics explained within the historical context of its discoverers and its times. I am not sure this is a book for a course, but if you need a reason to be excited or motivated about pure mathematics I strongly recommend it.

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  • $\begingroup$ The books by Dunham are uniformly outstanding in this respect $\endgroup$
    – vonbrand
    Jun 18, 2014 at 8:20
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Willard's "General Topology" has an extensive section devoted to historical notes, providing a background to each topic he goes through.

Goldblatt's "Lectures on the Hyperreals" has a section on historical background in the first chapter.

Unfortunately there doesn't seem to be an English translation, but Jürgen Elstrodt's "Maß- und Integrationstheorie" provides plenty of historical motivation as well as several short biographies of key mathematicians detailing how they helped to develop the field.

Generally speaking, maths books tend to steer clear of history (and quite often any sort of context whatsoever). To compensate I tend to browse Wikipedia (and specialised wiki's like nlab when available) to get some sense of background and how what I'm learning fits into the bigger picture. It's not enough on its own, but it helps to supplement the more streamlined textbooks.

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There's always Lawvere & Schanuel's "Conceptual Mathematics", which is an introduction to category theory aimed at undergraduates. It very much takes the approach of starting simple and motivating each next step.

To a lesser degree, Goldblatt's "Topoi" also attempts to motivate many of the ideas of topos theory, but he does so a bit more rapidly than the book above. This one is less for undergraduates, but is approachable with some determination. I can say I found the author's motivating comments helpful enough to make the material approachable at a time I had no more than a vague knowledge of set theory, so that's something.

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On wavelets, I liked "Analysis and Probability / Wavelets, Signals, Fractals" by P. Jorgersen. It comes with a lengthy "Getting started" and within it, a Glossary that explains -in a no linear way!- the interplay between mathematics (& probability), engineering and physics in the development of the subject over the last decades. The book is at graduate/advanced level tough.

And of course the classic "Ten lectures on wavelets" by Daubechies, Chapter 1; more suitable as text book imho.

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I haven't read too much literature about mathematics, but I read one very good book, called Mathematics: from the birth of numbers by Jan Gullberg, which I found very interesting. It covers everything from numbers to partial differential equations, while assuming no significant understanding of the topics. It also includes a few pages of history of the subject at the beginning of each chapter.

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How about Zermelo's Axiom of Choice: Its Origins, Developments, and Influence by Gregory H. Moore? The table of contents is shown here http://www.apronus.com/math/zermelos.htm.

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There are popular books by Georges Ifrah about the (pre)history of numbers. You cannot pick high math from them, though.

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I know most people on this forum already did Cal. But, I thought I might throw it out. James Stewart's Calculus textbook is awesome. It separates historical/theoretical and sometimes mixes them. Would recommend. (Anyhow it worked for me.)

I'm using Anton's Linear Algebra textbook. Suffice to say I am pretty bored.

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