By user3002473


2019-02-11 18:43:33 8 Comments

I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.

By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.

I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?

18 comments

@Alexandre Eremenko 2019-02-11 20:33:29

What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.

But still you may try these books:

  1. Michio Kuga, Galois' dream,

  2. David Mumford, Caroline Series, David Wright, Indra's Pearls,

  3. Hermann Weyl, Symmetry.

  4. Marcel Berger, Geometry revealed,

  5. D. Hilbert and Cohn-Vossen, Geometry and imagination,

  6. T. W. Korner, Fourier Analysis,

  7. T. W. Korner, The pleasures of counting.

  8. A. A. Kirillov, What are numbers?

  9. V. Arnold, Huygens and Barrow, Newton and Hooke.

  10. Mark Levi, Classical mechanics with Calculus of variations and optimal control.

  11. Shlomo Sternberg, Group theory and physics,

  12. Shlomo Sternberg, Celestial mechanics.

All these books are written in a leisurely informal style, with a lot of side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:

Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics.

@Amir Asghari 2019-02-11 22:04:55

Kirillov's is "What are numbers?"

@Alex M. 2019-02-11 22:07:57

To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.

@Alexandre Eremenko 2019-02-12 02:58:49

@Amir Asghari: thanks for the correction. I only have the Russian original.

@Alexandre Eremenko 2019-02-12 03:24:31

@Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).

@Tom Copeland 2019-02-12 21:19:17

Computing the Continuous Discretely by Beck and Robins.

Good intro to the interplay of analysis (Fourier analysis and number theory), geometry, and combinatorics.

Google books, pdf

@Tom Copeland 2019-02-18 11:17:04

A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…

@Jörg Neunhäuserer 2019-02-12 16:23:44

I think “Proofs from THE BOOK” of Aigner and Ziegler may be of some interest. Although the book is not primarily historical, it contains such aspects as well. The book is casual. I do not know if it is advanced or too elementary for you.

@Timothy Chow 2019-02-12 15:08:08

Based on your first paragraph, I would highly recommend the series What's Happening in the Mathematical Sciences. These provide excellent summaries of a wide variety of cutting-edge mathematics topics. The authors are mathematicians and so the accuracy of the discussion is very high and there is enough detail to satisfy the casual interest of a mathematician, but they also don't get bogged down in too much detail.

There have even been times when I wanted to gain a thorough understanding of a new and unfamiliar toipc, and I found that the introduction in What's Happening was better than any other introduction I could find. Of course I then I had to turn to more technical texts for more detail, but the overall perspective provided by the What's Happening article was invaluable.

@Moritz Raguschat 2019-02-12 14:44:39

I enjoyed Allen Hatcher's "Algebraic Topology" very much. It's free. You can find it online as a PDF on his university's webspace: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf Hope it will stay up there for long. He's been retired for a while now.

That was maybe the most enjoyable math book I've read.

@EFinat-S 2019-02-12 15:46:32

I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.

@Moritz Raguschat 2019-02-16 15:05:15

Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.

@Mike M 2019-02-12 10:29:03

James Gleick - Chaos: Making a New Science is a popular history I still remember from 25 years ago.

@Gerry Myerson 2019-02-11 20:46:13

Birth of a Theorem: A Mathematical Adventure, by Cedric Villani. I've linked to a review.

Also: Diaconis & Skyrms, Ten Great Ideas About Chance. I quote from the Preface:

This is a history book, a probability book, and a philosophy book. We give the history of what we see as great ideas in the development of probability, but we also pursue the philosophical import of these ideas....

At the beginning of this book we are thinking along with the pioneers, and the tools involved are simple. By the end, we are up to the present, and some technicalities have to be at arms length. We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper.

@kodlu 2019-02-12 09:04:49

Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase.

https://www.springer.com/gp/book/9783540852971

"A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin Gardner

@Jay 2019-02-12 03:59:34

For a casual book on cryptography and the mathematics behind it, I'll recommend Simon Singh's 'The Code Book'. The concept of modular mathematics and public key cryptography are explained beautifully.

@Conrad 2019-02-12 03:47:03

I would add 3 favorites:

The first two (more or less continuing one another) are by W Narkiewicz and are quite comprehensive combining history, some proofs, excellent references and state of the art results up to ~2010:

The Development of Prime Number Theory : From Euclid to Hardy and Littlewood

Rational Number Theory in the 20th Century: From PNT to FLT

Finally a superb exposition with proofs, explanations, history, from D Choimet and H Queffelec:

Twelve Landmarks of Twentieth-Century Analysis

@EFinat-S 2019-02-12 15:38:53

Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".

@HYL 2019-02-12 03:40:41

For a casual and beautiful promenade, there is A Singular Mathematical Promenade by E. Ghys.

For French readers, the collection Leçons de mathématiques d'aujourd'hui (there are now five volumes) presents a panorama of various subjects and research domains in (mostly pure) mathematics accessible to graduate or advanced undergraduate students.

@Crayoneater 2019-02-12 03:21:38

Silvanus P. Thompson's Calculus Made Easy is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by 2-3 decades.

@EFinat-S 2019-02-11 22:01:33

I think that the "Number Theory" series by Kato, Kurokawa, Kurihara and Saito fits here. They are beautifully written and require only some undergradute algebra and analysis. They were published (translated) in the AMS Translations of Mathematical Monographs:

Number theory. 1. Fermat's dream

Number theory. 2. Introduction to class field theory

Number theory. 3. Iwasawa theory and modular forms

@David Campen 2019-02-12 00:24:03

I find the Carus Mathematical Monographs to be in this category.

Also, by Julian Havil - "The Irrationals" and "Exploring Gamma".

@Alexandre Eremenko 2019-02-12 03:22:07

Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.

@Alexandre Eremenko 2019-02-13 18:19:38

The title is "Gamma. Exploring Euler's constant".

@Melquíades Ochoa 2019-02-11 22:10:29

In 'How to bake a π?', Eugenia Cheng provides a nice view of mathematics in general and category theory in particular that may fit your definition of 'casual'. Either way it is a very nice reading.

@Vidit Nanda 2019-02-11 22:09:08

Elementary Applied Topology by Ghrist does a fantastic job surveying recent trends in the application of (co)homological methods to practical science and engineering. It goes all the way from Euler characteristic to sheaf cohomology.

Oh, and the illustrations, depending on how far you get with deciphering them, are pretty cool too:

enter image description here

@Alex M. 2019-02-11 22:03:24

Regarding differential geometry and topology, there is the 3 volume "A Mathematical Gift - The Interplay Between Topology, Functions, Geometry, and Algebra" by K. Ueno, K. Shiga, S. Morita, T. Sunada. The level is "relaxed undergraduate mathematics". The book attempts to bring to the front the intuition behind some of the concepts encountered in geometry. (All the 4 authors are mathematicians working in Japanese universities.)

@Severin Schraven 2019-02-11 20:46:44

I personally enjoyed The KAM Story by H. Scott Dumas. It gives an overview of the history of KAM theory. Very enjoyable and yet informative.

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