Good "casual" advanced math books

Question

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?

Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that. — ssx, Feb 11 '19 at 18:48
This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: https://bookstore.ams.org/mbk-46/ — Qfwfq, Feb 11 '19 at 18:59
I really enjoyed History of Topology by I.M. James. It's a very casual read, but walks you through a lot of "advanced" mathematics. — Yousuf Soliman, Feb 11 '19 at 20:32
This describes history of mathematics rather than casual mathematics. For me "casual mathematics" would mean a causal, hand-waving, light-on-the-proof-details description of mathematical results. — Michael, Feb 12 '19 at 00:03
Not a book, but the ICM surveys (https://impa.br/icm2018/) are often great for getting a very basic idea of what's going on in various fields outside of one's own. — Sam Hopkins, Feb 12 '19 at 00:35
Not necessarily math but the book An Elementary Primer for Gauge Theory by K. Moriasu is a great read for anyone interested in Quantum Field Theory, mixing history, light hand-waving proofs/derivations, and physics. It's really short too. It's outdated now in light of more recent discoveries but still a great book. — Kai, Feb 12 '19 at 04:00
I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of. — Greg Friedman, Feb 13 '19 at 00:59
I loved The Calculus Gallery by William Dunham, which covers snapshots in the development of calculus: Newton, Leibniz, the Bernoullis, Cauchy, Riemann, Weierstrass, Baire, and Lebesgue. It's full of real maths — full-blown results and detailed proofs — but explained very clearly and easy to read. Accessible, enjoyable, fascinating, and educational! — gidds, Feb 13 '19 at 20:24
Cannot write an answer, but this MAA review indicates you might like M. Spivak's 'Comprehensive introduction to differential geometry', especially volume 2: "Once one starts talking about surfaces, however, Spivak really gets interesting. He takes a quasi-historical approach, meaning that he follows the historical order of progress when it is helpful and ignores it when it is not. After a brief chapter on the theory of surfaces before Gauss, what we get is a careful reading of Gauss's famous Disquisitiones Circa Superficies Curvas. (One of the innovations in this third edition is that [cont.] — , Feb 14 '19 at 12:55
[cont.] it includes the full text, with Spivak's comments on facing pages. Hooray!) Jumping off from Gauss, the chapter reworks the theory in modern terms. Then he does it again: starting from Riemann's famous essay "On the Hypotheses which lie at the Foundations of Geometry," Spivak develops the basic ideas in the geometry of manifolds, including a brilliant section called "The Birth of the Riemann Curvature Tensor." This is crucial: if one doesn't see how the curvature tensor is the natural way to generalize the notion of curvature, then all is lost. [...] Throughout all this, Spivak [cont.] — , Feb 14 '19 at 12:56
[cont.] maintains a sense of humor and a clear head."
https://www.maa.org/press/maa-reviews/a-comprehensive-introduction-to-differential-geometry-vol-ii — , Feb 14 '19 at 12:57
I have not slogged through much of it yet, but "Visual Complex Analysis" (Needham) seems like an interesting approach. — Spehro Pefhany, Feb 15 '19 at 16:02
"Everything and More: A Compact History of Infinity" by David Foster Wallace — jberryman, Feb 15 '19 at 22:33
Here's a great online textbook that covers the math behind deep learning. It's a good mix of background and math, and it's a great into to the technology. http://neuralnetworksanddeeplearning.com/ — Jay Speidell, Feb 18 '19 at 04:35
I think Morris Kline's "Mathematical Thought from Ancient to Modern Times" is what you may be looking for, e.g., https://archive.org/details/MathematicalThoughtFromAncientToModernTimes/page/n6 ( Possibly also, Saunders MacLane, "Mathematics Form and Function" https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function ) — John Forkosh, Feb 19 '19 at 07:20
I vaguely recall a book that I think may have been by John Milnor that might answer this. Is there someone here who is familiar with Milnor's books who can tell me if I'm deluded as to who it was? — Michael Hardy, Nov 12 '22 at 20:07

score 73 · Answer 1 · edited Nov 12 '22 at 20:09

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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:

Michio Kuga, Galois' dream,
David Mumford, Caroline Series, David Wright, Indra's Pearls,
Hermann Weyl, Symmetry.
Marcel Berger, Geometry revealed,
D. Hilbert and Cohn-Vossen, Geometry and imagination,
T. W. Körner, Fourier Analysis,
T. W. Körner, The pleasures of counting.
A. A. Kirillov, What are numbers?
V. Arnold, Huygens and Barrow, Newton and Hooke.
Mark Levi, Classical mechanics with Calculus of variations and optimal control.
Shlomo Sternberg, Group theory and physics,
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.

edited Nov 12 '22 at 20:09

Michael Hardy

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answered Feb 11 '19 at 20:33

Alexandre Eremenko

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Kirillov's is "What are numbers?" – Amir Asghari Feb 11 '19 at 22:04
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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. – Alex M. Feb 11 '19 at 22:07
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@Amir Asghari: thanks for the correction. I only have the Russian original. – Alexandre Eremenko Feb 12 '19 at 02:58
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@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). – Alexandre Eremenko Feb 12 '19 at 03:24

score 34 · Answer 2 · answered Feb 11 '19 at 22:09

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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:

answered Feb 11 '19 at 22:09

Vidit Nanda

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I just started looking through that book. Chapter 3 is amazing. – arsmath May 24 '20 at 11:30

score 32 · Answer 3 · edited Dec 31 '22 at 08:47

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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.

edited Dec 31 '22 at 08:47

Martin Sleziak

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answered Feb 12 '19 at 03:40

HYL

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Thanks for the recommendation of the book of Ghys. I'd put in near the top of my list if I knew about it! – Alexandre Eremenko Dec 05 '19 at 02:57

score 23 · Answer 4 · edited Feb 12 '19 at 18:37

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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.

edited Feb 12 '19 at 18:37

Nathan Hughes

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answered Feb 12 '19 at 16:23

Jörg Neunhäuserer

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score 21 · Answer 5 · edited Dec 31 '22 at 08:50

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

Chapter 10: Topology grows into a branch of mathematics in Never a Dull Moment: Hassler Whitney, Mathematics Pioneer by Keith Kendig**

Zeros of Entire Fourier Transforms by Dimitar Dimitrov and Peter Rusev

A long paper/short book on identifying polynomials and entire functions that have only real zeros and the influence of and applications to the Riemann hypothesis.

Learning Modern Algebra From Early Attempts to Prove Fermat's Theorem by Cuoco and Rotman.

Möbius and his Band: Mathematics and Astronomy in Nineteenth-century Germany edited by Fauvel, Flood, and Wilson.

A compilation of articles:

A Saxon mathematician by John Fauvel
The German mathematical community by Gert Schubering
The astronomical revolution by Allan Chapman
Möbius's geometrical mechanics by Jeremy Gray
The development of topology by Norman Biggs
Möbius's modern legacy by Ian Stewart

I had known of Möbius chiefly through the Listing-Möbius band, linear fractional transformations, and the Möbius function and inversion--all of continuing significance in modern mathematics. The articles cover more and provide a nice entre/appetizer for modern topics.

A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham/102712#102712 — Tom Copeland, Feb 18 '19 at 11:17
A brief, very readable, introductory survey of basic complex algebra on par with some other books mentioned in this stream is "The Skeleton Key of mathematics: A Simple Account of Complex Algebraic Theories" by Littlewood, with short shrift on the underlying history of the ideas but with mention of applications to relativity and quantum mechanics. — Tom Copeland, Nov 30 '22 at 19:29

score 18 · Answer 6 · edited Dec 31 '22 at 08:44

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 topic, 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.

score 12 · Answer 7 · edited Feb 12 '19 at 01:42

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

score 12 · Answer 8 · answered Feb 11 '19 at 22:03

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.)

score 12 · Answer 9 · answered Feb 12 '19 at 03:59

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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.

answered Feb 12 '19 at 03:59

Jay

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+1, and his book on Fermat last theorem is fantastic too, although they're maybe more elementary than what OP had in mind. – Adrien May 11 '19 at 08:46

score 12 · Answer 10 · answered Feb 12 '19 at 10:29

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

answered Feb 12 '19 at 10:29

Mike M

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score 12 · Answer 11 · answered May 17 '19 at 14:32

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I highly recommend "Office Hours With a Geometric Group Theorist", edited by Matt Clay and Dan Margalit, which is a series of essays on various topics in Geometric Group Theory written in a very informal style.

answered May 17 '19 at 14:32

APR

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score 11 · Answer 12 · answered Feb 11 '19 at 20:46

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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.

answered Feb 11 '19 at 20:46

Severin Schraven

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Gerry Myerson · Answer 13 · 2019-02-12T09:25:35.720

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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.

edited Feb 12 '19 at 09:25

answered Feb 11 '19 at 20:46

Gerry Myerson

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Have you read and enjoyed "Birth of a theorem"? – Alexandre Eremenko Dec 05 '19 at 02:51
@Alexandre, I read it, and enjoyed some of it. – Gerry Myerson Dec 05 '19 at 02:56
As I mentioned already earlier, I have read Villany's book and I suffered. – Joel Adler Sep 11 '21 at 09:13

score 10 · Answer 14 · edited Feb 12 '19 at 00:47

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

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

edited Feb 12 '19 at 00:47

Qfwfq

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answered Feb 12 '19 at 00:24

David Campen

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Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks. – Alexandre Eremenko Feb 12 '19 at 03:22
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The title is "Gamma. Exploring Euler's constant". – Alexandre Eremenko Feb 13 '19 at 18:19

score 9 · Answer 15 · answered Feb 12 '19 at 03:47

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

Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate". — efs, Feb 12 '19 at 15:38

score 8 · Answer 16 · answered Feb 11 '19 at 22:10

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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.

answered Feb 11 '19 at 22:10

Melquíades Ochoa

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score 5 · Answer 17 · answered Feb 12 '19 at 09:04

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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

answered Feb 12 '19 at 09:04

kodlu

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I remember trying to peruse Schroeder's book for an explanation of how he used quadratic residues to build diffusers. Maybe it's just me not understanding physics, but I still have no idea. My impression from some other sections of the book, though, is just that it is not well-written. E.g., can you tell from section 16.6 what the spirals on Figs. 16.1 and 16.2 have to do with the $S(m, k)$s? – darij grinberg Sep 11 '21 at 05:01

score 5 · Answer 18 · answered Feb 12 '19 at 14:44

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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.

answered Feb 12 '19 at 14:44

Moritz Raguschat

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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. – efs Feb 12 '19 at 15:46
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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: https://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.
– Moritz Raguschat Feb 16 '19 at 15:05
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It's admirable of him to provide free copies on the Net---worth a written thanks. – Tom Copeland Sep 11 '21 at 02:36

score 5 · Answer 19 · answered May 11 '19 at 01:41

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"Moonshine beyond the Monster" by Terry Gannon. I bought this book on a whim and then I couldn't put it down. Very good elementary overview of some deep mathematics.

answered May 11 '19 at 01:41

Drew Armstrong

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score 4 · Answer 20 · answered Feb 12 '19 at 03:21

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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.

answered Feb 12 '19 at 03:21

Crayoneater

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score 3 · Answer 21 · answered Sep 11 '21 at 04:24

I very much liked a book by David A Cox titled Galois Theory. The author writes in the section preface to the first edition

This book was written in an attempt to do justice to both the history and the power of Galois theory. My goal for students to appreciate the elegance of the theory and simultaneously have a strong sense of where it came from.

Happy reading!!

score 2 · Answer 22 · answered May 17 '19 at 14:45

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I like the books in the Student Mathematical Library that are published by the AMS. I am currently reading "Modern Cryptography and Elliptic Curves, A Beginner's Guide" by Thomas Shemanske. It is quite delightful.

answered May 17 '19 at 14:45

M. Khan

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score 2 · Answer 23 · answered Jun 21 '21 at 07:25

I would add in the recent book Change and Variations by Jeremy Grey. It provides a beautiful historical overview of the evolution from the beginnings of differential equations and calculus of variations to the modern theory. Though it’s written with a historical bent, it’s definitely rigorous and could easily be used for a course.

score 1 · Answer 24 · answered Nov 13 '22 at 03:43

I'm surprised that no one has yet mentioned The Symmetries of Things by John Conway, Heidi Burgiel and Chaim Goodman-Strauss. It's ostensibly an introduction to symmetry groups but touches into classifying hyperbolic tilings, the number of groups of a given order, 'infinite' polyhedra and several other fascinating related topics. It's casually enough written and well enough illustrated that it's easy to underestimate how deep you've gone until you take a look around.

score 0 · Answer 25 · answered Sep 11 '21 at 05:10

This is borderline advanced mathematics, but my impression is that

Andrew Granville, Number Theory Revealed: A Masterclass, 2020

could well be used this way. Many topics, some rather obscure, are covered, but the technical prerequisites are never particularly forbidding, nor is there much buildup; it's as bite-sized as a textbook can get.

Another field that has recently attracted semi-popular treatments are $p$-adic numbers. Here are two that come to my mind:

Finally,

Titu Andreescu, Gabriel Dospinescu, Problems from the Book, XYZ Press, 2nd edition 2010;
Titu Andreescu, Gabriel Dospinescu, Straight from the Book, XYZ Press, 2012

are meant as olympiad training texts but can be read profitably as light reasoning (avoiding the problems), as they contain a lot more theoretical discussion than the typical olympiad text (and both the theory and the problems are chosen with a lot of taste).

score 0 · Answer 26 · answered Nov 13 '22 at 08:55

Since no one has mentioned this I found the following trilogy by Ash and Gross fits your description: Fearless Symmetry , Elliptic Tales and Summing It Up.

I personally had enjoyed reading them and found the semi-technical but mostly conceptual writing to be truly enlightening in shaping my understanding of the maths fields in question when I had no formal training in them. In fact after reading them I kinda knowing how to approach themes of the books with further technicalities despite not being committed to do so.

Good "casual" advanced math books

26 Answers26

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