Ask HN: Math books you recommend just for the way they are written? | Hacker News Reader

Ask HN: Math books you recommend just for the way they are written?

97 pointscurious1672 comments

I have a favourite.

Probability Theory: The Logic of Science by E.T. Jaynes.

I just love it for the writing and the way it teaches probability. I haven't encountered any book in probability that is like this book. Every other book I have encountered are just axiom listing behemoths. This book have strengthened my understanding of probability.

Does any book come to your mind along these lines? Books that stop being pedantic where needed to first convey the topic to the reader. Then they worry about rigor.

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A few that come to mind immediately:

1. Books by VI Arnold. To get a sense of his style, see https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

2. Visual Complex Analysis by Tristan Needham is an absolute treat in visual intution.

3. The road to reality by Roger Penrose. This is an absolute masterpiece building up to cutting edge physics and all the necessary math from scratch. Reading the whole thing is probably a long-term project for most readers, but it can easily be read in chunks, and is an absolutely pleasurable experience (with most of the conceptual substance of a textbook, but without the dry rigor) and a fascinating taste of what/how Penrose sees.

4. From Mathematics to Generic Programming by Alex Stepanov. I've read only small bits from Stepanov, but I'm really looking forward to reading the whole book.

     Visual Complex Analysis by Tristan Needham is an absolute treat in visual intution

+1 !

>Books by VI Arnold

+1.

Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians by V.I.Arnold is one of my favourites.

Harold Edwards: Advanced Calculus: A Differential Forms Approach (https://www.maa.org/press/maa-reviews/advanced-calculus-a-di....)

Tristan Needham: Visual Complex Analysis (https://www.maa.org/press/maa-reviews/visual-complex-analysi...)

Tristan Needham: Visual Differential Geometry and Forms (https://press.princeton.edu/books/paperback/9780691203706/vi...)

Damn! You beat me to mentioning Harold Edwards' book, as it perfectly fits the description. The first three chapters are intuition and concept building, the next three chapters revisit the topics with more rigor and proofs, and then there are follow-on applications chapters with all sorts of nuggets. It is virtually mistake free and can be opened up and read at any point and still be understood, a stated goal of the author.

Thank you! I really loved Visual Complex Analysis..and started Needham's next book, which just arrived in the mail! :-) Worth it for all the other great books he mentions alone. It's also awesome, very much a sequel to VCA.

Concrete Mathematics, by Graham, Knuth, and Patashnik. Not only is the subject matter interesting, and, AFAICT not presented all together as a coherent body anywhere else, there are literally notes in the margins that the authors say reflect the comments of students who took courses using the book while it was being developed. Even if you don't care about the subject matter per se, the comments are well worth the read.

Similarly, Proofs from THE BOOK, by Aigner and Ziegler presents some interesting subject matter (short, elegant, and instructive proofs) all in one volume that you'd have to comb through large amounts of mathematical literature to encounter otherwise. The results themselves should be mostly familiar to any grad student or advanced undergrad in mathematics, but the proofs are sure to amuse as well as enlighten. The one unfortunate thing about this book is that the last edition was published almost 25 years ago.

My dad actually has a note in the margins of that book! Some joke about summations, if I recall.

I highly recommend the one that I am currently reading: Proofs: A Long Form Math Textbook by Jay Cummings [0].

There are two lists [1][2], one by Mark Saroufim, another one by Susan Rigetti that I have been following for quite some time now and intend to follow for the next months and years.

[0]: https://longformmath.com

[1]: https://marksaroufim.medium.com/technical-books-i-%EF%B8%8F-...

[2]: https://susanrigetti.com/math

"Calculus Made Easy" by Silvanus P. Thompson, now in public domain.

https://calculusmadeeasy.org/

The book is worth a peek just for this one chapter: https://calculusmadeeasy.org/1.html

Fantastic!

Turtle Geometry: The Computer as a medium for exploring Mathematics

By Harold Abelson

https://direct.mit.edu/books/book/4663/Turtle-GeometryThe-Co...

Gets to Curved Space-Time by chapter 9.

David MacKay: information theory, inference, learning algorithms.

It’s a classic in the field(s). Available as free pdf. Sets an example for how technical books should be written.

Came here to also say this, amazing book. There are also lectures on YouTube from the author.

Absolutely love this book. It’s the perfect bridge from physics to machine learning.

book pdf link: https://www.inference.org.uk/itprnn/book.pdf

website: https://www.inference.org.uk/mackay/itila/

Truly amazing books by Anthony Knapp (winner of Steele Prize for Mathematical Exposition):

Basic Algebra

Basic Real Analysis

Stokes's Theorem and Whitney Manifolds

Advanced Algebra

Advanced Real Analysis

All freely available from http://www.math.stonybrook.edu/~aknapp/download.html

I love everything by Richard Hamming. They're soaked with the enormous practical experience/expertise he has with what he writes about. They're not just about applied maths, but about the range of problems you run into when you try to apply them e.g.

Numerical Methods for Scientists and Engineers (1962)

Introduction To Applied Numerical Analysis (1971)

Digital Filters (1977)

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

You beat me to it!

In particular; Methods of Mathematics Applied to Calculus, Probability, and Statistics is one of a kind.

For the OP; there is also The Art Of Probability by Richard Hamming which you might find interesting.

And finally of course everybody needs to read; The Art of Doing Science and Engineering: Learning to Learn by Richard Hamming.

Or watch Hamming giving the Learning to Learn lectures in person! Awesome course in 31 parts:

https://www.youtube.com/playlist?list=PLctkxgWNSR89bl7hTOS3F...

It's also a summary of his career, explaining the basics and interesting parts of the various topics he's delved into, and how he thought about them.

Excellent!

Please make this as a independent top-level submission so that it gets the prominence it deserves.

> Methods of Mathematics Applied to Calculus, Probability, and Statistics is one of a kind.

I have tried finding this. Couldn't even find on libgen.

I see epub and mobi: https://libgen.rs/search.php?req=Hamming+methods+of

It is well worth getting the paper copy (rather thick book!).

"Numerical Linear Algebra" by Trefethen and Bau.

"Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory" by Bender and Orszag.

I'm not sure if they meet your definition of being well-written exactly as you say above, but these are extremely well-written math books.

This isn't a math book, but I was also a big fan of "Vector Quantization and Signal Compression" by Gersho and Gray.

Linderholm's Mathematics Made Difficult is a converse: a book that, despite utter lack of need, leaps into being pedantic. Nevertheless, it remains both amusing and instructive.

https://archive.org/details/mathematics-made-difficult

Larry Gonick's Cartoon Guides to Algebra, Statistics, Calculus http://www.larrygonick.com/

Yeah, the intersection of "serious maths book" and "great read" is pretty small.

Since you've already got Jaynes: David Mackay's inference book is also a good: http://www.inference.org.uk/itila/ even if not quite pure maths.

Strang's "Introduction to Linear Algebra". This book is a friend!

Garrity's "All the Math You Missed". Brings you up to speed fast, and has great references.

Strang's classes are also amazing http://mit.usiu.ac.ke/about/ocw-stories/gilbert-strang/. I haven't read Garrity going to grab it!

Came here to add Strang's. In the same spirit of Jaynes', a very good exposition for self-study students.

In logic, "Introduction to Logic: and to the Methodology of Deductive Sciences" - Tarski.

I wish that I started studying these three books before college, in order: 1) Tarski's 2) Strang's 3) Jaynes'

A few immediately come to mind:

1. Winning Ways for Your Mathematical Plays

2. A Singular Mathematical Promenade, available for free online: https://perso.ens-lyon.fr/ghys/promenade/

There are many other math books I really like - over the years I've collected a long list of reading recommendations: https://notzeb.com/rec.html (most of them won't fit as answers to this question)

Spivak's Calculus. Excellent first proof-based math textbook for anyone interesting in self-studying math.

Isn't that an Analysis book? ;)

No. But the way Calculus is taught in the US university freshmen, this book looks like a rigorous introduction to Analysis, even though it is not. It is just a sensible introduction to single variable calculus without hand wavy arguments.

Sorry, i was only joking :)

A beautiful book in nearly every way.

Byrne's translation of Euclid, you can find a digital version here: https://www.c82.net/euclid/

Winning Ways For Your Mathematical Plays by Berlekamp, Conway and Guy

Nonlinear dynamics and chaos by Strogatz.

He really takes the time to explain concepts clearly and unlike any other math book I've read, he proves certain ideas graphically and considers it good enough, without having to write out formal and stuffy sounding proofs

I'm also currently going through Applied Partial Differential Equations by Haberman. He explains the heat and wave equation from scratch and really makes a great effort to build up the complexity instead of just dumping it all in your lap.

+1 to Strogatz's Chaos book - this was the first textbook I ever read cover to cover, almost like a novel, because of how exciting it was.

I also really liked self studying Strang's Linear Algebra. Another one that comes to mind is Naive Set Theory by Halmos - it's a very short and well written introduction to set theory for practitioners but it still manages to get you thinking very deeply.

https://m.youtube.com/c/themathsorcerer for math book reviews

Proofs and Refutations by Imre Lakatos (https://en.wikipedia.org/wiki/Proofs_and_Refutations)

How to Solve it by George Polya (https://en.wikipedia.org/wiki/How_to_Solve_It)

A book in german. Elstrodt: Mass- und Integrationstheorie With notes on the people/history of the development of this field.

I particularly enjoy Ahlfors' writing in his complex analysis book.

An Introduction to Complex Analysis for Engineers by Michael Alder

https://cdn.preterhuman.net/texts/math/Mathematics%20-%20An%...

The Geometry of Physics: An Introduction by Theodore Frankel

A masterpiece that formalizes all of the handway bits you learned in physics. Will change your perspective forever.

Thermodynamics and an Introduction to Thermostatistics Herbert B. Callen

The only book that actually taught me thermo. Everything else was just confusing garbage.

Feynman & Hibbs was a joy. Physics rather than mathematics, but very enjoyable.

If it wasn't because it's a text introducing a somewhat niche approach to quantum mechanics that didn't gain broad traction, this book would totally have been the Kernighan & Ritchie of quantum mechanics.

Ah, that would've been one of the books I recommended. Shame that he passed with the book in a rather unfinished state.

> Does any book come to your mind along these lines? Books that stop being pedantic where needed to first convey the topic to the reader.

Mathematics books by physicists tend to be just that.

Homotopy Type Theory (Univalent Foundations for Mathematics)

A dry topic - but the authors manage to convey their enthusiasm and make a lot of effort to pick up the uninitiated

Lots of background and historical references

I love Atiyah & MacDonald’s “Introduction to Commutative Algebra”. I was always impressed by how it manages to be so extremely terse and yet so crystal clear and easily readable.

The swedish book Mot Bättre Vetande i Matematik by Andrejs Dunkels explains all the maths you need to know to begin studying at the university (meaning all you can learn in primary school + high school and then some) in less than 100 pages A5, no previous knowledge required, with some explanatory cartoons.

Cant recommend it enough, made me able to skip class in high school for about a year while still passing the tests with ease. (bad life hack ig)

The Art of Probability by Richard Hamming.

Not Math books, but David Griffiths Electrodynamics and Quantum Mechanics are such a pleasure to read for their style of writing.

Every student should read the following preparatory to "Higher Maths";

* Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George Simmons - Not a word wasted in less than 150 pages.

* Functions and Graphs by I.M.Gelfand et.al. - A must read (again, less than 150 pages) to build graphical intuition.

An Introduction to Probability Theory and its Applications

By William Feller, written in 1968 but feels very fresh even today.

David Williams’ Probability with Martingales is a personal favourite. And I’m not the only one.

"Récoltes et semailles" (Harvest and plantings) by Alexander Grothendieck.

Statistical Rethinking is really well written https://xcelab.net/rm/statistical-rethinking/ .

The Foundations of Statistical Inference by L.J. Savage

- Math with Bad Drawings by Ben Orlin

- Infinite Powers and The Joy of X by Steven Strogatz

- Godel, Escher, Bach by Hofstadter

(Avoiding text books as one's mileage might vary, and having fun depends on the readers' levels as well.)

Alas, Strogatz has made one of the (many edifying and entertaining) footnotes from Körner's The Pleasures of Counting obsolete:

> David Tranah ... suggested that a different title would help [sales]. For once, I did not take his advice and the title The Joy of x remains available.

(to understand this locution it may help to know that, in a more prudish century, The Joy of Sex made a big impression when it came on the scene)

Other maths books with similar puns in the title include:

* The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX Macro Package

* The Joy of Sets: Fundamentals of Contemporary Set Theory

* X and the City: Modeling Aspects of Urban Life

Div, grad, curl and all that - H.M. Schey Excellent intuitive and visual introduction to vector calculus.

Understanding Analysis by Stephen Abbott and Algebra by Michael Artin.

The Computational Beauty of Nature (1998) by William Gary Flake

Mathematics and its History by Stillwell.

"A Tangled Tale" by Lewis Carroll :-)

Steven Strogatz's Sync and Infinite Powers

Nonlinear dynamics and chaos also but Strogatz i highly recommend.

Measurement by Paul Lockhart

"An imaginary tale, the story of sqrt(-1)" -- it's a great romp through both the history of the imaginary unit and actually gently explains a fair bit of complex analysis to the reader, up to contour integration I think

Also, James Gleick's Chaos is a classic, as is the big original beast, Gödel Escher Bach by Douglas Hofstadter.

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