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Great books about mathematics

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151 pointsColinWright58 comments

Another great book, starting with the basics and then exploring a wide range of mathematical subjects, is The Princeton Companion to Mathematics:

http://press.princeton.edu/titles/8350.html

It's fun to just flip it open and start reading.

Yes and it's conveniently available on Safari [0] if the sticker shock of the print edition is too much. Also, the editor Tim Gowers wrote Mathematics: A Very Short Introduction [1].

0: http://my.safaribooksonline.com/book/math/9781400830398

1: http://www.amazon.com/dp/B000SEP2T2/

This is a wonderful book. A word of caution though: don't get the Kindle version from Amazon. I thought I'd get it for the Kindle, because the print version is huge and I sometimes like to read in bed and don't want to lug the dead tree version I own around, but I had to return it because of the standard problems with garbled notation (some symbols appearing as a square box or some other incorrect symbol) that affects every math book I've ever purchased on Amazon for the Kindle.

There error rate was much lower than in any other math book I've tried, but still much too high.

I clicked the link expecting to see this book. It's pretty impressive (and beautifully finished besides).

Agreed, well worth the price. The most useful books i have are these 2 Math for Physics undergrads books that capsulize the first 2 years of calculus, linear algebra etc that all math, physics, EE students go thru

http://www.scribd.com/doc/65695685/Mary-L-Boas-Mathematical-...

(older edition of Arfken et al

http://www.scribd.com/doc/84183760/Arfken-G-B-Weber-H-J-Math...

I really enjoyed the book on Fermat's last theorem by Simon Singh -

http://www.amazon.com/dp/0802713319

A long time ago, a friend's mother was complaining to me that (high school) Math is a dry subject and she doesn't blame her otherwise intelligent son for not being able get interested and do well in it. I wish I knew of this book then - I know it cranked up my interest in Math ever since.

I read this in a day in high school. It was great, and actually all of his books are great. The crypto one is really engaging, and a surprisingly easy while reasonably thorough read. The one about the big bang can get a little dry in spots, but it's 500 pages and by the end there are more than enough "wow, that's amazing" moments to make up for it.

"Proofs and Refutations" is a fantastic book, I really enjoy that style.

If someone is looking for a more lightweight introduction to Lakatos I can highly recommend "For and Against Method" which outlines his "arguments" with Feyerabend.

If you're not into science theory (imo) Lakatos is basically Popper++ (I think most people have heard of Popper)

Something else I just read which is related but lighter is How Mathematicians Think by William Byers.

Here is a great one: http://www.amazon.com/Introduction-Graph-Theory-Dover-Mathem...

"A stimulating excursion into pure mathematics aimed at "the mathematically traumatized," but great fun for mathematical hobbyists and serious mathematicians as well. Requiring only high school algebra as mathematical background, the book leads the reader from simple graphs through planar graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph, Euler walks, Hamilton walks, and a discussion of The Seven Bridges of Konigsberg."

Two additional suggestions: 1. 'A Mathematician's Lament' by Paul Lockhart (http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...)

2. 'How to Solve It' by G. Polya (http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp...)

May I present one of the greatest math books for general audience: Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev.

http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin...

Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come even close to this. Reading this book together with the The Princeton Companion to Mathematics was real treat.

I second the book as well as the above praise for the book as one of the greatest. I have read several chapters from the book and have loved them all.

Question: I haven't read The Princeton Companion to Mathematics. How would you compare these two?

The Princeton Companion is more like encyclopedia. Main part of the book is articles describing 100 or so mathematical concepts in alphabetical orders. Then it has articles describing major mathematical problems.

For a short, intuitive, practical, and affordable introduction to high school math and calculus, check out my book: "No bullshit guide to math and physics" http://minireference.com/

For a free alternative, check out: http://www.gutenberg.org/ebooks/33283

Anyone have good reading material about 'how to reason mathematically' on a basic level? I mean, not going too deep into any specific topic, but how to get better at interpreting equations and grokking relationships.

"How to Solve It" by G. Polya is an excellent guide on how to approach mathematical problems.

I think Gödel, Escher Bach does a good job imparting the "mathematical mindset", albeit indirectly. It's a great read even if it's quite long and a bit repetitive. Definitely worth a read if you're not already familiar with abstract mathematics and formal logic.

I have read that actually. I think it was much heavier on the 'logic' than the mathematics though, if you know what I mean. Great book though... love how it doesn't even really get started on its main topic (AI) until about three quarters in, having covered a vast range of supporting topics.

You may enjoy Mathematics for the Nonmathematician by Kline:

http://www.amazon.com/Mathematics-Nonmathematician-Dover-Boo...

I have not read it, but I have heard it recommended many times by knowledgeable educators.

Have you seen the coursera introduction to mathematical thinking [1]? That may be a good starting point.

[1]: https://www.coursera.org/course/maththink

My wife gave me this great book for Christmas:

  Love and Math: The Heart of Hidden Reality
  by Edward Frenkel

It begins with the author's struggle to learn the math behind quantum physics in spite of cold-war era soviet educational obstacles and leads bit by bit into the Langlands program, drawing connections between group theory, number theory and harmonic analysis.

It's definitely my favorite book of 2013.

http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465...

Yup. My thought as I read Love and Math a couple of months ago was "this would be great to give to a high school senior who is wondering whether to continue with mathematics." If you truly love doing math, you'll feel it when you read this book.

Polya's How to Solve It was recommended below. I also like How to Prove it by Daniel Velleman (http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...).

Check out 'The Joy of x' by Steven Strogatz.

http://www.stevenstrogatz.com/the_joy_of_x.html

Any artofproblemsolving.com book will be outstanding for this purpose. I suggest starting with Prealgebra by Richard Rusczyk.

You might enjoy one of the many books that exist for undergraduates to ease the transition into higher math classes, where there is a shift from the strong calculation-focus and rote-learning of most math teaching through calculus, to the more proof-centric and understanding-based approach that one finds in classes like abstract algebra, real and complex analysis, and other post-calculus math courses.

Here are some examples of the kinds of books I mean, and you can find others by following Amazon recommendations from those:

http://www.amazon.com/Nuts-Bolts-Proofs-Fourth-Edition/dp/01... http://www.amazon.com/Mathematical-Proofs-Transition-Advance...

With math textbooks especially, it pays to look for a previous edition, as the current edition can be ridiculously expensive, and the previous edition might be only 20% of the price, with no significant differences between the two.

Also, don't get them for the Kindle, as Amazon doesn't seem capable of publishing a math book with lots of notation that doesn't also have tons of errors where symbols get incorrectly imported. I've bought at least 20 and yet have to see one that didn't have lots of incorrect symbols.

How to Solve It, by Polya.

> I’ve been asked over and over for good books about mathematics for a layperson, someone who hasn’t taken advanced courses in university and is more simply interested in learning about what math is, and some of the more interesting historical figures and results from mathematics.

The book you want is "What is Mathematics?" [0] by Courant and Robbins.

[0] http://books.google.no/books?id=_kYBqLc5QoQC&printsec=frontc...

Does someone have a recommendation for a book that get deeper into mathematics by teaching actual mathematics rather than teaching about it? Many of the books I have come across leave me with more questions than answers [1].

I am interested in pure mathematics mainly including logic, set theory, category theory, etc.

[1] An excellent example not mentioned by others here is also The Road to Reality by Roger Penrose. The sections on mathematics are very good, but ultimately leave the topic open-ended too soon.

What are you looking for that isn't simply a math textbook? Honestly, that's how mathematicians learn "actual" mathematical subjects, too.

If you're comfortable with calculus as a subject, for example, and want a "pure mathematics" approach, I recommend Michael Spivak's Calculus. If you've never worked through a pure math textbook from start to finish, that's a good start.

There's not that much interesting in the three subjects you listed — logic, set theory, and category theory — that doesn't depend on other subjects or a prior level of mathematical maturity. Category theory was originally invented to solve and categorize problems in algebraic topology, for example.

For example, I rather like mathematical logic and model theory, but you're going to have a rough time if you don't have a visceral, intuitive understanding of countability arguments and at least a handful of subjects you'd be reasoning "about." Unless you know the standard model of arithmetic, for example, how can you think about non-standard models? Without that, important results like the Löwenheim–Skolem theorem will likely seem contextless.

Good points. I have already gone through several levels of engineering mathematics, so do understand differential and integral calculus well. I am assuming that is that Spivak's book is about -- please correct me if I am wrong; Amazon is not showing a preview of the book.

You are right in bringing countability arguments into the picture; I understand them only to some level. I would love to read a book that gives it a formal treatment. The following has been great for example:

https://news.ycombinator.com/item?id=6838917

Another one showed up on HN recently that I am still to read in full:

https://news.ycombinator.com/item?id=6966695

Thanks

Finally, cannot help but mention in praise, Colin Wright here on HN has been a good help before in clearing some of my doubts on the subject.

Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material.

Here are some exercises to give you a sense of the flavor. If you find these exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :)

These are all before we get to any "calculus." Here "function" means a function of the real numbers.

1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times.

2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times.

3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x.

4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such)

Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book.

The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).

Sounds good. Thanks!

"3. A set A of real numbers is said to be sense if every open interval contains a point of A."

For those googling: there's a typo: sense => dense (http://en.wikipedia.org/wiki/Dense_set)

Indeed! Me and my fat fingers. :)

> Spivak's Calculus is used as a first-year calculus textbook at lots of schools

Umm... where? Not at Stanford, where we used a mainstream, much easier book. So does Princeton. Harvard is famous for having developed a "touchy-feely" calculus book.

Perhaps abroad? It is typical of calculus courses in the US that the students come with fairly weak backgrounds, and a major purpose is to expose and patch holes in the students' backgrounds in algebra and trigonometry.

>Umm... where?

Lots of places http://www.math.uga.edu/~pete/MATH2400F11.html

If you took me to mean the "default" first-year calculus textbook then yes, that's not common. But it's definitely aimed at first-year college students, or at least people who haven't had prior exposure to rigorous mathematical thinking. Compare the style to, say, Spivak's Calculus on Manifolds to see what I mean.

(Edit: I just read your HN bio and know you know the stylistic differences, etc. Sorry!)

Spivak is the first-year Honors Calculus textbook at my alma mater, the University of Chicago. Harvard is also famous for having the most difficult first-year math classes that use even more advanced textbooks like Rudin's Principles of Mathematical Analysis.

My HS background in mathematics was definitely "weak," too. My senior year was the first year my school district ever offered calculus of any stripe in its entire history and I still managed to handle Spivak my first year of college. I took the AP Calculus test on my own and got a 4/5. It's not that crazy.

> Spivak's Calculus starts with a set of 13 axioms

I am reading third edition and it only has 12 not 13. Did that change in later editions?

I typed it from memory, so I could be wrong. I thought the 13th axiom was the least upper bound property.

Seconding the recommendation of Spivak's Calculus. That book lit a fire in me when I was 18. Be sure to work through the exercises, that's where all the fun is.

We just started a study group for that book at

http://www.reddit.com/r/calculusstudygroup/

if anyone is interested.

Since you mentioned you already have a strong applied math background, try Mathematics: Form and Function by Saunders MacLane for a broad portrait of mathematics that might get you excited to read more deeply into a given subtopic. Note that while this book surveys a broad swathe of mathematics, it depends on the reader already having a pretty high level of mathematical maturity.

Other than that, if you want to go deeper into mathematics, just choose a topic and read on it. If your experience is mostly applied, consider something like topology which is often presented more formally than earlier math courses, or go back to the stuff you've already studied, but in a more pure context (for linear algebra, Axler's Linear Algebra Done Right and Halmos' Finite-Dimensional Vector Spaces; for calculus, Spivak as mentioned already and then an analysis book).

Thanks a lot. All the recommendations look good (previewed each on Amazon). Saunders MacLane and Spivak sound to be good starts for me.

It's worth mentioning that Saunders MacLane, along with Samuel Eilenberg, invented category theory.

An enjoyable fiction that may help your kids get interested in maths is The Parrot's Theorem reviewed on a maths fiction site http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...

Your History of Mathematics is great, but I prefer the two volume work by D.E. Smith. Here's volume 1: http://www.amazon.com/History-Mathematics-Vol-Dover-Books/dp...

I read "The Mathematical Experience" when I was in high school. Unfortunately, I do not remember many details of the book (for me, "when I was in high school" is approaching 20 years ago) except I recall that I read it over and over again and thought it was excellent.

I agree with the author about History of Mathematics and Journey Through Genius, so I did what he said and bought The Mathematical Experience immediately.

of all the technical things I have learned over the years, higher mathematics (say calc 1-3 and diff. eqns.) is the topic I most regret brain dumping. I am not so much interested in the history, but can anyone recommend a streamlined primer?

The first few chapters of The Theoretical Minimum. There is a book and also a free set of online lectures.

Great list. I would also add:

What is Mathematics? http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...

#6. Euclid's Elements.

I'll second that with caveats. "The Elements" requires quite a bit of commentary to make sense and most editions seem to aspire more to translational accuracy than mathematical understanding. But if you can get past that it is the singular "must read" book in mathematics.

Awesome. Just what I've been looking for!

It's easy to find good programming e-books, but e-books about mathematics are harder to find. Does anyone know if there's a equivalent of O'reilly for math-books?

Springer-Verlag has ebooks but they're incredibly expensive, unfortunately.

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