Math books

Let's talk about math books. Interesting, fun, engaging math books. Books that keep you busy for months, if not years; books about deep esoteric topics, or new perspectives on old, familiar subjects. I have a few books on my list: + Calculus on manifolds - Michael Spivak + Topology from the differential viewpoint - John Willnard Milnor + Proofs and refutations - Imres Lakatos + Problems and theorems in Analysis - Polya & Szego Just to mention a few representatives of the class of books I'm thinking about. Anyway, what are your favorite math books? What are you reading right now? Any exercise giving you trouble? Tell me all about it!

Handbook of mathematics - Bronshtein

Numeric Methods. It's technically just an index of algorithms, which is why you should get the version which is translated into your most familiar programming language. (My copy is in FORTRAN since it was a gift from an old electrical engineer, but if I were to get a new copy I'd get the C++ version for readability.)

You have good taste OP. Some books I would recommend: Riemannian Geometry - John Lee Differential Forms in Algebraic Topology - R. Bott & L.W. Tu(I learned a ton of algebraic topology and manifold theory from this book, has a very unique approach that I like a lot. Took me almost a whole 6 months to read/work through 2/3 of it!) Dummit & Foote Mathematical Methods of Classical Mechanics - V.I. Arnold Basic Topology - Armstrong (this is the hardest and best intro topology book I know of) Currently my main book is "Differential Geometry" by a guy named Sharpe, going through very slowly due to reading time being taken up by other stuff but I anticipate having more reading time next month. It was a loan from a mentor. Very topological and global point of view which I appreciate. Also very advanced, not a first-time differential geometry text. Anyone got tips on good Dover books? They're really hit & miss, but they're so cheap I don't care. I have one called "The Variational Theory of Geodesics" which I am finding to be useful for applied differential geometry. "Differential Forms with Applications" by Flanders is also a classic, although it's extremely difficult.

Threads like this really make me wish we had a /Σ/ board. Math threads end up all over the place; sometimes here, sometimes /r/, /λ/, /Ω/, even /Δ/. I've recently taken a newfound liking to mathematics, and it's a bit of a mess to discuss and keep up with. The board probably wouldn't be very active, but it fits thematically with the other tech boards, and that it keeps popping up despite really belonging anywhere shows it wouldn't get zero traffic. Mods do as they will, but if I had my druthers I'd make it.

>>7784 I hear ya. I don't care about programming, just math. Wish there was a dedicated place to talk about math.

>>7794 > I don't care about programming, just math. Wish there was a dedicated place to talk about math. The best programming is just math tho. Install Octave and Haskell/ML.

>>7794 Mathematics is a small world. There are some forums devoted to mathematics, I know math-forums.com used to be The Spot but it looks pretty dead now. I have been active on reddit.com/r/math before, it's okay, but the majority of posts aren't very interesting. I suspect the vast majority of posts on any math forum are going to be devoted to students trying to get help on homework, or total novices who are confused about Godel's theorems or Cantor's theorem. Do do anything research-level you basically need to know people personally, or just write a blog and hope interesting/knowledgeable people find it and send you feedback. >>7796 >octave ew. I much prefer this tool https://www.jirka.org/genius.html Julia is also pretty nice but I've never done anything serious with it, just wrote a few simple fractal drawing functions.

my favorite math book so far has been A Book of Abstract Algebra by C. C. Pinter, which is exciting to read and clearly shows that the author is excited about math. i only read it in undergrad but it was engaging and honestly changed the way i think about textbooks, it also made me passionate about higher math as a whole. >>7794 back when kalyx was admin, we talked about and briefly created a board for math and science. a couple of the mods were opposed to it and it wound up not panning out because of the complaining and lack of availability for moderators. I'd be happy to see another science board, honestly..

>>7843 Good Book, been reading trough it. >>7797 >Mathematics is a small world. it truly is, most people get to know other people trough math competitions, things like the IMA(International Math Olympiad).

>>7745 The Art of Computer Programming surely qualifies, and also saves this thread from the abyss, for now.

>>7797 What do you think of nlab cafe ?

Read classics. The following are some classics. You can argue that something like Bourbaki's Elements of Mathematics, Serre's FGA & GAGA, Grothendieck's EGA & SGA are classics, but here are definitive ones: Euclid: Elements Archimedes: The Works of Archimedes Descartes: La Geometrie Euler: Element of Algebra (w/ Lagrange's notes) Introduction to Analysis of the Infinite Foundations of differential calculus Foundations of integral calculus Gauss: Disquisitiones Arithmeticae Riemann: On the Number of Primes Less Than a Given Magnitude On the Hypotheses which lie at the Bases of Geometry etc.

>>7745 hi frens (and lain). i'm in the early stages of self studying math right now. would my best course of action be following picrel in tandem with khan academy, or should i enroll in moocs like those suggested by ossu math?

>>9116 Don't watch lessons. Just read the books. Skip the Ash's book and substitute Apostol's calculus with Spivak's calculus. Because if you use Spivak's you don't need Ash's book anyway. Then study number theory and set theory at this point. I recommend Hardy's book for number theory and Pinter's book for set theory. You don't need most subjects in Lax's book. Use it as a secondary book or a reference. For linear algebra, I think Axler's book is good but it is quite biased. So supplement it with a general secondary book. For ODE, I really recommend Arnold's one but it's definitely not an intro book... so just read Tanenbaum first and read Arnold AFTER studying analysis. I think logically you can read right after, but you need to grow your mathematical maturity to read that book. For mathematical analysis, Rudin's PMA is the standard and good enough. But it lacks details. For details, Tao's book is really great. So read Tao's one first then PMA next. For algebra, I strongly recommend Artin (son) but it is a bit biased. Also Artin (father)'s Galois theory books (or notes?) are especially good. Dummit & Foote is not that bad though. For general topology, Munkres is the standard. And it is a good one. For probability theory, I recommend to study measure theory BEFORE studying probability theory. Because a probability function is just a measure with that the measure of whole space is 1. I recommend Tao's measure theory. For complex analysis, read Ahlfors. Read Ahlfors. Read Ahlfors.

>>9116 >>9118 I omited these: Spivak's calculus is awesome but it lacks vector calculus. For vector calculus, I recommend Hubbard's one. For multivariate analysis, I recommend Munkres' Analysis on Manifolds. Read Spivak's Calculus on Manifolds too if you have enough time. I think Spivak's book is a lot similar to Rudin's PMA. It is a good book but lacks details.

>>9118 >>9119 thanks very much, fren

I've recently read a lot more maths history/philosophy books. A good one was 'Makers of Mathematics' by Hollingdale. I'm currently reading 'The Mathematical Experience' by Davis & Hersh which is pretty interesting and has some selected topics in maths. Going through the further reading sections in each chapter is nice when I want to find more on specific things. I also have some books on recreational maths which are fun to go through. I found these, along with some others like Gödel Escher Bach, right at the back of my local library which has some pretty interesting reads that I haven't seen before. Like >>7797 and >>7794 said I wish there were more places to talk about it. I haven't seen https://mathchan.org/ mentioned but it is pretty dead/slow. Since there aren't many places to talk about maths I learned to be fine with threads like this that come once in a while.

>>9118 Imo, lectures or moocs are good for no proof based courses, those like calculus 1-3, linear algebra or differential calculus in MIT OCW or MITx (they teach those concepts faster and clearly). For pure math courses, books are the option.

>Let's talk about math books. Interesting, fun, engaging math books. I think you might like Differential Equations With Applications and Historical Notes by Simmons

>>9336 Proofs and refutations (mentioned in the OP) is a book you might enjoy. >>9734 Thank you, I've been looking for a book on differential equations.

>>7782 finished bott & tu about 4 months ago, without doubt my favourite math book i've read thus far.

>>9119 >Spivak's calculus is awesome but it lacks vector calculus. For vector calculus, I recommend Hubbard's one. In the third edition it covers vectors briefly, though I skipped it. I don't see the point when there are other books on vectors that are more detailed.

Book Of Proof by Richard Hammack (free) >https://richardhammack.github.io/BookOfProof/ >>7745 If you need to brush up on your precalculus, I highly recommend Precalculus Mathematics in a Nutshell by George F. Simmons because it's affordable and short book but it covers everything you need to know. It also has exercises.

I've read many math books but remember even less of the contents. From the top of my memory, some books that I enjoyed to come to mind. Serre - Linear Representations of Finite Groups. (The first half anyway) Pugh - Real Mathematical Analysis Aluffi - Algebra: Chapter 0

>>7745 Has anybody read Florian Cajori's work on mathematical notation? Was it good, interesting?

Manifolds, Sheaves and cohomology by Torsten Wedhorn is a very exposition of manifolds from a sheaves point-of-view. Also Basic Category Theory by Tom Leinster is the best introduction to category theory in my opinion, which can is open access (even open source) on arxiv: https://arxiv.org/abs/1612.09375 For those who interested in Category Theory with a fairly solid foundation I think Coend Calculus by Fosco Loregian, which is available at https://arxiv.org/abs/1501.02503

>>11252 Oh I forgot to add a few honorable mentions. Marker - Model Theory Lee - Smooth Manifolds Hamming - The Art of Probability For Scientists and Engineers Since there seems to be much talk concerning Category Theory in this thread, there's a new book by Birgit Richter that I recently bought called From Categories to Homotopy Theory. It seems to offer a very streamlined and rapid treatment of basic Category Theory to more advanced topics found in Homotopy Theory. It might be worth checking out.

>>11276 Oh. I have seen Wedhorn's book floating around my recommendations before. I think I even put it on my reading list at some point. Good to hear that it's worth a read!

man it's overwhelming knowing which books to buy, especially since some cost too much.

have any lains read "fractals everywhere"? Was thinking of picking it up

man I hope this thread never disappears. I love it.

>Let's talk about math books. Interesting, fun, engaging math books. Here are some of the classics: >An Introduction to Mathematics - Alfred North Whitehead >Mathematics: A Very Short Introduction - Timothy Gowers >Everything and more: A compact history of ∞ - David Foster Wallace >Mathematics For Human Flourishing - Francis Su >A Mathematician's Apology - G. H. Hardy >What is Mathematics?: An Elementary Approach to Ideas and Methods - Richard Courant, Herbert Robbins, Ian Stewart >Mathematics and its History - John Stillwell >Mathematics: It's Content, Method and Meaning - A. D. Aleksandrov et al. >A Mathematical Bridge: An Intuitive Journey in Higher Mathematics - Stephen Fletcher Hewson >The Main Stream of Mathematics - Edna Kramer. >The Nature and Growth of Modern Mathematics - Edna Kramer >Mathematics, Form and Function - Saunders Mac Lane >Mathematics: The Music of Reason - Jean A. Dieudonné >Mathematics and the Imagination - Edward Kasner & James Newman, with preface and review by Jorge Luis Borges >James R. Newman's The World of Mathematics >Timothy Gowers's Princeton Companion to Mathematics >The Mathematical Experience - Philip J. Davis, Reuben Hersh, Elena Anne Marchisotto >Calculus: A Liberal Art (a.k.a. Historical Approach) - William McGowen Priestley >Alice in Numberland: A Students' Guide to the Enjoyment of Mathematics - John Baylis & Rod Haggarty >Journey into Mathematics: An Introduction to Proofs - Joseph J. Rotman >Foundations and Fundamental Concepts of Mathematics - Howard Eves >Proofs and Refutations: The Logic of Mathematical Discovery - Imre Lakatos >Journey Through Genius: The Great Theorems of Mathematics – William Dunham >The Enjoyment of Mathematics - Hans Rademacher & Otto Toeplitz >Mathematics and Logic - Mark Kac, Stanislaw Ulam >The Pleasures of Counting - Thomas William Körner >Imagining Numbers (particularly the square root of minus fifteen) - Barry Mazur >The Education of T.C. MITS - Lilian Lieber, Barry Mazur >Infinity and the Mind - Rudy Rucker >Logicomix: An Epic Search for Truth by Apostolos Doxiadis & Christos Papadimitriou >Uncle Petros and Goldbach's Conjecture - Apostolos Doxiadis >Mathematics Made Difficult: A Handbook for the Perplexed - Carl E. Linderholm >Surreal Numbers: A Mathematical Novelette - Donald Knuth >Gödel, Escher, Bach: an Eternal Golden Braid - Douglas Hofstadter >Memorabilia Mathematica; or, the Philomath's Quotation-Book by Moritz >Récoltes et Semailles - Alexander Grothendieck

>>12328 Has an unordered, uncommented list of books ever been useful to someone?