## Cool math books

Straight-up math books:

- ‘Understanding Analysis’ by Abbott
- ‘The Art of Proof’ by Beck
- ‘Contributions to the Founding of the Theory of Transfinite Numbers’ by Cantor
- ‘What Is Mathematics?’ by Courant & Robbins
- ‘The Mathematical Experience’ by Davis and Hersh
- ‘Introduction to Mathematical Thinking’ by Devlin
- ‘Journey Through Genius: The Great Theorems of Mathematics’ by Dunham
- ‘One Two Three … Infinity’ by Gamow
- ‘Book of Proof’ by Hammack
- ‘Mathematics: a Human Endeavor’ by Jacobs
- ‘Mathematics and the Imagination’ by Kasner and Newman
- ‘Surreal Numbers’ by Knuth
- ‘The Pleasures of Counting’ by Korner
- ‘Proofs and Refutations: The Logic of Mathematical Discovery’ by Lakatos, Worall, and Zahar
- ‘The World of Mathematics’ by Newman
- ‘Numbers: Rational and Irrational’ by Ivan Niven (as well as other stuff in this series)
- ‘The Gentle Art of Mathematics’ by Pedoe
- ‘The Mathematical Tourist’ by Peterson
- ‘Mathematical Snapshots’ by Steinhaus
- ‘The Joy of x: A Guided Tour of Math, from One to Infinity’ by Strogatz
- ‘Mind Tools: the five levels of mathematical reality’ by Rucker
- ‘Math Girls’ by Yuki
- Math on Trial: How Numbers Get Used and Abused in the Courtroom by Leila Schneps

Other Resources:

- Jump Math (for kids)
- Khan Academy
- Keith Devlin’s Coursera course “Introduction to Mathematical Thinking”
- A math book list for kids
- Martin Gardner’s stuff (this for example)
- ‘Men of Mathematics’ by Bell, if you like soap operas featuring math
- ‘Gödel, Escher, Bach: an Eternal Golden Braid’ by Hofstadter, if you like philosophy
- ‘A Mathematician’s Apology’ by Hardy

Please comment on this page if you have something to add. Also feel free to give your opinions on the above, since I haven’t actually read them.

Morris Kline has a number of well-regarded overview books. I’m most familiar with the excellent ‘Mathematics in Western Culture’, but see also ‘Mathematics for the Nonmathematician’, which is a textbook for a liberal arts course in mathematics. For something more specific in focus (though still remarkably wide-ranging), Petr Beckmann’s ‘A History of Pi’ is highly recommended.

There are some marvellous books in the list already. ‘What Is Mathematics?’ is an excellent starting point for a self-learner. ‘Proofs and Refutations’ and ‘The Mathematical Experience’ should be required reading for all working mathematicians.

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It’s Morris Klines’ Mathematics and the Physical World for me. I read it in my second year at University. It was the closest I have come to a religious experience.

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How could I forget

Math Girls?! Anime-style story line, and real math. I plan to re-read it this December, to see if I can understand why the sum of the alternating harmonic series is pi2/6. I just looked up the website (bentobooks.com) and found out that Math Girls 2 will be out in December. Perfect holiday gift for me!LikeLike

added.

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The Joy of X is based on a New York Times series of a couple of years ago. I just received the book and haven’t gotten a chance to read it yet but did read the series. The series was great and I expect the book to be too. But in the interest of full disclosure, Steven Strogatz is a friend. A great book (not written by a friend—I don’t even know him) that I’m about a third of the way through is Math in 10 Lessons: The Grand Tour. It’s a pure mathematics book that tries to show those in the humanities, as well as scientists and other non- mathematicians, how mathematicians view their discipline.

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Ummm… thanks for the spelling correction, but surely you don’t have to be nasty. Khan Academy is an amazing site, and he really started something which I for one appreciate as a mother. My oldest son learned a lot of physics from him.

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Where was I nasty? Because I suggested that there are far better math vids and named some?

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“Surely you can do better” comes off, at best, as a bit haughty. Worse, it seems a bit snotty, condescending, patronizing, or all three. ( I suppose it could have been delivered innocently, by assuming the exact right “tone”, but the first reaction is to take it as described; your reply indicates that’s correct.)

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I LOVE LOVE LOVE KhanAcademy.org. I’m a neuropsychologist assessing children with learning problems…and having such high quality *tutoring* available for FREE is a Godsend for the kiddos I work with. My kiddos just want to know HOW to solve the problems on their homework!!!!

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“Meta math, the quest for Omega” by Gregory Chaitin is as mind bending than GEB

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Surreal Numbers was my very first cool math book, recommended by the same French mathematician who told me about the Nicolas Bourbaki. More recently I enjoyed reading Poincare’s Conjecture.

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Have you tried The Puzzling Adventures of Dr Ecco, by Dennis Shasha?

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I suggest you add “Who Is Fourier?: A Mathematical Adventure” to your list.

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Late to the party, I suggest “The enjoyment of Mathematics” by Rademacher&Toeplitz, maybe old fashioned but I remember having a lot of fun .

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Rademacher & Toeplitz is in the top 20. They are not afraid of tackling intricate topics and bringing them down to high school level (that is, good high school). But I must say that these expositions never get old fashioned – only Hollywood does.

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Do you have a recommendation for a good introduction to Bayesian statistics? I’m particularly interested in comparing Bayesian vs. frequentist statistics. And my statistical skills are pretty rusty ;-[

TIA, Pete Tillman

Consulting Geologist (retired), Arizona and New Mexico (USA)

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Try Gelman’s book, Bayesian Data Analysis. He also writes a blog .

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Thanks!

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Try Probability Theory: The Logic of Science by E. T. Jaynes. It’s a great read and has a very good introduction to probability. The math gets heavy pretty quickly but the text alone is worth the book.

John Kruchke’s book Doing Bayesian Anaysis is a first rate tutorial suitable for those of us who don’t think in intregals.

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Euler’s Gem: The Polyhedron Formula and the Birth of Topology by Richeson

The Fractal Geometry of Nature by Mandelbrot

How to Solve It by Polya

How to Lie with Statistics by Huff and Geis

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I think the best Book I’ve seen on teaching “proof” so far is “A Transition to Advanced Mathematics” by D. Smith et al. We used this for My “Foundations of Mathematics” course in college. And I still reference it! . What do you think?

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Books by John Allen Paulos (Innumeracy for example)

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What about “God Created Integers” by Stephen Hawking?

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You might also consider

From Zero To Infinityby Constance Reid.LikeLike

I would also add ‘A Radical Approach to Lebesgue’s Theory of Integration’ by Bressoud. This is a textbook on measure and integration, but the historical progress of the theory is told as the subject matter develops. A good complement to #3 might be ‘Georg Cantor’ by Dauben. I haven’t read #3, but this book gives a great detailed look into the work of Cantor, without being too technical, and also looks at his life.

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Yeah, the historical perspective — always missing in under/grad courses.

Typically, math concepts are presented as ex nihilo. But when you look more deeply, you see how it is cobbled together from bits of this and parts of that.

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Bob Moses & Charles Cobb. Radical Equations: Civil Rights from Mississippi to the Algebra Project. Beacon Press, 2002.

http://www.algebra.org/

Jeffrey Isaac. 1999. “The Algebra Project & Democratic Politics,” Dissent (Winter).

Highly recommended for trying to draw the analogy between the importance of verbal literacy to the struggle for voting rights in the 1960s and the importance of math literacy for poor/minority kids in contemporary America – if, of course, we hope them to have the opportunities of political & economic citizenship. Ella Baker meets quadratic equations.

The project has been running since the mid-1980s and raises all sorts of interesting political issues (hence the Isaac piece) but gets relatively little attention.

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“The Princeton Companion to Mathematics” edited by Gowers, Barrow-Green and Leader.

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I just came across your list (and site) and would like to suggest a few title I have enjoyed. One of the first books to get me interested in math was “Fermat’s Last Theorem” by Simon Singh, followed by several of Martin Gardner’s articles and books, “In Code” by Sarah Flannery, and several large reference-style compendiums that prove quite useful in trying to get a handle on new subjects, “Mathematics, From the Birth Of Numbers” by Jan Gullberg, “The VNR Concise Encyclopedia Of Mathematics” by W. Gellert et al, Martin Gardner’s :The Colossal Book of Mathematics:, and Dr. Richard Elwes’s “Mathematics 1001”.

Most recently, I just finisthed Nate Silver’s “The Signal and The Noise” and “Connected” by Nicholas A. Christakis.

I hope these suggestions help a bit and the site continues to provide such an interesting contribution to a fascinating subject.

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I love this book God Created Integers by Stephen Hawking

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This is my favorite – ‘A Million Random Digits with 100,000 Normal Deviates’. Not for the book but for the good read in Amazon’s comments about it.

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Donald MacKenzie, Statistics in Britian 1865 – 1930: The Social Construction of Scientific Knowledge (1981).

Galton, Pearson, and Fisher were all eugenists — it should never be forgotten that Galton’s statistics first emerged as a powerful rhetorical device for the advancement and promotion of the eugenics movement, and for the justification of mass sterilizations of inferior classes. Unfortunately, one group did in fact succeed on such a massive scale that it still horrifies and haunts humanity.

Does anyone remember Florence Nightingale’s rose chart? “In 1859 Nightingale was elected the first female member of the Royal Statistical Society and she later became an honorary member of the American Statistical Association” (wikipedia).

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Brian Butterworth, The Mathematical Brain;

Graham Farmelo, ed., It Must Be Beautiful: Great Equations of Modern Science;

Robert Kaplan, The Nothing that Is: A Natural History of Zero;

E. T. Jaynes, Probability Theory: The Logic of Science (corrected ed.);

David Acheson, From Calculus to Chaos.

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Glen, I remember Florence Nightingale’s rose chart. Brilliant innovation. You can find it along with other wonderful exhibits in Edward Tufte’s The Visual Display of Quantitative Information (2nd ed.).

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T. Korner has written several books other than “Pleasures of Counting” that are pretty accessible. His two Fourier Analysis books are written at an mid-undergraduate level, and his game theory book is at an advanced high school/early undergraduate level. Also both are full of fun stories. And very relevant since Fourier methods and their extensions have been so important over the past two centuries in so many fields, both pure and applied.

I wish I could suggest a good applied computational mathematics book. Sadly I don’t think there’s a great book out there yet. Also it’s a moving target, since the best languages, methods, libraries, and problems change from decade to decade.

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Thanks for the list. I emailed some links like Jump Math to my friends. Its really awesome. You did hard job to manage all these great resources here. Keep it up..upper…uppest! :p

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Back when the world was young, and I was even younger (about 10), I came across a book called Mathematician’s Delight by W W Sawyer. I can honestly say this book changed my life (very much for the better!).

I’m glad to see it’s still in print. It’s the kind of thing that should be left lying around innocently to catch the attention of inquisitive young minds, even if it does contain words (such as ‘slide rule’) that have disappeared from common usage!

Oh yes, the slide rule… another life changer…

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Thanks for the list of books. I had already read and enjoyed more than a few on the list and now I have several new titles to add to my reading queue.

One additional suggestion is “The Math Book” by Pickover. It’s more history than mathematics, but can serve as a great (albeit brief) introduction to numerous topics. The illustrations are wonderful. For bonus points read it concurrently with “The Physics Book”.

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I would agree with Dave about The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics by Clifford A. Pickover. It’s definitely worth a read and is a good one to recommend to people.

As is Math on Trial: How Numbers Get Used and Abused in the Courtroom. Which I found to be a fascinating read and the tension really grips you.

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I just came across this post. To the list I would add Conceptual Mathematics: A First Introduction to Categories:

http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&qid=1373909461&sr=8-1&keywords=conceptual+mathematics

No other book I know of gets students involved in the deep issues of math earlier than this one.

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W.K. Clifford’s _The Common Sense of the Exact Sciences_. In my previous existence as a high school math teacher I was always looking for ways to get beyond the purely manipulative approach featured in most textbooks to bring some insight into mathematical thinking. This book is one of the best.

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Please recommend an introductory textbook on statistics and probability. Many thanks, RS

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Wonderful – for general math problems, Knuth’s first volume (Computing?) has more problems with a breadth of difficulty, from 8th grade arithmetic to Fermat’s last theorem (unsolved when he published). For some philosophy with statistics & probability, Polya’s Mathematics and Plausible Reasoning volumes one and two (he show data proving that the different number of spots on a die “loads” it, I’ve forgotten who rolled 12 dice more than 23,000 times, but it’s covered with the requisite probability calculations – then he forgets to tell you why this was done in the first place. Then Gamma (Euler’s Constant) by Havil and The Story of sqrt(-1) by Nahim. The last two cover the history of both concepts (and Nahim debunks the importance of factoring x^2+1=0 as motivatomg “imaginary” numbers).

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‘What Is Mathematics?’ by Courant & Robbins – My all-time favorite.

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I was going to suggest WW Sawyer’s books as well but you beat me to it! I had a friend in high school who hated math and had a lot of trouble understanding the abstract stuff. He read “A Mathematician’s Delight” and magically fell in love with math! Hard to believe, but it’s a true story.

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Cathy – great blog

I am looking for a book on time series analysis for the layman. Can you recommend something?Thanks

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Have a look at http://www.amazon.com/Analysis-Time-Introduction-Chapman-Statistical/dp/0412716402/ref=sr_1_fkmr0_1?s=books&ie=UTF8&qid=1438522504&sr=1-1-fkmr0&keywords=statistics+engineering++time+series+chatfield

Everything by Chris Chatfield is a gem!

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The book that I like the most among those enlisted above is ‘What Is Mathematics?’ by Courant & Robbins. Thanks for sharing the links.

Khan Academy has always served me as a wonderful resource. I recommend it to my kiddos so often.

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I’d suggest adding “Wolfram MathWorld” — the online compendium on mathematics: http://mathworld.wolfram.com . Not entertaining, but amazingly thorough. And it’s free!

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Love and Math, by Edward Frenkel (just published, late 2013), intertwines an exciting autobiography (struggling through antisemitism, excluded from Moscow State U in the 1980s USSR, getting an exit visa to be a visiting prof at Harvard at age 21 while still an undergraduate!) with a serious effort to explain his research in the Langlands program for the general reader. Much more math than most popular math book, especially in the endnotes.

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What about Mathematics for the Millions by Hogben or Thompson’s Calculus for the common man. In fact the entire series of Books on Mathematics for the Common man are fantastic

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Kasner and Newman’s ‘Mathematics and the Imagination’ – old but good (and available as a free download at archive.org). Really inspired me as a kid.

Cundy and Rollet ‘Mathematical Models’. Ditto.

Fuse ‘Unit Polyhedron Origami’

Yandell ‘The Honors Class Hilbert’s Problems and Their Solvers’

Dars, Lesne and Papillault ‘The Unravellers Mathematical Snapshots’. This one is more about mathematicians than mathematics. It’s look at the mathematicians at the IHÉS: if you can’t go there, this is the next best thing.

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A few of my non-technical favorites that haven’t been mentioned:

Dunham’s “Mathematical Universe” and “Euler: The Master of Us All”

Mandelbrot: “The Fractal Geometry of Nature”

Derbyshire (yes, I know): “Prime Obsession” and “Unknown Quantity”

Eli Maor: “Trigonometric Delights”, “To Infinity and Beyond”, “e: The Story of a Number”

Kanigel’s Ramanujan biography

Edna Kramer’s “Nature and Growth of Modern Mathematics”

various titles by Raymond Smullyan and Ian Stewart

and a seconding for the works of Martin Gardner – I’m rereading his collection “Penrose Tiles and Trapdoor Cyphers” and it’s way cool

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Kalid Azad at BetterExplained.com does a great job of presenting the subject on an intuitive level.

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Courant and Robbins was my great introduction to math when I was in junior high. The Laktos book I read many years later. Although I was a math major, it discusses many topics in proving things that had worried me in my courses. I would add Gödel’s Proof by Nagel, Newman and Hofstadter. An update of a monograph from the ’60s, it presents a very understandable explanation of the First Incompleteness theorem. Another book, Incompleteness by Goldstein, covers much of the same ground but weaves in the philosophical implications of the theorem and some biographical background on Godel.

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Stanislaw Ulam’s “Adventures of a Mathematician” is one of the most fascinating books I ever read. Also Littlewood’s “A Mathematician’s Miscellany” is, in my opinion, rather more interesting than his friend Hardy’s “Apology”.

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Also Tim Gowers’s “A Very Short Introduction to Mathematics” is a tour de force, and beautifully written. He once amusingly commented that his Princeton Companion might also be titled “A Very Long Introduction to Mathematics”

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“Math on Trial” was written by Schneps and Colmez🙂

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How about Danica McKellar’s books?

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Symmetry by Hermann Weyl

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No one has suggested The Art of Mathematics by Jerry King. How odd.

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Another author of excellent general introductory books, Sherman K. Stein (emeritus professor at U.C. Davis) “Mathematics, the Man Made Universe” and “Strength in Numbers”. I strongly second the remarks about W.W. Sawyer. He went out from Cambridge to start a Department of Mathematics in the Gold Coast (now Ghana), and I believe wrote some of the books to foster an interest in mathematics there.

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You have a great list of math books many of which I have never read, and I plan to check them out. There is one book that stands out in my mind that is not on your list. The book “Five Equations that Changed the World” is an excellent read and I highly recommend it.

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I have a little math book designed for folks who don’t like math and have avoided the subject. In “Math for Grownups,” I take readers through the basic math they need on a daily basis, to manage their finances, garden, do DIY projects, and bake. Through funny and engaging real-world problems, readers are introduced to the concepts behind basic calculations. I would be honored to have you take a look and include it in the list. (My second book, “Math for Writers,” takes the same approach for journalists and freelance writers. But that’s a pretty niche readership.)

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I recommend “The Geometry of Multivariate Statistics” for those of you interested in gaining a deep conceptual understanding of applied statistics and linear algebra. Understanding the geometric interpretation of vector and matrix operations seems much more intuitive than thinking of the topic as an algorithm that blindly leads to a result.

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I like this book a lot:

http://press.princeton.edu/titles/9665.html

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Very good book.

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I will gladly plug mathmunch.org (which I co-author) and aperiodical.com and quantamagazine.org (which I do not).

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“The Joy of X” is really not a ‘cool math book’. Maybe it is just where I am now but I got no brain buzz from any of it. Moderately interesting but not cool.

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My suggestion: “How not to be wrong” by Jordan Ellenberg. Even though it’s aimed at a non-mathematical audience, explaining how various fields of math (with a focus on probability theory though) apply to everyday life, I as a mathematician also had great fun reading it.

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Shameless self-plug: for a precalculus, mechanics, and calculus all-in-one textbook, with zero bullshit in it, check out the noBS guide to math an physics http://noBSgui.de/to/MATHandPHYSICS/

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“Why is there a philosophy of mathematics at all?”, by Ian Hacking. This is very much a philosophy book, not a mathematics one, but it’s written to present the author’s personal viewpoint in the context of previous viewpoints rather than as a survey or textbook or even as an argument, so I found it much more readable than anything else on the subject.

I’d like to warn people away from “Godel, Escher, Bach”, because I think it’s wrong in a subtle way which is very hard for people who don’t have philosophy experience to realize the flaws, and Hofstader doesn’t give any pointers at all to arguments against his views.

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Good ideas! I’d like to shamelessly plug my book, Hacking Math Class with Python. All the code needed to explore many traditional math topics like solving equations and factoring polynomials to creating fractals and 3D Graphics. It could be an intermediate step to implementing your (or KW’s) excellent ideas on “The Future of Math Education.”

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I would recommend any book by either Ian Stewart or John Allen Paulos.

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Cathy, which books would you recommend for engineers who haven’t worked with math since college?

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I recommend “Road To Reality” by Roger Penrose; it’s somewhat advanced but is a great treatise on modern physics and the mathematics behind our current understanding of the laws of the universe.

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I’ve read some of these. Good list. Like to add another book which suggests how to fix an anomaly in the pipeline for getting more into STEM careers in general: “Long STEM Roads for Life”, subtitle Math Education and STEM Careers- see Amazon.com.

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Thanks for your terrific, informative appearance on the Brian Lehrer show today on WNYC, and congratulations on the publication of your book! I’m looking forward to sharing your blog with my 17 y/o daughter!

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Totally with you on the Cantor. But you left out Nagel’s “Godel’s Proof” which if I can understand it, is a clear exposition of the theorem. GEB was great, but doesn’t really give you the mathematical joy of the Gödel discovery.

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The Man who Loved Only Numbers by Hoffman about Erdos is a fascinating biography with some thought-provoking math problems that non-mathematicians can understand. On the kids’ list, I strongly recommend The Number Devil. Another good kids book is Sidesways Math From Wayside School.

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Anything by Raymond Smullyan, anything!!! 🙂

Thanks so much for this splendid blog. You make me happy. 🙂

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Your book is very important socially and relevant to what is happening all through the world of big data. People must understand how the Theory of Solving Ill-Posed Problems and VC theory applies to model making and also how important problem definition is to validation. We are building a world of self-fulfilling prophecy and true danger of information isolation and polarization. Prediction acts as a short circuit to debate; when the process for prediction development is misused, the prediction loses validity, yet few can detect that loss, especially in deep learning networks. Models are given authority they do not deserve!

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Not a technical book but I loved reading Fantasia Mathematica as a kid.

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