Suggestions for books on mathematics?

Jun 1, 2013


Discussion by: UncertaintyBlog

Does anyone have any recommendations for good introductory mathematics books? I'm trying to find something analogous to Richard Feynman's famous lectures on physics, but for math. Haven't thought much about it since high school, but would really like to review and go deeper. 

23 comments on “Suggestions for books on mathematics?

  • 2
    jburnforti says:

    Yes. Mathematics-a very short introduction by Timothy Gowers. He’s a Field Scholar -top of the tree and very good communicator. The book is excellent.



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  • 4
    Red Dog says:

    You have probably already thought of this but there are some excellent free online courses available. The first place I look is the iTunes Store (they have free stuff there as well). They have a whole section devoted to iTunes U. I took a fantastic class on Game Theory from Yale that way. The professor was fantastic and he put all the materials online including lecture notes, homework, answers, tests, etc. The quality is hit and miss though. I started a biology intro class but it was useless. Everything the Yale instructor did the bio teacher didn’t do, but there are definitely some great ones out there for free. i took a quick look and they have lots of Math classes from Algebra through Statistics, Calculus and some more advanced courses.



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  • 5
    phil rimmer says:

    I actually learned more maths, more readily from the Feynman Lectures than my Maths courses. He certainly sorted my divs from my grads and curls. I wish he had done a maths only course as well.

    A fun general read on maths is Marcus du Sautoy’s Music of the Primes. This manages to be about its specific issue and a general account of the history of mathematicians and mathematics.

    A useful gathering place for University courses available as videos online is the Android App., EduPort. Stanford, MIT courses and the like are collated here along with some popular materials.

    More concentrated Mathsoverflow collects a big long lists of good stuff from basic to advanced.

    From this list- Differential Equations at MIT shows the richness available.



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  • This may or may not be helpful, but Mathletics by John D. Barrow was definitely enjoyable in the Feynman sense. I’m not a mathematician by any stretch, but reviews promised to charm both lovers and haters of math. I like sports and this is a book about 100 different sports and the intrinsic math within them.

    Mike

    http://www.amazon.com/Mathletics-Scientist-Explains-Amazing-Things/dp/0393063410/ref=sr_1_1?s=books&ie=UTF8&qid=1370318095&sr=1-1&keywords=mathletics+barrow



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  • 8
    somebodysomewhere says:

    All the books suggested are good. For a first look at calculus I suggest Calculus Made Easy by Silvanus P. Thompson, updated by Martin Gardner. For a transition from the purely mechanical aspects of mathematics, calculation, I would suggest books like Introduction to Mathematical Thinking by Keith Devlin. I think Richard Feynman learnt about calculus from a book called Calculus for the Every Man, or Everyman’s Calculus or something. These books, like Martin Gardner’s update to the original by Thompson, are very good to get the basic details without drowning pages of explanations and confusion. It is always helpful to review everything, the very basics, first before jumping into more abstract ideas. The suggestion to take a foundation or basics course at college is a sound idea.



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  • I’m glad to hear you’re interested in maths. I thoroughly recommend John Stillwell’s ”Mathematics and its History”. This is a wonderful book. It introduces some of the most important and interesting developments in mathematics since the discovery of the Pythagrean Theorem. It is both a great history book and a great math textbook (complete with exercises). Here’s a link:
    http://books.google.co.uk/books/about/Mathematics_and_Its_History.html?id=V7mxZqjs5yUC&redir_esc=y

    For some math-lols, I also recommend ViHart’s YouTube channel:
    http://www.youtube.com/feed/UCOGeU-1Fig3rrDjhm9Zs_wg



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  • The Khan academy comes to mind:

    https://www.khanacademy.org/

    I’ve heard that it is a good resource, but I haven’t really had a look for myself.

    In reply to #4 by Red Dog:

    You have probably already thought of this but there are some excellent free online courses available. The first place I look is the iTunes Store (they have free stuff there as well). They have a whole section devoted to iTunes U. I took a fantastic class on Game Theory from Yale that way. The profes…



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  • Mathoverflow is kind of a math discussion board for researchers. They can be pretty harsh on questioners who don’t post research level stuff. But I do recommend perusing Bill Thurston’s (http://mathoverflow.net/users/9062/bill-thurston) and Terry Tao’s (http://mathoverflow.net/users/766/terry-tao) posts. They often make for great reading. The student version of overflow is mathstackexchange: http://math.stackexchange.com/

    In reply to #5 by phil rimmer:

    I actually learned more maths, more readily from the Feynman Lectures than my Maths courses. He certainly sorted my divs from my grads and curls. I wish he had done a maths only course as well.

    A fun general read on maths is Marcus du Sautoy’s Music of the Primes. This manages to be about its speci…



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  • MIT Opencourseware http://ocw.mit.edu/courses/mathematics/index.htm might be a good place to start.

    There’re a lot of mathematics videos on http://archive.org , though the ones I’ve seen are perhaps too basic e.g. http://archive.org/details/ClassicArithmeticMulitplicationAndDivision or too advanced e.g. http://archive.org/details/UC_Berkeley_Math_2011_08_17_Shaowei_Lin . I find using VideoLAN Player http://www.videolan.org useful as it allows one to play videos at high speed with correct pitch, which can save a lot of time.



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  • 14
    orbitfold says:

    I scoff at all the suggestions here. Mathematics is a far wider and more complex field than physics so there isn’t even a slim chance of a single book covering it all or even one fifth of it. If you want a good undergrad introduction to Calculus, for example, I think there is no better book than Spivaks Calculus (some may suggest Principles of Mathematical Analysis by Rudin but given the abysmal teaching of math in high schools these days it will probably be too much for most people). As far as linear algebra goes I think a good introductory book is Linear Algebra by Kenneth M Morgan and Ray Kunze. Then there are fields like topology, mathematical logic, theory of computation and the like. Maybe if you can say more specifically what area of mathematics interests you I can give a more specific recommendation. Since most people mean calculus when they say mathematics I would suggest Spivaks Calculus, it is really a good introductory text.



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  • 15
    k_w_kemp says:

    In reply to #5 by phil rimmer:

    . He certainly sorted my divs from my grads and curls. I wish he had done a maths only course as well.
    OK, you must be British, because you say “maths” not “math,” so can you translate this sentence into non-mathematical American for me? Thanks!



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  • 17
    phil rimmer says:

    In reply to #15 by k_w_kemp:

    In reply to #5 by phil rimmer:

    . He certainly sorted my divs from my grads and curls. I wish he had done a maths only course as well.
    OK, you must be British, because you say “maths” not “math,” so can you translate this sentence into non-mathematical American for me? Thanks!

    Sure. “He certainly sorted my div from my grad and curl.” 🙂

    or this

    Do you say “I study mathematic”? Its squiggly red line in my editor suggests not.



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  • 18
    PERSON says:

    In reply to #14 by orbitfold:

    I scoff at all the suggestions here. Mathematics is a far wider and more complex field than physics so there isn’t even a slim chance of a single book covering it all or even one fifth of it.

    I’m not sure why scoffing is required. That’s kind of obvious. The OP asked for an introductory book or set of lectures. You have to start somewhere. If you want something comprehensive and (as I’ve heard one professional mathematician put it) “anal”ytical, and happen to be interested in Calculus you could read Spivak or Apostol’s calculus, yes, but that’s rather dry and technical. Not to everyone’s taste or learning style.

    Surely a good introduction would give an overview of the fields that exist, and would allow one to then decide which seemed most interesting.



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  • 19
    orbitfold says:

    In reply to #18 by PERSON:

    Surely a good introduction would give an overview of the fields that exist, and would allow one to then decide which seemed most interesting.

    As far as science goes yes. You learn quite a bit if someone explains to you, say, the conservation of momentum. In fact with some explaining you can probably learn about it as much as current human knowledge goes. Same goes for example if someone explains to you the kinds of fossils found in the ground and how they are layered in time. However if someone states, for example, Euler’s formula I think you have learned nothing (as far as mathematics is concerned). Only when you understand the intricate machinery behind the formulas is when you have learned something comparable to what a mathematician understands. And there is no way to understand that machinery unless you have basic mathematical truths entrenched in your mind. That only comes with numerous exercises solved and statements rigidly proven. “There is no royal road to geometry” is still as valid today as it was back in Euclids time.



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  • 20
    PERSON says:

    In reply to #19 by orbitfold:

    In reply to #18 by PERSON:

    Surely a good introduction would give an overview of the fields that exist, and would allow one to then decide which seemed most interesting.

    As far as science goes yes. You learn quite a bit if someone explains to you, say, the conservation of momentum. In fact with som…

    There’s understanding and there’s an overview. You can get that Euler’s formula links geometry and algebra before you get into the gory details. Mathematicians seem to have a tendency to behave like Gauss and be “like the fox, who effaces his tracks in the sand with his tail.” This creates elegant proofs, I’m sure, and sorts the wheat from the chaff when evaluating students, but I don’t think it’s the only way mathematics can be learned or thought about. It may well be the most effective way, but it’s not necessarily the best in all circumstances.

    [T]here is no way to understand that machinery unless you have basic mathematical truths entrenched in your mind. That only comes with numerous exercises solved and statements rigidly proven. “There is no royal road to geometry”

    This is certainly true. But one can develop an awareness of what exists and decide in which topical direction to go. This is less necessary in a taught course I guess, since capabilities can be interpreted and advice given, but for self teaching it’s very important, I think.



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  • This is a very old book, but introductory math does not change that quickly, Mathematics for the Millions by Lancelot Hogben.

    If you have some money, visit the local university bookstore and see what texts they are using. If you don’t, see if you can find slightly out of date used copies.



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  • 22
    Phineas says:

    While it’s not a book, videos from the Khan Academy (searchable on Google or YouTube) explain maths with such clarity that it is exciting and enjoyable. Somewhat less poetic than Feynman, I should add.



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  • 23
    margana says:

    Check out https://www.khanacademy.org/, it is free, new, pretty good. It has a lot of math on it, from basic to advanced. Not a book, though.

    Here’s a book: Mathematics 1001 by Dr. Richard Ulwes. It’s more of a reminder of what you already know and an unstructured intro to new areas. Not really a learning book, but well-reviewed and I like it.



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