Why don't I understand "What is mathematics?" by Richard Courant I am 14 years old and can do A level maths happily. However, I want to further my knowledge of mathematics to pave my way for more discrete maths and harder analysis etc. So I decided to pickup "What is mathematics?" by Richard Courant because I heard it was very useful and it would be good for someone of my knowledge. 
I am having a lot of trouble understanding it though, and I'm not sure what it is that is stopping me. All I know is that it takes me a long time and some SE questions to understand even a page. It could be because of the way it is written which I am not used to?
If anyone has ever encountered this before then could you please offer me some advice and maybe a new way to approach it.
 A: Questions like this are very subtle, but I think I can probably highlight a few points.


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*Firstly, this book is rather elderly. It's been around for a while, and I've seen it described as old-fashioned and outdated. The outdated bit isn't such an issue, but the language used is perhaps a little stiff for an introduction to mathematics (though fairly true to what textbooks often look like). Looking quickly through a preview,

*Not everybody likes the same style of mathematics. There are several aspects to this. Firstly, you might not like the writing style/level of detail the author provides. Some people prefer terse books with the core concepts, so that you have to fill in the details; some people prefer very explanatory books which walk you through at least the first few calculations. Secondly, some people lean more towards applied mathematics, and away from pure mathematics. You might find yourself much more interested in how mathematical objects relate to concrete problems than in how the mathematical objects relate to each other. In this case, I still recommend learning more 'pure' mathematics as far as you can, but be aware that this is a very different ball game to A-level mathematics (which is more like the simple bits at the start of an engineering degree than a mathematics degree). You should try looking through books before buying them as much as you can in order to see whether or not they're the right kind of book for you.

*This is the first time you've tried reading something with any actual rigorous mathematics in it. This is always hard at first. It's a very different way of thinking to normal thought for most people, and that's fine. Work at it, and if you can, get people to critique your arguments. (One thing you might like to have a go at to supplement your learning is programming. Learning a little bit of how to solve simple problems on computers might help you with the mix of creative and practical logic you need for maths. Even just staring at some problems on Project Euler might help you think more clearly about what you're doing. That's just a random thought.)

*Don't be afraid to ask questions. It's perfectly normal to have lots of questions when you're first getting into gear with a new set of ideas. Particularly so when they're presented without clear motivation or a consistent theme - and "What Is Mathematics?" looks to me like it jumps around a lot! Talking new ideas through with fellow students is one of the best things about universities, in fact. You shouldn't always expect a pure mathematics definition, for instance, to be so obvious that you don't read it twice. Most of the time, you will find yourself reading a definition 3 or 4 times before you have any real sense of what's going on behind the scenes. It might take the best part of a year to get to grips with more abstract definitions, depending on how you think, and how much help you get.


In closing, you might want to try a book I've never read but which many people I know have: How to Prove It. This is an introduction to problem solving which you might find a lot more useful in figuring out how mathematicians think. As I said above, though, have a look through it before thinking about buying it.
(Have you ever read any Murderous Maths books, by the way? When I was younger I thought they were great fun, and probably helped to clear my thinking up a lot. The material is much more basic than A-level but I once loved them!)
A: I think it is profoundly important to distinguish (at least) three different types of mathematics writing: textbook writing (especially lower-division undergrad, but also introductory graduate level), "research papers" (in traditional refereed journals... the necessary professional purpose most often being personal advancement more than enlightening any reader), and ... "other": things written neither to sell textbooks nor to score status points. To confuse one with the other leads to bonus confusion about context.
I am not a fan of textbooks nor of highly-structured curricula, because such structure is inevitably contrived. The supposed necessity of weekly exercises (exams, grades) likewise contorts the appearance of mathematics. Similarly, but differently, many traditionally-published papers deliberately have an imposing facade, but not-so-much content... so that, in particular, it is far harder for a non-expert to read them than it's worth. Tsk. Sure, there's background required, but, also, there is a conflict of interest about making one's work appear simple.
If one's prior experience is with textbooks, and in a setting where one is pressured to treat textbooks as unquestionable authorities, one may have become too passive (if only to "survive" in a certain sense).
Books such as Courant's are not meant as textbooks, and, if interpreted as such, are incomprehensible. They are not "research", either, so are not designed to impress. They may treat substantial (if relatively elementary) mathematics, and may do so a bit carefully, but will not necessarily take up the pedantic-didactic style in which every smallest detail must be flogged (thus leaving the mass of details unfortunately undifferentiated).
In contrast to the way we are often told to read textbooks (line-by-line, not moving forward until everything is (allegedly) perfectly understood), it is smarter to read more lightly, to get a larger picture, to have some way to anticipate the relevance and necessary disposition of lower-level details. Try to imagine that no one will be playing "gotcha" in pop quizzes on what you've read!
It is legitimate, and desirable, to become aware of important mathematical ideas whose ramifications are only dimly glimpsed. All the better if a source such as Courant is provocative and disturbing, if that means that one is provoked and impelled to think more about what's going on.
In particular, there is no reason to require or expect anything like "mastery" of each chapter before looking at the next. This is an invidious myth. Instead, for a serious, thoughtful person, often skipping forward is more enlightening than sitting in one spot hoping for an epiphany. No compulsion to treat sources as sources for "tests"!!!
And, yes, ok to feel uneasy. Rereading later, in different contexts, is not merely fine, but highly desirable.
A: A couple of suggestions from someone who's been teaching college students for a long time :) First, learn to read math books with pencil and paper and try to figure things out as you go. Don't be passive. And yes, serious mathematics is hard to read casually.
A couple of suggestions of books to look at. A nicely written book that gives an introduction to proofs and mathematical thinking is How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, by Kevin Houston, Cambridge University Press, 2009. Also, check out any of Ian Stewart's books written for the neophyte mathematician; he has great stuff in there. You will see zillions of books on Amazon, but he will give you in many of his books a fascination with current mathematics, not just ancient stuff :). For something older but full of fascinating stuff, look at Hilbert and Cohn-Vossen, Geometry and the Imagination.
