TITLE: Why don't I understand "What is mathematics?" by Richard Courant
QUESTION [19 upvotes]: 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.

REPLY [6 votes]: 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.