How to read mathematics Unfortunately, most writers do not follow the above principles; consequently, mathematics is very hard to read. In general, mathematical literature should not be treated as relaxing armchair reading, but instead as a sequence of vigorous exercises which must be performed. Always read mathematics with a pencil and paper close at hand. Find concrete examples. Whenever you see a new deﬁnition or theorem, stop and imag- ine the simplest nontrivial example you can think of which satisﬁes the conditions of the deﬁnition or the statement of the theorem. In the discussion that follows (eg. the proof of the theorem), check each of the author’s statements against your example. Draw pictures. As new notation is introduced, draw pictures (like those in Figure ??) depicting the objects and their relationships. This will help you remember them later, and help you understand the author’s reasoning. Try to do it yourself. Before reading any proof, stop for a moment and try to prove it yourself. You probably won’t succeed, but the exercise will prepare you to understand the author’s proof. You will more clearly understand the meaning of the statement being proved, and the hurdles which must be overcome. Read iteratively. Mathematics cannot be read ‘linearly’ like other prose. Instead, it must be read in a sequence of increasingly detailed ‘passes’. First, skim the text to get the basic structure of ideas. Then, go back, and try to ﬁll in the details. If a particular statement doesn’t make sense, don’t get stuck on it; keep moving. It may make more sense when you come back to it again later, after seeing what follows. Expect to read a proof at least three or four times before it starts to make sense. Write in the book. Use highlighters (of one or more colours) to indicate where notation or terminology is deﬁned. This will help you later when you ask yourself, ‘What does f3 (x, y) mean again?’, or, ‘What is a bi-inﬁnite frobnitz?’ Use a pencil to indicate the logical structure of the proof (ie. underline important mile- stones, put boxes around submodules, etc.). Draw arrows to indicate that equation (17) is being implicitly invoked in the proof of Claim 23(b), etc. If you are uncomfortable writing in the original book, then photocopy the relevant pages and write on the photocopy instead. (Do not write in other people’s books.) Keep a notebook: When an author writes, ‘Clearly, statement XYZ is true,’, what she usually means is, ‘Claim: XYZ. Proof: Exercise.’ Thus, you should be prepared to work through these ‘exercises’ as you read. You will rapidly run out of room if you try to work these exercises in the margins of the book. Instead, keep a notebook handy, and cross-reference your scribblings with the text. Identify the logical superstructure: The logical superstructure of a proof is its hierar- chy of ‘modules’ and ‘submodules’ (see page ??). Most authors do not make this structure explicit (although they should), but it is there nonetheless. Try to pick out where one mod- ule ends and the next one begins, and identify the purpose of each. It may be helpful to indicate this with pencil marks in the text. Find the big idea: Most proofs are motivated by some intuition or picture in the author’s mind. Often, she keeps this picture secret. Try to ﬁgure out what this picture is. Once you have the picture, the mysterious technicalities of the proof may suddenly become transparent. Ask, ‘Is this necessary?’ There are three levels of understanding a proof: 1. Understanding, at a purely technical level, how each assertion logically follows from previous assertions. 2. Understanding the overarching strategy, the ‘big idea’ of the proof. 3. Understanding why the proof must be as it is, and couldn’t be simpliﬁed. Seek Level Three. At each stage, ask, ‘Is this necessary? Couldn’t this be done a simpler way?’ The answer is probably ‘no’, but in discovering why the answer is ‘no’, you will achieve a deeper mathematical understanding.