Ask HN: How or where to begin learning mathematics from first principles?
I absolutely, positively second the recommendation of "Real Mathematical Analysis" by Charles Pugh (don't miss the advice he relates from his colleague, on pages 9&10, with the heading "Metaphor and Analogy", which could easily form the basis for a dissertation on the psychology of mathematical intuition and inspiration). Pugh does an exquisite, uncommonly good job of avoiding a pitfall that >99.9% of mathematics authors fall into, making it more or less impossible for one to genuinely understand mathematics outside of a university. What pedagogy is it that Pugh (and Spivak) care enough to get right, where nearly all others fail? It is this: Pugh has carefully crafted his book into what I'd call a 'mind expansion tool': almost everything there is crafted to be read, internalized, and meditated on. By contrast, almost all other mathematics books read like a laundry list of theorems and proofs, with some discussion inserted as an afterthought.Let me tell you a dirty secret about mathematics textbooks: almost all of them are highly flawed and incomplete dialogs between the author and the supposed reader. The reason for this: the first and foremost purpose of almost all mathematics textbooks is to organize the AUTHOR'S conception of the subject (not the student's!), for the primary purpose of TEACHING a course on the subject. In other words, the book's primary purpose is NOT to be read directly. A given mathematics textbook represents a model of the way in which the author (casually) BELIEVES students of your level might try to reason about the subject, whereas in reality, the author has so long ago advanced beyond your level that s/he cannot even remember how difficult it was when s/he first learned the subject.If you attempt to read most mathematics texts directly, outside the context of a university course (and without having already gained a true understanding of mathematics), you will almost certainly reach a stage in your reading in which you have internalized a certain amount of verbiage (say, some theorems, maybe a proof or two, and some light discussion, with the pretense that the abstractions introduced are 'useful' for some unknown reason). Certainly, you are asked to do some problems at the end of the section, and this is in fact a somewhat reliable way to reach some kind of personal discovery, and hopefully at least some mild enlightenment about just what the section was really about. (A good textbook will have highly instructive problems; however, the difficulty is, it is virtually impossible to know just how worthy of your time they will be before you spend hours working on them.)However, even at university, I almost NEVER resorted to reading the textbook: careful attention paid to the lecture, copious notes, regular attendance of office hours, and most of all, intense thought about the problems SPECIFICALLY given (and hopefully invented) by the lecturer were all that I was ever inclined to pursue (and all that I ever needed to succeed). If I read the textbook at all, it was only ever sought as a reference, or to fill in the gaps of a lecture which I failed to understand completely. Which is precisely the reason most math texts read so poorly: they are supplementary material for university courses.Despite vouching for it, I do not recommend you only read Pugh--at least not right away, and not from cover-to-cover. If you must start from scratch, please start with Spivak's "Calculus", which is similarly excellent in directly addressing the pedagogical needs of an autodidactical learner. Please note that by far the most thing to learn when studying mathematics is something that is impossible to encapsulate in any specific result; I am talking about "mathematical maturity". If you only do a one or two problems in all of Spivak, but spend several hours thinking deeply about a specific aspect of a problem or passage that leads you to have new, creative thoughts, you will have learned more than you could have by merely working through it in a mindless fashion.If you do intend to make it through a significant chunk of Spivak, be prepared to spend an enormous amount of time at it. There are many, many difficult problems in it. In addition, you should be spending time and effort not only writing down the steps of your proofs, but trying to come to grips with the very definitions you are working with. In mathematics, definitions and assumptions are most important--and they are certainly more important than clever tricks. This is why graduate students in mathematics have to learn their subjects over again--most undergraduate subjects do not do a precise or complete enough job of completely stating all definitions needed to make the theory entirely clear.The greatest mathematician of the 20th century, Alexander Grothendieck (who recently passed away), was as productive as he was because of his uncanny skill in inventing definitions of mathematical objects which put the problem in a broader context. Raw mathematical power is available to mathematicians to the extent that they allow the context of ANY given problem which they attempt to expand in their mind, until it connects with the relevant intuition. Once this inspiration strikes, the answer becomes easy. To Grothendieck, solving a problem was more a test of his ability to create a useful theory, than an end to itself. This speaks volumes to the value of thinking abstractly and creatively, rather than just trying out hoards of problems and expecting things to magically line up in your brain, hoping for an answer to pop out. There are generally two kinds of problems in mathematics: those which simply require organizing the essential definitions and required theorems until the answer is obvious, and those which need a fundamentally new idea. In neither case will you be able to 'plug and chug'. A great deal of harm is done to students of mathematics in grade school, because the subjects are invariably taught by non-mathematicians, in a highly non-mathematical way--in fact, in a way that is antithetical to the very core of the subject. Please google and read Paul Lockhart's essay titled "A Mathematician's Lament" to see if you really understand just what mathematics is (or if deleterious notions from your schooldays are continuing to blind you from the simple beauty of pure mathematics). I will add as well the recommendation that you read G.H. Hardy's essay, "A Mathematician's Apology".Learning mathematics is so incredibly difficult for the novice because it is almost impossible to teach this process. One must fail over and over again. I cannot lie: mathematics will be probably be difficult and unnatural for everybody except those who allow themselves enough time to commit to thinking freely and creatively about it, until a point of 'accelerating returns' is reached. Attempting to proceed directly to applied problems will invariably fail. The counter-intuitive truth about applied mathematics is that studying pure mathematics is in fact far more practical than attempting to think about the problem directly. This is because an understanding of pure mathematics gives you the ability to CREATE. Alfred Whitehead said: "'Necessity is the mother of invention' is a silly proverb. 'Necessity is the mother of futile dodges' is much nearer the truth."I'll also leave you with a relevant quote from the great expository writer and mathematician Paul Halmos: "What does it take to be [a mathematician]? I think I know the answer: you have to be born right, you must continually strive to become perfect, you must love mathematics more than anything else, you must work at it hard and without stop, and you must never give up."And another, in which he tells you how you should read a mathematics text: "Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"I have heard professional mathematicians express themselves the difficulty that even they have in maintaining the attention span required to read a traditionally written, unmotivated mathematics textbook. One such mathematician said that he skipped directly to the theorems, and attempted to discover a proof for himself. This is another secret to mathematics: it is always better to invent proofs yourself than to read the ones given in the text. This may be counter-productive in the early stages of your learning, but it is something you should continuously challenge yourself to attempt. If the first steps of a proof do not come to mind automatically, cover up the proof given in the text, except for the first few words. Then try to prove it again from scratch, with the knowledge that the objects being used in just that initial part might be part of one possible proof. Repeat as necessary, until you have either discovered a proof for yourself, or you have uncovered the entire proof given in the text. In either case, you will have thought long and hard enough to never forget the definitions and ideas needed to write the proof you end up with, even if you forget the proof itself. Later, you will only remember the essential idea. Then, it is an excellent exercise to attempt to work out the details again.