Real Mathematical Analysis

What Hacker News says

1. harshrealityJan 2024
   I agree and I don't know why the myth persists that [baby] Rudin is the best way to learn analysis. It's certainly not so for a lot of people who could learn analysis perfectly well from friendlier (but not necessarily any less rigorous) sources. Rudin requires a lot of self-motivation.

   In addition to Abbott which you mentioned,

   "How to Think About Analysis" (Alcock), good gentle introduction for anyone who tries to dive into an analysis textbook and hits concrete.

   "Real Mathematical Analysis" (Pugh), good, rigorous approach that doesn't require quite as much effort to power through as Rudin does, although some of the exercises are tough.

   "Analysis I" and II (Tao), is good and starts from scratch building numbers from the Peano axioms, so there are no other unproven assumptions underlying everything later.

   "Counterexamples in Analysis" (Gelbalm & Olmstead), very helpful for understanding some pathological cases that break theorems one might intuitively believe are true.

   There's no reason today why someone can't grab pdfs of all of the above (even Rudin, although it's never been digitally typeset), from any friendly internet library, and use them all to build a better understanding rather than relying on one alone.

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2. phonebucketJan 2023
   A lot of recommendations depend on what you're trying to learn.

   But I've enjoyed the following texts to a larger extent than others:

   - Algebra: Chapter 0 (Aluffi)

   - Real Mathematical Analysis (Pugh)

   - Mathematics and its History (Stillwell)

   - An Introduction to Manifolds (Tu)

   - Gauge Fields, Knots and Gravity (Baez)

   - A First Look at Rigorous Probability Theory (Rosenthal)

   - All of Statistics (Wasserman)

   There are some authors I trust and am happy to buy so long as the topic vaguely interests me: VI Arnold, Tristan Needham and John Stillwell.

   I really like the list put out by @enriquto in a separate comment, but I've avoided duplicating those recommendations in the list above.

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3. amirkdvMar 2022
   (1) Real Mathematical Analysis, by Charles Pugh is a wonderful introduction to pure mathematics for a mathematically inclined engineer. The back cover starts like this

   > Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals?

   (2) The Man Who Mistook His Wife for a Hat, by Oliver Sacks. I think it's impossible to read this (or a number of other works of Sacks') and not be mesmerized by the workings of the human brain. Disclaimer: I wouldn't call neuroscience my "field of study" even though I did study it.

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4. laichzeit0Jun 2016
   Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....

   Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book).

   Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem.

   Step 4: Pick any of these books that interest you the most and do the same:

   - Calculus by Spivak

   - Algebra: Chapter 0 by Paolo Aluffi

   - Linear Algebra Done Right by Axler

   By then you should have enough mathematical maturity to know what to do next.

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5. WallWextraDec 2014
   I got started on "real" math with Spivak's Calculus. Some people start with Topology by Munkres, which is not a difficult book but is very abstract and rigorous so makes a good introduction. If you feel like you have ok calculus chops, maybe Real Mathematical Analysis by Charles Pugh. Other good books are Linear Algebra Done Right by Axler, or the linear algebra book by Friedberg, Insel, and Spence. Maybe even learn linear algebra first. It's so useful.

   Do plenty of exercises in every chapter, and read carefully. Count on about an hour per page (no joke). Plenty of math courses have their problem sets published, so you can google a course which uses your chosen book and just do the exercises they were assigned.

   If you don't feel comfortable with basic algebra and other high school math, there's Khan Academy, and some books sold to homeschoolers called Saxon Math.

   If you haven't had a course in calculus before, maybe you should skim a more intuitive book before or alongside reading Spivak. I don't know of any firsthand, but I heard Calculus for the Practical Man is good. Scans are freely available online (actually, of all these books) and Feynman famously learned calculus from it when he was 12.

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6. nicklafDec 2014
   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.

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