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Does someone have a recommendation for a book that get deeper into mathematics by teaching actual mathematics rather than teaching about it? Many of the books
by alok-g 13y ago
Does someone have a recommendation for a book that get deeper into mathematics by teaching actual mathematics rather than teaching about it? Many of the books I have come across leave me with more questions than answers [1].
I am interested in pure mathematics mainly including logic, set theory, category theory, etc.
[1] An excellent example not mentioned by others here is also The Road to Reality by Roger Penrose. The sections on mathematics are very good, but ultimately leave the topic open-ended too soon.
- jfarmer 13y agoWhat are you looking for that isn't simply a math textbook? Honestly, that's how mathematicians learn "actual" mathematical subjects, too. If you're comfortable with calculus as a subject, for example, and want a "pure mathematics" approach, I recommend Michael Spivak's Calculus. If you've never worked through a pure math textbook from start to finish, that's a good start. There's not that much interesting in the three subjects you listed — logic, set theory, and category theory — that doesn't depend on other subjects or a prior level of mathematical maturity. Category theory was originally invented to solve and categorize problems in algebraic topology, for example. For example, I rather like mathematical logic and model theory, but you're going to have a rough time if you don't have a visceral, intuitive understanding of countability arguments and at least a handful of subjects you'd be reasoning "about." Unless you know the standard model of arithmetic, for example, how can you think about non-standard models? Without that, important results like the Löwenheim–Skolem theorem will likely seem contextless.
- alok-g 13y agoGood points. I have already gone through several levels of engineering mathematics, so do understand differential and integral calculus well. I am assuming that is that Spivak's book is about -- please correct me if I am wrong; Amazon is not showing a preview of the book. You are right in bringing countability arguments into the picture; I understand them only to some level. I would love to read a book that gives it a formal treatment. The following has been great for example: https://news.ycombinator.com/item?id=6838917 https://news.ycombinator.com/item?id=6838917 Another one showed up on HN recently that I am still to read in full: https://news.ycombinator.com/item?id=6966695 https://news.ycombinator.com/item?id=6966695 Thanks Finally, cannot help but mention in praise, Colin Wright here on HN has been a good help before in clearing some of my doubts on the subject.
- jfarmer 13y agoSpivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :) These are all before we get to any "calculus." Here "function" means a function of the real numbers. 1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times. 2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times. 3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x. 4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such) Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book. The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).
- Someone 13y ago"3. A set A of real numbers is said to be sense if every open interval contains a point of A." For those googling: there's a typo: sense => dense (http://en.wikipedia.org/wiki/Dense_set http://en.wikipedia.org/wiki/Dense_set)
- jfarmer 13y agoIndeed! Me and my fat fingers. :)
- alok-g 13y agoSounds good. Thanks!
- Estragon 13y agoSeconding the recommendation of Spivak's Calculus. That book lit a fire in me when I was 18. Be sure to work through the exercises, that's where all the fun is.
- dominotw 13y agoWe just started a study group for that book at http://www.reddit.com/r/calculusstudygroup/ http://www.reddit.com/r/calculusstudygroup/ if anyone is interested.
- tokenrove 13y agoSince you mentioned you already have a strong applied math background, try Mathematics: Form and Function by Saunders MacLane for a broad portrait of mathematics that might get you excited to read more deeply into a given subtopic. Note that while this book surveys a broad swathe of mathematics, it depends on the reader already having a pretty high level of mathematical maturity. Other than that, if you want to go deeper into mathematics, just choose a topic and read on it. If your experience is mostly applied, consider something like topology which is often presented more formally than earlier math courses, or go back to the stuff you've already studied, but in a more pure context (for linear algebra, Axler's Linear Algebra Done Right and Halmos' Finite-Dimensional Vector Spaces; for calculus, Spivak as mentioned already and then an analysis book).