Calculus
> The above statement of zero vector is unique, I have no idea what is that means. In isolation, nothing. (Neither does the word “vector”, really.) In the context of that book, the idea is more or less as follows: Suppose you are playing a game. That game involves things called “vectors”, which are completely opaque to you. (I’m being serious here. If you’ve encountered about some other thing called “vectors”, forget about it—at least until you get to the examples section, where various ways to implement the game are discussed.) There’s a way to make a new vector given two existing ones…
Concrete Mathematics is pitched at graduate students in computing. Spivak’s Calculus is an introductory real analysis book pitched at undergraduates who have gone through a computational calculus course already and want to study the subject more formally and rigorously; it has many difficult problems and would generally benefit greatly from the structure and expert feedback of a university course. Jaynes’s book is probably most relevant to science students who are at least at the advanced undergraduate level. How to Solve It is a dictionary of heuristic problem-solving techniques which is…
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…
There are lots of cool options, depending on what you want to do with it. If you're interested in rigorously proving how calculus works, starting from the structure of the real numbers, I hear nothing but fantastic things about Spivak. I've not worked much of it myself, but the exercises seem very well chosen from the sections I've read. It exists in an interesting space between a calculus textbook and an introduction to real analysis, and you'll see stuff like the intermediate value theorem proven rigorously. If you want to practice computational applications, you may have to supplement it…
I don't understand why so many people recommend baby Rudin (Principles of Mathematical Analysis). The presentation in Rudin is not merely terse, but also quite dry and unmotivated. I suggest you avoid it--regardless of how much talent or maturity you have. There are plenty of more interesting texts which will teach you just as much: Spivak and Pugh are nice, I also recommend the recent two-volume work by Zorich. By the way, as you aquire experience you'll gain confidence and get over the urge to always check your answers. Here's a good exercise with a built-in answer key: When reading a…
Motivation and direction are important when starting out. I decided pretty early on that I wanted to be an algebraist, but would have to build some mathematical maturity before I could get there, so I had a rather shallow goal in the beginning: to be able to solve the previous years' math GRE papers. Off the top of my head, these were some of the books that I worked through: 1. Spivak's Calculus. 2. Johnstone's Notes on Set Theory and Logic. 3. Gamelin's Complex Analysis. 4. Hoffman & Kunz' Linear Algebra. 5. Dummit & Foote's Abstract Algebra; just the group theory. 6. Munkres'…
Spivak'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"…
What 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…
For a more conceptual introduction leaning on using computers, whose goal was getting STEM students up to speed to understand the context of work in their various fields, you might enjoy https://www.science.smith.edu/~callahan/intromine.html For something more traditional, take a look at textbooks by Piskunov, Courant, or Apostol. Spivak's Calculus has excellent problems if you are looking for something more abstract and rigorous (probably better after a first course). https://archive.org/details/n.-piskunov-differential-and-int... ;…
>One person's "droning" may be another person's "rare, illuminating presentation"; we are not all interested in the same things. You are absolutely correct in that regard. I've talked to many people who hated the textbooks I most liked, and preferred ones that I found unreadable. Just goes to show that what works for you may not work for me. You are very wrong on the other point: none of what I said is borne out of a view that mathematics should be difficult and "hard" an impenetrable (rather than soft and approachable). Indeed I view this sort of overlong prose as impenetrable and…