A transition to advanced mathematics

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1. blindmuteMay 2022
   In order:

   1. Book of Proof

   1a. How to Prove It: A Structured Approach

   2. Elements of Set Theory - Enderton

   2a. Naive Set Theory - Halmos

   3. Foundations of Analysis - Landau

   4. Basic Mathematics - Lang

   5. Principles of Mathematics - Allendoerfer & Oakley

   6. A Transition to Advanced Mathematics

   6a. Concept of Modern Mathematics - Stewart

   7. Calculus - Apostol or Spivak

   Don't skip any

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2. cg30eNov 2021
   For sure. I'll list some books for introduction to proofs, abstract algebra, real analysis, topology and category theory. These are not comprehensive, just listing books off the top of my head. I'll definitely be leaving off personal favorites other people have. You'll like some better than others. Some of these are beginner books and some are more advanced. A good tutor can help you get through the more advanced books. I tried to list the most beginner friendly book first in the list under each subject. Then the more advanced books later in the list.

   Introduction to Proofs:

   Just pick one of these that speaks to you the most. All three are good.

   Discrete Mathematics with Applications - Epp

   Discrete Mathematics and Its Applications - Rosen

   Mathematical Proofs: A Transition to Advanced Mathematics - Chartrand, et al.

   Abstract Algebra:

   How to Think about Abstract Algebra - Alcock

   Abstract Algebra - Pinter

   Abstract Algebra: A First Course - Saracino

   Algebra - Artin

   Abstract Algebra - Herstein

   Abstract Algebra - Dummit & Foote

   Linear Algebra:

   Maybe an engineering based book first if you haven't seen linear algebra in a while (e.g. Strang or Linear Algebra: Step by Step by Singh).

   Then:

   Linear Algebra - Friedberg, et al

   Linear Algebra Done Right - Axler

   Linear Algebra - Hoffman & Kunze

   Real Analysis:

   How to Think About Analysis - Alcock

   Understanding Analysis - Abbott

   Tao's Analysis text

   Principles of Mathematical Analysis - Rudin

   Topology:

   Topology - Munkres

   Topology A Categorical Approach - Tai-Danae Bradley, Tyler Bryson, and John Terilla

   Check out this list:

   http://pi.math.cornell.edu/~hatcher/Other/topologybooks.pdf for others.

   Category theory:

   Categories and Toposes: Visualized and Explained - Southwell

   Conceptual Mathematics: A First Introduction to Categories - Lawvere

   Category Theory for Programmers - Milewski (if you like functional programming)

   Programming with Categories - Fong, Milewski, Spivak (if you like functional programming)

   Category Theory in Context - Riehl

   There are a few others by Spivak which you may like.

   If you don't know category theory whatsoever then I like Southwell the best (pair them up with his youtube videos). Eugenia Cheng also has a nice set of lecture videos.

   If you already know math pretty well, then Riehl is a favorite.

   Hope that helps!

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3. whatever1078Sep 2021
   You need to transition to learning math.

   One possible starting point is Book of Proof by Richard Hammach[0]. It's free. It will show you the rudiments of how math is actually done and thought of. Another good choice is Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang. These books have no prerequisites(good to go if you know how to do arithmetic with fractions/decimals; not that these are strictly necessary, rather that's about the level of comfort with math one should posses). Search on Amazon and you'll see there are a ridiculous number of books on this subject of all varieties of ways of presentation. Almost every week a new book pops up, it seems. Note, intro-to-proofs books are not only for novices, though. There are kinds written for advanced undergrads and beginning grad students as well. For example: Mathematical Concepts by Jugen Jost. This speaks to the wealth of the kinds of math books available.

   Concurrently and/or after that you can study introduction to any of linear algebra, discrete math, group theory, analysis of reals, combinatorics, number theory, graph theory, algorithms, probability, category theory...whatever. Each of these are very deep subjects and there exist hundreds(if not 1000s of books from the past, current and future) of every imaginable level, depth and presentational quirk. Again check Amazon for titles. Check libge*n for actual files.

   The following intro books are so easy anyone and they momma can read these with utmost ease:

   Discrete Math by Susanna Epp

   How to Think about Analysis by Lara Alcock

   Linear Algebra: Step by Step by Kuldeep Singh

   Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings

   A Tour Through Graph Theory by Karin Saoub

   The above mentioned books are a few examples of elementary books that you can get started with right this minute.

   Note and remember the wealth of possible books, though. For example, consider the subject of math analysis.

   We can break it into three general parts: real analysis, complex analysis, functional analysis. Each can be further taken apart into measure theory, vector spaces, topology etc. Further, topology alone branches out into point-set topology, algebraic topology, differential topology etc. Each of these fly under many different names. Say, vector spaces is a huge part of linear algebra. For the beginners, it's usually the finite dimensional vector spaces (aka the title of a famous linear algebra book) version whereas the infinite version will take you into func analysis. Vector spaces are also usually included as a chapter in books on abstract algebra as they are an algebraic structure just like groups, rings, modules. But if there is linear algebra, there must be non-linear algebra too. Well, yes. Roughly, it corresponds to what's called algebraic geometry. It never ends.

   The point is every topic in math has loads of books dedicated to it. In turn every topic has a (sub)^{n}topic and there are tons of books on those. Each of the books come in elementary, quirky, armchair enthusiast, average undegrad student, middling grad student, kickazz graduate math aficionado, researcher, undergrad researcher wanna-be etc levels. There are also olden books, classics, those just published (say, at most 5 years ago) and up-and-coming ones. Also, each term in math has bagillion synonyms. So multiply the number of books above by n. I am not even gonna mention books written in different languages and lecture notes that are available for free online. A lot of published book often spend years as a free set of lecture notes before ending up on Amazon as a newly published book. There are also Olympiad books, proof compilations, capstone compilations, intro-to-research books, those that prove a single theorem and in the process go through tons of disparate math etc Take advantage of all of this.

   [0] https://www.people.vcu.edu/~rhammack/BookOfProof/

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4. uptownfunkJul 2020
   First read https://artofproblemsolving.com/news/articles/avoid-the-calc...

   By one of my early mathematics tutors in San Diego math circle

   Then buy something like: Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) (Featured Titles for Transition to Advanced Mathematics) https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_api_i_K4...

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5. ghufran_syedAug 2019
   I really like "Mathematical proofs - A transition to advanced mathematics" by Polimeni, Chartrand and Zhang If you're looking for a free option, "Book of proof" by Hammack also looks good, but I have less experience with it ( https://www.people.vcu.edu/~rhammack/BookOfProof/ )
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6. hnuser355Feb 2019
   I used the book “A Transition to Advanced Mathematics” in undergraduate and I loved it. Many but not all exercises have answers in the back
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