Reddit reviews How to Think About Analysis

You should read this. I found it here a month ago on this subreddit, and it really stuck with me. I love those stories that help round out abstract concepts I've been thinking of.

More generally though... simple algebra used to be for the greatest thinkers alive. 'Ars Magna'. "The Great Art" written by Geromalo Cardano in the 1600s or whatever was the first European mathematical work that advanced beyond what was known by the Greeks... it gave a partial solution to how to find solutions to homogenous cubic polynomials (ax^3 + bx^2 + cx + d = 0)

he solved it with hilarious methods. Galileo used some incredibly painful notation where you're juggling ratios instead of... you know. Doing algebra the way we think of it. Fibonacci tried to encourage a switch to our standard number system, because arithmatic is RADICALLY easier when dealing with a simple base ten system instead of whatever crappy roman numeral type language they were using before. Took them 400 years to adopt our modern number system from the time the 'better' alternative was introduced.

All this is to say... you're absolutely right. The crystal core of the ideas we use can radically change our reach. What was impossible with one way of working becomes elementary when you can look at it right. But you've got a few layers of problems here. First... what's the right way of looking at it? I just read Judea Pearl's "Causality", and it's fascinating seeing a branch of math that's still so young, that there are arguments about what the definitions and axioms should even be. It's still a bubbling cauldron of ideas more so than an established branch. But even once you've gotten the 'right' way of looking at things (often there are many possible ways, you need to pick the right one for the job) now you're left with the arguably harder task of communication. How do you build a bridge to efficiently transmit a new way of thinking? I love 3blue1brown just because his whole shtick is finding new ways to graphically describe concepts that most people only vaguely understanding. The article I linked above (Ars Longa, Vita Brevis: 'long art, short life') breaks down the emergence of an art as being in 3 tiers... the inventors, the teachers, and the teacher teachers. The 'best' teachers I think are what you're asking about partly, but the right 'inventors' (what is the perfect framing that should be taught?) is part of the problem too.

Anyway, a related article you might also enjoy... thought as technology. A cool little exploration by Michael Nielson about the fact that 'how to think about things' is itself a technology, just one that's a pain in the ass to pass on compared to physical goods. He had some cool things to say on the topic you might also enjoy.

Also also... from a math perspective, I highly recommend you check out Alcock's how to think about analysis if you're looking for something fun to read. It's a very, very light introduction to real analysis, looking at the foundations of calculus, limits, series and convergence and so on. If you're interested in the 'heart' of what it means to learn math, I think you'll find that to be a pretty fun, approachable little book. You'll be able to blow through it in a couple weeks, but it'll give you some good framing for continuing the journey, if you're interested in doing so.

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.

I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/

https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful

If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2

And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X

How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.

How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).

I never understand the voting on this sub. Some unrelated posts are upvoted, but tangentially related posts downvoted. Hell, even two similar topics on the main page of this sub get different votes.

@OP, How To Think About Analysis by Lara Alcock.

I didn't struggle with Real Analysis mostly because it addresses all your "why?" questions from the get-go.

How to Think About Analysis by Lara Alcock is a nice book that walks you through the Analysis skeleton in a very short time especially if you have no problem with quantifiers.

I am struggling with Linear Algebra right now because of high school style "shut up and do these useless exercises" attitude of most LA books.

I found a book on LA:Real Linear Algebra by Fekete that seems to kick ass(deals with most of your "why" questions) if you are struggling with that as well. I found a free copy online, but it's shit quality, so I had to buy it for that exorbitant price. I think it's worth it since I am tired of crap Linear Algebra books.

...here's a book I recommend

https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539

I know someone else on /r/math has met the author

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

In the grand scheme of math: jack shit. But who's to stop you after 2 months of studying?

What do you know so far? Are you comfortable with inequalities and math induction?

Check out the books below for a nice intro to Real Analysis:

How to Think About Analysis by Lara Alcock.

A First Course in Mathematical Analysis by D. A. Brannan.

Numbers and Functions: Steps to Analysis by R. P. Burn.

Inside Calculus by George R. Exner .

Discrete And Continuous Calculus: The Essentials by R. Scott McIntire.

Good Look.

Lara Alcock How to think about Analysis, How to study as a Mathematics Major

There is actually a book called How to Think About Analysis which you might find useful. I have not read it myself, but I have read the author's other book and highly recommend her as an author.

In case you also want some intro to Analysis(Calculus made a bit more rigorous), here's some:

How to Think About Analysis by Lara Alcock.

A First Course in Mathematical Analysis by Brannan.

I have a friend who swears by this book

https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539

Yep, the stuff is quite hard and requires a lot of thinking about examples and counterexamples to understand what things mean. And you need time. You just can't learn this stuff in a cram session before an exam. A resource you might find helpful is

https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539

I am sure this is the book you're referring to https://www.amazon.ca/Think-About-Analysis-Lara-Alcock/dp/0198723539

https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539

Alcock is a Math Ed researcher with a huge focus on proofs in undergraduate mathematics.

I have Abbott's and Charles Pugh's books. Both excellent and probably in your reserve library. There's another book I noticed on Amazon, I've never heard anybody on reddit or math.stackexchange mention, probably worth $20: https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539

Also Spivak, Apostol, other books: https://www.reddit.com/r/math/comments/3drlya/what_mathematical_analysis_book_should_i_read/

There's lots of other threads here and math.SE that're helpful. Maybe looking thru Courant/Robbins What is Math witht he mindset that it's an enjoyable read