What is Reddit's opinion of Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)?

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I used this book in my first year of university for a class on writing proofs. I think it is an excellent book and has helped me in developing my proof strategies for Real Analysis and Abstract Algebra.

In a math course I recently took that was basically an introduction to math proofs we used Mathematical Proofs: A Transition to Advanced Mathematics which I found to be a great text. It begins by going through the language and syntax used in proofs and slowly progresses through theory, different types of proofs, and eventually proofs from advanced calculus. There are so many examples that are very well laid out and explained. I would highly recommend it for learning proofs from scratch.

Well, since your question did not ask for an opinion as to whether or not to study Mathematics, but rather, what book to read in order to learn it, that is all I shall tender. If you want a decent introduction to theoretical math, then I suggest this. It has decent sections on symbolic logic, set theory, a pretty decent introduction to number theory, and great exercises. The thing about learning about mathematics is that it isn't about calculus, or algebra, or linear algebra, those are just tools (alongside coffee) used by mathematicians to prove theorems. Best of luck to you, enjoy your journey down the rabbit hole my friend.

You should read Mathematical Proofs by Chartrand and work through the problems.

The first chapters are about set theory and logic, which are very easy. You'll get it right away. The rest of it teaches you proof techniques. The book is about how to think mathematically. You don't have to know anything beyond algebra with the exception of the very last chapter (which is about calculus).

I couldn't recommend it more highly. It's an amazingly readable book, which is really helpful when you're trying to self-teach. It explains everything clearly and simply, with examples. You can't put it down. :) I really wish I had read this before I went to college.

EDIT: I just noticed the amazon description says this book is for after calculus, but that is silly. You do not need to know any calculus for this book.

Possible path:

Learn to think like mathematicians because you'll need it. For example, Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al is a good book for that. When you got the basics of math argumentation down, it's time for abstract algebra with emphasis on vector spaces(you really need good working knowledge of linear algebra). People like Axler's Linear Algebra Done Right. Maybe, study that. Or maybe work through Maclane's Algebra or Chapter 0 by Aluffi.

After that you want to get familiar with more or less rigorous calculus. One possibility is to study Spivak's Calculus, then pick up Munkres Analysis on Manifolds.

Up next: differential geometry which is your main goal. At this point your mathematical sophistication will have matured to the level of a grad student of math.

Good luck.