Proofs

This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own.This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor.This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.After every chapter are "pro-tips," which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more.

What Hacker News says

seanhunterJul 2024

One of my favourite acknowledgements in a book is from "Proofs: A long-form mathematics textbook" by Jay Cummings[1]. Which as it sounds like is a maths textbook. Anyhow the acknowledgement reads:

   "To my loving wife,
   who read this entire book,
   apart from the math parts."

[1] Which in my opinion is an excellent book if you want to learn how to read and do maths proofs btw.

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beryilmaJul 2024

As a PSA for anybody who wants a very readable introduction to "real" math, check out Jay Cummings' two books: Proofs and Real Analysis at https://longformmath.com/

Each paperback book costs less than $20 on Amazon.

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PatentlyDC123Apr 2024

Real Analysis. I know Baby Rudin is probably the gold standard, but Jay Cummings' Proofs and Real Analysis have been great for people like me delving deeper into mathematics.

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beryilmaApr 2024

One the same subject and as accessible, I love the two books from Jay Cummings: Proofs and Real Analysis. Each just $16 on Amazon. It is a joy to read these books and try some of the exercises. I wish PDF versions were also available...

For those interested: https://webpages.csus.edu/jay.cummings/Books.html

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nicfApr 2024

I'm a mathematician, currently working as a private tutor for adults who want to learn proof-based math. I had a quick glance through this book and it seems to me like a pretty nice version of this "intro to proofs" sort of book. This is a topic that's done well in a lot of different books, though, so if you really want to dig into this topic I'd maybe recommend looking at a couple different ones and finding the one that agrees with you the most.

Right now I have a student working on this material and we're using "How to Prove It: A Structured Approach" by Daniel Velleman, which so far I'm finding decent. Some others I've seen (but that I haven't looked at in as much detail) are "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings and "Book of Proof" by Richard Hammack.

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__rito__Nov 2022

I highly recommend the one that I am currently reading: Proofs: A Long Form Math Textbook by Jay Cummings [0].

There are two lists [1][2], one by Mark Saroufim, another one by Susan Rigetti that I have been following for quite some time now and intend to follow for the next months and years.

[0]: https://longformmath.com

[1]: https://marksaroufim.medium.com/technical-books-i-%EF%B8%8F-...

[2]: https://susanrigetti.com/math

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