What textbooks would you recommend to learn proof writing?

Question

I am a high school student and want to pursue an undergraduate degree in mathematics in the near future. My teachers always emphasize the importance of mathematical maturity and the ability to write precise, accurate and clear proofs. I have no experience in proof-writing at all and have just learnt proof by induction and contradiction in my class. What textbooks would you recommend where I can learn proof writing, and start to understand books such as Calculus by Michael Spivak, Linear Algebra by Sheldon Axler, Principles of Mathematical Analysis by Walter Rudin and every single introductory abstract algebra book in existence?

Answer 1

There are a couple of textbooks I found helpful when I was transitioning away from computation focused mathematics courses. Most of these text will have nearly the same material, so I recommend sampling the texts if you can and see if you prefer the author's style and exposition. Also, keep in mind that practice is the ultimate instructor:

• Mathematical Proofs: A Transition to Advanced Mathematics - Gary Chartrand, Albert D. Polimeni, Ping Zhang. I found this particular text very helpful. It has a lot of examples and a lot of practice problems. It also covers proofs in a few different areas of mathematics (such as number theory, combinatorics, calculus, linear algebra) after the general material. https://www.pearson.com/us/higher-education/program/Chartrand-Mathematical-Proofs-A-Transition-to-Advanced-Mathematics-4th-Edition/PGM1763590.html
• The Book of Proof - Richard Hammack. More concise than the first, but freely available. https://www.people.vcu.edu/~rhammack/BookOfProof/ .
• Writing Proofs in Analysis - Jonathan M. Kane. This text focuses on analysis and provides a very structured approach to proofs which can be helpful. https://link.springer.com/book/10.1007/978-3-319-30967-5

You mentioned Calculus by Spivak, and I would say don't be afraid to try that too. It is an excellent book and one that I wish I had come across earlier in my studies. Hope this helps some and good luck!

Answer 2

Solving Mathematical Problems: A Personal Perspective by ${{\color{gold}{\textbf{Terence Tao}}}}$

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"There are a handful of really wonderful books that can introduce a young high-school student to the beauty of mathematics. This is definitely one of them. Besides, this book is probably going to be known as the first book written by one of the best mathematicians of the twenty-first century." ― Mihaela Poplicher, MAA Reviews

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Highly Recommend from my side.

Answer 3

I'm also a future university student. So take my recomendations with a bit of salt but I'd have to recommend

1. Sheldon Axler's Linear Algebra Done Right
2. The first few chapters of Terence Tao's Analysis I (mainly chapter 2 and 3 but chapter 4 and 5 also are great it just gets really hard to follow for me so I haven't been able to go very far with it so I'd only recomend going so far with it.
3. My top suggestion: Keith Devlin's An introduction to Mathematical thinking (This is a coursera course that you can take for free btw)
4. Ive also heard of How to prove it by Daniel J. Velleman
5. and finally A concise Introduction to logic by Patrick J. Hurley

althought I can't attest to how difficult or good those last two are, similarly with spivak's calculus. But I have heard its a great intro to college level mathematics from others. Those are my suggestions from others and myself