I am an engineer with tons of math courses behind me. However, all of them were applied wherein instead of giving us intuition, they gave us a toolbox. (If you encounter this then use that to solve.)

For my research (and as a requirement for grad school), I am required to be able to write and read proofs. These can range from elementary to SIAM journal level.

My research is fairly computational (Optimization and Linear Algebra) and I need proofs only to be able to understand which algorithms work, their convergence and to write proofs for my algorithms if I ever construct one.

I have 2 options now:

  1. Read a book like How to Prove it or How to read and do proofs.

  2. Read Baby Rudin and end it all forever. (Since, I believe that should be enough of proof doing for the rest of my career)

I don't see Baby Rudin's (with all its topology & calculus) helping me in my objectives. Am I missing something?

Which way should I go?

Owing to severe time constraints, I can do only one.

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    $\begingroup$ In the words of the El Paso commercial, porque no los dos? $\endgroup$ – Gerry Myerson Mar 10 '12 at 5:24
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    $\begingroup$ Huh? Google translate converts is to Because both. If you intend to say that "do both", refer edit. $\endgroup$ – Inquest Mar 10 '12 at 5:26
  • $\begingroup$ Do both. Option 1 will give you some basics of doing proofs, but you'll still need some practice with problems which are more advanced, which is where something like Baby Rudin (option 2) comes in. $\endgroup$ – KReiser Mar 10 '12 at 5:30
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    $\begingroup$ As a hispanic I will translate Gerry Myerson's comment: Why not both? $\endgroup$ – Daniel Montealegre Mar 10 '12 at 5:32
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    $\begingroup$ Why not read a book about optimization and linear algebra? You seem to have the impression that Baby Rudin is by far the most rigorous undergrad math book. If so, you're wrong. Also, Rudin puts a lot of effort in making his proofs short and elegant, which means you can read them and verify them without really understanding what is going on. For your purpose you want your proofs to be detailed and straightforward. $\endgroup$ – Stefan Mar 10 '12 at 6:25

Doing proofs is a skill to be developed, not a set of knowledge to learn. You can't become good at doing proofs by reading a book. You need to practice proving things yourself.

Books like Rudin can be useful as a source of little problems to prove (eg, work some of the exercises and attempt to prove some of the theorems on your own before reading the proof in the book). However if you read passively you will not improve very much. Even if you know the all the ins and outs of every definition and theorem, and can recall every proof, you will still be less well-equiped to face new proofs than someone who reads less carefully but trys to prove things on their own.

I recommend "How to Solve It" by George Polya, "Solving Mathematical Problems: A Personal Perspective" by Terro Tao, and "The Cauchy Schwarz Master Class" by Michael Steele.

  • $\begingroup$ But, can I replace Rudin with any other book for proofs? Why is Real Analysis treated as the course for learning proofs? $\endgroup$ – Inquest Mar 10 '12 at 8:37
  • $\begingroup$ Random convenience based on scheduling? Real analysis is often the first course where math majors branch off from general courses taken by science and engineering majors. It's not that way everywhere - at a lot of universities the "learning proofs" course is abstract algebra. Where I went to college it was a discrete math class. $\endgroup$ – Nick Alger Mar 10 '12 at 9:47
  • $\begingroup$ So, I can pick up any book with a theoretical flavor and convert it to my "Lets learn proofs" course? $\endgroup$ – Inquest Mar 10 '12 at 10:00
  • $\begingroup$ Sure, although I would emphasize that it's not about reading the book. Theres a lot of masochism in the math community about reading the "hardest books" and I think this is counterproductive. It's very similar to rock climbing - you don't read a technical book by the best rock climber and expect to go out and start climbing 5.12 routes. It's unlikely you could even understand what they are saying. You just try easier stuff 5.7, 5.8, and slowly build basic skills as you progress, perhaps incorporating things from books as you go. $\endgroup$ – Nick Alger Mar 10 '12 at 10:23
  • $\begingroup$ I couldn't appreciate the rock climbing example but I got your point :) $\endgroup$ – Inquest Mar 10 '12 at 10:25

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