Which do you recommend for learning how to write proofs — How to Prove it by Velleman, or How to Solve it by Polya?

Question

Which of these two books is suited for a student looking to learn how to write proofs? I have a working knowledge of calculus and linear algebra, but I'm not good at writing proofs. My intention is to learn proofs in general, not necessarily for the calculus and linear algebra.

How to Prove it by Daniel Velleman
How to Solve It by George Polya

I ask because the latter is suggested on a highly voted question here, but the former has a more apt name. The reviews aren't helping. You can suggest other books.

Perhaps this will be an unpopular answer, but I found the best way to learn how to write proofs was to read them and imitate them. — Holdsworth88, Jul 04 '12 at 16:23
See also Book about technical and academic writing and posts which are shown there among linked questions. — Martin Sleziak, Dec 21 '14 at 09:51
@Holdsworth88: I always tell that to my students, and they always look at me like "Nobody ain't got time for that!" — Raskolnikov, Dec 21 '14 at 10:58

André Nicolas · Accepted Answer · 2012-07-04T16:30:25.450

15

Velleman's How to Prove it is quite a bit more relevant to your needs. It is organized like a conventional text, and pays a lot of attention to proof writing.

Polya's book focuses on problem-solving. One can view it as a better book, certainly a historically far more important book. But it focuses on how one finds the idea that will crack a concrete problem.

There is quite a bit of material in Velleman that is useful for writing proofs in linear algebra, in particular on how to proceed from definitions. There is none of that in Polya. There is also essentially nothing in Polya on basic analysis. Polya beautifully accomplishes his aims: they just happen to be different from what you said you wanted.

edited Jul 04 '12 at 16:30

answered Jul 04 '12 at 15:32

André Nicolas

507,029

http://math.stackexchange.com/questions/7743/getting-better-at-proofs suggests otherwise hence the doubt. – Inquest Jul 04 '12 at 15:35
There will inevitably be differences of opinion. But I am quite convinced that Velleman will be far more useful to you in writing proofs associated with your courses. If you were a young person getting into contest-type mathematics, my advice would be different. – André Nicolas Jul 04 '12 at 15:43

score 10 · Answer 2 · answered Jul 04 '12 at 15:18

10

The Polya is more advanced than I think you are looking for. It is a well-known classic, but assumes the reader already knows how to write proofs.

I haven't seen it, but the Velleman should be good for you. Another one I have seen is by Solow: How to Read and Do Proofs

answered Jul 04 '12 at 15:18

GEdgar

111,679

Can you elaborate on more advanced? I am a programmer and can work with matrices, scipy, numpy and all. Enough? – Inquest Jul 04 '12 at 15:20
1

Edited. It assumes the reader already knows how to write proofs. – GEdgar Jul 04 '12 at 15:23
Assumes reader knows to write proofs? To what extent? – Inquest Jul 04 '12 at 15:24

score 5 · Answer 3 · answered Jul 20 '13 at 14:27

5

I know it's not one of the two you referred to, but you might want to look at How to Think Like A Mathematician by Kevin Houston.

answered Jul 20 '13 at 14:27

score 4 · Answer 4 · answered Jul 20 '13 at 15:22

4

Another book that you should check out is Gary Chartrand's Mathematical Proofs: A Transition to Advanced Mathematics. It has an instructor solutions manual.

answered Jul 20 '13 at 15:22

Which do you recommend for learning how to write proofs — How to Prove it by Velleman, or How to Solve it by Polya?

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