What is the Best Book for Learning Mathematical Proofs?

[The_Brain]

Unfortunately, most of us have been taught math by simply watching the teacher derive a forumla and then memorizing it. I really feel that it is important to be able to be able to write and solve proofs as that is how new things are discovered. I am looking for a good proof book to introduce to me to this. My mathematical background is up to the equivalent of 2 semesters of an honors Calculus course so I'm not looking for some advanced book on Analysis or whatnot. Maybe the best way is to go through the book that MOP'ers do on proofs - I don't know - that's why I am asking you!

 

[Dr.Brain]

GO for::::

GN BERMAN ...A PROBLEM BOOK ON MATHEMATICAL ANALYSIS

OR IA MARON ..PROBLEMS IN CALCULUS

REALLY GOOD BOOKS..I HAVE SOLVED BOTH OF THEM

 

[maverick280857]

Berman is good for problems not proofs (its not a textbook). Maron is very good but proofs are rare and they aren't written in a manner that you can understand if you're starting out.

So I suggest you get hold of Thomas and Finney and/or the Schaum series on Calculus (textbooks).

Cheers
Vivek

 

[mathwonk]

It is interesting you recommend Thomas and Finney as it is a book I avoid now as having essentially no proofs. I taught from the 9th edition and found it very clear and excellent, with certain interesting cases of each theorem proved and others omitted, which seemed a good approach.

Unfortunately next time I tried to use it, it was in the 10th edition and I noticed that it now omitted ALL cases of the proof I was interested in, leaving them to the appendix, where no one in my class ever goes.

So if you really want to learn proofs, I would recommend an honors level book like Apostol. But that is quite hard and intended for the most advanced calculus courses at the best places, like Chicago.

I myself started, actually in high school, with a wonderful little book called Principles of Mathematics, by Allendoerfer and Oakley. It is out of print but some copies may be available on used book websites like abebooks.com.

There is also a terrific high school geometry book by Harold Jacobs that teaches proofs and logic very well. I would like to use it in my college level proofs course some time.

 

[mathwonk]

Try this one: at $6 it seems hard to beat. The 1st edition had chapters on logic and proof, boolean algebra and switching theory, groups rings and fields, complex numbers and trig, analytic geometry, statistics and calculus.

Allendoerfer & Oakley, Cletus, Carl B.
Principles Of Mathematics
McGraw-Hill. New York 1963*HB 8vo VG/-- Index. 540 pages. 2nd ed. Glossy beige blue cloth boards with white letters. Slight sunning along top edge of covers + spine; tight; clean.
Bookseller Inventory #8659
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Price:*US$*6.00 (Convert Currency) Shipping:*Rates & Speed


Bookseller:*Lee Madden, Book Dealer, P.O. Box 6161, Brattleboro VT 05302

 

[maverick280857]

Oops..my mistake in recommending you Thomas and Finney. I actually have the 10th edition and a very old edition (70s or 80s maybe). The older one has proofs though you might not be impressed by the emphasis given to them. It is after all, a book that will teach you how to do things.

If I understand correctly, The_Brain is beginning theoretical calculus. So I do not think Apostol would be a very good first book. It could be a bit too advanced and abstract (the real math) for beginners. If you want all the theorems in one place these books will be good--and I do not mean to discourage you from reading them; its just that they might be a bit terse for first-timers. Perhaps a first time reading from Thomas and Finney coupled with reference sessions using books like Apostol is a good idea. It really depends on your ability to catch things.

 

[graphic7]

I like Apostol, myself. I just got my hands on a copy of it yesterday. An instructor of mine had a copy - first edition. I briefly read through the first chapter last night. Apostol's text seems to be readable and concise at the same time. I was disappointed in a few areas where we he didn't give proofs to the reader. One area that a proof would have been nice is that e^x is *always* irrational; however, the case where x = -1 was satisfying. I'm looking forward reading it.

 

[matt grime]

"One area that a proof would have been nice is that e^x is *always* irrational"

but it isn't, x=0, x=log(any rational)

for analysis proofs proper you do a lot worse than Kelley's general topology if you can get hold of a copy.

the books mentioned like finney et al are not books to learn mathematics from as a mathematician does, they are books to learn how to do engineering problems. what level of mathematical rigour do actually want to study?

 

[courtrigrad]

I like Courant. He provides a good balance between theory and mathematical rigor.

 

[mathwonk]

Interesting you mention Kelley's general topology, Matt. That is a good suggestion. I read most of that in the summer after my senior year in college and liked it a lot.

However later, when trying to learn some algebraic topology, e/g/ how does a sphere differ from a torus, I began to feel Kelley's book had remarkably little content.

I did know general topology quite well, but it just doesn't seem to provide more than a language. I.e. you learn a language, but do not hear anything interesting spoken in that language. Its like learning grammar instead of reading literature. Or maybe the good stuff was in the back, and I didn't get that far.


Of course most proof books emphasize formality over content. So maybe Kelley is in keeping with most traditional instruction in proofs. If you want a book that is very formal, and gives detailed proofs, and practice in a useful language, probably Kelley is one of the best alternatives. It is certainly written by an expert and a master teacher. (He even taught abstract algebra on television in the 60's.) The language it teaches is also universally used. I would suggest his abstract algebra book from that continental classroom course maybe as an even more interesting and useful book, especially for beginners. It specifically covers logic and proof and matrix algebra and complex numbers as I recall. So it is a useful precaculus book.

I would suggest though that it is more useful to learn mathematical content than formal language, and like to encourage learning the proof of an actual interesting mathematical fact, rather than just how to prove things formally.

For instance even calculus can be used to prove interesting things, like the fundamental theorem of algebra using Green's theorem, or that the fact there is no never zero vector field on a sphere using Stokes theorem.

As to e^x always being irrational, graphic7, wouldn't that contradict the intermediate value theorem? I.e. if e^x ever takes two different values it has to take also all values between those two. Do you think there exist two different numbers with no rational values between them?

Here is a chance to use some proof theoretic reasoning: can you prove that if e^0 = 1 and e^1 = e > 2, then e^x there are an infinite number of x's between 0 and 1, such that e^x is rational? (Read the intermediate value theorem.)

If you will read Apostol carefully, I think you will learn how to prove things.

 

[matt grime]

I think something important to bear in mind about Kelley's general topology is that it was written in 1955 (my copy is 43 years old), and if you view it with a modern take on what topology is, then it isn't going to be what you expect. Its modern title perhaps ought to be point set topology; there is no reference to categories and so on, so that's an important warning from mathwonk there.

(Somewhere I have a differential geometry book from 1920 that has to be seen to be believed.)

 

[graphic7]

You know, you're right. I phrased that terribly and wasn't considering all the other cases in which e^x could be rational (intermediate value theorem). I was targetting, more specifically, the irrationality of e^1, itself. It just seemed natural that after Apostol proved the irrationality of the sqrt(2) that he should have proved the irrationality of e^1 (I would've enjoyed it), next. This is all just my opinion; I thought the proof of e^-1 was rather misplaced.

Apostol had me hooked last night. I believe I stayed up till 3 in the morning reading it and doing the exercises for the first chapter; my brain is rather cooked.

Thanks mathwonk and Matt for refreshing my knowledge about the intermediate value theorem.

 

[Muzza]

Doesn't it follow that if 1/x is irrational, then x is irrational?

 

[vsage]

If you want a ridiculously hard one, I suggest the USSR Mathematical Olympiad Problem Book: Selected Problems and Theorems from Elementary Mathematics. It was the first book I looked at that dealt exclusively with proofs, but it made me a lot more able in that particular field. It's tough though: No Joe Citizen could solve any of the 320 problems so it's not exactly a confidence-booster. After three months of working on it a few hours a day I'd say I've only finished 20 or 30 satisfactorily, and maybe double that with just answers.