What are some recommended resources for learning proofs in mathematics?

[StilloftheNite]

Ok I am sort of a math noob so bare with me. First and foremost, I am thinking about changing majors from boring Accounting to Mathematics and I now have a whole slew of questions. If I was initially a math major I would try to understand everything, just so you know.

When you take the standard undergrad beginning math such as Calc 1-3, differential equations, linear algebra, are you expected to understand any of the proofs? Reason I ask is because I took Calc I at a community college and I feel it was made too easy to pass and I now feel I am missing what should have really been learned.

Instead of going back and trying to figure out what I missed is there any specified courses that deal specifically with proofs and theories so that I can brush up on in the future what I maybe should have known?

We never touched any proofs at all...not even epsilon delta proof, not even mentioned by teacher. So that is what I mean...is there any higher level courses that deal with it specifically?

Are you expected to understand how to prove chain rule, power rule etc. in Calc I? Is this specifically dealt with in any other math courses?

 

[middleCmusic]

When you take the standard undergrad beginning math such as Calc 1-3, differential equations, linear algebra, are you expected to understand any of the proofs?

We never touched any proofs at all...not even epsilon delta proof, not even mentioned by teacher. So that is what I mean...is there any higher level courses that deal with it specifically?

Are you expected to understand how to prove chain rule, power rule etc. in Calc I? Is this specifically dealt with in any other math courses?

Hey StilloftheNite,

Sorry no one has gotten around to your question yet. I will do my best to give you an answer based on my experiences doing math so far.

A Bit About the Math Curriculum

Because Calc I-III, Linear Algebra and Differential Equations are also required courses for physics students, the level of rigor will vary from university to university. In general, you ARE most definitely expected to understand proofs as a math major, but sometimes they are under-emphasized or even omitted in Calculus I or II. Usually, a proof will be given in those courses, even if you aren't required to be able to regurgitate it.

Many colleges offer a "bridge" course which is supposed to teach you proof techniques and the fundamentals of math (e.g. set theory, basic logic, epsilon and delta proofs, construction of the real numbers, etc.). Sometimes these courses are taught alongside learning either abstract algebra or analysis. If you want an okay book which is specifically geared toward teaching you how to prove things, try Transition to Higher Mathematics by Dumas and McCarthy. It's nothing special, but it gets the job done. You could also try How to Prove It by G. Polya. I've never read it, but I've heard it's good for beginners.

After the beginning courses, most math courses are very proof heavy. There are many fewer computational exercises (sometimes none) and your homeworks will consist of proving things using theorems and definitions from the book. Sometimes on a test, you may be ask to reprove a result which you proved in class or was proved in the book, while other times it will be something new, based on the previous material.

The first real proof heavy course that most math undergraduates take (besides that bridge course) is either Abstract Algebra or Real Analysis. Abstract Algebra is just what it sounds like - an abstraction of the more familiar algebra we were taught in school - and Real Analysis is a more rigorous, more modern, proof-based version of Calculus.

Applied Math vs. Pure Math

Now, depending on whether you intend to study applied math or pure math, your life after undergrad will be more or less proof-heavy. However, even the applied math journals are full of proofs. The difference is that pure math people tend to find more satisfaction in the theory, while the applied math people find more satisfaction in its applications to real-life problems in economics, physics, biology, etc.

I should add that most good calculus and linear algebra texts do have plenty of proofs in them, and at the school I went to, we did most of our proof-learning in linear algebra. However, as I said earlier, because of the relationship between physics and math (and economics and math), the relative focus on proofs in those beginning courses will vary from school to school. As you advance in math, the more mundane applications to physics, economics, and engineering decrease, and the rigor increases in kind. But even very abstract math topics like topology and group theory have many applications to the sciences - but usually you only find math majors actually taking the courses. The scientists (usually) just use the results.

Math and Your Career

If you're thinking about jobs, you should know that people who study pure math typically don't make very much money when they first start out - new professors tend to be making between $30K and $50K when they first start out, even if they're geniuses. Applied math is probably a little better, and if you just want to major in math because it's fun, and then get a job in accounting or some other analytical job at a corporation, you could probably make anywhere from $60K - $100K+, even sometimes at entry-level. Engineering and computer science can be very lucrative, because the job opportunities are huge and there aren't enough engineers and programmers to go around. I'm sure this will change someday, but at the moment, you can probably do really well. (Ironically, a plumber would probably make more money than any of the people on this list if he/she plays his/her cards right. Haha... oh boy...)

But in the end, do what's interesting to you, and the money won't be an issue.

Hope this helped!
A junior studying mathematics at BU

P. S. I would try reading through Michael Spivak's Calculus if you're looking for a proof-heavy version of Calculus. If that's a little intimidating, you could also try finding The Calculus by Louis Leithold (particularly the older versions), which I found to be a marvelous exposition of Calc I-III. It doesn't contain any vector calculus though. For that, you would want something like Advanced Calculus by Wilfred Kaplan. And despite all the hate he gets - and deserves - for publishing a gazillion versions of his famous book, extorting more money from students, you could try looking at the Calculus books by Stewart. The earlier versions sell used really cheap (the material hasn't changed in the last 50 years) and they contain everything from Pre-Calc through Vector Calculus. If I remember correctly, they also contain many proofs, although some of the more difficult ones may be relegated to an appendix.

 

[micromass]

I certainly do think you should be able to understand proofs in calculus. The issue doesn't seem to be that you weren't able to understand them, but that you weren't taught them. Sadly enough, most calculus courses are severely dumbed down and just give the result instead of proving it.

I think it might be worthwhile if you self-studied proofs for a while. Of course, you can get a proof book an read it. But I'm not a fan of proof books. In my opinion, you can only learn proofs in a very specific setting such as calculus or linear algebra. Learning artificial proofs doesn't help you. Furthermore, to really learn proofs, you need some person to go over your proofs and basically rip them apart. Most proofs that people do for the first time are completely awful. That holds for everybody. My first proofs were total garbage. And learning a proof book doesn't help. Because in the proof book, you usually learn to do proofs in a way that doesn't exactly correspond to how you do proofs in later courses.

My advice to you is to get Spivak's calculus and Lang's linear algebra (if you never done linear algebra before, then you should get Lang's first course in linear algebra). Start from the beginning and work through most (if not all) of the problems. Furthermore, always present your solutions to somebody else to check. For example, you can use our homework forum: https://www.physicsforums.com/forumdisplay.php?f=156 Be sure to include in your post that you want people to be very pedantic and to completely rip the proof apart. They shouldn't only give criticism on the content, but also on the style you presented it. This is the only way to learn how to prove.