Expert: Sherry Wallin - 1/24/2009

Question
QUESTION: I'm a young woman studying Honours Math, but originally was focussed in Biology in high school.  I have had a good background in computation, but I feel underprepared for my honours courses because I've done very little proof-work on my own.  I'm not really sure of how to develop this ability independently, but I'd like to start. (There aren't any math circles at my university, I don't have friends that I'm sufficiently comfortable with in my classes to learn from...  I'm trying to learn from my professor, but we are given derivations in class rather than focusing on the grammar and technique of proof in the context of our courses.)  In the summer, I tried just cracking open Baby Rudin, and got stuck on the first page for about a month before my classes started!  I had the chance to talk to a PhD student at one point, who said he'd read thousands of math texts to develop his ability in proof.  

How can I become fluent in proof?  I don't expect it to be easy, but I figure there's a way to do this.  Trying to prove existing theorems is frightening, and doing problems in the texts is frustrating as there aren't often solutions.

Do you know of a book where theorems are listed in some order of increasing difficulty?  Perhaps I could start with something like that!

ANSWER: Hi Claire~
   A good beginning is to learn the basic rules of logic usually found in a symbolic logic course. It could be offered as a math, philosophy, or computer science course. Understanding valid and invalid statements is necessary as well as when different types of statements are forced to be true. After that it is trial and error, seeing others work and just trying one method and if it doesn't take you where you need to go try another. Good algebraic skills is a must. Taking a high school geometry class where you are first taught to do a two column proof would be helpful also. Usually using a theorem already proved, a corollary, or a definition is a good starting place to begin any proof.

I hope this has been helpful. Also Number Theory is generally the easier of the upper division math courses one could take where you write proofs and can get some practice.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Thank you.  As I've been dragged through an abstract algebra course and one analysis course already, I'm very comfortable with logic alone, and in using symbols to clean up my proofs.  Part of my problem can be likened, I think, to not always thinking like a computer.  The proofs in analysis can get very fundamental, and I often found myself skipping over certain things, like introducing certain restrictions, or accidentally using a variable for two different uses as I used it for one thing in one part of the proof and subconsciously switched in another section to using it for something slightly different.  A computer would catch such things, because otherwise, the program would crash.  Richard Feynmann supposedly said that he tried to think like a Martian when solving problems... I think one part of my problem is thinking like a computer and reading a proof line by line in my "editing" stage, accepting information linearly and directly from the written information.  Similarly, as I try to convert my solution into precise mathematical language, I find myself bogged down by the formalities of proof (which are actually important, but to my non-computer mind, seemingly excessive).  So, maybe programming would help as the computer could initially catch certain things and help me to think a bit more linearly...  Though I don't know what I would try doing as exercises.

I often find myself trying to start proofs from definitions and then manipulating them, and I've quickly seen this is generally quite unwise.  It seems to lead to me committing to strategies that will result in dead-ends, without considering more options.

I thought it might be helpful to learn by seeing a theorem in a book without looking at the proof, then attempting to prove it, and comparing it to the formal proof in the book.  For analysis, I find that in, say, Rudin, I don't have any idea of how to attempt the proofs on my own, and I've only just begun to identify common techniques in analysis (familiar manipulations of inequalities, familiar uses of epsilon...).  I thought starting even from geometry might help, though I don't feel like I have the time to do such things.  Perhaps Number Theory is a good choice for some independent study, though I guess I should be focusing on my analysis right now.

As an illustrator, I like visiting different artists' websites and looking at their work, and seeing how each of them uses colour, how each draws their lines, and how each draws noses, and so on.  It seems very difficult to find an analogy for proof: a collection of proofs at varying stages of formalism, where you could compare the techniques of the various people, to learn inductively. I would love to see all of my colleagues' homework each week, but we don't get that at my university!

So, I'm past the logic stage, I think.  I'm naturally, at the moment, a more global thinker, and could benefit from training in linear thinking, and I'd really just like to see how other people do things, so that I could develop my own aesthetics concerning proof.  It is happening, but considering how I was graded last term in both of these proof courses, I sure wish it could happen more quickly.

Well, please let me know what you think, and then I'll try my best.  Thank you very much!

Answer
Wow Claire, I commend you for your well thought out and clear prose. I also admire your drive to become more proficient. Sometimes taking the longer path proves to be the most efficient in the long run. I seemed to do everything backwards and made my work harder than it had to be. For example I took calculus without ever taking a high school geometry course. As you well know calculus is analytical but relies heavily on geometry. After obtaining my BA in Math I went back and took a geometry course and well of course I got an easy A and asked questions that the other students thought I was from Mars or some other planet but I still feel I learned a lot anyway. I took abstract algebra before ever taking any course on techniques of proof writing. After my BA I took a proof writing course and then took a graduate level abstract algebra course and wondered why I thought abstract algebra was so difficult before my BA. I hope you get the drift, the shortest path is not always the best path. I am sure I am much much older than you and I went back after 20 years and re took all 3 calculus courses, linear algebra, differential equations, unix, visual basic, and java as well as statistics and discrete math. I went incognito under my pen name at the same college I taught at. It was interesting to observe the younger students perception of me. If you are interested I can share a story or two there. But again the bottom line is retake your abstract algebra or analysis course from a different teacher, different university, and do share ideas with fellow students because the more different perspectives you can get the richer your experience will be.

Math Prof