Modern Algebra & Real Analysis: Learn Proofwriting?

[blueberryfive]

Hello,

I just took ordinary diff eq and I've had calc III and linear algebra, but I'm worried about taking Modern Algebra or Real Analysis next semester because I have no experience writing proofs. The linear algebra class was all computation on tests and homework (we did see some proofs on the blackboard). Is one supposed to learn proofwriting in real or modern? I haven't had logic either, which I assume is important. I'm also switching institutions.

 

[gb7nash]

Real analysis will be very difficult for you if you haven't been exposed to proofs. Sorry.

However, intro-level modern algebra is a good class to take to get experience in writing proofs. For elementary group/ring theory, you practically don't need any background. You should be fine taking it.

 

[blueberryfive]

Great, thanks a lot for your candor. Just a side question: did you have experience writing proofs before you took modern algebra? like I said, I'm transferring and I want to gauge my mathematical competency going into this new school. If anything, is there a short text you know of that could introduce me to proofs? I have about a month before I start classes. Thanks again for your suggestion though, I think I'll definitely go with modern then.

 

[Robert1986]

At my college, Linear Algebra is included as a part of Calc. II so everyone who takes Calc II (which is almost everyone) knows a little linear algebra. We math majors take a course called Abstract Vector Spaces. This is nothing really more than Linear Algebra though we don't limit ourselves to working with R^n but we do only deal in vector spaces over the reals or complexes. The point of this class is two fold: 1) Give math majors a more theoretical understanding of Lin. Alg. and 2) It is the first proof-based class we take so part of the class is spent teaching us how to do proofs. IMHO, more colleges should do this. It is 100% absurd to throw students into a proof-based class with no proof-writing experience. This is kind of like taking Calc I without knowing (non-abstract) algebra. We used the book Linear Algebra by Tom Apostol and here are some lecture notes on writing proofs: http://people.math.gatech.edu/~heil/handouts/proofs.pdf.I do disagree slightly with nash. I don't think that Analysis will much more difficult for you than algebra.

The reason is that the stuff in algebra will be 100% completely new. If you are like me (and I imagine most people) algebra will be unlike ANYTHING you have EVER seen in ANY math class, at first, and then you'll start making connection. On the other hand, in analysis, you will be doing more familiar stuff. Granted, it will be a little insane at first and it might seem like you are grasping at straws, but it will be much more comfortable for you than will algebra, IMHO. However, this level of comfort can be your undoing and this is where the lack of proof-writing can kill you. You will be tempted to use "obvious" ideas in your proofs when the "obvious" idea you are using is also the one you are proving. So, use experience as a guide, but when you are writing your proofs, forget anything you knew coming into the class. That being said, there is one VERY good thing about these classes. When I took algebra, I knew that I wanted to try to become a mathematician. It is a very wonderful subject to learn. If you don't like proofs, you shouldn't be a mathematician. If you like doing proofs, then you should try to become one.

 

[blueberryfive]

Thanks a lot Robert, the notes were helpful. I understand everything except the very last proof, but I didn't really look at it too long. How do you know that's the correct term for n+1? if you put (n+1) in place of n in n(n+1)/2 you don't get the term he said was the correct one for n+1...maybe I'm missing something, maybe I'll never be a math major...In an case, I do like this stuff a lot, and I put a lot of effort into my coursework. That being said, I don't want to get murdered in modern or analysis because of my limited proof-writing experience. I did have very shaky algebra skills going into calculus though, and I ended up doing really well. Thanks for your words.

 

[blueberryfive]

Nvm got it

 

[blueberryfive]

It also gives me some relief that you compare proof-writing to algebra because ostensibly that means that it's something one can practice and get good at as opposed to something one has to be brilliant at from the very start.

 

[Robert1986]

Even at my college, where anyone taking analysis or algebra is expected to have taken that Abstract Vector Spaces class, the profs spend a pretty good deal of time going over how to write good proofs. So, hopefully the profs at your school will do the same. I don't know for certain, but I would imagine that most good fiction writers didn't just wake up one morning and decide "I'm going to write, today." No, rather they (might have) gone to college, they read good writers and they practice, practice, practice and write many drafts. This is what we must do as mathematicians.See, this is one thing that the world-at-large does not understand about real mathematics. As an example, when I told a lady at work that I was majoring in math, she said "Oh math was always my favorite subject in college." She said she liked it because there were clear steps she could "follow" to get an answer. She then said that she hated "word problems" because she wasn't any good at them. Later at lunch she said that she could do just as well in my math classes as I do if she was in the class because that we she would "learn the steps" to solve problems.Now, this particular lady is one of my favorites at work and she is really representative of how most people think of mathematics. They know that when they were in school, the were brutally forced to memorize algorithms for solving problems, so they think that upper-level mathematics is all the same but with more complex algorithms. However, this is not at all true. As you go on in math (and this is coming from a guy with 1.5 to 2 years more experience than you have so you'll learn this real fast and you have already learned some of this) you will see that mathematics becomes more like writing poetry in the sense there is no algorithm for writing poems, otherwise, we wouldn't need poets. Similarly, there are no algorithms for proving things, if there were, mathematica would have proven the Riemann Hypothesis by now.

The problems you are doing aren't "integrate so-and-so" but you will be proving things about (at times) very abstract things. I can't explain myself fully, but you will understand. But, let me leave you with some advice that I wish I had heard when I first started doing proof-based stuff. You see, I had terrible troubles understanding proofs in some of my books. I would read literally hours on perhaps one page. I thought I was an idiot for not being able to digest the stuff a lot faster. To be sure, a genius could get it much faster than I could, but I (and I am not saying this to show off) really think I am a good math student, and I realize now that it is OK to take a while to get through some proofs. You really want to understand EVERY concept before you tackle the next one.When I read a math book, I usually follow these steps:

1)Sit outside (or some other place where you can't really write anything) and read through a section of your book. Don't write anything down, and don't read any proofs or examples.

2)Do something for 20 minutes or so and try to think about why some of the lemmas/theorems you read might be true. Don't actually come up with proofs in your mind, just try to convince yourself that it is plausible that the stuff you are reading is correct.

3)Sit at a desk with your book and your paper (and, if you really want to be like a mathematician, some coffee.) Start reading the section again. When you come to a lemma/theorem, attempt to prove it yourself (don't spend too long on this if you can't come up with anything.) Then read the proof, and this time, understand every word of it before you go on. For me, it is helpful to re-write the proof, but in my own words and usually explaining in greater detail exactly what is happening.

4)In a few days (preferably after you have done the above process for the next section or two) go over the section and try to re-construct the proofs (not from memory, but using what you have learned.) Now, doing this takes some discipline (something which I lack) and there are probably people with much more experience than I that would say this is all wrong. But, when I do it, I really feel that I have learned the material. That being said, I don't do this nearly as much as I should. I leave you with a quote by the great analyst Paul Halmos from his wonderful book: I Want To Be A Mathematician: An Automathogrophy.

"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"

 

[blueberryfive]

It was super encouraging to read something like this from someone who knows the ropes. Really, it made a huge difference--I'm not nearly as intimidated to go into this stuff now, and it's gotten me pretty pumped. Thanks.

 

[Robert1986]

Very good! I'm glad to see you are already hooked :)