Reading Mathematical Theorems And Their Proofs

[Bashyboy]

Lately, I have been experiencing a sort of anxiety over not understanding some of these proofs in my calculus textbook. I just finished a calculus I course, and we did not spend any time learning them; so, I thought, since it is summer time now, I would go back over and try to learn those different theorems and their corresponding proofs. But I just don' seem to understand them. Is this normal, to not learn the proofs to different theorems in a Calculus I class? One person who works as a tutor at my school said that I shouldn't worry all that much about learning these proofs, especially since I haven't taken a logic course (I believe that is what see referred to it as). Is this true? Should I just continue to learn new material, get a better intuition for calculus, and then eventually go back and learn the proofs?

 

[shinwolf14]

Generally speaking, most teachers don't teach the proofs because they are complicated and they don't have time. Most calc 1 students do not have the background to fully grasp the proofs. This is why analysis was developed. If you want to learn the proofs, you should take an analysis course.

EDIT: if you want to help yourself, go back and make sure your foundation is strong with limits/derivatives/integrals. There are theorems that are important (MVT, FTOC) but they will have no immediate impact on learning Calculus 2

 

[Bashyboy]

Oh, I would love to take that. When would I be able to take that course? So, I don't have to worry about not understanding them now? Should I at least try reading them, and if I don't understand them after reading it a several times, just move on?

 

[micromass]

Many people who get confronted by proofs find them very difficult at first. It's a big roadblock that you need to pass. If you're a math major, then proofs are essential. In that case, I would try your hardest to understand them anyway. It will pay off later.
If you're not a math major, then you can probably safely skip them.

What theorems are we talking about anyway??

Perhaps you could start a thread in the math forum asking for assistence with the proof?? I'll be happy to help you!

 

[shinwolf14]

Oh, I would love to take that. When would I be able to take that course? So, I don't have to worry about not understanding them now? Should I at least try reading them, and if I don't understand them after reading it a several times, just move on?

Usually Analysis is a junior/senior level math course.

I am not saying you shouldn't understand them. By all means, if you can follow the proofs then go right ahead. What I am trying to say is there is a whole branch of mathematics devoted to studying Calculus. Also most of the time the proofs in the textbooks are just terrible.

You should definitely know what the theorem says but proving the theorem at this point in your academic career has very little benefit (as compared to solidifying your differentiation/integration skills)

 

[Mholnic-]

I'm only just entering the calculus sequence, but as I understand it a solid understanding of proofs is necessary for later courses. Depending on how 'theoretical' your future calculus courses are you might encounter them there, but you'll see them in Linear Algebra and if you take any Analysis courses. If you're considering a Mathematics degree you'll definitely want to become familiar with proofs, the earlier the better.

A few of the suggestions I've seen and heeded are as follows:
For a general introduction to proofs, try 'How to Prove It: A Structured Approach' by Velleman. For a 'proofy' treatment of calculus (great for aspiring math majors, with REALLY hard problems) try Calculus by Michael Spivak. I'm working through these now, the proof book is easy enough to follow on my own but Spivak.. well, I can follow the chapters but most of the exercises are above my ability (so far).

I think Spivak would be great for you, you can 'review' the calculus you've already learned in a new light, acquire some skills at proving/analysis and perhaps dip into calc II. If you want intuition, well, you acquire that by understanding theory behind the math. At least that's how I've always seen it. There isn't much intuition to learning formulaic shortcuts if you don't understand what is really going on in the background.

 

[Fredrik]

As shinwolf already indicated, you will encounter the same theorems and their proofs again, if you take a more advanced course. These usually have "analysis" in the name instead of "calculus" for some reason. So I think it's OK to not understand all the proofs now. But you should at least make an effort to understand the easier ones.

It's hard for everyone to understand proofs in math books. I don't think the problem is that you haven't taken a full course on logic. It would however help to understand truth tables of logical operations, so that you understand things like that the statements
$A\Rightarrow B$
$\lnot B\Rightarrow\lnot A$
$\lnot(A\land\lnot B)$
are equivalent. The best way to see that is to write down their truth tables. If you do, you will see that they're all the same.

I don't think it would help much to study more logic than this. You just have to study more examples of proofs, and practice doing proofs on your own. If you want to be good at proofs, I would recommend that you do something like this: Study the proof in the book first. Then try to prove the theorem without looking at the book. Then do it over and over, while imagining yourself explaining it to someone else. Keep doing it until you don't see a way to improve your explanation. If it's a difficult proof, it can take many attempts to get to that point.

 

[micromass]

As shinwolf already indicated, you will encounter the same theorems and their proofs again, if you take a more advanced course. These usually have "analysis" in the name instead of "calculus" for some reason. So I think it's OK to not understand all the proofs now. But you should at least make an effort to understand the easier ones.

It's hard for everyone to understand proofs in math books. I don't think the problem is that you haven't taken a full course on logic. It would however help to understand truth tables of logical operations, so that you understand things like that the statements
$A\Rightarrow B$
$\lnot B\Rightarrow\lnot A$
$\lnot(A\land\lnot B)$
are equivalent. The best way to see that is to write down their truth tables. If you do, you will see that they're all the same.

I don't think it would help much to study more logic than this. You just have to study more examples of proofs, and practice doing proofs on your own. If you want to be good at proofs, I would recommend that you do something like this: Study the proof in the book first. Then try to prove the theorem without looking at the book. Then do it over and over, while imagining yourself explaining it to someone else. Keep doing it until you don't see a way to improve your explanation. If it's a difficult proof, it can take many attempts to get to that point.

In addition to this wonderful post, I would add that mathematics is not a spectator sport. You learn by doing and trying it yourself, not by reading things in a book.
For example, proving something by induction is something you should have done quite a lot of times yourself before you really get what's going on. The same holds with proofs by contradiction, which seem quite exotic at first.

So in addition to trying to understand the proofs yourself, try to prove some easier things first. Then let somebody (perhaps on this forum) read your proof and indicate what's wrong about it. These are the only ways to really get it.

 

[Bashyboy]

Wow, well I am greatly appreciative for all of the advice. Thank you everyone. As I've mentioned before, we seriously need a thank you button.