If you read a proof but cannot repeat it afterwards, did you learn much?

[andytoh]

If you read a proof of a theorem (e.g. a long and/or difficult one), and you fully understood every step perfectly while reading it, but cannot repeat it afterwards, would you say that your math ability has really improved much through reading it? Should you read proofs of theorems only if you know that you can repeat it afterwards, otherwise you are simply wasting your time and are better off doing exercises using the theorems? What are your opinions?

 

[rudinreader]

In my opinion the most important ability is what is implied in Rudin's format. You are asked to read 10-20-30-40 pages or material, then solve problems at the end of the chapter based on that material - and the fact of the matter is that there may be one or more theorems in the book that you won't really need (or need much).

Once you solve problems, write them down in a place where you can reference it 1 month/year/ or decade later. Note that the theorems in question are already written down in the book... (Personally speaking, I prefer doing this on a computer - and backing up the data - but I use html-math (http://dionysia.org/html/entities/symbols.html) along with inserting tex images (http://www.sitmo.com/latex/) manually into a web page).

If the book author did a good job in choosing problems, then you will most definitely be needing to look carefully at certain proofs to find a various trick that was used, and reapply it to the problem -- because it's in the very nature of advanced math books that the problems are not going to be rote one line applications of a theorem (unless it's a clever use of such..).

I don't particularly like Spivak's Calculus on Manifolds format, where there is a really short material section followed by very tough problem sets (however I must admit that when I read through chapters 1-4 I was indeed very impressed at how much information you could convey in such a small space, so it's still a very good book).

Actually there is a lot to say about the open ended question of "memorization of proofs" in Math, but I think the simplest answer is that there are some proofs that you would really want to know because they are particularly easy: such as a continuous image of a compact set is compact, a compact subset of a Hausdorff space is closed, etc..

But it's probably not economically optimal to try to memorize them once and for all the first time you encounter them (especially if you don't continue in math). But instead as you continue in math, and you run into those "facts" in more advanced problems, go ahead and take the time to try to resolve(reprove) them.. After you do that enough times you tend to remember them.

But the other point is that a really juicy source of learning "how to do math" is that when you find that you need an answer to a problem you solved 6 months ago, and you look at your old solution, and you really find it was poorly written... It teaches you to write down solutions clearly the first time and do not abuse the terms such as "obviously...", "it then follows that..." especially when you are rereading your own work and finding that what you said was obvious is not even obvious to yourself anymore!

Anyways, that's my two cents on this question...

 

[HallsofIvy]

"Reading" a proof, by itself, doesn't do much for you. To read mathematics, in general, you must read it several times, then put the book aside and try to go through it from memory. In a proof expecially, you should try to work out the "details", that may be passed over quickly, yourself.

 

[andytoh]

"Reading" a proof, by itself, doesn't do much for you. To read mathematics, in general, you must read it several times, then put the book aside and try to go through it from memory. In a proof expecially, you should try to work out the "details", that may be passed over quickly, yourself.

Hmm... In that case, I should do what rudinreader has suggested, and rewrite the entire proof myself (peeking when necessary) and fill in all the details omitted in the book's proof, even type it out so that I can save it for future rereading.

 

[Defennder]

I'm not a math major myself, but there was a number of times when I sought the help of some maths graduate students for some particular problems. When I asked for a proof of some of the theorems they used, they would think for a while, then say "Well I can't give you one now, but in general for maths proofs, there are theorems out there where you just read and understand it just once in the course of your studies, then basically it can be forgotten as long as you understand its implications and applications".

It kind of made me wonder for a math major at least, to what extent is being able to reproduce a proof without reference to a text or even a sketch of a strategy of how to execute it actually important when learning mathematics on its own? (ie. not because one needs it for physics, computing courses etc.)

Is this mindset typical of maths majors or are these people just lazy to explain to me how to prove them?

 

[HallsofIvy]

From the point of view of a professional, "memorizing" proofs is not terribly important (although it can save time looking them up!) but knowing the general concepts of the proof is- since that may suggest techniques for a new problem. On the other hand, studying a proof carefully (not necessarily "memorizing") because you can learn methods that will apply to other proofs.

And, of course, there is the classic (probably apocryphal) story about the oral exam: A professor asks the student to prove a well-known theorem. The student answers that he hasn't memorized the proof but " I know where to look it up if I need it". To which the professors reply is "You need it it now!"

 

[disregardthat]

Knowing the proof of a theorem is satisfying when you solve a problem involving it.

 

[Werg22]

I think that the content of a proof is as important as its result. Proofs constitute a great resource for learning new notions, techniques or bringing to your attention things you never quite noticed, even within concepts you were long familiar with. If you cannot repeat the proof but have absorbed new ideas from it, you definitely have learned something.

 

[Gib Z]

It's usually not best to venture out trying to memorize the proof, but rather the key techniques or devices used in that proof. It saves a lot of brain memory, you can remember it better, and if you've actually learned something, you can go through the whole proof with just those few devices.

For example, when using Riemann sums to derive the formula for the integral of a function x^n, n being a positive integer, you can either try to remember every step in the 2 or 3 page derivation, or just remember the key step; subintervals of (b/a)^{1/n}.

 

[rudinreader]

Hmm... In that case, I should do what rudinreader has suggested, and rewrite the entire proof myself (peeking when necessary) and fill in all the details omitted in the book's proof, even type it out so that I can save it for future rereading.

This is not what I was suggesting... I was actually saying you don't even need to look at the proofs (or the chapter content) if 1) you can solve the problems and 2) you aren't interested in them. But I was saying you should write down the problems because in general they aren't written in the book so if you need to reference them later, it really sucks if you solved it once, then look at it again, and not know the answer..

But ultimately to 1) solve the problems in a good book you end up needing to read most or all the proofs carefully -- or at least a part of them.

I will give an example from personal experience, as a case in point, in L'Hospital's rule.

I still haven't memorized the proof, and I won't even look it up on wiki for the sake of this thread. But what I do remember about it is that I didn't like the proof in baby Rudin, and I prefer to prove L'Hospital using sequences. (I.e. to show f -> L as x -> c you can take an arbtrary sequence p_n (!= c) that converges to c).

The other point is that you need to apply some sort of multiplication/division trick to change the infty/infty and 0/0 terms using the mean value theorem into f'/g'..

It's actually a poor memory, and in fact in the future eventually I am going to try to be able to remember more.. But still I know the statement, that if f/g gives inf/inf, then you can take the limit as f'/g'.

 

[mathwonk]

if you learn to ride a bike and cannot ride one afterwards, have you learned a lot?

 

[Gib Z]

if you learn to ride a bike and cannot ride one afterwards, have you learned a lot?

Yes! Don't crash your bike!

 

[makc]

if you learn to ride a bike and cannot ride one afterwards, have you learned a lot?

but if you can ride it, but don't know the math behind its dynamics/kynematics/whatever, you still learned enough to ride it.

 

[Defennder]

if you learn to ride a bike and cannot ride one afterwards, have you learned a lot?

I don't think this is an appropriate analogy. After all, the skill of riding a bike is due to procedural memory, while reading and understanding is declarative memory. Hence someone who is extremely forgetful may be able to learn to ride a bike but probably cannot remember enough to follow through a long and difficult proof:

http://en.wikipedia.org/wiki/Procedural_memory
http://en.wikipedia.org/wiki/Declarative_memory

 

[disregardthat]

I believe that understanding a proof is much more important than remembering a proof. And if you continously flip back some pages to take a look at the proof for a theorem as you are using it you will become familiar with the proof, and perhaps enough to know the proof step by step. The worst thing you can do is to skip the proof of a theorem you are going to use.

 

[mathwonk]

some of you obviously have never understood a proof well enough to have it in your molecules. you can't ride a bike by memorizing the rules either.

 

[rudinreader]

some of you obviously have never understood a proof well enough to have it in your molecules. you can't ride a bike by memorizing the rules either.

In the example of L'Hospital's rule, it's actually a pretty hard theorem in comparison to other differential calculus theorems. So I think it partially depends on a particular theorem whether or not a graduate student should be condemned for not "having it in his/her molecules". In my original post I was being very extreme in theory but not in practice about "not needing to read" etc.., but ultimately I still think the best approach is to "write it down". As for L'Hospital, this is the second time I have written (part of) it down. After finding my old solution, I didn't really like it that much (I hate when that happens!), and in retrospect I wish to banish the idea of using sequences. Yet still, I found it easier to read my solution than the one in baby Rudin.. so it helped me reprove L'Hospitals more quickly than if I hadn't written it..


So here goes (only one case addressed)... One of the things that makes L'Hospital tricky is that it addresses multiple cases.. So let's look at a single case, and try to be content with a (smaller) theorem that f,g -> \infty, and f'(x)/g'(x) -> L implies f/g -> L. (We assume f,g are differentiable and nonzero in a region for the theorem to make sense):

Suppose f(x) -> \infty, g(x) -> \infty, and f'(x)/g'(x) -> L (finite) as x -> \infty.

Verify the equality f(x)/g(x) = f(y)/g(x) + (1-g(y)/g(x))*(f(x)-f(y))/(g(x)-g(y)).

Fix e > 0. Fix M > 0 such that x >= M implies |f'(x)/g'(x)-L| < e. Since f(M), g(M) are fixed, we can choose M1 > M such that x >= M1 implies |f(M)/g(x)|,|g(M)/g(x)| < e.

With this set up apply the CMVT, for any x > M, choose Cx in (M,x) such that f'(Cx)/g'(Cx) = (f(x)-f(M))/(g(x)-g(M)).

Then for all x >= M1,
|f(x)/g(x) - L|
= |f(M)/g(x) + (1-g(M)/g(x))*f'(Cx)/g'(Cx) - L|
<= |f(M)/g(x)| + |f'(Cx)/g'(Cx) - L| + |g(M)/g(x)*f'(Cx)/g'(Cx)|
< e + e + |g(M)/g(x)|*(|L| + e)
< 2e + e(|L| + e) = e(|L| + 2) + e^2.

It follows that f(x)/g(x) -> L.



Well, that in my opinion is the essense of what makes L'Hospital tick, although I am going to be lazy and not look at the other cases yet because I have some stuff I need to do..

(I will get back to this later to finish off the other cases... oh well)



My old L'Hospital's Rule.

1-a) Suppose f, g are differentiable in (a,infinity), g'(x) != 0 on (a,infinity), 
 f'(x)/g'(x) -> L in R as x -> infinity, and g(x) -> infinity as x -> infinity.
Then f(x)/g(x) -> L as x -> infinity.

proof:

The trick is to use the equality
 f(x)/g(x) = f(y)/g(x) + (1-g(y)/g(x))*(f(x)-f(y))/(g(x)-g(y)).

Fix e > 0.  Let x_n be a sequence in (a,infinity) such that x_n -> infinity.  Fix N0 large enough so that f'(x_n)/g'(x_n) is in (L-e,L+e) for n >= N0.  Then fix N1 >= N0 large enough so that x_n > x_N0 for n >= N1, and both f(x_N0)/g(x_n), g(x_N0)/g(x_n) are in (-e,e) for n >= N1.

For each n >= N1, choose E_n in (x_N0,x_n) such that
 f'(E_n)/g'(E_n) = (f(x_n)-f(x_N0))/(g(x_n)-g(x_N0)).

Then for all n >= N1,
 |f(x_n)/g(x_n) - L| 
  = |f(x_N0)/g(x_n) + (1-g(x_N0)/g(x_n))*f'(E_n)/g'(E_n) - L|
  <= |f(x_N0)/g(x_n)| + |f'(E_n)/g'(E_n) - L| + |g(x_N0)/g(x_n)*f'(E_n)/g'(E_n)|
  < e + e + |g(x_N0)/g(x_n)|*(|L| + e)
  < 2e + e(|L| + e) = e(|L| + 2) + e^2.

The right hand side can be made arbitrarily small, so we conclude f(x_n)/g(x_n) -> L as n -> infinity.  Since {x_n} was chosen arbitrarily, we are done.


1-b) Suppose f, g are differentiable in (a,infinity), g'(x) != 0 on (a,infinity), 
 f'(x)/g'(x) -> infinity as x -> infinity, and g(x) -> infinity as x -> infinity.
Then f(x)/g(x) -> infinity as x -> infinity.

proof:

Again, use the trick
 f(x)/g(x) = f(y)/g(x) + (1-g(y)/g(x))*(f(x)-f(y))/(g(x)-g(y)).

Fix M > 0.  Let x_n be a sequence in (a,infinity) such that x_n -> infinity.  Fix N0 large enough so that f'(x_n)/g'(x_n) > M for n >= N0.  Then fix N1 >= N0 large enough so that x_n > x_N0 for n >= N1, and both f(x_N0)/g(x_n), g(x_N0)/g(x_n) are in (-e,e) for n >= N1.

For each n >= N1, choose E_n in (x_N0,x_n) such that
 f'(E_n)/g'(E_n) = (f(x_n)-f(x_N0))/(g(x_n)-g(x_N0)).

Then  for all n >= N1,
 f(x_n)/g(x_n)
  = f(x_N0)/g(x_n) + (1-g(x_N0)/g(x_n))*f'(E_n)/g'(E_n)
(use small e < 1)
  > -e + (1 - e)M.

The right hand side can be made arbitrarily large, so we conclude f(x_n)/g(x_n) -> infinity as n -> infinity.  Since {x_n} was chosen arbitrarily, we are done.


2-a)  Suppose f, g are differentiable in (a,b), g'(x) != 0 on (a,b), f(x) -> 0, g(x) -> 0 as x -> a, and f'(x)/g'(x) -> L in R as x -> a.  Then f(x)/g(x) -> L as x -> a.

Proof:

First expand f, g continuously (or redefine them) so that f(a) = g(a) = 0.  Fix e > 0.  Let x_n be a sequence in (a,b) such that x_n -> a.  For each n >= N0, choose E_n in (a,x_n) such that f'(E_n)/g'(E_n) = (f(x_n)-f(a))/(g(x_n)-g(a)) = f(x_n)/g(x_n).  Clearly E_n -> a.  Thus f(x_n)/g(x_n) = f'(E_n)/g'(E_n) -> L as n -> infinity.

 

[rudinreader]

Actually, I wrote the above reply kind of without concern of how it sounded.. because I doubt wonk was aiming his comment at me (though some of my remarks were kind of weak when taken to the extreme..)..

Since I'm not good at number theory (as of yet) I woud have also have trouble proving 937*234 = 219258 (on the spot, If it was a homework assignment, I could make some arguments).. In fact I am not expecting that in my future grad studies that I will ever be able to prove such arbitrary statements on the spot.. But I don't expect that was what the point was..

I have finished a L'Hospital proof here: https://www.physicsforums.com/showthread.php?t=214203.

But to add to wonk's comments... There was a point in time (after I had already got the epsilon-delta experience) where I was considering switching from math into to engineering.. Ultimately I decided to pursue the math Phd.

After learning how to make simple epsilon-delta arguments, I would have been MUCH better at the math in engineering than without it, despite whether or not engineering(or physics?) students probably sense that you just need to remember a handful of formulas..

I could go on and on... oh well...

 

[stefandxm]

if you learn to ride a bike and cannot ride one afterwards, have you learned a lot?

It will give you the confidence to be able to try it again. And fail, and then learn again. I never really did understand this memorize this and that hype that surrounds the math community.

-- DISCLAIMER: IAM NOT A MATHEMATICIAN AND NEVER WILL BE --
However, I do find the question a bit off. Say you read a calculus proof, and its carried out with algebra. Just because you understand the algebraic operations it doesn't mean you know the analytic part - the important part. However, if you understand both and then just "forget" it, no harm done, you probably know it and will be able to use it without rabbling the theorem all over again. (who cares about people having to rabble theorems to be able to state a valid point anyways?)

 

[wildman]

I think just reading proofs has value. As long as you understand each step and don't desire to be an expert in that subject matter. If all you want to do is use the formula, then reading and understanding the proofs once is a good exercise so that you KNOW that the derivations are valid. An engineer doesn't need to understand in detail (or even can understand) every mathematical tool he uses, BUT he has to have gone through and understood the proofs some time in his career.

 

[slider142]

I usually find that attempting a proof before reading the author's proof is much more enlightening, as you can usually see (from hitting dead ends yourself) why the author made certain constructions that would otherwise seem to come out of thin air if you had not attempted the proof yourself. If you do manage to prove the theorem yourself, then you have learned a great deal more than any reading could have done.