How important is memorizing proof?

[CuriousBanker]

Hello all,

I'm still a math newbie. I'm just about finished with part 1 of Spivak's calculus; this is my first experience with more advanced math / deeper understanding (in high school and college I did well in calculus because I could memorize to perform operations, but I never had to understand why the formulas were the way they were; no proofs were mentioned).

Anyway, in doing some of Spivak's proofs, I am 1) Understanding and memorizing the results...for instance, I am rembering if g is continuous at a, and f is continuous at g of a, then f(g(a)) is continuous 2) Most of the results are intuitive to me 3) I am understanding the proofs as I go through them...I am able to logically understand all that he is saying.

What I am not able to do, however, is memorize or recreate all the proofs. Some of them are very long and complicated and frankly I don't know how anybody came up with them in the first place. Especially the proof of why a function can't have to limits; the lemma he provides + the proof derived from the lemma are 2 pages of incredibly confusing work arounds to get to the results. And this is only part 1 of spivaks calculus...im sure this stuff is a cake walk for most math majors and it will only get harder from here on out.

So my question is, do you math guys all remember/can recreate most proofs? I am not talking about proving something like the triangle inequality, I'm talking about the really complex ones. Is having an intuitive understanding, remembering the outcomes, and following the proofs when I go through them enough?

 

[jgens]

Eventually you should be able to figure all of those proofs out on your own, but that is probably a bit much to be asking on a first pass through the material. If you can do most of the exercises and have a good intuitive understanding of things, then that should be fine for now. As you get more comfortable with the ideas, the trickier proofs should come to you, but it's okay if they seem elusive at the moment.

 

[CuriousBanker]

I think eventually I will be able to do them all on my own.

but I have no idea how, when asked to prove that a function cannot have two limits at one point, I am supposed to bust out this:

 

[jgens]

That lemma is unnecessary to show that limits are unique. Instead it is used to establish rules for computing limits of sums, products, and quotients of functions.

Edit: As to how you would figure out the proof for that lemma anyway, the basic idea is that you write out the desired estimate and then work backwards to figure out what works.

 

[Mandelbroth]

I think eventually I will be able to do them all on my own.

but I have no idea how, when asked to prove that a function cannot have two limits at one point, I am supposed to bust out this:

I'd be lazy and say something like "The space is Hausdorff. Halmos."

You should be fine. Take your time, learn the math, and read proofs. I wouldn't memorize them, but at least read them and understand them.

 

[lavinia]

I suggest trying to prove things yourself before you even read the proof. When you clearly know what you are missing in order to complete the proof then look back to find it in the text.

Memorizing in my opinion does not help you to think. Rather you want to know what must be right and then derive the proof.

 

[HallsofIvy]

Far more important than memorizing a proof is learning it! That is, understanding exactly why each statement follows from the previous one. That way you can reproduce it when you need.
(This is basically what lavinia said.)

 

[CuriousBanker]

Far more important than memorizing a proof is learning it! That is, understanding exactly why each statement follows from the previous one. That way you can reproduce it when you need.
(This is basically what lavinia said.)

I agree. Just not sure how anybody would be able to recreate a proof requiring a lemma that was specifically created for that proof.

 

[AlephZero]

Just not sure how anybody would be able to recreate a proof requiring a lemma that was specifically created for that proof.

Mathematics doesn't get invented in the same order that it gets published (either in papers or textbooks). If you want to find a proof of something, you start with the big ideas and then fill in the details. That's what other posts mean by "learning" a proof - i.e. learning how the the big ideas fit together. if you know that, you can work out the details without having to memorize everything.

if some of the "details" need a long (but maybe straightforward) proof, you split them off as a separate theorem. A "lemma" is just another name for a "theorem that you only use a few times, to prove something else".

 

[CuriousBanker]

I get you.
Can you help me with one last thing? I have two things left on my agenda before moving on to part 2 of spivak.
This isn't in his book but I just wanted to know it anyway but can't figure it out on my own (currently I suck at coming up with proofs on my own although I have understood almost all of them so far when I read them)
Prove that the function f(x)=x^n is continuous everywhere.
Though incredibly intuitively obvious, I for some reason cannot prove it

 

[jgens]

Prove that the function f(x)=x^n is continuous everywhere.

The identity function is continuous as are products of continuous functions. This means that the n-fold product of the identity is continuous and the proof is complete.

 

[Mandelbroth]

I get you.
Can you help me with one last thing? I have two things left on my agenda before moving on to part 2 of spivak.
This isn't in his book but I just wanted to know it anyway but can't figure it out on my own (currently I suck at coming up with proofs on my own although I have understood almost all of them so far when I read them)
Prove that the function f(x)=x^n is continuous everywhere.
Though incredibly intuitively obvious, I for some reason cannot prove it

I think you mean "Prove that the function $f$ such that $f(x)=x^n$ for all $x\in\mathbb{R}$ is continuous everywhere."

As jgens says, finite products of continuous functions on $\mathbb{R}$ are continuous. You should be able to show that the identity function is continuous, and thus that finite products of the identity function are continuous.

 

[WannabeNewton]

So my question is, do you math guys all remember/can recreate most proofs?

Not only is this pointless but also futile. Open up a text on functional analysis and try to memorize the proofs: you will find yourself drowned under a sea of inequalities that you just won't be able to keep track of in your head if you tried to memorize it all (unless you have an exceptional memory but that's an outlier case!).

Hell it's even worse in differential geometry if calculations are done using indices. No one would even want to memorize the calculation showing that $\xi^c \nabla_c (\xi^b \nabla_b \eta^a) = R^{a}{}{}_{bcd}\xi^b \eta^c \xi^d$ because there are too many indices to keep track of in the calculation. Plus what would memorizing it even accomplish? If you truly understood the concepts and/or techniques introduced in the chapter or section or what have you then you would be able to do the proofs/calculations by yourself without any need for memorization.

 

[R136a1]

I'd be lazy and say something like "The space is Hausdorff. Halmos."

Can we stop showing off and actually help the OP? Thanks.

 

[Jorriss]

I'd be lazy and say something like "The space is Hausdorff. Halmos."

Okay, well then he needs to prove R is hausdorff and that limits are unique in hausdorff spaces. Doesn't save that much work does it?

 

[CuriousBanker]

Not only is this pointless but also futile. Open up a text on functional analysis and try to memorize the proofs: you will find yourself drowned under a sea of inequalities that you just won't be able to keep track of in your head if you tried to memorize it all (unless you have an exceptional memory but that's an outlier case!).

Hell it's even worse in differential geometry if calculations are done using indices. No one would even want to memorize the calculation showing that $\xi^c \nabla_c (\xi^b \nabla_b \eta^a) = R^{a}{}{}_{bcd}\xi^b \eta^c \xi^d$ because there are too many indices to keep track of in the calculation. Plus what would memorizing it even accomplish? If you truly understood the concepts and/or techniques introduced in the chapter or section or what have you then you would be able to do the proofs/calculations by yourself without any need for memorization.

So how do I improve at proving things? Is there a book or site which really nails it into your head? for a lot of simple proofs I can come up with something...but it seems as if I am going to need the skills if I am in pursuit of theoretical math type subjects rather than just being able to compute things on a calculator...it probably makes more sense to get good now and do everything right then just get the overview and comeback later to do the proofs myself (I assume)

 

[CuriousBanker]

The identity function is continuous as are products of continuous functions. This means that the n-fold product of the identity is continuous and the proof is complete.

Ah, this makes perfect sense. Somebody tried proving it to me with the binomial theorem but I wasn't able to connect the last step...what you said makes much more sense.

I thank you all in this thread. Maybe I should just go to school for this stuff...but I keep telling myself I can do it all on my own if I try hard enough. It has to be possible, especially with resources like this website, so thank you all.

 

[Mandelbroth]

Okay, well then he needs to prove R is hausdorff and that limits are unique in hausdorff spaces. Doesn't save that much work does it?

My remark on Hausdorff spaces was meant as a joke. Hence the tongue face. It was not meant to show off. I'm sorry if it was interpreted as such. My intent was to exploit a silly shortcut in a lighthearted manner.

Maybe I should just go to school for this stuff...but I keep telling myself I can do it all on my own if I try hard enough.

I guarantee you that it is possible to learn all of this without going to school and taking a specific class on it. Not just because some long-dead geniuses invented this stuff and many of them learned from, essentially, scratch, but because there ARE people who learn this on their own. Keep going, and I'm pretty sure you'll learn it.

The best way to learn proofs is to do proofs. That's the best thing you can do.

 

[HallsofIvy]

If you want to prove '$f(x)= x^n$ is continuous for all x' using the basic definition of continuous, then you must prove that $\lim_{x\to a} x^n= a^n$ for all real numbers a. To do that you have to look at $|x^n- a^n|= |x- a||x^{n-1}+ x^{n-2}+ \cdot\cdot\cdot+ 1|< \epsilon$ for any $\epsilon> 0$. Essentially, you now have to find an upper bound on $x^{n-1}+ x^{n-2}+ \cdot\cdot\cdot+ 1|$.