Comment by zackattack | Hacker News Reader

Comment by zackattack | Hacker News Reader

after review, i'm not sure that this post answers your question, and because i'm very tired it may simply be self-serving and totally useless to you. however, i am taking a gamble that it could answer your question and am posting it anyway.

it's normal to not be able to derive proofs on your own without seeing them beforehand. you don't want to "rote up stuff"? then don't do math, bitch.

seriously.

the only way that you're going to develop the ability to prove things on your own is by reading over and memorizing a lot of other proofs. math is a skill to be built and you will notice the repeating patterns in proof techniques, and then you will be able to prove new solutions yourself.

i totally killed all my math classes in college because i wrote down the proofs on index cards and drilled myself until i memorized them. i spent tons of time in the library doing ROTE MEMORIZATION of all the various proofs. does that make me a "boring tool" who can't "think creatively" and only relies on "lame rote repetition"? no, not at all, i was consistently a creative problem solver. WHY?? because i built the base of explicit knowledge upon which to draw, and THEN i was able to apply it in novel ways. DOING math is INTUITIVE. that it means it's SKILL-BASED and NOT RATIONAL. FOLLOWING a proof, on the other hand, requires LOGIC.

if your problem is understanding proofs, then, just make sure you have a thorough understanding of the building blocks and concepts.

do you know how i made it through abstract algebra? before school started, i sat down at the bookstore and read through the first 5 chapters of the book. then, when we discussed the chapters in class, it was my second exposure and it was much easier to memorize. consequently, i was way ahead of everyone and did minimal work. use your prefrontal cortex and think long-term.

and if you're having trouble visualizing things, try to reduce them into their smallest possible visual elements. for example, if you're trying to understand a proof of something in R^n, then start by understanding how to do it in R^2. Then R^3.

always try to look for concrete examples first. it's often much easier to go from concrete -> abstract than abstract -> concrete. it's how the human brain works. our consciousness is just a series of metaphors for real-world operations.

Aside from the part about "bitch", I agree with pretty much everything zackattack said. Key points:

1. On your second exposure, things will make MUCH more sense. It's just how I've found the mind works. The first time you read something, you're reaction is: what??? Come back to it in a few days and you feel more comfortable with it.

2. When I first started reading proofs, I felt so clueless as to _why_ they were doing the steps that they were. Basically, I felt like I was trying to scale a sheer cliff that was smooth as glass, with no handholds. I had no idea where to _start_ if I wanted to prove something. However, after reading a lot of proofs, you start to understand some tricks, concepts, and approaches on how to tackle them.

3. Going from concrete --> abstract (i.e. generalizing) is the way to go. As zackattack said, always try to work out cases in lower dimensions, or even "trivial" cases. After doing one or two of those, it's a lot easier to guess at the general form and verify that, than to derive the general form from the get-go. Also, doing the lower-dimension case will help you build _intuition_. Mathematics is all about formalizing intuition. The symbols and equations will often obscure that, but always try to understand the concept behind what the equation means. That will make most proofs much more amenable to understanding.

> it's normal to not be able to derive proofs on your own without seeing them beforehand

I STRONGLY disagree.

Let's address this with an example: "Two planes are either parallel or they intersect in a line" Someone who just finished high school should be able to "prove" it informally without even opening a text book. Of course high school education doesn't give you the tools to write a formal proof, but if you think about it for a while, then write an informal proof, when you finally open your book and read the formal proof it will be pretty obvious to you. Once you learn proper maths in University, you should be able to sketch a formal proof very quickly.

Obvioulsy you cannot reinvent all the math from the last 5000 years all by yourself, so you need massive amounts of effort and study to learn (a part of) it. In my view, learning the basics of your problem space, then thinking informally about it, then (informally) writing your own theorems and proofs is the best way of learning. You won't come up with everything, but when you go to your book and read that all-important theorem you will think: "Aaaahhh! of course! how didn't I think of it!?"

my point was that he should not trip about it, but instead be learning more techniques. in the future when he has sufficient skill, then OBVIOUSLY he should attempt to prove them on his own.

I am a she by the way. :/

There's a cultural issue in affect over here. From where I come from all I am expected to do is rote stuff up, not understand it. My teachers explicitly say that I'm wasting my time, but I frankly disagree. The why is not important for this discussion.

So a lot of math is fairly non-intuitive for me. My problem wasn't understanding the proof, but visualizing it and deriving it on my own without any aid.

The last bit of your comment is spot on though. I do have tons problems in visualizing stuff, but there's more to it than that. It's a learned skill that depends upon the knowledge of the fundamentals I have and how I have to apply them. I lack that at some level and I am trying to understand how to correct it.

Thanks for replying!

I am telling you. You will develop intuition, and the ability to derive proofs on your own, only after you spend lots and lots of time reviewing other proofs. Memorizing them and their techniques will accelerate your learning process. If you are ever in doubt about a technique used in a proof, study it further until you are confident you understand.

> does that make me a "boring tool" who can't "think creatively" and only relies on "lame rote repetition"?

I strongly disagree. It does make you such. Memorization is overrated.

And I strongly suspect that you actually don't know what you're talking about. (How far did you get in math?)

Indeed. I never memorized anything, then I got to uni. It's just a waste of time rederiving 5000 years worth of mathematics for the fun of it, and pretty much impossible to do in exam conditions. You need to know the theorems backwards in order to be able to intiutively apply them.

It's exactly the same as how basic calculus is taught...integration at low levels is a HELL of a lot easier when you know that say, 1/n integrates to ln n, and you can then just use basic algebra to get your integral into that form. Without having that core intuitive knowledge obtained by practicing and learning, you can't make those jumps.

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