
Ask HN: How Do I Get Math? - todayiamme
I just graduated out of high school and I want to learn loads of stuff in math, but there's a problem. I don't get it. Where I live it is expected of me to mindlessly crunch questions and output solutions by rote#. Practice is an important part of learning anything, and I know that I have to put my hours in. However, I <i>need</i> to understand why I am doing what I am doing.<p>Proofs of key concepts are like magic to me. I can follow the logic, but I cannot derive them on my own without seeing them before hand. I know that I lack understanding somewhere down the line, but I don't know where. Moreover, I don't know how to cure it.<p>I don't want to just rote up stuff. I want to appreciate the beauty of what I am learning, but I simply don't know how.<p>Any suggestions?<p>Thank you in advance.<p># Most teachers tell me to practice in order to memorize "problem solving techniques". I don't want to do that. I want to see the logic on my own, follow it through and think about what I am solving. I want to <i>see</i> stuff for what it is, and engage it on that basis.
======
Vargas
Pick something "easy" (high school trigonometry did the trick for me) and try
to understand it completely. Don't look at hte formulas or the proofs, just
look at the problem space itself.

You have a circle with radius=1 centered in (0,0). Draw a line from the center
of the circle to any point in the circle. Now you build a triangle using this
line as your hypotenuse. Inmediately, you will see that the heigth and the
width of the triangle will be in (-1, 1). That is why -1 <= sen(x) <= 1 and -1
<= cos(x) <= 1. You can see that your triangle is always a rigth triangle so
you can apply Pitagoras. That is why sen^2(x) + cos^2(x) = 1. Keep thinking
about it for a few weeks and you will realize that all the formulas you
memorized in high school are just common sense, you can deduce all of them
just by drawing a circle with a triangle in your mind.

Try to do the same with other stuff (set theory is a good second thing to look
at). You will discover that math notation is just notation and that many
proofs (at high school level) are just common sense written in a very formal
language.

Think about what the maths are about, not about formulas or notation.

~~~
todayiamme
Thanks a lot!

You nailed it. This is exactly what I want to do!

Can you suggest some good resources and a sort of rough path for me out of
your experience? I know that it may not be possible for you to do so and it is
quite okay if you can't do the latter.

Thank you for replying.

~~~
Vargas
Out of my experience, you only 'get' math when your boss gives you a paper on
a subject you have absolutely no clue and says: "I want this implemented in
our system by next week" :-P

On a more serious note: Other people have recommended very good books that
will teach you problem solving so I would like to give you a feeling of what
"real" math is. In high school and in most University degrees you only do
applied math => how to use math to solve this or that problem. The more math
you know, the more tools you have to solve your problem. For a given problem:
could I solve it with a partial differential equation? with a Bayesian
approach? etc. You can see how the more math you know, the wider your search
for a solution can be.

However, most people never get to do "pure" math => to create your own new
math. When you read a book or attend to a class you only get to see the
finished product. When you see a finished building, they have removed all the
scaffolding, the cranes, and all the other tools that were used to build it.
If you spend some time creating your own new math you will get a very deep
understanding of what math is and it will widen your learning abilities.

Of course I don't expect a high school student to create something truly new,
this is just a learning exercise. I just want to provide a 'feeling', I won't
be formal or rigorous.

Definition 1: An object of class TIAM is composed of an integer and a set. The
set can be empty or contain letters from the English alphabet with no
duplicates.

Definition 2: Operation "+" This operation will take two TIAM objects and
produce a new TIAM object. The integer part of the new object will be the sum
of the cardinalities of the sets. The set part of the new object will be the
union of the two sets with all the letters shifted as many places as specified
by the integer part of the second TIAM object.

Example 1: [1, {a,b,c}] + [2, {c, d}] = [cardinal({a,b,c}) + cardinal({c,d}),
{a,b,c}U{c,d}>>2] = [3+2,{a,b,c,d}>>2] = [5, c,d,e,f]

Example 2: [0,{z}] + [1, {}] = [cardinal({z}) + cardinal({}), {z}U{}>>1] =
[1+0,{z}>>1] = [1,{a}]

* TIAM objects are closed under +. (See <http://en.wikipedia.org/wiki/Closure_%28mathematics%29>). Do you understand why? Can you prove it?

* Is the operation + conmutative or non conmutative? (See <http://en.wikipedia.org/wiki/Commutativity>) Why? Can you prove that it is / it isn't?

* Can you find a neutral element? (See <http://en.wikipedia.org/wiki/Identity_element>) I mean, can you find a TIAM object (I will call it TIAM-ZERO) so that TIAM-A + TIAM-ZERO = TIAM-A for any given value of TIAM-A?

Etc.

This is just an example, try to build your own stuff and reason about it. Ask
yourself questions about it and answer them. Once you get used to this kind of
thinking, you will learn faster and deeper.

~~~
todayiamme
I am sorry if I make a fool out of myself, and I am really sorry if they are
some mistakes over here (it's 4 AM and I need to sleep), but I'll try to do
this.

#TIAM objects are closed under + as it is a union of two sets and we are not
including any element outside that set. So basically I get a bigger set that
includes all the elements of the previous set sans repetition. Now, since I am
including all the unique elements of the previous set this means that no new
symbol is over there.

However, the last operation (the shift one) isn't one with closure.

#It is commutative as it is the addition of the two sets with an algebraic
operation. So it doesn't matter what order in which I add the stuff together.
The results always remain the same.

#Yeah,

[0, {}]

~~~
hollywood77s
4) Can you find an identity element TIAM-ZERO' such that TIAM-ZERO' + TIAM-A =
TIAM-A for any TIAM-A

5) Is question 4 different than question 3 and why or why not?

~~~
todayiamme
4) The identity element is the same as that in question 3.

5) It's the same question as you've asked me to come up with something that
when it's put through a machine that computes the '+' operation with another
thing it comes out the same on the other side. In this operator it's limited
to this {0, []}. I don't know how I know this, and I don't know if I can prove
this but I think that this is the answer.

~~~
wcarss
Are you sure?

When I think on this, let's say TIAM-A is

[1, {a}], and you add [0, {}]

that's [card({a})+card({}), ({a}U{})>>1], or [1,{b}]

so that can't be the neutral element.

I think that there must not be an identity element, because the >> operator
forces a change under any circumstance which is dependent upon TIAM-A's
contents.

Similarly (I haven't thought about this enough yet), I suspect that the
operation is non-commutative, because unions can change the number of elements
(deletion of duplicates), and the shift is involved (which changes elements,
as opposed to just combining them). So let's say you shift and then union, you
could get a different result than if you union and shift.

I'm gonna see if we can get a concrete example here...

at somewhat random:

TIAM-A (now referred to as A): [0, {a,b,f}]

TIAM-B (now referred to as B): [1, {b,c}]

TIAM-C (now referred to as C): [2, {d,e}]

A+B+C=T1, A+C+B=T2, if T1 != T2, not commutative (pardon me as I riff on
notation for ease of writing)

A+B = [0, {a,b,f}] + [1, {b,c}]

A+B = [3+2, (abf U bc)>>3+2]

A+B = [5, (abcf)>>5]

A+B = [5, fghk]

A+B+C = [5,fghk] + [2, de]

A+B+C = [4+2, (fghk U de)>>4+2]

A+B+C = [6, (defghk)>>6]

A+B+C = [6, jklmnq] = T1

then:

A+C = [0, {a,b,f}] + [2, {d,e}]

A+C = [3+2, (abf U de)>>3+2]

A+C = [5, abdef>>5]

A+C = [5, fgijk]

A+C+B = [5, fgijk] + [1, {b,c}]

A+C+B = [5+2, (fgijk U bc)>>5+2]

A+C+B = [7, (bcfgijk)>>7]

A+C+B = [7, ijmnpqr] = T2

T1 = [6, jklmnq], does not equal T2 = [7, ijmnpqr]

hence, non-commutative - but I could have misunderstood something entirely
here

~~~
Vargas
Hi, yes, I made it non-conmutative to make it a little bit more difficult, I
think you don't get to see a non conmutative algebra in high school, it was a
way to take her out of her comfort area.

However you made a sligth mistake with the shifting part:

"The set part of the new object will be the union of the two sets with all the
letters shifted as many places as specified by the integer part of the second
TIAM object."

I think you are shifting a little bit too much.

~~~
wcarss
aha!

I interpreted "the second TIAM object" as "the new TIAM object", as opposed to
the second of the original objects. Thank you for the clarification :)

~~~
todayiamme
I learned something just the same. Thanks a lot!

------
thefool
Take a proof based math class in college. They will likely start at a low
enough level to make you happy.

Once you get to the point where the subjects start dividing, you'll see that
the real skill that math teaches you is to look at a problem, and create a set
of definitions which frame all the properties of the problem and make them
make sense in that framework.

You get this skill by both playing with problems and trying to define
frameworks (theorems, lemma's, axioms, ect) and by reading about the
techniques that others have developed. It isn't about rote memorization, its
about playing with abstract concepts till you have a way of understanding
them. One thing that playing will teach you is that it will show you that the
path you take to understanding something is not the same as the simplest path
for explaining something once you understand it.

In contrast, in things like physics you are generally given the frameworks (by
the mathematicians) and the cleverness comes from understanding how particular
situations can be described by the existing (pretty abstract) frameworks (like
newtons laws).

If you are bothered by how some of the ideas seem to appear from nowhere,
notice that there is a lot of math out there that you don't know, and even
basic statements about open and closed sets build on a lot of more basic set
theory and group theory.

~~~
todayiamme
Thanks a lot for commenting. Your comment was pretty insightful to me and
validated a few of the assumptions I had made.

Although, it is highly unlikely that I will end up at a college which offers
such a class (indian education pretty much sucks) I'll search OCW and iTunes U
until I find something like that.

I want to play with abstract concepts like you've said, but I have internal
problems which I need to overcome before reaching that goal. I can't sit down
at stuff at any lengths of time like that until I am really engrossed stuff
from my past comes up and I tend to get emotionally strained whenever this
happens. I remember stuff that shouldn't be there in my mind in the first
place and it is a constant battle to get stuff done.

So, is there any structure you use to get stuff done?

~~~
pmb
Don't sell Indian universities short. It is very hard to get a job in
mathematics, which means that even at "bad" universities the mathematics
faculty can be quite good.

~~~
todayiamme
That's the problem you see.

No one takes a job in mathematics, down here the trend is to do engineering
get a MBA and work in Goldman Sachs. I wish this wasn't so, but it's quite
true. There are precious few souls who try to buck the norm, but they are far
and few in between.

Most mathematicians end up in coaching institutions which exist for one thing;
to "crack" the IIT-JEE for the aforementioned rat race. I really wish I could
sell them short, but outside the IISc and TIFR along with a few other research
organizations there is hardly any innovation taking place. Most universities
are intellectually dead and students study only to get "placements" in some
company so that they can prepare for their MBA later on.

I know that this sounds like an excessive generalization, but I live near one
of those prestigious IITs and I've worked with a PhD student as well as
solicited the advice of a few professors. You know what shocked me? One of
them, who is one of the original brilliant researchers and an ex-dean, told me
point blank to get the hell out of this country and that my future isn't over
here.

I really want to change things, but since I am in this system I simply can't
do it. However, at least I can speak the truth. Please try to understand that
I am not judging anyone in this context. It isn't anybody's fault. It's just a
reflection of how the majority of this society thinks.

~~~
pmb
You misunderstand me. I am not saying that Indian Universities properly
prepare you for a lifetime of success. I am saying that it is certain you can
get the mathematical knowledge you want from them. As to whether that is the
_career_ you want or whether you want to be a mathematician - I have no idea.
But if you want to learn and understand proof-based math, Indian universities
are as good as everyone else's, and the good ones are quite good.

------
silentbicycle
You might find Jan Gullberg's _Mathematics: From the Birth of Numbers_ helpful
- it's a big book, but it's fairly light reading. Instead of a dense book of
problems, proofs, exercises, etc., it gives an overview of the different kinds
of math, their historical context, what kinds of problems they're good at
solving, etc. More the 'what' and 'why' than 'how'.

It might help you get a better sense of what you're more interested in
learning. (You might really like probability and statistics but not trig, for
example.) You can probably find a copy for under $10, and a decent public
library is likely to have it as well.

I've had a lot of fun with Project Euler (<http://projecteuler.net/>), a math
puzzle website for programmers, though it's not clear from your post whether
you're also into programming or not.

Also, another hard part of learning math is the notation. There's no getting
around that. The Gullberg book will introduce it, but it won't drill it into
you the way working through exercise problems will.

------
jcdreads
You may find that undergraduate physics is a good match for your learning
style. There is plenty of ridiculous math, but _all_ of it is based in simple
(!) models of the way the world actually works. Differential equations and
vector calculus are relatively easy for visual thinkers, and the need to get
the math to tell you precise answers drives you pretty naturally to some
reasonable approximation of mathematical rigor. Where heavy mathematics
demands insane amounts of rigor to make progress, physics is merely
descriptive, so you can enjoy learning about the symmetry and beauty of things
(like how all of chemistry falls out of quantum mechanics) without (all of)
the mindlessly mechanistic formality (like proving that legendre polynomials
are really a complete basis set).

Also know that math (and physics for that matter) is almost universally poorly
taught, so don't get discouraged by that.

------
Arun2009
I am NOT a Mathematician, but FWIW, here are my antidotes to the crisis in
meaning that you are facing:

1\. You most likely wouldn't have had a problem solver's education. So the
first thing to do would be do start understanding how problem solvers approach
Mathematics. I suggest two books:

\- The Art and Craft of Problem Solving by Paul Zeitz

\- How to Solve It by George Polya

There are also others, e.g., [http://www.amazon.com/Math-Olympiad-
Resources/lm/1SWDJ5NA047...](http://www.amazon.com/Math-Olympiad-
Resources/lm/1SWDJ5NA04748)

Keep in mind that most proofs you see are highly polished for the sake of
clear presentation. The route Mathematicians take to reach the proofs are
almost always messy. And, unfortunately, they like to clear their tracks.

A great way to learn how to solve problems is to find a good problem solver
and asking them to think out loud while they are solving problems. Also, try
to find alternative proofs of any proof you learn.

2\. Understand the _history_ behind what you're studying. How did the ideas
you're trying to learn (Abstract Algebra, Topology, Vector Analysis, etc.)
come about? This will add an immense amount of meaning to your subjects.

I recommend the following books:

\- "Men of Mathematics" by ET Bell. Also "The Development of Mathematics" by
the same author.

\- "A Concise History of Mathematics" by Dirk Struik,

\- "A History of Mathematics" by Boyer

\- And of course, the internet.

3\. Read a few "big picture" books side-by-side. A few suggestions:

\- "Foundations and Fundamental Concepts of Mathematics" by Howard Eves

\- "Concepts of Modern Mathematics" by Ian Stewart

4\. IMHO - and this is just my preference - study application of Mathematics
to Physics and Engineering. Physics especially was an inspiration for a lot of
Mathematics, and in addition, this will also let you solve concrete problems
using the tools you learn.

5\. During my undergraduate and graduate periods, I found several Schaum's and
Dover books helpful. They're usually short and pretty cheap.

6\. Initially, do not let rigor get in the way of understanding the content.
Fourier, Newton, Euler, etc. weren't all that rigorous by modern standards.

~~~
kenjackson
Great suggestions!

Two more must read books: 1\. "The Mathematical Experience" by Davis and Hersh
([http://en.wikipedia.org/wiki/The_Mathematical_Experience#cit...](http://en.wikipedia.org/wiki/The_Mathematical_Experience#cite_note-1))
2\. "I Want To Be a Mathematician" ([http://www.amazon.com/I-Want-Be-
Mathematician-Automathograph...](http://www.amazon.com/I-Want-Be-
Mathematician-Automathography/dp/0387960783)).

Probably the key thing for me in helping my math ability was to actively try
to prove theorems. Before reading a proof, I always try to solve the theorem
myself first. And then after reading the actual proof... try to prove it
again. You'll be surprised how many times you can't prove a theorem for which
you just read the proof!!!

But this will help you get better at doing proofs, and understanding math. And
it will also help you appreciate good proofs, because you would have already
tried to solve it. You'll say, "Ahh... I didn't even think to try that, but
that was exactly the step I would have needed".

Lastly, as someone else mentioned -- the proofs you read in texts are polished
proofs. Often those theorems proved have been attempted by famous
mathematicians who failed to prove it in their lifetimes. Take your time, be
rigorous, and thoughtful. If you do that, you come out ahead regardless.

------
Ben_Dean
I was in the same position as you recently (and still am to some degree)
though in worse shape, since I decided I needed math at the end of college.
The points people made previously about working at it and repetition are
critical. I absolutely understand that you don't want to do things by rote,
but what's important to realize is that like ANY other skill, there is one
best way to improve, and it's called practice. You'll just need to do exercise
after exercise, and read and re-read proofs, even when you don't feel that you
"get" it. First you do it, then you'll get it.

The other thing that works the same for every other skill is to teach it. Get
a friend who feels the same, and teach each other math. This works pretty well
when you're not even talking to a real person, even. Just having to formulate
your understanding solidly enough to convey it to someone else reveals where
you get it and where you don't.

The personal trick that will help though I find is the hardest to do
consistently is to be OK with not getting it. You'll spend a lot of time not
understanding, so get comfortable with that fact and keep going. For myself, I
tend to back off the subject when I don't get it right away, but I learn much
faster when I stick with the stuff that makes me uncomfortable.

Prescriptively, try looking at How To Prove It for an explanation of what's
going on in proofs, and how to engage with them.

------
Open-Juicer
I've got lots of good math books to recommend. However, giving beginners too
much information would make them more confused.

So, only two recommendations for you, one book, one website:

1) If I'm only allowed to recommend one math book to beginners, It'll
definitely be:

What is Mathematics - by Richard Courant and Herbert Robbins

Take a look at the review by Albert Einstein. Yes, Albert Einstein!

[http://www.amazon.com/Mathematics-Elementary-Approach-
Ideas-...](http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-
Methods/dp/0195105192)

2)BetterExplained

This site explains math intuitively unlike the traditional formal approach.

<http://betterexplained.com/archives/>

------
Tichy
Practice and have some patience. I still remember the first year of my maths
degree. The exercises seemed rather hard. A couple of months later, they
seemed easy (the ones from the early months, not the new ones from the later
months).

I don't think rote learning is the solution, but there are common problem
solving techniques.

Also, my experience was that you don't understand maths so much as that you
get used to it. I think I know how you feel - I also expected to logically
process a section of the maths book and then having understood. In practice it
didn't work that way. I remember at some point I suddenly had a grasp of
"Gaussian Manifolds" , which seemed completely intractable at first glance
(don't ask me now - it might not have been 'Gaussian Manifolds' but something
else, I forgot the name and the concept).

I think in reality as a mathematician you just have to get used to feeling
stupid.

it also frustrated me how little we really understand. For example sometimes
we prove something by some really clever way, some border case contradiction
or whatever. But to me that was not understanding - understanding would be
"seeing it", not proving it.

~~~
todayiamme
Wow. Thanks a lot for that insight.

You know I still think sometimes that I am a retard and I can't possibly
understand stuff. I know it's rather self-defeating, but when I used to see a
few teachers handle stuff with amazing dexterity I used to just gape at the
whiteboard. Aware that something beautiful was going on, without being able to
put my finger on it.

Whenever I used to talk about this to people they would say that I don't work
hard enough. Or, that I am a fool to expect anything more than just get the
marks which are expected of me. Or, that this is not the time or place to do
such stuff. When I told my mother right now about this amazing place called HN
and this thread she told me to shut up and stop being so impractical (don't
judge her on this she's quite amazing).

Long story short, in the end I assumed that the problem was with me and that I
was deficient in some way or the other. Now, I'm learning that is not so and
hence this thread. So, your comment was an eye opener.

I am so sorry that I put proofs in there. To me that was the biggest symptom
of not seeing stuff, you know. So, I thought that people may understand from
that. My bad.

~~~
Tichy
Nothing against proofs, of course proofs are the basis of maths. What I meant
is that typically I couldn't just read a mathematical text, follow every step
logically, and then have an understanding of that mathematical subject.
Rather, the understanding would sink in after a while of trying to follow the
proofs and solving exercises.

Could you give examples of the kind of problems that stump you, and the things
you try to do? I am sure a lot of people, including me, would be glad to help.

To me it just sounds as if you are trying to take too many steps at once, if
you can not follow some of the proofs.

~~~
todayiamme
For example a few days ago this teacher wrote this problem in class;

(x^2 - 2x + 2^|a|)/(x^2 - a^2) > 0

I had no clue how to do it. When he wrote the solution on the board only then
could I follow it. I didn't understand why I couldn't see the solution on my
own. It looks so simple later on and I just can't understand why I couldn't
get it on my own in the first place. Can you help me out with that?

Thank you.

~~~
Tichy
Here's hoping I won't embarrass myself too much...

Since presumably your teacher already wrote the solution onto the board, I'll
just talk about how I would arrive at a solution.

So first of all, it is a very common problem. We must have solved countless
polynomial problems at school. So usually, seeing this kind of problem should
already ring several bells.

So what I associate with polynomials (from school day) is

\- zero points

\- factorization

\- quadratic equation

\- "curve discussion" (don't know the english name for it, determining minima
and maxima by calculating differentials)

Did you have the same associations? If not, then that is simply a matter of
experience that would come with practice. Without that experience, naturally
arriving at a solution would be hard.

It should be immediately obvious that we can factorize this polynomial without
problems: simply because we know that we can factorize every second degree
polynomial, thanks to the quadratic equation. This knowledge is basically
"rote learning", except that you'll probably pick it up without decidedly
doing rote learning. The polynomial presented here is already factorized into
two second degree polynomials, so no problem there - do you agree so far?
(Bonus points for seeing immediately that (x^2-a^2) = (x+a)(x-a) - this is a
really common and basic formula that you should probably just "know". However,
in principle you can also derive it from the quadratic equation, knowing this
formula is just a shortcut).

So, are these zero points of use after all? Yes!!! We can simply check some
values of the function between the zero points and determine if they are
greater or less than zero. Since the function can only "switch sides" at the
zero points, we then know it has to have the same sign (negative or positive)
everywhere between those zero points (and same for outside of all the zero
points).

Do you agree so far? If not, where is that you were stuck?

With that idea, let's just determine the zero points. From the (x+a)(x-a) part
we already know that a and -a are zero points.

Using the quadratic formula on x^2 - 2x + a^|a| I get -1-sqrt(1-2^|a|) and
-1+sqrt(1-2^|a|) - just ask if that is too fast, also be aware that I am
_very_ prone to make errors in such calculations.

From there on, I admit it is not much fun. We have to somehow order the zero
points, so that we now when the sign could change. So we have to make case
distinctions.

In fact, I haven't yet thought further than this, and now I am looking for
ways to make it easier (I hate case distinctions). Laziness is the main
motivator for maths, therefore I am not that much in favor of the rote
learning approach...

So one step back: we have a products of two polynomials, poly1*poly2 with
poly2 = x^2-a^2. That product is > 0 if poly1 > 0 and poly2 > 0 or poly1 < 0
and poly2 < 0 (if this is not obvious, please ask).

It is clear that poly2 < 0 for |x| < |a| and > 0 for |x| > |a|.

So let's look at poly1 = (x+1-sqrt(1-2^|a|))(x+1+sqrt(1-2^|a|)) (if my
calculation was correct).

Since 1-sqrt(1-2^|a|) < 1+sqrt(1-2^|a|) we get

poly1 < 0 for -(1+sqrt(1-2^|a|)) < x < -(1-sqrt(1-2^|a|)) and > 0 for x <
-(1+sqrt(1-2^|a|)) or x > -(1-sqrt(1-2^|a|))

So alltogether, the (intermediate) solution is

(-(1+sqrt(1-2^|a|)) < x < -(1-sqrt(1-2^|a|)) and |x| < |a|) or (x <
-(1+sqrt(1-2^|a|)) or x > -(1-sqrt(1-2^|a|))) and |x| > |a|

I have a feeling this should be simplified further, but I'll post it like this
because it becomes a little bit intractable in the small comment box of HN.
Maybe I'll find time to finish later.

Anyway, I probably made mistakes, and overlooked some much easier approach.
The main point is that most of the steps are very common approaches, and I
think/hope I demonstrated dividing the problem into smaller sub-problems.

One advice I would give for maths is to not be intimidated - you need a little
bit of faith... If you fear the equations, you'll be too paralyzed to find a
solution.

Also, I used to cheat a little bit on the problems in school: usually chances
are high that the problem at hand has something to do with a recent thing the
teacher has taught us. So it pays to look for ways to use the knowledge from
the latest lessons.

Also, this problem seems comparatively ugly.

Sorry that this got rather complicated. Let me know how I can help better.
I'll try to create a shorter summary of the steps involved. (Meanwhile, please
post the shorter solution if there is one and you find it).

~~~
todayiamme
Awesome reply.

Thank you.

I'll reply when in the evening. Right now I need to crash and it's 4 AM and
I'm mentally impaired.

------
ajdecon
The kind of understanding and appreciation you're talking about, where you "
_see_ " the math, can be hard to get to. For me I never found it until I
started doing math in applied contexts such as physics or computer science.

My "eureka" moment was in a physics course where we were looking at the
derivation of the differential equation for a damped harmonic oscillator. One
moment the equation was something I had seen many times and knew how to solve
problems with; the next, something "clicked" and I _understood_ it, where the
terms came from, why it all made sense, in a way that's hard to describe.
Visualizing the oscillator and knowing the equation were the same thing in a
weird way.

After that, and a couple more similar moments, I found it much much easier to
understand and appreciate not only applied math but also purer concepts.

------
pragmatic
I suggest Project Euler <http://projecteuler.net/>. There are a series of
(interesting?) problems to solve that will require you to learn a little math
and a little programming.

For me, the hardest part of Math is the Byzantine formula. My brain just
doesn't look at them and go "oh, that's how it works." Instead, I need to
translate that into some nice executable code (Python, etc).

I stumbled through Calculus not really getting what it was about. Physics on
the other hand, was real world, here is the formula; here is how to apply it.
That made it click for me.

TL;DR: Physics and Project Euler.

~~~
todayiamme
>>>For me, the hardest part of Math is the Byzantine formula. My brain just
doesn't look at them and go "oh, that's how it works." Instead, I need to
translate that into some nice executable code (Python, etc). I stumbled
through Calculus not really getting what it was about. Physics on the other
hand, was real world, here is the formula; here is how to apply it. That made
it click for me.<<<

I know exactly what you mean. I can see stuff in physics and I intuitively
know what is going to happen when I take a metallic hollow piston with air
inside of it and place it partially in a magnetic field. It's observation of
Solenoids and other stuff that allows me to do that, but I really couldn't do
the same with math, and now I am trying to do that.

------
cvore
When I came out of high-school some years ago I had the same problem. I really
wanted to learn the math needed to do 3d-programming so I could to do fancy
demo-scene stuff (see www.scene.org), but the high-school level math just
didn't cut it and I didn't manage to learn it by my self. I started at a
university to study CS, and took some calculus courses there. It was a fairly
advanced calculus course which focused on proofs, and I worked really hard on
it through cooperation with others and the help of great teachers. Although
this experience lead me to end up doing a PhD in math, it was this first more
advanced mathematical experience which made me able to really _get_ it. It
took a lot of hard work, and after helping some younger people with high-
school level stuff, I now see that the math-books at this level often suck.
The "proofs" are more heuristic justifications that unfortunately often simply
hide the real point of what is going on. My advice would be to take a proof-
oriented math-course at university level. Even if the curriculum is not the
math you would like to learn (for example a calculus course might be ideal
even though you would like to do discrete math stuff for CS), I do not think
that mathematical maturity needed to go on by your self is possible to get in
any other way (except if you are a genius).

------
sammyo
Lemma: More coffee.

Derived from the theorem that a mathematician is a machine that transforms
coffee into theorems.

Ok other thoughts. Go listen to some hard stuff. I loved a talk by Ron Graham
about problems that would never be solved by computation. I remember feeling
like I followed but all that remains is awe and images of infinite arrows.

Knowing I'll never be a mathematician but intrigued by topology, I thought
about trying to write “Idiots guide to Algebraic Topology” as a way to push
beyond mobius strips and klein bottles. I've made it through an intro and some
insights seem sort of trivial, more along the lines of how to write down
precisely an action like a twist. We know intuitively what a twist is, what a
bounce is, getting it exact is hard work. But it relates to a lot of stuff we
grasp as denizens of a 3D world.

Draw lots of pictures, get really good a graphing. Graham seemed to dash off
quick graphs that gave quick insight, I expect he has incredible amounts of
practice.

Reading some history of math has helped me.

I also think talking/arguing/chatting/ranting with peers that are very
interested in topics in the general range will go along way.

------
will_critchlow
Pick something elementary to prove (by which I mean requires little prior
knowledge) - like the fact that the square root of 2 cannot be expressed as a
fraction and look up the proof. Spend time understanding not only the proof
itself, but also, crucially, why it is important to prove things in this kind
of formal way. Here lies the beauty of mathematics at first. Later you will
find exciting results, at first, there are only beautiful proofs.

You are then probably going to have to go back and do some rote-work. I think
you probably need to crunch some calculations to get comfortable enough to
ever get really good.

The good news is that with the excitement in hand, you should be able to carry
on through the rote stuff and then start on some undergrad courses (plenty are
published for free online) that have no prerequisites. Logic and number theory
are good for lack of existing knowledge - both come from sets of axioms.

[Obligatory, 2. ???, 3. Profit]

------
dododo
maths can be like programming: it can seem pointless without a purpose. too
much is made of proofs and too little of intuition.

so pick a project, something you'd like to know. then apply math as, and when,
needed.

what are you interested in? (other than learning math) if we know, then maybe
someone can suggest a project related to something you'd care about :)

~~~
todayiamme
Awesome. You nailed how I wanted to approach it. I am sorry if this comment
sounds like a bunch of gibberish, but I am half asleep down here.

I actually want to learn maths because I want to make this beautiful thing I
have in my head. It's basically a user friendly data mining app that learns as
you go along and it gives you the stuff you need. _Not_ what I think you need.

Basically I am trying to make a system that can collect data, and analyze it
for someone who doesn't know what parameters to set in the first place. After
that I want to apply what I learn from those interaction to trim the excess
fat or give anything more if required. I know that this is a really lofty
goal, but if I make it then I would be the happiest person on earth. As this
is a part of an even bigger thing I want to make, which is brewing in my head.
:)

I haven't been able to get much mileage as I have concentration issues and
there are environmental pressures which expect me to conform to the rat race
of indian society. So, I figured that this was the best way to kill two birds
with the same stone; I master my fundamentals as well as create the foundation
for my real life later on. I won't give up on my idea though and I try to
force stuff through my head at each sane moment of time I get.

Any suggestions?

Thank you for commenting.

~~~
dododo
often the simplest things in machine learning/data mining work best---so maybe
come up with a very simple way of achieving what you want, try it out. work
out what's not so good, and improve. repeat.

an important thing is to work out what maths you can just rationalise and
ignore the details of, and what maths you need to know in detail. for example,
you probably at this stage don't need to know how to prove the central limit
theorem, but it might help you to know the intuition. it might be a good idea
to know, in detail, where linear regression comes from though.

warning: in my experience, data mining books tend to provide poor explanations
of the mathematical justifications of what's going on. if you can, get to a
library and get some more theoretical books on machine learning or statistics.
in stats, larry wasserman's all of statistics is great, in machine learning,
hastie et al's the elements of statistical learning, mackay's information
theory book, bishop's machine learning book, etc, etc...

with any text book, don't read them cover to cover. just get what you need and
move on. if it's not obvious what you need from the book, try another one and
go back. books can be expensive, but time is more precious. it's more
important to do math than read math.

------
jacquesm
<http://news.ycombinator.com/item?id=755043>

~~~
todayiamme
I got a lot of mileage out of that thread thank you, but my other problem
still remains; how do I _see_ it and appreciate it?

There's something beautiful going on over here and I want to be able to peel
it apart like I do with my ideas. It's like I can learn the formula of force
and how it is applied day to day, but I don't necessarily _see_ it around me
unless I can visualize it and internalize what it means.

I know I am quite dumb and somehow there are like layers in my understanding.
I can't vocalize it but there is a huge difference to me between learning
about stuff the way I am supposed to and learning it the way I want to.

Thanks a lot for replying! :)

~~~
jacquesm
Appreciation comes with understanding, there is not much I can do about that
for you. I'm not too good at math (I know a lot of tricks to use a computer to
work around my lack of math skills though ;) ), so plenty of the beauty of
math is hidden from me too, but the root of it for me is that math is an
absolute, something that stands all by itself and that does not need a
physical universe to manifest itself in order to be 'true', even though some
of the mathematical constants do seem to be rooted in our physical
environment.

(pi for instance).

I don't like the 'I am quite dumb' assessment you give yourself, everybody is
born 'dumb', it's those that spend the time to educate themselves that rise
above that and you seem to be willing to do so.

Everybody, no matter at what level of knowledge they currently reside has the
power and the potential to rise above that level, maybe not much, but you can
always improve.

I'd say work through some of the stuff mentioned in that thread and see if it
helps you to gain understanding, you're almost certain to pick up something
and that in return should help with your other problem, the appreciation.

Best of luck with all this, think of it as enjoying the journey, not
necessarily to only enjoy it all upon reaching a destination.

~~~
todayiamme
Thank you for replying! It's always awesome to talk to you.

>>>I don't like the 'I am quite dumb' assessment you give yourself, everybody
is born 'dumb', it's those that spend the time to educate themselves that rise
above that and you seem to be willing to do so.<<<

You're right I am really sorry. It's just that old habits and learned
helplessness dies hard.

------
hackinthebochs
Speaking as someone who always "got" math, the understanding really needs to
come from you. You can't expect to get it from a book or from the teacher. I
had two tricks that has served me well: always ask yourself "why" until you
reach that critical "a-ha" point, and secondly, visualize visualize visualize.

What I always did for new concepts was to ask myself "why". And I'd just think
about it and visualize it until a reasonable explanation came to me. It
doesn't necessarily have to be completely "right", but it just needs to make
sense to you.

As an example from high school, everyone seemed to struggle learning basic
algebra. The basic idea to learn from algebra is that you can do the same
thing to both sides of an equation and the "truth" of it remains in-tact. This
fact covers about 90% of everything you'll learn in Algebra I and II. The rest
is basically just learning tricks for different situations.

The problem is you can't really be made to "get" this critical idea, the
understanding has to come from you. The takeaway from all this is to focus on
finding that a-ha moment for every new concept you learn.

As far as proofs go, again you can't be taught to do them well. You have to
find your way on your own. The first step is to just convince yourself what
you're trying to prove is true. Either by working out many different special
cases and finding some pattern, or by visualizing whatever process you're
trying to prove. Then you "formalize" your intuitive understanding by writing
a proof. Convincing yourself that something is true is the crucial part.

------
pmb
You want to go take a proof-based college math class. Or sit in on one. Or
find videos of one. It is hard to guarantee that a course will be like that,
unless its name is either "Analysis" or "Abstract Algebra". Don't sell Indian
universities short - it's so hard to get a good job in mathematics that you
have to go very far down the hierarchy to find a place with bad
mathematicians.

And you do need to memorize techniques - the only issue is that the techniques
you need to memorize are at a much higher level than you are currently being
taught. Techniques include "induction", "contradiction", "diagonalization",
"construction", etc. Mathematics is a toolset, and the techniques and facts
you learn are the tools required. They are beautiful in themselves, and can
bootstrap to better tools, but if you aren't a mathematician then you should
be memorizing solution techniques, just at a higher level than you currently
are.

------
sp332
You will have to learn some problem solving techniques. In fact, the more the
better. You will find that some of them will be intuitive for you, and many of
them won't. Don't think of these as "rote". It's training the basic mental
skills you'll need to call on without thinking about it. It's like learning to
walk: the exploration isn't in finding new ways of walking, but in finding new
places to do the same old "left, right, left, right".

I wouldn't worry too much about not being able to derive a proof yourself. At
least you can follow a proof - that's still a useful skill. Deriving a proof
is much, much harder.

------
golwengaud
I've found Terry Tao's blog (in particular the stuff linked at
<http://terrytao.wordpress.com/career-advice/> ) fascinating and helpful.

------
chunkbot
I'm going to play devil's advocate and say that you should listen to your
teachers.

I've seen plenty of students "diagnose" themselves, only to neither mindlessly
crunch questions nor appreciate the beauty of mathematics.

~~~
todayiamme
My teachers tell me to just solve questions and get good marks so that I can
move ahead in the rat race of life.

They are quite awesome, but unfortunately my paradigm doesn't exactly
superimpose upon theirs. So, I don't get the mileage I should be getting from
their advice. My goals and aspirations are simply too different. The ones that
do understand, smile and tell me to just tow the line.

I don't want a job in some big company. Neither do I want truckloads of money.
I just want to create beautiful things.

------
sandGorgon
Start with this - <http://www.geniebusters.org/Riemann_intro.html> and then
[http://www.schillerinstitute.org/fid_97-01/982_Gauss_Ceres.h...](http://www.schillerinstitute.org/fid_97-01/982_Gauss_Ceres.html)

It follows Vladimir Arnold's method of re-coupling physics with mathematics as
the original way of learning it... since observation of planetary physics was
how a lot of mathematics was derived.

It is a very, very interesting read.

------
scotty79
Watch these videos and understand them: <http://www.khanacademy.org/>

You can start at as basic level as you like.

Expose yourself to lost of proofs of simple theorems. Try to reconstruct proof
after single reading (with understanding why steps follow one another).

After a while you will see how information flows within a proof.

Basically expose yourself to a lot of math proofs and be sure that you
understand why every element of each single proof has to be where it is.

------
ad93611
Yes, there is beauty in math and in proofs. I was able to see this beauty
after reading the wonderful book by George Polya called "How to solve it?" I
understood math intuitively, after learning many proofs like you, but the key
point that I learnt from the book was "review your work". When you review, you
reinforce how a particular problem was solved and learn from the method that
was used. This forms as the basis for your next problem and so on.

------
jonsen
Since you are also into programming, I would recommend studying some discrete
math. _Discrete Mathematics and Its Applications_ by Kenneth H. Rosen is
rather friendly and instructive on how to do proofs.

------
BrentRitterbeck
Get "How to Prove It" by Daniel Velleman.

------
rortian
I'm going to try to give you something totally off the wall. Don't forget that
mathematicians were people. Here's three biographies of some of my favorite
mathematicians:

[http://www-history.mcs.st-and.ac.uk/Biographies/Poincare.htm...](http://www-
history.mcs.st-and.ac.uk/Biographies/Poincare.html)

[http://www-history.mcs.st-and.ac.uk/Biographies/Peirce_Charl...](http://www-
history.mcs.st-and.ac.uk/Biographies/Peirce_Charles.html)

<http://www-history.mcs.st-and.ac.uk/Biographies/Smale.html>

Also, please check out some of the giants on that site and wikipedia: Gauss,
Laplace, Euler. Why were they interested in what they were? What techniques
did they develop? A very crucial question that very few ever ask: Would they
have done much differently had they had access to a computer? (Of course).

Focusing on proofs to learn math is very perverse. A proof in a textbook is a
proof that has been refined an almost ludicrous number of times. Definitions
and axioms have even been optimized for the sake of elegance of said proofs.
Don't get me wrong, a connoisseur appreciates what has been done. However,
removed from the problem that motivated this way of thinking, mathematical
techniques can often confuse more than enlighten.

What would you like to learn exactly?

~~~
todayiamme
In this thread I realize that I have been a bit vague. I want to develop the
skills of visualization and in depth understanding of math for a reason. I
want to learn how to make systems that exploit machine learning and use
statistical techniques to infer patterns from data.

In the future some day I want to work in A.I. and create something beautiful
with the knowledge I have gained. This is why I want to lay down the
foundations to understand the beautiful advances in it I see around me.

Thanks a lot for commenting!

~~~
rortian
Cool, although I'd suggest not getting to excited about AI itself. It's cool
your already focusing on machine learning.

A great way to get going in that direction is to check out the state of the
art as expressed by AT&T labs Netflix Prize solution. To get very far you'll
want to have some background in linear algebra. However, theory isn't the way
to go (although you'll probably need it eventually). Do a little digging and
you'll find code for svds, the simplex method and all sorts of super important
algorithms that are used all over the place.

You'll find if you ask questions like: How does x work in y algorithm? people
will have a better idea of where to start. When you actually are coding and
able to see numbers for a particular problem, you'll be able to wrap your mind
around what's going on.

~~~
todayiamme
Thank you.

------
zackattack
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.

~~~
impeachgod
> 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.

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

~~~
hitonagashi
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.

------
HilbertSpace
What you got about the role of rote learning from your teachers is bad and
appears not to be from teaching in the US.

The 'understanding' you explained you wanted want in math IS the right
direction. The proper role for 'rote' learning is meager.

To get anything new, or even new to you, in math, first essentially have to
guess it. For this you need some intuition, that is, some ways to guess. You
should be free to take wild flights of intuition where only one in a million
guesses is correct. Proofs, then, add discipline to your intuition. Slowly
your intuition becomes more accurate.

E.g., given a guess, do some 'thought checks': So, check some extreme cases.
Guess what some of the consequences might be and see of those, first-cut, seem
to be true or false. Then will have guesses (1) very likely true, (2) very
likely false, (3) don't much know yet. Once you narrow down to what you need,
you will be well on your way to success.

Actually for anything new, the harder work is guessing it's true and being
fairly sure, based mostly on intuition and your checks, it is. Then usually a
proof is the less difficult work. That is, by the time you start a proof, you
might have some quite good intuitive evidence that what you are trying to
prove is true and, also, some good hints on how to prove it.

For more, read the now famous comment by A. Wiles on how he wanders in dark
rooms in an old mansion, bumps into furniture, slowly figures out where
everything is, finally finds the light switch, then clearly sees everything in
the room, and then starts on another dark room. Practiced intuition provides
the guesses, and proofs provide the correctness.

What I am saying here about guessing, Wiles, etc. is good for research, e.g.,
your Ph.D., and some of the more difficult text book exercises. Mostly in the
exercises can do well enough with less than a million wild guesses!

Yes, there is some 'sense' to at least the 'analysis' (i.e., calculus and
beyond) part of math. E.g., 'half' of calculus is finding areas under curves.
Okay, what curves have properties sufficient actually to HAVE area well
defined in this sense? Okay, the now classic answer is that the curve has to
be 'continuous'. So, when you learn about continuous curves, you are learning
about the classic hypothesis that makes area under the curve well defined.

Okay, why? Well, that's because are trying to find the area under the curve
over a 'closed' interval such as all the numbers x so that 0 <= x <= 1, that
is, the interval [0,1]. Okay.

What's so great about such a close interval? Uh, it's 'compact' so that any
infinite subset has a limit point -- that is, if take infinitely many points,
they MUST bunch up at least somewhere. We know that [0,1] is compact because
it is closed and bounded. Also, a continuous curve on a compact set is
'uniformly' continuous, that is, is highly restricted in how fast it can
'wiggle'. So, it has to be somewhat 'well behaved'. Also, that curve has to be
bounded -- can't run off to infinity, either positive or negative. Also the
curve has to actually have a largest value and a smallest value.

Then, consider all such continuous curves on [0,1]: Let the 'distance' between
any two be the absolute value of their maximum difference, which we now know
has to exist. All those curves form an infinite dimensional vector space, and
that distance is a 'norm'. Next, if a sequence of such curves appear to
converge, then they really do, that is, that vector space is 'complete' (much
as you learned for the real numbers) in that vector space and, thus, is a
Banach space. Now you have a nice list of surprising Banach space properties
-- Hahn-Banach, open mapping, closed graph, uniform boundedness.

Now you have some of the high points, intuitively, of Baby Rudin and some what
makes 'sense' there.

I very much disagree with most of the advice in this thread: E.g., I read
Bell's 'Men of Mathematics' and CANNOT recommend that book: (1) It won't give
you the understanding you said, appropriately, you wanted. (2) It's a BIG
distraction, misleading, even dangerous, from anything like math now. Polya is
okay but a bit off the track. Maybe read Polya about half way through your
college level studies.

Instead, for what you want, the usual courses are FINE and are:

HIGH SCHOOL:

first year algebra

plane geometry (with all the emphasis on proofs)

second year algebra

trigonometry (with all the emphasis on proving the identities)

solid geometry (with all the emphasis on proofs), optional but good if you can
get it.

COLLEGE:

freshman and sophomore calculus from any of the famous books, e.g., Protter
and Morrey, Thomas. Here seek to get some of the understanding you want via
pictures and applications to physical science and engineering.

abstract algebra -- emphasis on proofs and a really good start on the
understanding you want.

linear algebra -- emphasis on proofs.

Baby Rudin (W. Rudin, 'Principles of Mathematical Analysis') -- calculus with
the proofs.

ordinary differential equations -- with Baby Rudin and linear algebra, can do
well with nearly all the proofs. I like Coddington's book. Then have some fun
with deterministic optimal control, A/C circuit theory, and more.

advanced calculus -- various applied topics in 'analysis', often without the
proofs, e.g., Hildebrand's book long popular at MIT.

BIG Note: One of the most important applications of calculus in several
variables is to Maxwell's equations. For that, you need Stokes' theorem in 1,
2, and 3 dimensions. For that, I STRONGLY recommend that you get an
appropriate, OLD treatment. The best I know of is

Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced
Calculus', Addison-Wesley, Reading, Massachusetts, 1957.

He has a more recent book where he left out the good stuff; f'get his more
recent book.

What you want from that old book is only about 30 pages. I got the whole book
used -- no, I won't sell it! So, find the book in a library and make copies of
the pages. Then can cover those pages in one very pleasant evening. Then will
have quite well the calculus of several variables you need, and that nearly
all of the physical science community STILL uses, for Maxwell's equations and
more. For the math stuff with exterior algebra, etc., that's mostly for later
and mostly for modern approaches to relativity theory.

Uh, Stokes' theorem done that way is just some simple pictures and a simple
application of the fundamental theorem of calculus you learned in freshman
calculus and proved in very fine detail in Baby Rudin.

By then you should have all you want and more.

A high school plane geometry course with all the emphasis on proofs is one of
the best starts for the understanding you want.

In the books where the emphasis is on proofs, nearly all the exercises are
proofs. The easy exercises you should be able to do easily enough, and as you
work on the more difficult exercises you will get good at proofs. By the time
you can write a good proof of, in any separable metric space, each closed set
is the union of a perfect set and a set that is at most countable, you will be
well on your way.

Generally, as you start to write proofs, you should have at least a semester
where a good mathematician reads and corrects your proofs. The usual place for
this help is in the course on abstract algebra.

