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November 14th, 2009, 09:18 AM   #1
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What is the most efficient way to gain mathematical insight?

Hi everyone.

Little bit of my background:
Im currently studying mathematics in the third year at university (or maybe you call it college in your country), and I have two more years left (at least (= ). I feel that I have affinity for physics and maths, and it is something i really want to master. My grades in these subjects is good - but my way (the easiest way ... ) to obtain good grades has been to practice problem solving techniques, rather than obtaining the deep mathematical insight Im wanting. As of now it's difficult for me to apply my current mathematical skills to new subjects independently, meaning that if Im facing problems i have not seen before, there is a great possibility that i will manage not solve it. I want that kind of insight that allows me to apply my math skills to any problem faced!

My question is therefore: How do I obtain this kind of mathematical knowledge? Do you have any books to recommend? How should i use the books? How should i work with the topics? Im interested in all topics, and its up to you to recommend in what order those topics should be studied. Im willing to lay down some hard work.. Both general tips, and specific study-tips (like write down the teorems on a sheet as you read) is of great help.

Any reflections on this matter is truly appreciated.

Thank you.

-k4ff3
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November 14th, 2009, 03:19 PM   #2
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Re: What is the most efficient way to gain mathematical insight?

Mathematical insight is not something that can be learned through books or the sort. It is gained by taking the creative approach to mathematics. For example, instead of learning about something online, try and figure it out yourself. This way, you'll be "thinking outside the box".
So the next time your professor lectures about something, try and think beyond what he or she is teaching. This will help you gain insight.
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November 14th, 2009, 03:26 PM   #3
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Re: What is the most efficient way to gain mathematical insight?

I'd recommend looking at "A Course in Pure Mathematics" by G.H. Hardy. It's an old book (just republished last year for its 100th anniversary), but in my opinion it will help you develop insight into the reasoning of calculus.
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November 20th, 2009, 07:48 AM   #4
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Re: What is the most efficient way to gain mathematical insight?

Gaining this sort of insight isn't really something that can be "taught", which makes it difficult.

I'm not sure exactly what you feel you're struggling with. But overall, there are a couple things to be thinking about:

* Nothing is "obvious".
Try to be extremely formal with all of your proofs. Make sure your steps all follow immediately from previous steps, definitions or theorems. Spend some time proving the "really basic" properties that follow immediately from applying the definition. Also, ask yourself what sort of objects satisfy certain properties, and which don't. Eg. For complete metric spaces, come up with a "canonical" example of a complete metric space, a "canonical" incomplete metric space, and a degenerate example of each. For example, the discrete metric is complete (if you know about metric spaces, you may want to prove this), but it really doesn't match our intuition for what a complete metric space "should be."

On that note, try to understand what the intuition for a property or object is-- what does it "mean" for a set to be a group under an operation? Also, try to keep track of where intuition departs from math-- For example, we like to think of topological spaces geometrically, but there are some very non-geometric topological spaces.

* Rewrite the same thing as many different ways as you can.
For example, if the problem asks a question about a normal subgroup, you should be thinking of all the characterizations of normality-- It's the kernel of a homomorphism, it's invariant under conjugatian (which really is the same as its left and right cosets are the same), if a and b are in the same coset of N, then a-b is in N.

* When working on a proof, pay attention to everywhere you use your assumptions.

* After writing a proof, make sure the result seems to make sense.
Does it match up with intuition? If not, figure out why. If the problem is with your intuition, try to figure out what you are assuming to be true, and make a note of it.

Are any basic examples of the structure a counter-example to your "theorem"? Does each step follow from the last? Are you sure?
(I have a friend who has written 3 or 4 wrong proofs this semester, and every time, he realized it was wrong based on these checks, although normally I had to pick out the false step for him )

* Learn to look for counter-examples.
If you're asked to prove something wrong, look at some basic examples of the structure you're looking at. Does the statement hold for them? If so, can you see what properties make it work? If so, try to come up with an example where that property doesn't hold. Does the statement fail now? Rinse and repeat.

* Rewrite your assumptions. Rewrite them in different words. Rewrite them with the definitions of any terms you are uncomfortable with.

* Look for connections.

* Rewrite any objects you're looking at in terms of other objects. The complement of an open set is closed. The complement of a closed set is open. A connected space has proper (non-empty) clopen sets. g is in the Center of G means gh=hg for any h.

* State the obvious. Often. And then state it again.

* Ask stupid questions. Then answer them.
Is R complete? Why is a polynomial continuous? Is Z abelian? Finitely generated? What about Z^n? What does Abelian mean anyway?

* Don't be afraid to ask someone else stupid questions.

* Don't be discouraged when you sit for hours without understanding what to do; let the gears keep grinding.
Put on some music and rock out while you think. Rewrite the assumptions. Try to do something. When you get stuck, try to figure out why that doesn't work. Does it get you anywhere at all?

* Don't be afraid to go do something else for an hour or 2 and then come back to work on a problem.
This is when some of the best insights happen-- go make some tea, read a book, watch a movie, get coffee with a friend, do something. Then come back and start again. Sometimes it'll be hard to get back in the zone-- redo some easier problems: Try to reword your argument or try to find a cleaner argument.

* Work on a simpler problem.
Need to separate two compact sets? Don't! separate a compact set from a point. Can you use this same argument again? Will a similar argument work for two sets?

* Work on a more general problem.
Don't show that n is divisible by 3, show that all numbers of a certain form are divisible by 3. Then show that n has that form.

Hope these give you something useful to think about.

Cheers,
Cory

Edit: Also, don't underestimate the importance of problem-solving techniques. It really is important to have some "tricks" up your sleeve.
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November 25th, 2009, 09:38 AM   #5
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Re: What is the most efficient way to gain mathematical insight?

Thank you cknapp for an awesome reply! It makes perfect sense, but it seems that hard work is essential here.

Do you have any particularly good books to recommend?
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November 25th, 2009, 04:05 PM   #6
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Re: What is the most efficient way to gain mathematical insight?

Very good advice, cknapp, especially the idea of re-proving the basics. (It does help.)

Quote:
Originally Posted by cknapp
Also, try to keep track of where intuition departs from math-- For example, we like to think of topological spaces geometrically, but there are some very non-geometric topological spaces.
The good news here is that the more you work with things the more your intuition matches. Also, learning things from different perspectives helps. I learned point-set topology, so I have no geometric insight into topological spaces -- but the discrete metric isn't unusual to me as a result.

Quote:
Originally Posted by cknapp
Does it match up with intuition? If not, figure out why. If the problem is with your intuition, try to figure out what you are assuming to be true, and make a note of it.
Sometimes you see one thing that doesn't match your intuition and you make a note (usually mentally). But when the second and third ones roll around, that usually means you need to re-think your conception of the objects. Once you do, it will be easier to think about them. Depending on the context, you might see "f(x) = 3x^2 - 7" as a curve, a map, a set of ordered pairs, a member of R[x], or a collection of symbols. If you're thinking about it the wrong way for the context you'll make it harder for yourself.

Quote:
Originally Posted by cknapp
* Ask stupid questions. Then answer them.
Is R complete? Why is a polynomial continuous? Is Z abelian? Finitely generated? What about Z^n? What does Abelian mean anyway?
I don't really have anything to add here, just pointing out how important this is. If the teacher says, "It is obvious that ___", always ask yourself "Why is that obvious?".

Quote:
Originally Posted by cknapp
* Don't be discouraged when you sit for hours without understanding what to do; let the gears keep grinding.
Put on some music and rock out while you think. Rewrite the assumptions. Try to do something. When you get stuck, try to figure out why that doesn't work. Does it get you anywhere at all?
This step depends on your style and preferences, of course. I can't do math to music, and I can hardly program without it.
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April 11th, 2013, 10:15 PM   #7
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Re: What is the most efficient way to gain mathematical insi

Excellent question. Fortunately Nature has given us the answer.

Evolution.

It's not how can I find the answer, but how can I evolve an answer. Evolution is a feedback mechanism. Give a few basic rules, the animal breed offsprings and repeat the game. Imagine the game is carried out to infinity, or a very long time, and viola, you have evolved a solution.

An abacus becomes a computer.
A bicycle becomes a car
orphan Coco Channel was taught to sew by nuns and built a fashion empire.

etc . . .

The answer is "how can I evolve the solution?"

A classic example is the Fibonacci series. It starts out very crude, but quickly approaches Phi in just a few steps.

Phi:
http://www.flickr.com/photos/85937466@N ... 2637573906
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April 12th, 2013, 06:20 AM   #8
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Re: What is the most efficient way to gain mathematical insi

Quote:
Originally Posted by long_quach
Excellent question. Fortunately Nature has given us the answer.

Evolution.
You could use evolution. You'd need a way to judge how much mathematical insight a person has, and ensure that their ability to reproduce was positively related to this insight. If the connection was strong enough you might see serious improvement in just a few hundred generations. But that doesn't help a person gain insight, just their distant descendants.
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