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Questions about Problem Solving and Original Ideas

What's up yall. So let me explain a bit. I'm currently not taking any College classes. But I've been self-studying. I just work on whatever, and when I start a new topic, I'll play with it some.

After learning some of the basics, I'll try to figure out some more complex ways to use the idea. Or even take a more advanced concept and then try to figure out how it got to that point. "Fill in the gaps" with my own creativity. Sometimes it works, other (more often) times it doesn't. It is quite time consuming and stressful, but when I do get some "original ideas" it can be a confident boost.

Is this a good way to approach the subject? Even though everything I've been working on (Calc 1-2) has been worked out and proven time after time. Should even be wasting time doing things like this or wait until I have to write a research paper? haha

Speaking of research, how is it done? Specifically Math and Physics. Is it only one person working on a problem? Or is there a lot of collaboration?

Thanks all.

<< a few small edits by berkeman to remove the gangsta' speak >>
 
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Tom Mattson

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Is this a good way to approach the subject? Even though everything I've been working on (Calc 1-2) has been worked out and proven time after time. Should even be wasting time doing things like this or wait until I have to write a research paper? haha
You won't be able to write a research paper if you don't cut your teeth on re-working simple exercises that have already been done ad infinitum. That's because you have to build up your own intuition, no one can do it for you. So a proof may have been done many times before, but if you haven't proved it yourself it's new to you. Another thing you need to do if you're going to be good at mathematics is build up a repertoire of counterexamples. This is best done by studying what's been done already.

Since you're at the level of Calc 1-2, I would try to prove the theorems in the book. If it's a good book then at least some of the proofs will be there so you can check your work. I would also recommend getting a copy of "How to Prove It" by Velleman. It will expose you to some good proof techniques that you can get a surprising amount of mileage out of.
 

berkeman

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What's up yall. So let me explain a bit. I'm currently not taking any College classes. But I've been self-studying. I just work on whatever, and when I start a new topic, I'll play with it some.

After learning some of the basics, I'll try to figure out some more complex ways to use the idea. Or even take a more advanced concept and then try to figure out how it got to that point. "Fill in the gaps" with my own creativity. Sometimes it works, other (more often) times it doesn't. It is quite time consuming and stressful, but when I do get some "original ideas" it can be a confident boost.

Is this a good way to approach the subject? Even though everything I've been working on (Calc 1-2) has been worked out and proven time after time. Should even be wasting time doing things like this or wait until I have to write a research paper? haha

Speaking of research, how is it done? Specifically Math and Physics. Is it only one person working on a problem? Or is there a lot of collaboration?

Thanks all.

<< a few small edits by berkeman to remove the gangsta' speak >>
Welcome to the PF, mathG. Self-study can be problematikc. I've done a fair bit of it, and to be honest, the pace is slower than a real course, and when you get stuck, it's hard to figure out what is going on.

On the flip side, you can get instructors in courses who really are not good at conveying their grasp of the subject matter, and that can be a bad situation too. The best scenario is when you take a class that you are very interested in, from a talented instructor who is good at passing along their mental images and knowledge and techniques....

If self-study is your only option right now for whatever reason, I'd suggest looking for a book or other learning materials that are geared for self-study. And when you end up stuck on a problem in your studies, by all means use the Homework Help forums here at the PF. We require that you post the full problem statement and the relevant equations, and that you show your attempt at a solution. But as long as you do that, you will be able to tap a vast pool of knowledge here -- people who really do have a strong knowledge in the subject matter, and who are good at transferring their mental images and knowledge to you in their tutoring of you in helping you to figure out the problem.
 
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When you self study, you will always run across something that you can't understand. It is a natural tendency to sink tons and tons of time into this one little piece, and it can be extremely demotivating if you can't figure it out. To avoid these problem, a good strategy is this: first make a good effort to figure it out on your own, but don't be excessive. If you cant figure it out after this, make a mental note of exactly what the problem is and move on. Later when you learn new things, always be looking to fill in all those unanswered questions you had from before.

This will make you a true baller.
 
Thanks for all the answers. Even though my screen name is Math-related, I don't plan on major in Math. I plan to major in Physics and Chemistry with maybe a Math minor.

With that being said, and sorry to change the subject but, should I be concerned with Math proofs?
 
Nah, don't worry about proofs. Things are just true in math, proofs are kind of a waste of time. They feed academics, but don't really answer any questions at all, especially not ones like "why is this true?" or "is it really true?" or "how did I discover this was true?".
 
Nah, don't worry about proofs. Things are just true in math, proofs are kind of a waste of time. They feed academics, but don't really answer any questions at all, especially not ones like "why is this true?" or "is it really true?" or "how did I discover this was true?".
I can't tell if you're being sarcastic. Proofs answer all of those questions.
 
"Proofs answer all of those questions. "

No they don't. Let me provide a proof by contradiction.
Assume they did. Then they would be answers, not proofs. But proofs are proofs ... a tautology. Therefore, our assumption that proofs answer questions leads to a contradiction, ergo, proofs don't answer questions.

QED.

That last part, by the way, is from the latin "If you don't agree with me, you're a dummy." I wonder if studies correlating the appearance of "QED" below a proof to the correctness of the proof have been done... I would enjoy reading such a study.
 
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Nah, don't worry about proofs. Things are just true in math, proofs are kind of a waste of time. They feed academics, but don't really answer any questions at all, especially not ones like "why is this true?" or "is it really true?" or "how did I discover this was true?".
It EXACTLY answers those three questions. That's what a proof is, if you happen to read it carefully.
 
I'm 90% sure he's being sarcastic
 
What's the other 10%?
 
"Proofs answer all of those questions. "

No they don't. Let me provide a proof by contradiction.
Assume they did. Then they would be answers, not proofs. But proofs are proofs ... a tautology. Therefore, our assumption that proofs answer questions leads to a contradiction, ergo, proofs don't answer questions.

QED.

That last part, by the way, is from the latin "If you don't agree with me, you're a dummy." I wonder if studies correlating the appearance of "QED" below a proof to the correctness of the proof have been done... I would enjoy reading such a study.
I'll provide a proof by contradiction as well.

Assume that proofs do not answer those questions. Then, since those are the kinds of questions mathematicians ask, they would not know those answers, since they devised proofs in order to answer those sorts of questions. But, they do. Therefore, proofs answer those questions. Q.E.D.

I'll also provide an example: Why does 1 + 1 = 2? To answer this, you would need to read a proof on how basic arithmetic is derived from the ZFC axioms. Therefore, that proof answers your question.
 
on research... in physics, especially if you're going to go into experimental research. people usually do it in collaborations.
 
Monocles:

There is a flaw in your proof. I'm quoting where it occurs:

"they would not know those answers, since they devised proofs in order to answer those sorts of questions."

'they' refers to the proofs, in the way you have set up the sentence. "Devising proofs" isn't a well-defined operation on the set of proofs. I think there is a misunderstanding of the grammar you're using to generate your language.
 
Yeah, I didn't bother putting much effort into my proof since this is a rather silly argument, since I already know that I'm right since proofs have answered all of those questions for me, and I'm not out to change any anonymous internet people's minds, I'm just having fun. If they haven't for you, then that's too bad. Allow me to point out a flaw in your proof:

You are using this proof to answer my question of how come proofs don't answer these questions. However, according to your own proof, that proof is then an answer, and not a proof. Therefore, your proof is not a proof, because it is an answer.
 
"However, according to your own proof, that proof is then an answer, and not a proof. Therefore, your proof is not a proof, because it is an answer. "

Although had it really answered your question, you wouldn't still be talking. Ergo, not an answer. By process of elimination, it is a proof.
 

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