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jedishrfu

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Consider the series 1+3, 1+3+5, 1+3+5+7 and revelation that they sum to squares

4, 9, 16 ...

Noticing the pattern one can deduce a theorem that describes the behavior seen.

But it is also true that while we may see a pattern, it may not truly exist And that’s why proofs are so important.

Ramanujan was famous for discovering a great many patterns like this and in many cases they had been discovered before, in some cases they simply weren’t true and in others they were magical and never before seen.

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I think given an example in abstract advanced Math is more difficult?

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wrobel

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fortune, intuition

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jedishrfu

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I knew an artist who did beautiful works in wood where he'd find an old knarly looking block of wood and hew it with a chainsaw and then a chisel and finally polished it to a fine sheen. He said he followed the grains of the wood and they told him the pattern that he sculpted.

Basically, there is no recipe anyone can follow. One can only learn by trying to reproduce the proofs of others and from there learn how to create your own theorems.

- #7

martinbn

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What do you mean by "guess a theorem"?

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You mean create something new don't you. I studied Euclidean geometry two years ago (later in life). It wasn't hard. No, not the geometry, that was hard. I mean coming up with something novel. It's like a skater dude once said, "if you're not passionate about something, it's really hard to be good at it." That's the key I think: passion. Recently I watched a movie about Steven Jobs. He said it too: the passion keeps you going, keeps you digging, keeps you perservering.

Jacob Bronowski said, "if a man doesn't take fire in what he does, he'll never create anything new at all."

Find the fire.

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- #9

FactChecker

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If anything, this might be the easiest. Abstract math usually has many specific, concrete examples that led to the abstraction.I think given an example in abstract advanced Math is more difficult?

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The guessing mean propose a theorem

- #11

FactChecker

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All in all, we're talking about creativity; a most misterious gift. Do you feel like matching Euler?

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I humbly suggest none of us do.

- #15

trees and plants

Perhaps trying to connect the objects someone wants in the form of a theorem, then trying to prove it and if the proof leads somewhere else he makes changes to the initial statement of the conjecture?

I have not tried this method in advanced abstract math where scientific research is done but it could work i think.

I think that most of the times if not all the proof could lead to a theorem by perhaps changing the initial statement of the theorem.

So i think it could work.

- #16

trees and plants

Perhaps it can take i think most of the times from one day, one week, two weeks, a month to solve a relatively difficult conjecture.There are other more difficult problems too, which can take more time to be solved.

Think of Fermat's last theorem or Poincare's conjecture as examples. Or the not yet solved Riemann hypothesis.

- #17

Vanadium 50

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I have not tried this method in advanced abstract math where scientific research is done

@trees and plants , with all due respect, your proof skills are nowhere near that of an advanced researcher. You abandoned the "all numbers are either even or odd" proof before you were successful. Pretending you are something you are not isn't good for you or anyone else.

It's probably a good idea to learn how to do proofs yourself before giving advice to others.

- #18

trees and plants

That statement i think is not covered in the material given at my math department. I learned a little about the successor and Peano axioms though in the past. I do not think that means i can not do proofs in other areas of math or other math or physics topics.@trees and plants , with all due respect, your proof skills are nowhere near that of an advanced researcher. You abandoned the "all numbers are either even or odd" proof before you were successful. Pretending you are something you are not isn't good for you or anyone else.

It's probably a good idea to learn how to do proofs yourself before giving advice to others.

How can someone improve his proving abilities?By learning proofs, theorems, definitions and trying to solve problems?Which problems should he choose to solve and if he gets stuck what should he do? Thank you.

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Vanadium 50

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trees and plants

I have not done any scientific research in advanced abstract math yet. A professor in my department i think said that proofs and theorems are produced at the same time usually. So that is mostly how i concluded the rest of what i said about making conjectures and then making changes to the statement of the conjecture according to the proof someone has made. I think this is logical. Why would it not be?

- #21

Mark44

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A theorem and a proof cannot be created at the same time. The statement of the theorem comes first, and then a proof of that statement comes later.I have not done any scientific research in advanced abstract math yet. A professor in my department i think said that proofs and theorems are produced at the same time usually. So that is mostly how i concluded the rest of what i said about making conjectures and then making changes to the statement of the conjecture according to the proof someone has made. I think this is logical. Why would it not be?

I don't see that your approach of making a conjecture, and then adjusting the conjecture to match the proof makes any sense. Perhaps you can give an example of whether this might work...

I also agree with @Vanadium 50's point that your inability to complete a very simple proof doesn't give you much credibility to advise anyone on how to create a theorem.

- #22

Vanadium 50

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I do not think that means i can not do proofs in other areas of math or other math or physics topics.

It kind of does. If you can't do an elementary proof, you have little hope of doing an advanced proof. And if you require that someone else show you how to do a proof before you can prove it, you're not proving anything. Yo are merely repeating.

I am also with @Mark44. I don't see that your approach of making a conjecture, and then adjusting the conjecture to match the proof makes any sense. If you have an example, that would be clearer.

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If a proof attempt yields some kind of truth, it's progress already. But is what's revealed somehow useful to us? Said revelation could lead to other conjectures, too. Don't get stuck on just proving one or two statements. You might also want to try and generate counter-examples.Perhaps trying to connect the objects someone wants in the form of a theorem, then trying to prove it and if the proof leads somewhere else he makes changes to the initial statement of the conjecture?

I'm not sure why the specification 'scientific' is present, but I assure you, research is done in many disciplines, not just math. As for whether it can work: yes, it Can. But will it work? Who knows..I have not tried this method in advanced abstract math where scientific research is done but it could work i think.

Well, sort of.. we don't just arbitrarily juggle with statements and see if something works out, though, that can be a part of the process, but we do have educated guesses about what might work. And it's not only about whether it works. A given conjecture often is somehow connected to the theory we're studying, a missing piece that makes something else work out. It's quite involved ..I think that most of the times if not all the proof could lead to a theorem by perhaps changing the initial statement of the theorem.

- #24

trees and plants

So, what i said perhaps works?Is this used in scientific research or generally research?If a proof attempt yields some kind of truth, it's progress already. But is what's revealed somehow useful to us? Said revelation could lead to other conjectures, too. Don't get stuck on just proving one or two statements. You might also want to try and generate counter-examples.

I'm not sure why the specification 'scientific' is present, but I assure you, research is done in many disciplines, not just math. As for whether it can work: yes, it Can. But will it work? Who knows..

Well, sort of.. we don't just arbitrarily juggle with statements and see if something works out, though, that can be a part of the process, but we do have educated guesses about what might work. And it's not only about whether it works. A given conjecture often is somehow connected to the theory we're studying, a missing piece that makes something else work out. It's quite involved ..

- #25

trees and plants

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