How can we create a theorem?

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In Elementary Geometry we can use drawing figure to guess the geometry theorem.How can we guess a theorem in Math in general?
 

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  • #2
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Through examples that imply the theorem.

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.
 
  • #3
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I think given an example in abstract advanced Math is more difficult?
 
  • #4
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It is easy to give a theorem if we can use "induction" reasons.But what guess us about the theorem in genereal case?
 
  • #6
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There is a lot of insight that one gains when you research a subject where you might see a pattern and then try to express it as a theorem. In some ways, you are asking how does one compose music , sculpt or paint.

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"?
 
  • #8
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In Elementary Geometry we can use drawing figure to guess the geometry theorem.How can we guess a theorem in Math in general?

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
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I think given an example in abstract advanced Math is more difficult?
If anything, this might be the easiest. Abstract math usually has many specific, concrete examples that led to the abstraction.
 
  • #10
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The guessing mean propose a theorem
 
  • #11
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I would think that one likely reason to propose a theorem is that you are working on something and a long calculation suddenly reduces to a very clean and simple answer. You would think that there is something fundamental going on that made the final answer so simple. Another reason would be if you are working on something, worried about a possible complication, and suddenly realize that the complication can not happen in that case. That might lead you to propose a theorem.
 
  • #12
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Some phenomena occur accross many examples and that leads to conjecture and hopefully..proof. Other times, there is intuition involved. Having worked on similar problems for a long time, you kinda know (but might not) what you are looking for. In short, I don't know of any theorem-generating algorithm, myself.
 
  • #13
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All in all, we're talking about creativity; a most misterious gift. Do you feel like matching Euler?
 
  • #14
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I humbly suggest none of us do.
 
  • #15
trees and plants
This is my personal opinion.

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.
 
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  • #16
trees and plants
So, the question is how to do the proof? I think this needs for the person doing the proof to try problem solving and knowing theorems,proofs from others, definitions at least and try to make the proof.

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
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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
@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.
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.

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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  • #19
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I'm not saying an inability to do elementary proofs makes you a bad person. I am saying that it doesn't put you in a good position to give other people advice on how to do more advanced proofs, and that you probably shouldn't imply you are doing "scientific research" in "advanced abstract math".
 
  • #20
trees and plants
I'm not saying an inability to do elementary proofs makes you a bad person. I am saying that it doesn't put you in a good position to give other people advice on how to do more advanced proofs, and that you probably shouldn't imply you are doing "scientific research" in "advanced abstract math".
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
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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?
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 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.
 
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  • #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.
 
  • #23
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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?
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 have not tried this method in advanced abstract math where scientific research is done but it could work i think.
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 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.
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 ..
 
  • #24
trees and plants
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 ..
So, what i said perhaps works?Is this used in scientific research or generally research?
 
  • #25
trees and plants
Some people want to prove already expressed conjectures or already expressed problems or questions, i think this is ok in the sciences. What happens if we want to make the conjecture and lead our own ways and make our own directions and prove them? How often do scientists do this?
 

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