Can we discover new theorems by analyzing their proofs?

[fxdung]

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

 

[jedishrfu]

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.

 

[fxdung]

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

 

[fxdung]

It is easy to give a theorem if we can use "induction" reasons.But what guess us about the theorem in genereal case?

 

[wrobel]

fortune, intuition

 

[jedishrfu]

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.

 

[martinbn]

What do you mean by "guess a theorem"?

 

[aheight]

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.

 

[FactChecker]

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.

 

[fxdung]

The guessing mean propose a theorem

 

[FactChecker]

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.

 

[nuuskur]

Some phenomena occur across 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.

 

[Gordianus]

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

 

[nuuskur]

I humbly suggest none of us do.

 

[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.

 

[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.

 

[Vanadium 50]

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.

 

[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.

 

[Vanadium 50]

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".

 

[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?

 

[Mark44]

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.

 

[Vanadium 50]

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.

 

[nuuskur]

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 ..

 

[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?

 

[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?

 

[nuuskur]

So, what i said perhaps works?Is this used in scientific research or generally research?

It might work, I don't know. I'm not aware of any set methods to conduct research. Nobody tells you "this is how you conjecture or prove theorems". There are some general guidelines and, perhaps, pointers passed from supervisor to student. We learn as we do, I suppose.

You're kind of in over your head. You try to insert yourself into a world you know very little about and expect to understand or worse, expect someone else to explain it to you in a few passages of text.

Keep in mind that it's fine if you don't understand. But instead of big leaps, try small steps first :)

 

[Mark44]

Some people want to prove already expressed conjectures or already expressed problems or questions, i think this is ok in the sciences.

And in mathematics, where students are learning how to construct logical arguments.

What happens if we want to make the conjecture and lead our own ways and make our own directions and prove them?

This is foolish if one doesn't understand how to construct a proof of a simple statement.

You're kind of in over your head. You try to insert yourself into a world you know very little about and expect to understand or worse, expect someone else to explain it to you in a few passages of text.

Amen to that.

Keep in mind that it's fine if you don't understand. But instead of big leaps, try small steps first :)

You (trees and plants/infinitely small/universe function) really should go for the small steps first. In previous posts you have mentioned that you have failed a number of math classes, and have a difficult time doing assigned homework problems or problems on tests. That should be a signal to you that you need to be focusing on the work in your classes rather than trying to come up with whole new areas of mathematics.

From a thread you posted in back in November:

I am an undergraduate math student at university.Unfortunately I have delayed my graduation and stayed some more years at the math school.I feel bad about it.I have problems with solving exercises in math and physics.I memorise things theorems proofs and other things this is not my problem I think.Solving exercises is the problem.I know some classical mechanics from an introductory book to this topic, I try to learn on my own other math and physics from textbooks or articles and journals on my free time.

A major problem I see is that you are relying too much on memorization of theorems and proofs, but with not enough emphasis on working problems. My advice again is that you spend much less time on "other math and physics" and much more time on the classes you are actually taking.

 

[S.G. Janssens]

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.

I largely agree with what others have said (my summary: learn the basics and get them right through lots of exercising before pondering about all kinds of meta-questions).

With that understood, I actually do think that there is some sense in what you write in the quote above, but mostly at a research level or when answering open-ended questions. (Maybe the sort of question where you are asked to study a pattern, formulate a conjecture, and prove it.)

Namely, when doing research, one usually does not know the exact formulation of the theorem one is trying to prove. There is an idea of what the theorem should look like. For a nice example, look up "Utopian Theorem" in:

J.C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, CUP, 2001.

Usually this initial idea is, strictly speaking, false, as is discovered in the process of proof, after which the theorem is adjusted (weakened, usually, sometimes completely changed) until a proof becomes possible.

Of course, knowing when and how to adjust, so as to obtain sharp results, requires understanding the basics and practical experience doing actual textbook proofs yourself.

 

[trees and plants]

Of course, in order to know when and how to adjust, so as to obtain sharp results, requires understanding the basics and practical experience doing actual textbook proofs yourself.

I think a mistake i made so far is that i read the proofs without trying to see how the person who did the proof thought to do the proof. Proofs in my textbooks or pdfs have their order of presenting themselves but i think someone needs most of the times if not all to find how the proof should start, continue and end according to the thought process needed for the theorem. Is this correct?

 

[S.G. Janssens]

I think a mistake i made so far is that i read the proofs without trying to see how the person who did the proof thought to do the proof. Proofs in my textbooks or pdfs have their order of presenting themselves but i think someone needs most of the times if not all to find how the proof should start, continue and end according to the thought process needed for the theorem. Is this correct?

If you mean that getting the idea of the proof is often the most time-consuming part, then I think you are correct. It is not always true, however: Sometimes it is not very difficult to understand what must be the broad lines of a proof, while it is much more difficult to overcome certain technical details.

In any case, I don't think you are making a mistake, as you write in the above quote. There are multiple levels on which to understand a proof. Upon a first reading, it is not always necessary to understand how the writer got to the idea of the proof. As long as you can understand the steps and how they lead from premise to conclusion, you are making progress.

To test this, make a lot of exercises, solve a lot of problems from the same book. If the book is of good quality and you show enough patience and perseverance, then the exercises will be just difficult enough to test your understanding, without making it impossible to proceed.

 

[nucl34rgg]

"Necessity is the mother of invention"

Most theorems are invented to be used to prove something so that ultimately, people can use it for something else (maybe more proofs or physics/engineering).

 

[mathwonk]

One of my professors suggested analyzing the proof of a theorem you have learned, and noticing whether it really uses all the hypotheses it has. If not, you can prove a more general theorem by omitting that unnecessary hypothesis. Sometimes your theorem will prove less, but still be new.

Here is an example: George Kempf proved that when one considers the birational map from the symmetric power of a curve to the theta divisor in its jacobian, the tangent map is a birational surjection from the normal bundle of the fiber, to the tangent cone of the theta divisor.

https://www.jstor.org/stable/1970910?origin=crossref&seq=1

My colleague and I noticed that in the case of a Prym variety, where one again has a map from a divisor variety to the theta divisor of the Prym variety, even though it is not birational, the tangent map on the normal bundle is sometimes still a surjection onto the tangent cone of the theta divisor. This allows one to understand the tangent cone of the theta divisor almost as well as in the birational case. The essential similarity was the fact that in both examples, Jacobian and Prym, the fibers of the original parametrizing map were smooth subvarieties, i.e. the derivative of the parametrizing map vanished only along tangents to the fiber. Using that in the case where the Prym divisor variety was also smooth, which always holds for Jacobians, gave the new result, by essentially the same proof as the old one. I.e. the key point in finding a new theorem, was noticing that the key property was not one of the overtly mentioned hypotheses, but one of the facts hidden in the proof. In the words of my professor, "try to find a theorem whose proof proves more than it claims to."

https://www.math.uga.edu/sites/default/files/inline-files/sv2rst.pdf

It is not at all easy to find interesting new theorems however by this or any other strategy. This example is several decades old, and even with the collaboration of brilliant colleagues, I contributed to producing less than one paper a year during my career. Kempf's beautiful theorem itself built on insights he unearthed in a wonderful paper of Andreotti and Mayer, and others shown him by David Mumford.

That history is discussed in the appendix to this paper:
https://www.math.uga.edu/sites/default/files/inline-files/onparam.pdfanother idea that relates to your post #29, is to read the statement of a theorem in your book and then close the book and try to prove it yourself. This forces you to think of the idea for the proof, or at least how to begin. This exercise is usually not completely successful, but even if you only get the very faintest first idea, you already have jumped the hurdle of finding out how to begin. And if you in fact prove it yourself, often by a somewhat different proof, you may have already proved a new result, maybe a slightly stronger version. I read this advice in a famous article of Zariski, a pioneer of american algebraic geometry.