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fxdung
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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?
fxdung said:In Elementary Geometry we can use drawing figure to guess the geometry theorem.How can we guess a theorem in Math in general?
If anything, this might be the easiest. Abstract math usually has many specific, concrete examples that led to the abstraction.fxdung said:I think given an example in abstract advanced Math is more difficult?
trees and plants said:I have not tried this method in advanced abstract math where scientific research is done
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.Vanadium 50 said:@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.
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?Vanadium 50 said: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".
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.trees and plants said: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?
trees and plants said:I do not think that means i can not do proofs in other areas of math or other math or physics topics.
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.trees and plants said: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.
So, what i said perhaps works?Is this used in scientific research or generally research?nuuskur said: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 ..
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.trees and plants said:So, what i said perhaps works?Is this used in scientific research or generally research?
And in mathematics, where students are learning how to construct logical arguments.trees and plants said:Some people want to prove already expressed conjectures or already expressed problems or questions, i think this is ok in the sciences.
This is foolish if one doesn't understand how to construct a proof of a simple statement.trees and plants said:What happens if we want to make the conjecture and lead our own ways and make our own directions and prove them?
Amen to that.nuuskur said: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.
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.nuuskur said:Keep in mind that it's fine if you don't understand. But instead of big leaps, try small steps first :)
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.trees and plants said: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.
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).trees and plants said: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 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 said: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.
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.trees and plants said: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?
Analyzing the proofs of theorems allows us to gain a deeper understanding of the underlying concepts and principles that lead to the theorem. It also helps us to identify any potential flaws or errors in the proof.
Yes, analyzing proofs can lead to the discovery of new theorems. By examining the steps and techniques used in a proof, we may be able to generalize them and apply them to other problems, leading to the discovery of new theorems.
Analyzing proofs is an essential part of the process of proving and verifying mathematical theorems. By understanding the logic and techniques used in proofs, we can build upon existing knowledge and make new discoveries, contributing to the advancement of mathematics.
No, it is not necessary to analyze proofs in order to understand a theorem. However, analyzing proofs can provide a deeper understanding and appreciation for the theorem and its implications.
While analyzing proofs can lead to the discovery of new theorems, it is not the only method for doing so. Some theorems may require new techniques or approaches that cannot be derived solely from analyzing existing proofs. Additionally, not all theorems may have proofs that are easily analyzed or understood.