Proving Math Theorems: Standard Procedure & Fermat's Last

[JizzaDaMan]

What is the standard procedure for proving or disproving a mathematical theorem? for example Fermat's last theorem?

 

[HallsofIvy]

Stare at a blank sheet of paper until blood comes out your forehead!


Seriously, there is no "one" way to prove theorems. The most basic concept, which you will typically see in secondary school geometry, is to find a series of "bridges" from the hypotheses to the conclusion. You are given hypothesis "A" and you want to prove conclusion "Z". So you look for definitions, axioms, theorems, etc. that will lead to "islands" between A and Z. That is, you look for some statement, B, such that you know some definition, axiom, or theorem, etc., that will let you go from A to B and you hope you will be able to go from B to Z. And unless it the theorem is very simple, you look for some statement, C, so that there is a definition, axiom, or theorem, etc. that will let you go from B to C and, hopefully, then you can go from C to Z.

How do you know, ahead of time, that C will let you get to Z? You don't. You try it and see what happens. If you know a fair amount of information about "C" and "Z", you might recognize that "C" is, in some sense, "closer" to "Z" than was "B" and so there is a good chance you can go from "C" to "Z", but there are no guarentees. If you hit a road block, you go back and try something else.

(That is one reason why mentors here get so frustrated with people who say the 'can't do a problem' or 'don't know where to start'. You seldom know that a method you are using will solve the problem. Just try. If it doesn't work, try something else.)

But, as I said, that is just one way, the most basic, of proving things. Depending on the situation, 'proof by induction' or 'indirect proof' might be more useful. Basically, you think about everything you know related to the problem, including the precise words of definitions, theorems, etc. and try to apply them to find relationships between parts of the problem.

 

[JizzaDaMan]

Thanks for the reply :) i always wondered how so many of these theorems are proven because the proofs just seem so random, and it appears that they are just that :D

 

[chiro]

Thanks for the reply :) i always wondered how so many of these theorems are proven because the proofs just seem so random, and it appears that they are just that :D

One thing about a lot of mathematics is that all other people see is the final piece of work and not all those scraps of paper that are filling up the bins in every single room with scribbles all over them and all kinds of attempts that capture the history and the context of the proof.

 

[HallsofIvy]

Thanks for the reply :) i always wondered how so many of these theorems are proven because the proofs just seem so random, and it appears that they are just that :D

Frankly, that's a lot like asking, "When you are writing a book, how do you know what words to use?"

 

[arildno]

Thanks for the reply :) i always wondered how so many of these theorems are proven because the proofs just seem so random, and it appears that they are just that :D

Maths is an art, not a train to steer along a pre-determined track.
Do you think an artist knows if his next painting will be good enough?

But, even so, talent and exercise are critically important success factors, both for the mathematician and the artist.

 

[uperkurk]

(That is one reason why mentors here get so frustrated with people who say the 'can't do a problem' or 'don't know where to start'. You seldom know that a method you are using will solve the problem. Just try. If it doesn't work, try something else.)

I'm sorry I have to disagree, I myself am terrible at maths and have only just started learning algebra, but if you don't know how to work out an equation, then you can never know what the correct answer is.

I had an algebra question earlier and I did it two ways, I was not sure which way was correct and I ended up with -4 for the first way, and 8 for the second way.

Turns out the correct answer was actually -8 so I had it completely wrong. Sometimes you just need to be given a little push in the right direction or someone to show you WHY you're going wrong.

 

[HallsofIvy]

I'm sorry I have to disagree, I myself am terrible at maths and have only just started learning algebra, but if you don't know how to work out an equation, then you can never know what the correct answer is.

I had an algebra question earlier and I did it two ways, I was not sure which way was correct and I ended up with -4 for the first way, and 8 for the second way.

Turns out the correct answer was actually -8 so I had it completely wrong. Sometimes you just need to be given a little push in the right direction or someone to show you WHY you're going wrong.

What are you disagreeing with? Having said that you have little experience with mathematics, why do you feel that you can disagree about how you do mathematics?

"If you don't know how to work out an equation, then you can never know what the correct answer is" is certainly not true! I don't know how to solve $x^6- 3x^5+ 5x^3- 15x+ 20=0$ but I certainly can evaluate that for any given number and determine whether it is a solution or not. It is, typically, much easier to show that something is a solution, or is not, to a problem than it is to solve the problem initially.

You are certainly welcome to say that you do not know how to do something. But it doesn't follow that others, in particular those who have studied that particular topic, must not know either.

 

[uperkurk]

Obviously my maths skills are nowhere near as advanced as others on this board. What I mean by never know the answer is you may try working out the answer multiple ways and coming up with multiple answers, all of which are different. Ok let me use the example that you posted.

I am nowhere near skilled to answer this and I don't even know what type of algebra equation this is but nevertheless I will try to solve it.

I won't bother to write the steps but here is the furthest I can get and whether or not this is even the correct way to work out such problem I have no idea but...

$$15x-x^2+20=0$$

Now I could sit and shuffle these numbers around all day and because I don't know what the answer should be, I will never know if I have solved it or not, do you see what I mean?

 

[pwsnafu]

Obviously my maths skills are nowhere near as advanced as others on this board. What I mean by never know the answer is you may try working out the answer multiple ways and coming up with multiple answers, all of which are different.

In that case there really is only two possibilities:

1. Each answer is a partial solution and you need the information from all of them, or
2. You are careless.

I think you are number 2.

Ok let me use the example that you posted.

I am nowhere near skilled to answer this and I don't even know what type of algebra equation this is but nevertheless I will try to solve it.

I won't bother to write the steps

Very bad idea. You should always write all your working. The reader can't read your mind.

but here is the furthest I can get and whether or not this is even the correct way to work out such problem I have no idea but...

$$15x-x^2+20=0$$

How did you get that? Bad news: something went very wrong.

Now I could sit and shuffle these numbers around all day and because I don't know what the answer should be, I will never know if I have solved it or not, do you see what I mean?

Part of doing mathematics is understanding how to study the properties of the solutions without knowing what those solutions are! You seem to be just manipulating symbols without thinking about what you are doing.

 

[uperkurk]

I thought I was simplifying :P in all fairness 3 days into teaching myself algebra I think I'm allowed to go very wrong on such an equation. Come to think of it is that a quadratic equation?

 

[chiro]

One thing about mathematics for the new students is that the whole thing should be self-consistent.

If you get a solution, plug it back into the constraints you had before: if they match up then this is a good sign and if they don't then go back and look at is up.

Using the system as a way to cross-check your solutions since everything should simply re-inforce the constraints you already start off with: if they don't then see what is going on and address it.

It doesn't matter if you are finding the roots to a quadratic, doing a hypothesis test, or doing a proof, all of these things are similar to many different formulations of the same thing and if you use those similar things to check your work, then not only can you double check your results, but also get a feel for whether something "seems" right or seems "way off".

 

[Millennial]

Obviously my maths skills are nowhere near as advanced as others on this board. What I mean by never know the answer is you may try working out the answer multiple ways and coming up with multiple answers, all of which are different. Ok let me use the example that you posted.

I am nowhere near skilled to answer this and I don't even know what type of algebra equation this is but nevertheless I will try to solve it.

I won't bother to write the steps but here is the furthest I can get and whether or not this is even the correct way to work out such problem I have no idea but...

$$15x-x^2+20=0$$

Now I could sit and shuffle these numbers around all day and because I don't know what the answer should be, I will never know if I have solved it or not, do you see what I mean?

As HallsOfIvy said, nearly %99 of the proofs we have in mathematics use the principle of "Start from something you know and see what happens!" There are many conclusions derived just using that, for example Riemann's functional equation, the quadratic formula and many more.

For your problem, you can use the quadratic formula that I just mentioned. It is derived by purely playing around with the equation you gave and using some tricks in mathematics (like completing the square.)

As long as your way of working is correct and has no errors, any method should do fine. The Basel problem, for example, can be tackled in so many ways that I'd bet my money on nobody knows them all. You can start from a square integral, a Fourier transform and an infinite product and end up with the same result. It doesn't matter where you start from as long as it is correct. I can't prove anything by starting from 0=1, because it is simply not correct.

 

[micromass]

I can't prove anything by starting from 0=1

Quite the contrary: you can prove everything by starting from 0=1! But all those things are wrong since you started from a false statement.

Sorry, I had to make a comment about that

 

[HallsofIvy]

Obviously my maths skills are nowhere near as advanced as others on this board. What I mean by never know the answer is you may try working out the answer multiple ways and coming up with multiple answers, all of which are different. Ok let me use the example that you posted.

I am nowhere near skilled to answer this and I don't even know what type of algebra equation this is but nevertheless I will try to solve it.

I won't bother to write the steps but here is the furthest I can get and whether or not this is even the correct way to work out such problem I have no idea but...

$$15x-x^2+20=0$$

Now I could sit and shuffle these numbers around all day and because I don't know what the answer should be, I will never know if I have solved it or not, do you see what I mean?

Without knowing what kind of mathematics you have taken, it's impossible to respond. Have you actually taken a course that involved quadratic equations? If so you could think about trying to factor and, if you can't try "completing the square" or the "quadratic formula". But certainly, if you think some number is a solution, you can replace x by that number and then it is just simple arithmetic to check it.

 

[uperkurk]

I have been teaching myself algebra a couple of hours a night for the past week. I have no studied quadratics yet I have only done things like simplifying, expanding, factorial and stuff like that. I can do basic problem solving like where I get given an equation and I have to find what x is but as far as quadratics go I've merely heard of the word and never learned it yet.

 

[Millennial]

Quite the contrary: you can prove everything by starting from 0=1! But all those things are wrong since you started from a false statement.

Sorry, I had to make a comment about that

Yes, but I suppose you understood what I meant there. 0=1 pretty much implies everything possible in mathematics, because if 0=1, practically everything is equal to everything. The reason I can't prove anything is that it is false. Your original assumption must be true if your proofs based on it will always produce valid results (on the other hand, it isn't necessary for the assumption to be true if it only produced a valid result once or twice. For example, 0=1 implies 1=1, which is true!). I just classify it as "not being able to prove" if your original assumption is false.