Math is fascinating to me, though I struggled with math in high school and only took the minimum I needed. (Age changes things, man.) I’m reading a book on Wiles’ proof for Fermat’s Last Theorem and got curious about proofs.
At what point does something move from an assumption with examples (well yeah. Look at this) to a full proof?
Simple example that came to mind:
For any number n, where n is a prime >2, the sum of the factors of n cannot be odd.
In: Mathematics
Essentially, mathematical proofs are all kind of the same– you take what you know, and you use that information to get to a new conclusion that must be true based on what you know.
As an example, let’s prove a statement– “An even number plus an even number is an even number”
How do we prove this? Well, what do we know? We know that an even number is divisible by 2, since that’s the definition of an even number. In other words, we can think of any even number as “2 times another number.”
So let’s say that I have an even number x, and an even number y. In this case, we can say that x=2n, and that y=2m. In this case, by definition,
x * y = 2n * 2m = 4 * n * m = 2 * (2 * n * m). And since 2, n, and m are all numbers, then 2 * n * m must also be a number. And look, we’re back at the definition of an even number.
This is a brief example of how we can use a list of assumptions and create a new statement using those assumptions.
Note however, that in your post you say
>At what point does something move from an assumption with examples (well yeah. Look at this) to a full proof?
*I did not start this proof by making assumptions.* I began this proof by making statements of fact and definition, and all I did was provide examples to work with in order to prove my statement.
You could also do what is called a “proof by contradiction.” In a proof by contradiction, you assume the opposite of what you’re looking for, and then you prove that by making that assumption, you lead yourself to a logical fallacy.
For example:
Assume that x and y are even numbers, and that x * y is odd. In this case, x * y must be able to be written as 2(n+1), the definition of an odd number. Note that n cannot be odd. If n were odd, then n+1 would be even, and since 2 is even, 2 * (n+1) would have to be even (we just proved this), but this is impossible because our assumption is that 2(n+1) is odd. Thus, n must be even, and n+1 must be odd. But if n+1 is odd, then it isn’t divisible by 2. Thus, the only way to factor n+1 and 2 into x and y, is for x= 2, and y= n+1. But then y has to be odd, which violates an initial assumption.
This proof takes an assertion– 2 even numbers multiplied together can be odd– and proves that it’s impossible by going through the logical conclusion of what must be the case.
These are just two quick methods that can be used to prove a statement. There are other methods too, like induction, that I didn’t get into, but I hope that you can see the use of such techniques.
Happy to answer any questions.
Proofs basically work this way:
1. Say something you know is true.
2. Then, say the same thing, but slightly differently. It must still be true.
3. So, whatever you come towards in the end must also be true.
Let’s see an example. I will postulate that x + (-x) < 1.
Let’s start by saying something we know to be true:
1. 1 – 1 = 0
2. Let’s multiply by x:
3. 1(x) – 1(x) = 0
4. Now, we also know that 0 < 1. So if x < 0, x < 1!
5. So, 1 – 1 < 1.
6. But, then, also 1(x) – 1(x) < 1.
Really ,we never actually said anything new at any point. We stated something we knew was true, and kept reformulating it until we got what we wanted.
I make a statement. I say something like “2+2=4”
The proof then breaks down the definition of every single part of that statement. “Two is a number, countint up, 0, 1, 2. It is two. We have to agree on that to continue.” Okay everyone agrees that 2 is an integer number. “+ means addition. Addition is when you sum two values. Summing is taking the values the result is their combined value in such a way that it is their sum. We must agree on what + means to continue.” “= means that the two sides of the equation are of equal value. We must agree on this to continue.”
Basically proofs break down every single part of the theorem, and reduce them into the basics (or frequently other proofs) that other people generally agree to be true. This way it shows that every aspect of the theorem has been thoroughly thought through and more importantly, provided all the building blocks of the proof are true, then the theorem itself must be true.
Proofs aren’t really about examples and assumptions. We might start a proof with an assumption, but that’s not the proof – that’s the statement we’re trying to prove. And if we give an example, that’s not a proof either, until we can show that it’s true for every example you can imagine. If I show you a hundred examples of something and say “See? That’s proof that it’s true,” all I’ve proved it that it’s true for those hundred examples. You only need to show me *one* example where it’s not true to show I’m wrong.
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