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
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.
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