– what does it mean for a theory to be “proven” mathematically?
In: Mathematics
We can postulate many things about math, without being able to prove them. We make observations of the world, and note certain patterns and we might ask “Does this pattern occur in all cases?”. We can assume it does, but until we actually have a way to prove it holds, its merely a theory.
A classic example of a mathematical proof goes as follows. Given a number x in the set of all integers (meaning given a number without a decimal), if that number is even than x times itself will also be even. This makes sense, you can test it with a few numbers, but we want to prove its always true. Thats a bit harder, but we can do it.
We can start from somewhere we know, how we define an even number. Its a number that can be divided by 2. So we say that x = 2n where n is any number. Lets square both sides and see what happens.
`Square both sides`
`x^2 = (2n)^2.`
`x^2 = 4n^2`
`rewriting to be of the form x=2n`
`x^2 = 2(2n^2)`
Look at that, in our last step we see that our right side is of the from 2 * some n. This is the same as our definition of even number! We have used whats called a direct proof to show that an even number, times itself is also even. There are other ways, but this is one of the easiest examples I could think of.
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