[RexyRexFE]

Hey guys, sorry for the long read, just thought you'd enjoy this.

[Brian]

Good job with this paper!

[AATW]

The trouble with your proof that 2=1 is you haven't actually broken maths, you've made a mistake.
In one step you've divided by (a-b). But your initial assertion is that a=b. If a=b then a-b = 0. So you've divided by 0.
And you can't divide by 0 in maths, the answer is actually undefined. There's a very good explanation why here:



So yeah, any proof where you're dividing by 0 is breaking the rules of maths and thus invalid.

[cravingTD]

"Excluded Middle...proves that either one thing, or the other, is true." 

What you were suppose to say is the law of the excluded middle states that a logical statement is either true or its negation is true. It defiantly doesn't mean you can "prove anything you want" to be true.

Taking your example and applying LEM. "The sky is green or the sky is purple" is true or its negation is true, that is "The sky is not green and the sky is not purple".  The first statement is false so the former must be true by LEM.

https://en.wikipedia.org/wiki/Law_of_excluded_middle
https://en.wikipedia.org/wiki/De_Morgan%27s_laws