Comment by nobodyandproud
Comment by nobodyandproud 5 days ago
Not like 5, but High-school Geometry.
If you remember Geometry, there are two ways to prove something:
- By making it (constructing)
- By contradiction (reductio ad absurdum)
During the late 1800s to early 1900s, when math was becoming more formalized, a group of mathematicians had issues with the second method.
From their point of view if you can’t show how to make it, then you’ve not proven that it exists.
Now it turns out that indirect proofs like contradiction requires the law of excluded middle: If something isn’t true, then it must be false (or vice versa).
It turns out that AoC is needed/implied, for the law of excluded middle; hence the objection to AoC; and enables these non-constructive proofs.
https://en.m.wikipedia.org/wiki/Law_of_excluded_middle
Another AoC proof: Prove that an irrational number to a irrational power can be rational.
sqrt(2)^sqrt(2) : If rational, then done.
Else (sqrt(2)^sqrt(2))^sqrt(2) = 2.
QED (and non-constructive).
Note that this proof doesn't require the axiom of choice, only excluded middle.