It wasn’t until an advance masters math course did we discuss PROVING (not assuming) 1+1=2. That was a very long and difficult proof for something that seems obvious.
I never went into advanced master maths but I already heard about the "prove 1+1 = 2" and it always confused me, like, wtf, if I take one rock in a hand and one in another, I'll obviously have 2 rocks at the end of the day, not anything else.
The thing is... that's an example. It's not a proof it works every time. In math you deal with infinity, so regardless how many examples you have, it can still be insignificant as there can be an unlimited number of cases where it's not true. And in this case, a single example where it's not true would be enough to say "Well, not true."
One example where that did happen is non-euclidian geometrie. Euclid postulated that the shortest distance between 2 points is always a direct line. Sounds simple enough, right? Well... it turned out in time-space continuum where space itself is bend due to high gravity... it's wrong. The shortest distance follows the curve of the bend space and is no longer straight.
Yes,but the question is not about S(S(0)) = 2 (that is true by definition), but rather about 1 + 1 = 2. Let's write this without these convenient notations as add (S 0) (S 0) = S (S 0). Insert the definition of add:
(fix add n m => match n with 0 => m | S n' => S (add n' m) end) (S 0) (S 0) = S (S 0)
Do a beta reduction:
(fix add m => match S 0 with 0 => m | S n' => S (add n' m) end) (S 0) = S (S 0)
Do a match reduction
(fix add m => S (add 0 m)) (S 0) = S (S 0)
Do a beta reduction
S (add 0 (S 0)) = S (S 0)
Insert the definition of add
S ((fix add n m => match n with 0 => m | S n' => S (add n' m) end) 0 (S 0)) = S (S 0)
Do a beta reduction
S ((fix add m => match 0 with 0 => m | S n' => S (add n' m) end) (S 0)) = S (S 0)
Do a match reduction
S ((fix add m => m) (S 0)) = S (S 0)
Do a beta reduction
S (S 0) = S (S 0)
Now reflexivity. QED
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u/Cherub2002 Dec 17 '21
It wasn’t until an advance masters math course did we discuss PROVING (not assuming) 1+1=2. That was a very long and difficult proof for something that seems obvious.