K-97 said
Can anyone explain why 1 + 2 + 3 + 4 .... and so on to infinity = -1/12?
Analytical continuation of the Riemann Function. Not that I can actually explain any of that, without pretty much parroting what other people said. But
this guy is pretty alright, if you have 20 minutes to spare.
TheMadAsshatter said
Given that you can make loopholes like these in math, is it possible that math is complete bullshit and the smart guys are wrong? If not, is it not true that math is flawed?
Those aren't the loopholes you are looking for, math isn't complete bullshit, but number systems do have at least one inherent flaw in them that we know of.
Godels incompleteness theorems. Essentially, for any set of axioms, there will exist some true statement expressible within them that cannot be proved by them while remaining consistent. The proof for this is along the lines of the liar paradox.(i.e: This sentence is false.)
---Dubious stuff ahead, take with grain of salt and plenty of self research. Actually, just assume everything I say is total bull. :^)---
Say you have some theory T, and some statement G that is "G cannot be proven by T". Should you try to disprove G within T, you will arrive at a contradiction. Which means "G can be proven by T", but that means that you'd have to prove it can't be proven within T, but that means you wouldn't be able to prove it, but to prove that you'd have to prove you could but then you couldn't but had to but couldn't but could but couldn't... and so on. Which means G actually is True, and unprovable.
As a result, it is impossible to prove everything in number theory, at least, not without coming to a contradiction at some point. It's still possible to prove stuff, and it is still possible to add stuff whenever you get to a problem you can't solve with the tools you have available, but never will you be able to prove everything. Even with an infinite amount of axioms, you'd still have something you couldn't prove.
This is the root of key stuff like the halting problem, and the set paradox.
--Also, apologies to Halo--