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bigjeff5 comments on How to Convince Me That 2 + 2 = 3 - Less Wrong
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There is an example that often floats around, where it is 'proven' mathematically that 1=2 (or some other such equality, by the principle of explosion it doesn't really matter). The trick is that at some step in the proof, a non obvious division by zero occurred.
I imagine it's the same proof that makes 2+2=5. There is a point in the proof where the correct result is obviously 0=0 (though never explicitly written), yet it continues as though it didn't happen.
It's an example of making the problems so complex that you make a mistake, it's not a valid proof.
The proof for 2+2=4, incidentally, is almost 400 pages long. The simplistic versions most use take for granted many things for granted (like + and = and 2) that the actual proof does not.