I did not in fact mean that 2 is defined as 1+1; you (and several others) are correct in saying that 1+1=2 requires proof for all reasonable definitions of +. 0

I did not in fact mean that 2 is defined as 1+1; you (and several others) are correct in saying that 1+1=2 requires proof for all reasonable definitions of +. 0^0 = 1 follows much more directly from the definitions in which it is true. My overall point has been articulated by arketyp in a sibling post to tomp's; like ^, + is "just a definition" which we arranged so as to line up with our intuitions. I disagree with your assertion that 1+1=2 and 0^0=1 are of completely diferent kinds. That depends on what intuitive angle you approach them from; my intuition for ^ is that it is the iteration of multiplication, and in this context the only sensible definition is that x^0 = 1 (the unit of multiplication), for any x. The generalization to continuous bases and exponents should preserve this property. But this is certainly a matter of taste, and certainly you'll find more disagreement here than in the matter of 1+1=2. What this tells us about human nature, I think, is not much: humans will disagree more when things get more complicated (^ is more complicated/abstract than +) and there is more room for disagreement.