Well of course we made it up. No one handed us any stone tablets with 0^0 on them.
Mathematicians are always making definitions, and working out which of them should be kept and which should be discarded. We keep the definitions that make the most sense, that make our lives the easiest, that make theorems easy to state, that give math a sense of being natural. Indeed, in the early days of algebraic geometry there were big debates over which definitions to adopt.
It is like deciding on a convention when you design a new programming language. In this case, experience has shown that it is pretty much always better to say that 0^0 is 1 and not 0. Among other reasons, there is exactly one map from the empty set to the empty set.
But if you say 0^0 = 0, you don't get math blowing up in some big contradiction. You just get a little more kludge here and there, a few extra special cases of lemmas that have to be spelled out in more detail. Nothing too awful.
Mathematicians are always making definitions, and working out which of them should be kept and which should be discarded. We keep the definitions that make the most sense, that make our lives the easiest, that make theorems easy to state, that give math a sense of being natural. Indeed, in the early days of algebraic geometry there were big debates over which definitions to adopt.
It is like deciding on a convention when you design a new programming language. In this case, experience has shown that it is pretty much always better to say that 0^0 is 1 and not 0. Among other reasons, there is exactly one map from the empty set to the empty set.
But if you say 0^0 = 0, you don't get math blowing up in some big contradiction. You just get a little more kludge here and there, a few extra special cases of lemmas that have to be spelled out in more detail. Nothing too awful.