By Art Duval, Contributing Editor, University of Texas at El Paso
How is \(0^0\) defined? On one hand, we say \(x^0 = 1\) for all positive \(x\); on the other hand, we say \(0^y = 0\) for all positive \(y\). The French language has the Académie française to decide its arcane details. There is no equivalent for mathematics, so there is no one deciding once and for all what \(0^0\) equals, or if it even equals anything at all. But that doesn’t matter. While some definitions are so well-established (e.g., “polynomial”, “circle”, “prime number”, etc.) that altering them only causes confusion, in many situations we can define terms as we please, as long as we are clear and consistent.
Don’t get me wrong; the notion of mathematics as proceeding in a never-ending sequence of “definition-theorem-proof” is essential to our understanding of it, and to its rigorous foundations. My mathematical experience has trained me to ask, “What are the definitions?” before answering questions in (and sometimes out of) mathematics. Yet, while we tell students that the definition needs to come before the proof of the theorem, what students apparently hear is that the definition needs to come before the idea, as opposed to the definition coming from the idea.