Quote:
Originally Posted by wblipp
Look up the integral definition of gamma. Something like this is necessary because Gamma is defined for reals (except negative integers). Your proposal would destroy the elegance of this definition. Your perception of elegance comes from only knowing the factorial correspondence. The gamma function has many other uses, and deserves elegance within its own domain.

Although to be fair, the mathematical community wrestled with this question of convention for a long time.