The answer is that the gamma function is the unique logarithmically convex extension of the factorial function.
The properties of the standard gamma function are great, but some might prefer the properties of Hadamard's or Luschny's alternative gamma function.
Like many arbitrary extensions in mathematics -- be it the factorial function or the division function which deals with 1 / 0 differently -- it's a matter of taste and convenience!
Interestingly the arbitrariness of some mathematical choices seem to unnerve folks on an almost existential level. My guess is that it conflicts with their expectation of capital T truth from mathematics.
I don't think that's exactly the same problem.
It is perfectly valid to define an operation on a particular domain. Defining an add operation on even numbers is perfectly fine (If you had said odd, well...). Factorial on real numbers is more or less the Gamma function. The problem here is that the operation is defined on one domain...
> We can say that division’s domain is all the real numbers except zero.
...but then, for some reason, it is decided that will will be applied on a different domain. That is, that we're going to apply on all real numbers including zero. So now we need to modify the definition of the operation. And the thing is that, the new definition, created to accommodate the new domain, usually tends to produce friction precisely on the new elements added to the domain. We excluded zero initially because that way the operation was defined in a simpler way. Now, having to consider zero implies that our definition becomes more "complex".
Now, we can change the definition in different ways. Is any one of those objectively "better" or "worse" than the rest? Well, that's what the discussion is all about.
The issue with special definitions is you can use them to say anything you want, turning regular, common operations into weirdness.
What does it mean to take a factorial on the real numbers, or to add only on the even integers? In both cases, we're twisting what are generally accepted mechanics and domains into something else to get a funny result, like 1/0 = 0.
He gets to that point here:
" We can say that division’s domain is all the real numbers except zero. This is what we do in our day-to-day lives, and the way that Agda and Idris handle division. We can choose some value that isn’t a real number, such as “undefined” or infinity, and say x/0 = <whatever>. This is what some mathematicians do with the Riemann sphere. We could choose some real number, like 19, and say that x/0 = 19. This is what Isabelle, Lean and Coq do.
"
So, tiny change in definition --> big change in outcome.
*I'm an undergrad