You’ve produced an interesting answer, but there are definite problems with it.
This is an 11 year old thread. There’s probably not much interest in it anymore.
You’ve defined an illegitimate inverse. The factorial is a function which maps integers to integers. As such, any potential ‘inverse’ function must also map integers to integers. Instead, you have a function which maps integers to rationals.
For example, you have the inverse of 5! as 2+5/6. What is (2+5/6)! ? It should produce 5, yet it doesn’t, because factorials on non-integers are not well defined.
Even if we used the generalization of factorials to the real numbers - the Gamma function - we still have a problem. The gamma function has no inverse and can’t have one.
Basically, you have a function which acts as a pseudo-inverse for those integers which are the result of a factorial. For other integers, it produces some output that is essentially arbitrary.
That’s sort of the point though. Of course the other integers would be arbitrary, however I have defined these numbers in such a way to make the function definable, otherwise there would be no definable function. As I said in the beginning these values are based solely on logical reasoning. Basically to say that they are arbitrary. Also, it’s not incredibly intelligent to say that an inverse of a function is limited to mapping in the same aspect as the function itself. It would lead to Mathematics inheriting no evolution to it’s design. If we limited ourselves to certain “laws” of mathematics, we would never have calculus.
Lol also, this was the only intelligible website I found, where people actually knew what they were talking about. I just wanted to get it out there to get some feedback. Thanks!
That’s in the definition of an inverse. If you want to redefine it, ok, but then you’re answering a different question.
There’s no need to appeal to “evolution of Mathematics” (whatever that means). It’s ok if you want to define your own version of an ‘inverse’, but it won’t be the same animal.
Of course, the problem with using a home-grown version is that it doesn’t behave the way the original does, which can be a problem if somebody doesn’t realize it’s not the same thing. That’s why definitions are important and you should be careful with saying it’s an ‘inverse’ when it doesn’t have the properties we expect.
As an example, a function f has an inverse f’ if and only if f(f’(n)) = f’(f(n)) = n. But in your case, this fails. If f(n) = n!, then f(f’(5)) = (2+5/6)!, which has no definition. You skirt this by defining your own “inverse”, but it obviously doesn’t satisfy this very basic property of traditional function inverses.
I’ll add a note that even here, the inverse is limited over a range. The discontinuities ensure the general function has no inverse.
The above is probably the closest thing to an inverse of factorials, if we extend the definition of factorial over reals.
ETA: On re-reading the thread, all these points have actually been brought up before, and we’re just re-hashing them again. Zombie ideas/thread, indeed.
