Harristotelian Logic

I like to separate mathematical development out into generations, where first generation approaches were covered by the traditional mathematics that most people think of when they think of "mathematics" or "math", followed by the 2nd generation of problems that can be covered by what I call extreme mathematics, followed by a 3rd generation of problems to be handled by generations of mathematicians yet unborn.

The first generation of mathematicians dealt with functions with simple arguments, like f(x) or f(x,y), etc. and only came close to more complex arguments with things like e^{-t^2}, while the 2nd generation comes out of the box with more complex arguments, like in my prime counting function in its various forms where you have functions with arguments like P(x,sqrt(y)) and P(y,sqrt(y)), which probably creates confusion for people brought up on the more primitive techniques, so this post is about understanding functions from 2nd generation math problems.

A function takes an argument and carries it to something else where that something else can be an explicit expression, for instance:

f(x,y) = x + y

carries arguments to the explicit expression x+y, where the first argument goes into x, and the second goes into y, so also

f(x,y^2) = x + y^2

as the more complex—2nd generation—argument has simply been carried to the explicit expression.

That's it. That's how more complicated arguments are handled in more advanced mathematics.

The argument itself is simply like an instruction to take this thing and place it inside of some other thing in a special way.

And you can still manipulate the functions using algebra like always as notice you can subtract the first from the second to get

f(x,y^2) - f(x,y) = y^2 - y

without any problem.