Evidently @Ed and you both care about the topic of whether functions and numbers are essentially different. I think it’s kind of a distraction but just for sake of discussion I’ll play.
First, a function goes from any set to any other set. We always denote a function as (f : X \to Y ), meaning that (f) is a function that inputs an element of a set (X) and outputs an element of a set (Y).
Since everything in math is a set (in the standard set-theoretic formalism), a function can input anything and output anything.
For example a function can input and output functions. A familiar example is the derivative operator in one real variable. We have a function (D) that inputs a function of one real variable and outputs another. For example (D x^2 = 2x). (D sin
x = cos x), etc.
Or a function could input a set and output a number; for example the function that counts the number of elements of a finite set, and outputs -1 if the set’s not finite. That’s a perfectly valid function from the proper class of all sets to the natural numbers.
So functions can be completely arbitrary in terms of what they input and output. I don’t see how this sheds light.
Well numbers have properties too. Everything has properties. So that doesn’t distinguish functions from numbers.
Conceptually maybe not, but formally functions are often numbers. For example in mathematical logic, we use Gödel numbering to represent a function or a formula by a specific number.
Another example would be to use the fact that there are as many continuous functions from the reals to the reals as there are reals. So in principle there’s a mapping that inputs a continuous function and outputs a real number that can be used as a proxy for it. Instead of saying cosine we can just say #45.3. Every function has an associated number. So again, the distinction between numbers and functions is less clear to me than it is to you and @Ed.
Well … I question the relevance or point of the observation, since it’s not clear to me that it’s true, and it’s definitely clear to me that it’s a red herring in the .999… discussion. I don’t get the bit about numbers and functions. Set theory doesn’t distinguish between numbers and functions, they’re both different types of sets. So I honestly don’t know what point is being made here.
Oh but of course it is. Every decimal expression is a function (d : \mathbb N_+ \to D) where (D) is the set of decimal digits (D = {0,1,2,3,4,5,6,7,8,9 } ). That’s what a decimal expression is. You give me the number 47, I give you back the 47-th decimal digit. (Just referring to the digits to the right of the decimal point, we can patch up the idea to account for the leftward digits if needed). You give me the number 545535 and I return that digit. That is exactly what a decimal expression is, a function from the set of positive natural numbers to the set of digits.
I’m using (\mathbb N_+) which is the set 1, 2, 3, … so that the first place to the right of the decimal point is 1 and not 0 for convenience. I hope that’s clear.
You see this, right? (\pi ) is a function, (\sqrt 2 ) is a function. [After we deal with the pesky leftward digits].
What function represents (\pi - 3 )?
f(1) = 1
f(2) = 4
f(3) = 1
f(4) = 5
f(5) = 9
etc.