Quote:
Originally Posted by Michali
Quote:
Originally Posted by naturalist.atheist
Quote:
Originally Posted by Michali
Seebs mentioned a use of the equals sign where it is used to describe a relationship where one thing "yields" another thing, and this is different than something being identical with another thing. Probably rightly so.
Now, firstly, let's analyze this concept and maybe it will shed some light on the whole ordeal.
First question:
Is it that functions yield such and such, or is it that such and such yields a function? In other words when we notate f(x)=y... how do we linguistically express this expression?
|
In the mathematical sense a "function" is a mapping with specific properties such as "one-to-one" and "onto". However in this age of computers the concept of a function and the "=" gets a bit muddy, with the "=" and a function becoming a kind of operation as opposted to a mathematical identity. |
It seems like this mapping description is something like "x -> y" where this "->" is meant to be interpreted as what exactly? So let's just assume that X and Y have this mapping between them, but what is the best way to describe this relationship?
In other words, let's say I have x=(1234) and y=(abcd).
You might suggest f(x)=y could be represented with the following:
1-a
2-b
3-c
4-d
What exactly will the concept of function relate to in the above diagram? Is it the hyphen(abstract) playing the role of function, or are 4 hyphens together playing the role of function. Is the entire diagram playing the role of function? Or are only the principle features playing the role of function, and if so, what are they.
In other words, how should we use "function" in English? That's what I think I'm searching for. |
One way to understand it is to consider a sub-class of functions called discrete functions.
Given a discrete set called
X which contains the set of integers
I there is another set
Y which also contains a set of integers where there is a mapping between a member in the set
X to a member in the set
Y. You could think it as something like this,
For a range of integers that enumarate the integers in
X, X(1), X(2), X(3), ...... and X(1) could be 1, X(2) is 2 and so forth.
There is a pairing between the two sets created by a mapping
X(1) -> Y(1) [ 1 maps to Y(1)]
X(2) -> Y(2)
X(3) -> Y(3)
And so on.
In the continuous case the sets become all numbers. How the numbers map specifically defines the specific function. The mapping or function is
onto if there is a member of X for every member in Y. A function is
one-to-one if for a given X(n) there is always only one member of Y that it maps to (not two or more values). This allows us to assume that if Y(n) = Y(m) then n = m.
Quote:
Quote:
Quote:
Second question:
Is this relationship anything like the "emergence" concept? For instance, are the pieces of a puzzle yielding the entire picture? What are the specifications surrounding that?
|
Mathematically the answer is no since "=" is not a relationship. It is an "identity". Mathematically when we say f(x) = y we identify f(x) as y but we are none the wiser unless we can define y in terms of x.
|
Right I agree that is less of a relationship and more like identity myself. But reading your sentence here closely, and it seems as though you don't think f(x) yields y, but that f(x) is identical to y. Am I right?
|
You are right. Mathematically when you use the "=" it means that the left side is identical to the right side. The "=" is also used widely in computer programming but it is not the same thing as the "=" in mathematics. When the "=" is used in programming it is essentially a memory operation not a mathematical identity.