[Math] Is argument of a function and independent variable essesntially the same things

Question

From the top of this page of Wikipedia-

In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.

From the same page-

A mathematical function has one or more arguments in the form of independent variables designated in the function's definition.

Again from the same page-

The independent variables are mentioned in the list of arguments that the function takes.

So, I gets confused about the difference of these terms. It seems to me that they should be different but Wikipedia is not agreeing. Can you help me with this?

Answer 1

A function has arguments, one or more : "square root" is a unary function (it has one argument), $+$ is a binary function (two arguments), and so on.

A variable is a syntactical object (a symbol) used in the expression of a function to designate an argument place: it is a "place holder".

We can write e.g "$f$( __)" to designate the fact that the function $f$ has one argument, like an "empty slot" to be filled with an input value in order to "calculate" the corresponding value of the function: the output.

With more arguments, it can be misleading to use "slots" :

$+$(_ , _)

and thus the use of variables for denoting argument-places has been adopted :

$+(x,y)$

i.e. by convention : $x+y$.