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According to one way of understanding the term, there is no object called a “variable” unless that object is a sign, and so the name “variable name” is redundant.  Variables, if they are anything at all, are analogous to numerals, not numbers, and thus they fall within the broad class of signs called ''identifiers'', more specifically, ''indices''.  In the case of variables, the advice of nominalism, not to confuse a variable name with the name of a variable, seems to be well taken.

According to one way of understanding the term, there is no object called a “variable” unless that object is a sign, and so the name “variable name” is redundant.  Variables, if they are anything at all, are analogous to numerals, not numbers, and thus they fall within the broad class of signs called ''identifiers'', more specifically, ''indices''.  In the case of variables, the advice of nominalism, not to confuse a variable name with the name of a variable, seems to be well taken.

If the world of elements appropriate to this discussion is organized into objective and syntactic domains, then there are fundamentally just two different ways of regarding variables, as objects or as signs.  One can say that a variable is a fictional object that is contrived to provide a variable name with a form of objective referent, or one can say that a variable is a sign itself, the same thing as a variable name.  In the present setting, it is convenient to arrange these broad approaches to variables according to the respective norms of significance under which one finds them most often pursued.

If the world of elements appropriate to this discussion is organized into objective and syntactic domains, then there are fundamentally just two different ways of regarding variables, as objects or as signs.  One can say that a variable is a fictional object that is contrived to provide a variable name with a form of objective referent, or one can say that a variable is a sign itself, the same thing as a variable name.  In the present setting, it is convenient to arrange these broad approaches to variables under the NOSs where one finds them most often pursued.

1. The IL approach to the question takes the "objective construal" of variables as its most commonly chosen default.  The IL style that is used in ordinary mathematical discussion associates a variable with a determinate set, one that the variable is regarded as "ranging over".  As a result, this NOS is forced to invoke a version of set theory, usually naive, to account for its use of variables.

# The informal language approach to the question takes the objective construal of variables as its most commonly chosen default.  The informal language style that is used in ordinary mathematical discussion associates a variable with a determinate set, one that the variable is regarded as “ranging over”.  As a result, this norm of significance is forced to invoke a version of set theory, usually naive, to account for its use of variables.

 

# The formal language styles are manifestly varied in their explanations of variables, since there are many ways to formalize their ordinary uses.  Two of the main alternatives are:  (a) formalizing the set theory that is invoked with the use of variables, and (b) formalizing the sign relations in which variables operate as indices.  Since an index is a kind of sign that denotes its object by virtue of an actual connection with it, and since the nature and direction of these actual connections can vary immensely from moment to moment, a variable is an extremely flexible and adaptable kind of sign, hence its character as a “reusable sign”.

2. The FL styles are manifestly varied in their explanations of variables, since there are many ways to formalize their ordinary uses.  Two of the main alternatives are:  (a) formalizing the set theory that is invoked with the use of variables, and (b) formalizing the sign relations in which variables operate as indices.  Since an index is a kind of sign that denotes its object by virtue of an actual connection with it, and since the nature and direction of these actual connections can vary immensely from moment to moment, a variable is an extremely flexible and adaptable kind of sign, hence its character as a "reusable sign".

# The computational language styles are also legion in their approaches to variables, but they can be divided into those that eliminate variables as a primitive concept and those that retain a notion of variables in their conceptual basis.

 

3. The CL styles are also legion in their approaches to variables, but they can be divided into those eliminate variables as a primitive concept and those that retain a notion of variables in their conceptual basis.

a. An instructive case is presented by what is the most complete working out of the computational programme, the "combinator calculus".  Here, the goal is to eliminate the notion of a variable altogether from the conceptual basis of a formal system.  In other words, it is projected to reduce its status as a primitive concept, one that applies to symbols in the object language, and to reformulate it as a derived concept, one that is more appropriate to describing constructions in a metalanguage.

a. An instructive case is presented by what is the most complete working out of the computational programme, the "combinator calculus".  Here, the goal is to eliminate the notion of a variable altogether from the conceptual basis of a formal system.  In other words, it is projected to reduce its status as a primitive concept, one that applies to symbols in the object language, and to reformulate it as a derived concept, one that is more appropriate to describing constructions in a metalanguage.