Changes

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 according to the respective norms of significance under which one finds them most often pursued.

# 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.

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# 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 &ldquo;reusable sign&rdquo;.

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<li>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 &ldquo;ranging over&rdquo;.  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.</li>

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.

 

<li>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 &ldquo;reusable sign&rdquo;.</li>

 

 

 

<li>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.</li>

b. In CL contexts where variables are retained as a primitive notion, there is a form of distinction between variables and variable names, but here it takes on a different sense, being the distinction between a sign and its HO sign.  This is because a variable is conceived as a "store", a "component of state" (COS) of the interpreting machine, that contains different values from time to time, while the variable name is a symbolic version of that store's address.  The store when full, or the state when determinate, constitutes a form of numeral, not a number, and so it is still a sign, not the object itself.  This makes the variable name in this setting a type of HO sign.

<li>In computational language contexts where variables are retained as a primitive notion, there is a form of distinction between variables and variable names, but here it takes on a different sense, being the distinction between a sign and its higher order sign.  This is because a variable is conceived as a &ldquo;store&rdquo;, a ''component of state'' of the interpreting machine, that contains different values from time to time, while the variable name is a symbolic version of that store's address.  The store when full, or the state when determinate, constitutes a form of numeral, not a number, and so it is still a sign, not the object itself.  This makes the variable name in this setting a type of higher order sign.</li></ol></ol>

It is not just the influence of different conventions about language use that forms the source of so much confusion.  Different conventions that prevail in different contexts would generate conceptual turbulence only at their boundaries with each other, and not distribute the disturbance throughout the interiors of these contexts, as is currently the case.  But there are higher order differential conventions, in other words, conventions about changing conventions, that apply without warning all throughout what is pretended to be a uniform context.

It is not just the influence of different conventions about language use that forms the source of so much confusion.  Different conventions that prevail in different contexts would generate conceptual turbulence only at their boundaries with each other, and not distribute the disturbance throughout the interiors of these contexts, as is currently the case.  But there are higher order differential conventions, in other words, conventions about changing conventions, that apply without warning all throughout what is pretended to be a uniform context.