Changes

Another issue on which the three styles of usage diverge most severely is with respect to a crucial problem about the status of variables.  Often this is posed as a question about the ''ontological status'' of variables, what kinds of objects they are, but it is better treated as a question about the ''pragmatic status'' of variables, what kinds of signs they are used as.  In this section, I try to accommodate common practices in the use of variables in the process of building a bridge to the pragmatic perspective.  The goal is to reconstruct customary ways of regarding variables within a overarching framework of sign relations, while disentangling the many confusions about the status of variables that obstruct their clear and consistent formalization.

Another issue on which the three styles of usage diverge most severely is with respect to a crucial problem about the status of variables.  Often this is posed as a question about the ''ontological status'' of variables, what kinds of objects they are, but it is better treated as a question about the ''pragmatic status'' of variables, what kinds of signs they are used as.  In this section, I try to accommodate common practices in the use of variables in the process of building a bridge to the pragmatic perspective.  The goal is to reconstruct customary ways of regarding variables within a overarching framework of sign relations, while disentangling the many confusions about the status of variables that obstruct their clear and consistent formalization.

Variables are the most problematic entities that have to be dealt with in the process of formalization, and this makes it useful to explore several  different ways of approaching their treatment, either of accounting for them or explaining them away.  The various tactics available for dealing with variables can be organized according to how they respond to two questions:  Are variables good or bad, and what kinds of things are variables anyway?  In other words:

Variables are the most problematic entities that have to be dealt with in the process of formalization, and this makes it useful to explore several  different ways of approaching their treatment, either of accounting for them or explaining them away.  The various tactics available for dealing with variables can be organized according to how they respond to two questions:  Are variables good or bad, and what kinds of things are variables anyway?  That is: (1) Are variables a good thing to have in a purified system of interpretation or a target formal system, or should variables be eliminated by the work of formalization? (2) What sorts of things should variables be construed as?

 

# Are variables good things to have in a purified system of interpretation or a target formal system, or should variables be eliminated by the work of formalization?

# What sorts of things should variables be construed as?

The answers given to these questions determine several consequences.  If variables are good things, things that ought to be retained in a purified formal system, then it must be possible to account for their valid uses in a sensible fashion.  If variables are bad things, things that ought to be eliminated from a purified formal system, then it must be possible to "explain away" their properties and utilities in terms of more basic concepts and operations.

The answers given to these questions determine several consequences.  If variables are good things, things that ought to be retained in a purified formal system, then it must be possible to account for their valid uses in a sensible fashion.  If variables are bad things, things that ought to be eliminated from a purified formal system, then it must be possible to “explain away” their properties and utilities in terms of more basic concepts and operations.

One approach is to eliminate variables altogether from the primitive conceptual basis of one's formalism, replacing every form of substitution with a form of application.  In the abstract, this makes applications of constant operators to one another the only type of combination that needs to be considered.  This is the strategy of the so called "combinator calculus".

One approach is to eliminate variables altogether from the primitive conceptual basis of one's formalism, replacing every form of substitution with a form of application.  In the abstract, this makes applications of constant operators to one another the only type of combination that needs to be considered.  This is the strategy of the so called "combinator calculus".