12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Wednesday December 04, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 19:44, 8 August 2012
, 19:44, 8 August 2012→6.14. Issue 3. The Status of Variables: markup
If it is desired to retain a notion of variables in the formalism, and to maintain variables as objects of reference, then there are a couple of partial explanations of variables that still afford them with various measures of objective existence.
If it is desired to retain a notion of variables in the formalism, and to maintain variables as objects of reference, then there are a couple of partial explanations of variables that still afford them with various measures of objective existence.
In the ''elemental construal'' of variables, a variable <math>x\!</math> is just an existing object <math>x\!</math> that is an element of a set <math>X,\!</math> the catch being “which element?” In spite of this lack of information, one is still permitted to write <math>{}^{\backprime\backprime} x \in X {}^{\prime\prime}\!</math> as a syntactically well formed expression and otherwise treat the variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> as a pronoun on a grammatical par with a noun. Given enough information about the contexts of usage and interpretation, this explanation of the variable <math>x\!</math> as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math>
In the ''elemental construal'' of variables, a variable <math>x\!</math> is just an existing object <math>x\!</math> that is an element of a set <math>X,\!</math> the catch being “which element?” In spite of this lack of information, one is still permitted to write <math>{}^{\backprime\backprime} x \in X {}^{\prime\prime}\!</math> as a syntactically well-formed expression and otherwise treat the variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> as a pronoun on a grammatical par with a noun. Given enough information about the contexts of usage and interpretation, this explanation of the variable <math>x\!</math> as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math>
In the ''functional construal'' of variables, a variable is a function of unknown circumstances that results in a known range of definite values. This tactic pushes the ostensible location of the uncertainty back a bit, into the domain of a named function, but it cannot eliminate it entirely. Thus, a variable is a function <math>x : X \to Y\!</math> that maps a domain of unknown circumstances, or a ''sample space'' <math>X,\!</math> into a range <math>Y\!</math> of outcome values. Typically, variables of this sort come in sets of the form <math>\{ x_i : X \to Y \},\!</math> collectively called ''coordinate projections'' and together constituting a basis for a whole class of functions <math>x : X \to Y\!</math> sharing a similar type. This construal succeeds in giving each variable name <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> an objective referent, namely, the coordinate projection <math>x_i,</math> but the explanation is partial to the extent that the domain of unknown circumstances remains to be explained. Completing this explanation of variables, to the extent that it can be accomplished, requires an account of how these unknown circumstances can be known exactly to the extent that they are in fact described, that is, in terms of their effects under the given projections.
<pre>
<pre>
As suggested by the whole direction of the present work, the ultimate explanation of variables is to be given by the pragmatic theory of signs, where variables are treated as a special class of signs called "indices".
As suggested by the whole direction of the present work, the ultimate explanation of variables is to be given by the pragmatic theory of signs, where variables are treated as a special class of signs called "indices".
