Changes

The two main operators that we've examined so far are the enlargement or shift operator <math>\operatorname{E} : X^\circ \to \operatorname{E}X^\circ</math> and the difference operator <math>\operatorname{D} : X^\circ \to \operatorname{E}X^\circ.</math>  The operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> act on propositions in <math>X^\circ,</math> that is, propositions of the form <math>f : X \to \mathbb{B}</math> that are said to be ''about'' the subject matter of <math>X,\!</math> and they produce extended propositions of the forms <math>\operatorname{E}f, \operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in <math>X.\!</math>

The two main operators that we've examined so far are the enlargement or shift operator <math>\operatorname{E} : X^\circ \to \operatorname{E}X^\circ</math> and the difference operator <math>\operatorname{D} : X^\circ \to \operatorname{E}X^\circ.</math>  The operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> act on propositions in <math>X^\circ,</math> that is, propositions of the form <math>f : X \to \mathbb{B}</math> that are said to be ''about'' the subject matter of <math>X,\!</math> and they produce extended propositions of the forms <math>\operatorname{E}f, \operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in <math>X.\!</math>

Here we have need of visual representations, some array of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us before we try to climb any higher into the ever more rarefied air of abstractions.

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us as we venture higher into the ever more rarefied air of abstractions.

 

One good picture comes to us by way of the ''field'' concept.  Given a space <math>X,\!</math> a ''field'' of a specified type <math>Y\!</math> over <math>X\!</math> is formed by associating with each point of <math>X\!</math> an object of type <math>Y\!</math>  If that sounds like the same thing as a function from <math>X,\!</math> to the space of things of type <math>y\!</math> &mdash; it is nothing but &mdash; and yet it does seem helpful to vary the mental images and to take advantage of the figures of speech that spring to mind under the emblem of this field idea.

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In the field picture, a proposition f : X -> B becomes

In the field picture, a proposition f : X -> B becomes

a "scalar" field, that is, a field of values in B, or

a "scalar" field, that is, a field of values in B, or