Changes

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.

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.

In the field picture, a proposition <math>f : X \to \mathbb{B}</math> becomes a ''scalar field'', that is, a field of values in <math>\mathbb{B}.</math>

In the field picture a proposition <math>f : X \to \mathbb{B}</math> becomes a ''scalar field'', that is, a field of values in <math>\mathbb{B}.</math>

Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure&nbsp;22-a.

Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure&nbsp;22-a.