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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
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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.
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> — it is nothing but — 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> — it is nothing but — 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 f : X -> B becomes
a "scalar" field, that is, a field of values in B, or
Let us take a moment to view an old proposition
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> that is depicted in Figure 22-a.
in this new light, for example, the conjunction
pq : X -> B that is depicted in Figure 22-a.
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Figure 22-a. Conjunction pq : X -> B
Figure 22-a. Conjunction pq : X -> B
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Each of the operators E, D : X% -> EX% takes us from considering
Each of the operators E, D : X% -> EX% takes us from considering
propositions f : X -> B, here viewed as "scalar fields" over X,
propositions f : X -> B, here viewed as "scalar fields" over X,