Changes

| <math>\ldots\!</math>

| <math>\ldots\!</math>

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The Table exhibits a sample of likely parallels between the real and boolean domains.  The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table.  These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that followe their courses through the states of an arbitrary space <math>X.\!</math>  Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

The Table exhibits a sample of likely parallels between the real and boolean domains.  The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table.  These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math>  Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math>  In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math>  The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones.  Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k \in \mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math>

It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math>  In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math>  The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones.  Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math>

===Theory of Control and Control of Theory===

===Theory of Control and Control of Theory===