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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 21:20, 29 August 2013
, 21:20, 29 August 2013→Analytic Series : Coordinate Method: TeX formatting + fix transcription errors in table 50
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And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could “just as easily” be, and to attach no more importance to what is than to what is not.
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|+ Table 50. Computation of an Analytic Series in Terms of Coordinates
|+ Table 50. Computation of an Analytic Series in Terms of Coordinates
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<br>
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The first six columns of the Table, taken as a whole, represent the variables of a construct that I describe as the ''contingent universe'' [''u'', ''v'', d''u'', d''v'', ''u''′, ''v''′ ], or the bundle of ''contingency spaces'' [d''u'', d''v'', ''u''′, ''v''′ ] over the universe [''u'', ''v'']. Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:
by the following equations:
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<math>\begin{matrix}
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\[8pt]
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\end{matrix}</math>
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These relations correspond to the formal substitutions that are made in defining E''J'' and D''J''. For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.
<br>
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.
The five columns to the right of the double bar in Table 50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math>
The five columns to the right of the double bar in Table 50 contain the values of the dependent variables {<math>\epsilon</math>''J'', E''J'', D''J'', d''J'', d<sup>2</sup>''J''}. These are normally interpreted as values of functions W''J'' : E''U'' → '''B''' or as values of propositions in the extended universe [''u'', ''v'', d''u'', d''v''], but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, say, ‹''u'', ''v'', ''u''′, ''v''′›.
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we “share structure” in the Table by listing only the first entries of each constant block.
The column for <math>\epsilon</math>''J'' is computed as ''J''‹''u'', ''v''› = ''uv''. This, along with the columns for ''u'' and ''v'', illustrates the Table's ''structure-sharing'' scheme, listing only the initial entries of each constant block.
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math>
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<math>\begin{matrix}
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)
& = &
J(u + \mathrm{d}u, v + \mathrm{d}v)
& = &
J(u', v')
\end{matrix}</math>
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This makes it easy to determine E''J'' by inspection, computing the conjunction ''J''‹''u''′, ''v''′› = ''u''′ ''v''′ from the columns headed ''u''′ and ''v''′. Since all of these forms express the same proposition E''J'' in E''U''<sup> •</sup>, the dependence on d''u'' and d''v'' is still present but merely left implicit in the final variant ''J''‹''u''′, ''v''′›.
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This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math>
* NB. On occasion, it is tempting to use the further notation ''J''′‹''u'', ''v''› = ''J''‹''u''′, ''v''′›, especially to suggest a transformation that acts on whole propositions, for example, taking the proposition ''J'' into the proposition ''J''′ = E''J''. The prime [′] then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character, and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.
Given the values of <math>\epsilon</math>''J'' and E''J'', the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation D''J'' = <math>\epsilon</math>''J'' + E''J''. The first order differential d''J'' is found by looking in each block of constant ‹''u'', ''v''› and choosing the linear function of ‹d''u'', d''v''› that best approximates D''J'' in that block. Finally, the remainder is computed as r''J'' = D''J'' + d''J'', in this case yielding the second order differential d<sup>2</sup>''J''.
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math>
====Analytic Series : Recap====
====Analytic Series : Recap====