Changes

We arrive at the last of Peirce's statements about the "number of" map that we singled out above:

We arrive at the last of Peirce's statements about the "number of" map that we singled out above:

'''NOF 4'''

'''NOF 4.1'''

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<p>and there are just as many <math>x\!</math>'s per <math>y\!</math> as there are, ''per'' things, things of the universe, then we have also the arithmetical equation:</p>

<p>and there are just as many <math>x\!</math>'s per <math>y\!</math> as there are ''per'' things, things of the universe, then we have also the arithmetical equation:</p>

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| align="center" | <math>[x][y] ~=~ [z].</math>

| align="center" | <math>[x][y] ~=~ [z].</math>

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Peirce is here observing what we might call a ''contingent morphism''.  Provided that a certain condition, to be named in short order, happens to be satisfied, we would find it holding that the "number of" map <math>v : S \to \mathbb{R}</math> such that <math>v(s) = [s]\!</math> serves to preserve the multiplication of relative terms, that is to say, the composition of relations, in the form:  <math>[xy] = [x][y].\!</math>

Peirce is here observing what we might call a ''contingent morphism''.  Provided that a certain condition, to be named in short order, happens to be satisfied, we would find it holding that the "number of" map <math>v : S \to \mathbb{R}</math> such that <math>v(s) = [s]\!</math> serves to preserve the multiplication of relative terms, that is to say, the composition of relations, in the form:  <math>[xy] = [x][y].\!</math> So let us try to uncross Peirce's manifestly chiasmatic encryption of the condition that is called on in support of this preservation.

So let us try to uncross Peirce's manifestly chiasmatic encryption of the condition that is called on in support of this preservation.

The proviso for the equation <math>[xy] = [x][y]\!</math> to hold is this:

 

Proviso for [''xy''] = [''x''][''y''] —

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there are just as many ''x''’s per ''y'' as there are ''per'' things[,] things of the universe …

<p>There are just as many <math>x\!</math>'s per <math>y\!</math> as there are ''per'' things, things of the universe.</p>

 

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I have placed angle brackets around a comma that CP shows but CE omits, not that it helps much either way.  So let us resort to the example:

Returning to the example that Peirce gives:

 

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<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>

<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>

: <p>[''t''][''f''] =  [''tf'']</p>

<p>(Peirce, CP 3.76).</p>

 

<p>holds arithmetically.  (CP 3.76).</p>

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The 2-adic relative term ''t'' determines a 2-adic relation ''T''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', where U and V are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively.  To make the case as simple as we can and still cover the point, let's say that there are just four people in our initial universe of discourse, and that just two of them are French.  The bigraphic composition below shows all of the pertinent facts of the case.

The 2-adic relative term ''t'' determines a 2-adic relation ''T''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', where U and V are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively.  To make the case as simple as we can and still cover the point, let's say that there are just four people in our initial universe of discourse, and that just two of them are French.  The bigraphic composition below shows all of the pertinent facts of the case.

<pre>

<pre>

T_1    T_32  T_33    T_64  T_65    T_96  T_97    T_128

T_1    T_32  T_33    T_64  T_65    T_96  T_97    T_128

     J            K            L            M

     J            K            L            M

</pre>

</pre>

Here, the order of relational composition flows up the page.  For convenience, the absolute term ''f'' = "frenchman" has been converted by using the comma functor to give the idempotent representation ‘''f''’ = ''f'', = "frenchman that is ---", and thus it can be taken as a selective from the universe of mankind.

Here, the order of relational composition flows up the page.  For convenience, the absolute term ''f'' = "frenchman" has been converted by using the comma functor to give the idempotent representation ‘''f''’ = ''f'', = "frenchman that is ---", and thus it can be taken as a selective from the universe of mankind.