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Revision as of 16:56, 4 January 2008
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===Commentary Note 11.20===
===Commentary Note 11.20===
We arrive at the last, for the time being, of Peirce's statements about the "number of" map.
We arrive at the last, for the time being, of
Peirce's statements about the "number of" map.
<blockquote>
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<p>Even ordinary numerical multiplication involves the same idea, for 2 × 3 is a pair of triplets, and 3 × 2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives.</p>
<p>If we have an equation of the form:</p>
: <p>''xy'' = ''z''</p>
<p>and there are just as many x's per y as there are ''per'' things, things of the universe, then we have also the arithmetical equation:</p>
: <p>[''x''][''y''] = [''z''].</p>
<p>(Peirce, CP 3.76).</p>
</blockquote>
Peirce is here observing what we might dub a "contingent morphism"
Peirce is here observing what we might dub a "contingent morphism" or a "skeptraphotic arrow", if you will. Provided that a certain condition, to be named and, what is more hopeful, to be clarified in short order, happens to be satisfied, we would find it holding that the "number of" map ''v'' : ''S'' → '''R''' such that ''vs'' = [''s''] serves to preserve the multiplication of relative terms, that is as much to say, the composition of relations, in the form: [''xy''] = [''x''][''y''].
or a "skeptraphotic arrow", if you will. Provided that a certain
condition, to be named and, what is more hopeful, to be clarified
in short order, happens to be satisfied, we would find it holding
that the "number of" map 'v' : S -> R such that 'v's = [s] serves
to preserve the multiplication of relative terms, that is as much
to say, the composition of relations, in the form: [xy] = [x][y].
<pre>
So let us try to uncross Peirce's manifestly chiasmatic encryption
So let us try to uncross Peirce's manifestly chiasmatic encryption
of the condition that is called on in support of this preservation.
of the condition that is called on in support of this preservation.
