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

 

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

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<p>The conception of multiplication we have adopted is that of the application of one relation to another.  …</p>

<p>The conception of multiplication we have adopted is that of the application of one relation to another.  &hellip;</p>

<p>Even ordinary numerical multiplication involves the same idea, for 2&nbsp;&times;&nbsp;3 is a pair of triplets, and 3&nbsp;&times;&nbsp;2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives.</p>

<p>Even ordinary numerical multiplication involves the same idea, for <math>~2 \times 3~</math> is a pair of triplets, and <math>~3 \times 2~</math> 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>If we have an equation of the form:</p>

 

: <p>''xy'' = ''z''</p>

| align="center" | <math>xy ~=~ z</math>

 

<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>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>[''x''][''y''] = [''z''].</p>

 

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

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

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

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