We come to the last of Peirce’s statements about the “number of” function that we singled out in Comment 11.2.
NOF 4.1
The conception of multiplication we have adopted is that of the application of one relation to another. …
Even ordinary numerical multiplication involves the same idea, for 
If we have an equation of the form
and there are just as many 
![[x][y] ~=~ [z].](https://s0.wp.com/latex.php?latex=%5Bx%5D%5By%5D+%7E%3D%7E+%5Bz%5D.&bg=ffffff&fg=000000&s=1 "[x][y] ~=~ [z].")
(Peirce, CP 3.76)
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 
![v(s) = [s]](https://s0.wp.com/latex.php?latex=v%28s%29+%3D+%5Bs%5D&bg=ffffff&fg=000000&s=1 "v(s) = [s]")
![[xy] = [x][y].](https://s0.wp.com/latex.php?latex=%5Bxy%5D+%3D+%5Bx%5D%5By%5D.&bg=ffffff&fg=000000&s=1 "[xy] = [x][y].")
The condition for the equation ![[xy] = [x][y]](https://s0.wp.com/latex.php?latex=%5Bxy%5D+%3D+%5Bx%5D%5By%5D&bg=ffffff&fg=000000&s=1 "[xy] = [x][y]")
There are just as many
(Peirce, CP 3.76)
Returning to the example that Peirce gives:
NOF 4.2
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
![[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7Bt%7D%5D%5B%5Cmathrm%7Bf%7D%5D+%7E%3D%7E+%5B%5Cmathit%7Bt%7D%5Cmathrm%7Bf%7D%5D&bg=ffffff&fg=000000&s=1 "[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]")
holds arithmetically.
(Peirce, CP 3.76)
Now that is something that we can sink our teeth into and trace the bigraph representation of the situation. It will help to recall our first examination of the “tooth of” relation and to adjust the picture we sketched of it on that occasion.
Transcribing Peirce’s example:
![\begin{array}{ll} \text{Let} & \mathrm{m} ~=~ \text{man} \\[8pt] \text{and} & \mathit{t} ~=~ \text{tooth of}\,\underline{~~~~}. \\[8pt] \text{Then} & v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++%5Ctext%7BLet%7D++%26++%5Cmathrm%7Bm%7D+%7E%3D%7E+%5Ctext%7Bman%7D++%5C%5C%5B8pt%5D++%5Ctext%7Band%7D++%26++%5Cmathit%7Bt%7D+%7E%3D%7E+%5Ctext%7Btooth+of%7D%5C%2C%5Cunderline%7B%7E%7E%7E%7E%7D.++%5C%5C%5B8pt%5D++%5Ctext%7BThen%7D+%26++v%28%5Cmathit%7Bt%7D%29+%7E%3D%7E+%5B%5Cmathit%7Bt%7D%5D+%7E%3D%7E+%5Cdisplaystyle%5Cfrac%7B%5B%5Cmathit%7Bt%7D%5Cmathrm%7Bm%7D%5D%7D%7B%5B%5Cmathrm%7Bm%7D%5D%7D.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{ll} \text{Let} & \mathrm{m} ~=~ \text{man} \\[8pt] \text{and} & \mathit{t} ~=~ \text{tooth of}\,\underline{~~~~}. \\[8pt] \text{Then} & v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}. \end{array}")
That is to say, the number of the relative term 
The dyadic relative term 
To make the case as simple as possible and still cover the point, suppose there are just four people in our universe of discourse and just two of them are French. The bigraphical composition below shows the pertinent facts of the case.
 |
(52) |
In this picture the order of relational composition flows down the page. For convenience in composing relations, the absolute term 
By way of a legend for the Figure, we have the following data:
![\begin{array}{lllr} \mathrm{m} & = & \mathrm{J} ~+\!\!,~ \mathrm{K} ~+\!\!,~ \mathrm{L} ~+\!\!,~ \mathrm{M} \qquad = & \mathbf{1} \\[6pt] \mathrm{f} & = & \mathrm{K} ~+\!\!,~ \mathrm{M} \\[6pt] \mathrm{f,} & = & \mathrm{K}\!:\!\mathrm{K} ~+\!\!,~ \mathrm{M}\!:\!\mathrm{M} \\[6pt] \mathit{t} & = & (T_{001} ~+\!\!,~ \dots ~+\!\!,~ T_{032}):J & ~+\!\!, \\[6pt] & & (T_{033} ~+\!\!,~ \dots ~+\!\!,~ T_{064}):K & ~+\!\!, \\[6pt] & & (T_{065} ~+\!\!,~ \dots ~+\!\!,~ T_{096}):L & ~+\!\!, \\[6pt] & & (T_{097} ~+\!\!,~ \dots ~+\!\!,~ T_{128}):M \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blllr%7D++%5Cmathrm%7Bm%7D+%26+%3D+%26++%5Cmathrm%7BJ%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BK%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BL%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BM%7D+%5Cqquad+%3D+%26+%5Cmathbf%7B1%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bf%7D+%26+%3D+%26+%5Cmathrm%7BK%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BM%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bf%2C%7D+%26+%3D+%26+%5Cmathrm%7BK%7D%5C%21%3A%5C%21%5Cmathrm%7BK%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BM%7D%5C%21%3A%5C%21%5Cmathrm%7BM%7D++%5C%5C%5B6pt%5D++%5Cmathit%7Bt%7D+%26+%3D+%26+%28T_%7B001%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cdots+%7E%2B%5C%21%5C%21%2C%7E+T_%7B032%7D%29%3AJ+%26+%7E%2B%5C%21%5C%21%2C++%5C%5C%5B6pt%5D++%26+%26+%28T_%7B033%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cdots+%7E%2B%5C%21%5C%21%2C%7E+T_%7B064%7D%29%3AK+%26+%7E%2B%5C%21%5C%21%2C++%5C%5C%5B6pt%5D++%26+%26+%28T_%7B065%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cdots+%7E%2B%5C%21%5C%21%2C%7E+T_%7B096%7D%29%3AL+%26+%7E%2B%5C%21%5C%21%2C++%5C%5C%5B6pt%5D++%26+%26+%28T_%7B097%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cdots+%7E%2B%5C%21%5C%21%2C%7E+T_%7B128%7D%29%3AM++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{lllr} \mathrm{m} & = & \mathrm{J} ~+\!\!,~ \mathrm{K} ~+\!\!,~ \mathrm{L} ~+\!\!,~ \mathrm{M} \qquad = & \mathbf{1} \\[6pt] \mathrm{f} & = & \mathrm{K} ~+\!\!,~ \mathrm{M} \\[6pt] \mathrm{f,} & = & \mathrm{K}\!:\!\mathrm{K} ~+\!\!,~ \mathrm{M}\!:\!\mathrm{M} \\[6pt] \mathit{t} & = & (T_{001} ~+\!\!,~ \dots ~+\!\!,~ T_{032}):J & ~+\!\!, \\[6pt] & & (T_{033} ~+\!\!,~ \dots ~+\!\!,~ T_{064}):K & ~+\!\!, \\[6pt] & & (T_{065} ~+\!\!,~ \dots ~+\!\!,~ T_{096}):L & ~+\!\!, \\[6pt] & & (T_{097} ~+\!\!,~ \dots ~+\!\!,~ T_{128}):M \end{array}")
Now let’s see if we can use this picture to make sense of the following statement:
NOF 4.3
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
holds arithmetically.
(Peirce, CP 3.76)
In statistical terms, Peirce is saying this: If the population of Frenchmen is a fair sample of the general population with regard to the factor of dentition, then the morphic equation,
![[\mathit{t}\mathrm{f}] ~=~ [\mathit{t}][\mathrm{f}],](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7Bt%7D%5Cmathrm%7Bf%7D%5D+%7E%3D%7E+%5B%5Cmathit%7Bt%7D%5D%5B%5Cmathrm%7Bf%7D%5D%2C&bg=ffffff&fg=000000&s=1 "[\mathit{t}\mathrm{f}] ~=~ [\mathit{t}][\mathrm{f}],")
whose transpose gives the equation,
![[\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{f}]}{[\mathrm{f}]},](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7Bt%7D%5D+%7E%3D%7E+%5Cdisplaystyle%5Cfrac%7B%5B%5Cmathit%7Bt%7D%5Cmathrm%7Bf%7D%5D%7D%7B%5B%5Cmathrm%7Bf%7D%5D%7D%2C&bg=ffffff&fg=000000&s=1 "[\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{f}]}{[\mathrm{f}]},")
is every bit as true as the defining equation in this circumstance, namely,
![[\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7Bt%7D%5D+%7E%3D%7E+%5Cdisplaystyle%5Cfrac%7B%5B%5Cmathit%7Bt%7D%5Cmathrm%7Bm%7D%5D%7D%7B%5B%5Cmathrm%7Bm%7D%5D%7D.&bg=ffffff&fg=000000&s=1 "[\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.")
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