ONT Re: Logic Of Relatives
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LOR. Note 62
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We arrive at the last, for the time being, of
Peirce's statements about the "number of" map.
NOF 4.
| 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 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs,
| where "triplet of" and "pair of" are evidently relatives.
|
| If we have an equation of the form:
|
| xy = z
|
| 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:
|
| [x][y] = [z].
|
| C.S. Peirce, CP 3.76
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 '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].
So let us try to uncross Peirce's manifestly chiasmatic encryption
of the condition that is called on in support of this preservation.
Proviso for [xy] = [x][y] --
| there are just as many x's per y
| as there are 'per' things<,>
| things of the universe ...
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:
| 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:
|
| ['t'][f] = ['t'f]
|
| holds arithmetically. (CP 3.76).
Now that is something that we can sink our teeth into,
and trace the bigraph representation of the situation.
In order to do this, it will help to recall our first
examination of the "tooth of" relation, and to adjust
the picture that we sketched of it on that occasion.
Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".
Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect
human dentition, the number of the relative term "tooth of ---" is equal to
the number of teeth of humans divided by the number of humans, that is, 32.
The 2-adic relative term 't' determines a 2-adic relation T c U x 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.
T_1 T_32 T_33 T_64 T_65 T_96 T_97 T_128
o ... o o ... o o ... o o ... o U
\ | / \ | / \ | / \ | /
\ | / \ | / \ | / \ | / 't'
\|/ \|/ \|/ \|/
o o o o V = m = 1
| |
| | 'f'
| |
o o o o V = m = 1
J K L M
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.
By way of a legend for the figure, we have the following data:
| m = J +, K +, L +, M = 1
|
| f = K +, M
|
|'f' = K:K +, M:M
|
|'t' = (T_001 +, ... +, T_032):J +,
| (T_033 +, ... +, T_064):K +,
| (T_065 +, ... +, T_096):L +,
| (T_097 +, ... +, T_128):M
Now let's see if we can use this picture
to make sense of the following statement:
| 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:
|
| ['t'][f] = ['t'f]
|
| holds arithmetically. (CP 3.76).
In the lingua franca of statistics, Peirce is saying this:
That if the population of Frenchmen is a "fair sample" of
the general population with regard to dentition, then the
morphic equation ['t'f] = ['t'][f], whose transpose gives
['t'] = ['t'f]/[f], is every bite as true as the defining
equation in this circumstance, namely, ['t'] = ['t'm]/[m].
Jon Awbrey
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