ONT Re: Differential Logic -- Series A

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DLOG.  Note A12

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It is common in algebra to switch around
between different conventions of display,
as the momentary fancy happens to strike,
and I see that Peirce is no different in
this sort of shiftiness than anyone else.
A changeover appears to occur especially
whenever he shifts from logical contexts
to algebraic contexts of application.

In the paper "On the Relative Forms of Quaternions" (CP 3.323),
we observe Peirce providing the following sorts of explanation:

| If X, Y, Z denote the three rectangular components of a vector, and W denote
| numerical unity (or a fourth rectangular component, involving space of four
| dimensions), and (Y:Z) denote the operation of converting the Y component
| of a vector into its Z component, then
|
|     1  =  (W:W) + (X:X) + (Y:Y) + (Z:Z)
|
|     i  =  (X:W) - (W:X) - (Y:Z) + (Z:Y)
|
|     j  =  (Y:W) - (W:Y) - (Z:X) + (X:Z)
|
|     k  =  (Z:W) - (W:Z) - (X:Y) + (Y:X)
|
| In the language of logic (Y:Z) is a relative term whose relate is
| a Y component, and whose correlate is a Z component.  The law of
| multiplication is plainly (Y:Z)(Z:X) = (Y:X), (Y:Z)(X:W) = 0,
| and the application of these rules to the above values of
| 1, i, j, k gives the quaternion relations
|
|     i^2  =  j^2  =  k^2  =  -1,
|
|     ijk  =  -1,
|
|     etc.
|
| The symbol a(Y:Z) denotes the changing of Y to Z and the
| multiplication of the result by 'a'.  If the relatives be
| arranged in a block
|
|     W:W     W:X     W:Y     W:Z
|
|     X:W     X:X     X:Y     X:Z
|
|     Y:W     Y:X     Y:Y     Y:Z
|
|     Z:W     Z:X     Z:Y     Z:Z
|
| then the quaternion w + xi + yj + zk
| is represented by the matrix of numbers
|
|     w       -x      -y      -z
|
|     x        w      -z       y
|
|     y        z       w      -x
|
|     z       -y       x       w
|
| The multiplication of such matrices follows the same laws as the
| multiplication of quaternions.  The determinant of the matrix =
| the fourth power of the tensor of the quaternion.
|
| The imaginary x + y(-1)^(1/2) may likewise be represented by the matrix
|
|      x      y
|
|     -y      x
|
| and the determinant of the matrix = the square of the modulus.
|
| C.S. Peirce, 'Collected Papers', CP 3.323, (1882).
|'Johns Hopkins University Circulars', No. 13, p. 179.

This way of talking is the mark of a person who opts
to multiply his matrices "on the right", as they say.
Yet Peirce still continues to call the first element
of the ordered pair (i:j) its "relate" while calling
the second element of the pair (i:j) its "correlate".
That doesn't comport very well, so far as I can tell,
with his customary reading of relative terms, suited
more to the multiplication of matrices "on the left".

So I still have a few wrinkles to iron out before
I can give this story a smooth enough consistency.

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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