Monday, 5 June 2017

Peirce on Quaternions

Had it not been for my discussions with Peter Rowlands at Liverpool University, I wouldn't know what a quaternion was. That I took it seriously was because it plays a vital role in Rowland's physical theory which unites quantum and classical mechanics, and my interest in this has evolved through a desire to tackle the nonsense that is talked about in the social sciences about sociomateriality, entanglement, etc. But then there is a another coincidence (actually, I'm more convinced there is no such thing - these are aspects of some kind of cosmic symmetry). I got to know Rowlands because he is a friend of Lou Kauffman, who has been one of the champions of Spencer-Brown's Laws of Form.

One of the precursors to Spencer-Brown's visual calculus is contained in the existential graphs of Charles Sanders Peirce. So on Saturday, I went looking in the collected writings of Peirce for more detail on his existential graphs. Then I stumbled on a table of what looked like the kind of quaternion matrix which dominates Rowlands work. Sure enough, a quick check in the index and Peirce's work is full of quaternions - and this is a very neglected aspect of his work.

To be honest, I've never been entirely satisfied with the semiotics. But the mathematical foundation to the semiotics makes this make sense. It situates the semiotics as a kind of non-commutative algebra (i.e. quaternion algebra) - and in fact what Peirce does is very similar intellectually to what Rowlands does. It means that Peirce's triads are more than a kind of convention or convenience: the three dimensions are precisely the kind of rotational non-commutative symmetry that was described by Hamilton. I'm really excited about this!

So here's Peirce on the "Logic of Quantity" in the collected papers (vol. IV), p110:

The idea of multiplication has been widely generalized by mathematicians in the interest of the science of quantity itself. In quaternions, and more generally in all linear associative algebra, which is the same as the theory of matrices, it is not commutative. The general idea, which is found in all of these is that the product of two units is the pair of units regarded as a new unit. Now there are two senses in which  a "pair" may be understood, according as BA is, or is not, regarded as the same as AB. Ordinary arithmetic makes them the same. Hence, 2 x 3 of the pairs consisting of one unit of a set of 2, say I, J, and another unit of a set of 3, say X,Y,Z the pairs IX, IY, IZ, JX, JY, JZ, are the same as the pairs formed by taking a unit of the set of 3 first, followed by a unit of the set of 2. So when we say that the area of a rectangle is equal to its length multiplied by its breadth, we mean that the area consists of all the units derived from coupling a unit of length with a unit of breadth. But in the multiplication of matrices, each unit in the Pth row and Qth column, which I write P:Q of the multiplier coupled with a unit in the Qth row and Rth column, or Q:R gives:
      (P:Q)(Q:R) = P:R
or a unit of the Pth row and Rth column of the multiplicand. If their order be reversed,
      (Q:R)(P:Q) = 0
unless it happens that R = P.

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