Laws of Form

Laws of Form is a book published by G Spencer Brown, I think in around 1972. It starts off pretty straightforwardly by introducing a form of boolean algebra using one symbol a bit like an L rotated through 180 degrees. The key feature of this symbol is that it makes a distinction between an inside and an outside so we can replace it with parentheses. If I write (A) then A is inside the parentheses and in ()A the A is outside. This distinction between inside and outside is in fact what he names his symbol, a 'distinction'.

We can write a distinction on its own as (). Now we can introduce some simple rules. A distinction inside a distinction isnothing. Ie. (()) =. I haven't left out the right hand side. As it might be confusing I'll use the symbol 1 to represent a blank space. So (()) = 1 and trivially (1)=() and ()1=() as 1 is just another way of not writing anything at all. Brown also introduces the rule ()() = (). These rules now have an interpretation in boolean algebra:

1 -> false
() -> true
(a) -> not a
a b -> a or b

This is a nice notation for boolean algebra and may originally have been invented by C S Peirce.

But after a few pages I find Laws of Form to be completely incomprehensible. Brown suddenly makes a leap from simple Boolean algebra to completely bizarre operations that result in the description of some kind of binary sequential circuits related to solutions to recurrent equations. I more or less dismissed Brown as a crackpot.

Curiously, however, Louis H Kauffman, a well known Knot Theorist among other things, has taken to Laws of Form with zeal. He sees a commonality with the types of diagrams that he uses for Knot Theory. But he goes further and sees connections with the properties of spacetime. Kauffman has a Paper here as well as some material in his great book Knots and Physics. Still, when you look closely the comments are all pretty vague and really don't go beyond the simple boolean algebra that Brown discusses towards the beginning of Laws of Form.

So even though there's at least one reputable mathematician who's interested in his work he still seemed like a crackpot. Take a look at USENET discussions about the book and you'll see endless quasi-mystical discussions about the book with very little meat - all tending to confirm the crackpot hypothesis. And if you wanted any kind of confirmation that he was a crackpot you just have to make note of the fact that a conference on Laws of Form was held at Esalen.

But then I came across Mark W Hopkins' USNET posts on the subject, in particular here. For the first time in my knowledge a mathematician has attempted to redescribe the later parts of Laws of Form (what Hopkins calls LOF2 and LOF3) in a non-trivial way. I haven't read the posts yet but judging by the quality of many of his other posts it should be interesting.

My question now is this: who is Mark W Hopkins. I've learned a ton of stuff about finite state machines and regular languages from his posts but his web presence is pretty ephemeral and seems to end a few years ago.