More Laws of Form

I mentioned earlier that I was taking a fresh look at G Spencer Brown's Laws of Form because I found a document by Mark W Hopkins that redescribes the content of this book in more traditional mathematical notation.

Hopkins starts by converting Brown's spatial notation to a linear notation using 1 to represent the empty space and () to represent distinction. We have the usual transformation rules:

(()) -> 1 and ()() -> ()()

He places this in the framework of a transformational category which is esentially a category in which the objects are expressions and the arrows are applications of transformation rules. It's pretty obvious that the above system is isomorphic to the usual boolean algebra (0,1,\/,/\,~) with

1 -> 0
() -> 1
(a) -> ~a
a b -> a \/ b

Not much of interest there.

Brown also describes expressions with feedback. But these are nothing other than sequential logic circuits, ie. circuits where the state at time t+1 is a boolean function of the state at time t. Hopkins formalises these as simple finite state automata. Nothing new there either. I certainly don't see any advantage over a language like Verilog which is designed for sequential logic.

Now, suppose A= 1. Then ((A)A)->1. Similarly, if A=() then ((A)A)->1. So we have the rule ((A)A)->1. So if we extend the language of Laws of Form to include variables we'd like to be able to deduce ((A)A)->1 within the language. We certainly can't deduce it with the rules introduced above. So we must introduce some new transformation rules. One of them is ((A)A)->1. Mark Hopkins describes two such systems and shows how they are complete in the sense that we can make all of the deductions we expect to be able to make in the presence of variables. This is vaguely interesting but really we're just learning how to manipulate boolean expressions, nothing profound.

Unfortunately I don't have a copy of Laws of Form to hand so I can't check Hopkins description against Brown's. My last reading of the book certainly didn't make as much sense as Hopkins explanation so I'm anxious to compare. But if this really is the content of Laws of Form then there really isn't much to it. However, I didn't see any discussion of imaginary logic values in Hopkins' document. It may be that these values are simply oscillating states in sequential logic circuits. I really must obtain the book again.

Anyway, it's fun to explore the Laws of Form web site. Apparently a group at the now defunct Interval Research company was trying to make use of Laws of Form in reconfigurable computing. I also came across this. This last document claims to simplify mathematics but all it seems to do so introduce a cumbersome notation. I'd rather use lambda calculus or combinatory logic.

Update: I just found this published critique of Laws of Form.

Primer

At the weekend I watched the movie Primer. An interesting time travel story made on a budget of $7,000. I mention it because unlike movies like Back to the Future it seems to have a certain rigour. There are no people slowly fading out as characters fail to achieve their goals in another 'timeline' and apparently no people remembering things that aren't in their own histories.

Or at least I think this is the case. Unfortunately the rigour of the movie might be its downfall - it makes no concessions to the audience to make the story easy to follow. When we see one of the characters in a scene it's not easy to tell which of the scenes, either earlier or later in the movie, are in that characters history at that time. And of course by time a potential candidate to be in his history does appear later in the film, we no longer remember the details of the earlier scene. Memento was a straightforward film shown in reverse. (And also inconsistent, as if reversing the story was too much for the scriptwriter to handle.) Primer forks, loops and twists like no movie I've ever seen. As a topological space it has a high genus.

Unfortunately, it was too much of a challenge for me. I really have no idea what happened, although there appears to be an interesting sequence where one of the characters repeatedly travels back in time, each time making slight modifications to his actions, in an attempt to iteratively optimise an outcome. But what that outcome was, and why he tried to achieve it, is beyond me. I guess I'll have to watch it a few more times. It's very rare that a movie has provided as much of an intellectual challenge.

And one point in its favour - not only was the technobabble not excruciatingly bad, it was even believable in places. Primer is one of the few truly Science Fiction movies I have seen.

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.

I'm a descendant of Bessel and Gauss

Via Artin in fact.

No, we don't share any genes, but instead I have inherited their memes. See here for details.