by George Spencer-Brown

isbn: 0963989901
subject: Nonfiction, Mathematics
finished: 1/1/2002

It's often interesting to read a controversial book, if for no other reason than it forces one to make up one's own mind about the content. Laws of Form seems to be especially controversial: it's either a ground-breaking work of genius or the output of a crackpot, depending on who one asks. I picked it up since the controversy was intriguing and because I've always enjoyed learning about formal systems.

There are good and bad things about this book. The best thing about it is that it introduces an interesting new notation for representing propositional logic that permits any proposition to be constructed out of different arrangements of a single symbol. Unfortunately, the bad things about this book outweigh the good things; I'll try to enumerate some of them. The primary technical mistake is a failure to distinguish between a formal system and its interpretation. In other words, a new notation for propositional logic is only that, and it cannot tell us anything new about the logic: it merely makes certain formulae more convenient to express (and, in general, certain formulae less convenient). The second mistake Spencer-Brown makes is to spend the first ten chapters of the book deriving nearly trivial theorems, while describing these theorems in such a way that non-mathematicians are likely to be confused into thinking they have some significance. In general, they don't -- propositional logic is such a simple, inexpressive formal system that it is nearly useless on its own. The third mistake is the tremendously bizarre Chapter 11, which introduces a logical equivalent of imaginary numbers. This is an interesting idea that arguably has the potential to make Russell's paradox into a non-issue, but the chapter never goes anywhere: it's confused and mercifully brief. The fourth mistake is the language used in the book: rather than using established terminology, Spencer-Brown introduces a large number of colorful new terms -- in some cases this is merely inconvenient but in others it feels deceptive. And this leads to the final mistake made in Laws of Form, which is that it hopelessly oversells itself. It claims to be a breakthrough, a generalization of existing formal systems that has profound implications across a wide range of disciplines. It isn't -- it's a bunch of flowery language layered on top of a formal system that freshmen in college quickly get bored with.

To focus on individual mistakes that Spencer-Brown made is to ignore the larger point that this book contains a lot of text but very little content. It's like a large false diamond: an attractive curiosity; worthless, but still alluring to the unwary. There are plenty of good books about logic out there; for example, I recommend Howard DeLong's A Profile of Mathematical Logic -- it's not any harder to find than Laws of Form, and is incomparably better.