Laws of Form

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Summary: Reader be Alert.
Review: Chapters 5-10 of this book are elementary Boolean algebra, reworked using a notation C S Peirce proposed in papers written in 1886 but not published until 1993. The result is called the primary algebra, which features a curious little twist: complementation with an empty scope can be interpreted as Boolean 0 or 1. The notation has a great virtue: self-duality. It has a drawback: you can't write it at the computer without a good equation editor. All the benefits of Spencer-Brown's notation carry over to an alternatie notation that uses only keyboard characters; EMail [e-mail] for specifics.

The primary algebra is also isomorphic to Peirce's alpha existential graphs, a remarkable formalism Peirce derived in the 1890s. Even though this work was republished in his 1933 Collected Papers (cited in LoF), Peirce's logical graphs have been subsequently ignored by all but a handful around the globe.

Chpts. 1-4 make for hard enignmatic reading, as Spencer-Brown seems indifferent to, or ignorant of, definitional conventions in the math and logic literatures. These chapters convey a new variety of Boolean arithmetic, called the primary arithmetic, going somewhat beyond Peirce. There are two sets of metatheorems: one shows that logical equivalence is an equivalence relation, the other that logical equivalence partitions all primary arithmetic formulae into two equivalence classes. The terminology here is that of standard math and does not appear in LoF.

Appendix 1 relates the primary algebra to the Sheffer stroke. Appendix 2 interprets the primary algebra for zero-order logic, monadic logic, and syllogisms. These appendices are unnecessarily enigmatic in parts, but do make some exciting points. In particular, Spencer-Brown's treatment of the syllogism
is by far the simplest and most elegant I know.

Chpt. 11 makes the highly speculative claim that certain recursive Boolean equations not solved by either 0 or 1 can be seen as having an "imaginary" solution. The discussion here is not at all informed by the substantial literature on recursive and primitive recursive functions. The author claims, elsewhere, that proofs of the Four Color Map theorem, Fermat's Last theorem, and Goldbach's conjecture are impossible without a logic incorporating imaginary truth values. This claim is refuted by the subsequent proof of 2 of these 3 conjectures, using conventional mathematics grounded in classical logic.

This is a book about Boolean algebra, presented without set theory. Hence all claims that the formalism of this book renders the Theory of Types unnecessary are wholly irrelevant. The Theory of Types requires a background logic including a domain of discourse and predicate letters; in other words, first order logic. Bertrand Russell missed this fact because he was 95 years of age and out of it when he met with Spencer-Brown.

That this book excites many people is evidence that Boolean algebra, and the remarkable logic of Charles S Peirce, are much less known than they deserve to be.

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Summary: Pancake Demo, "The Laws" calculus of indications (Windows)
Review: I hope to foment a revolution in grades 4-12 math teaching based on this brilliant intellectual achievement of G.Spencer-Brown's. Get it at any price. My review is mainly in the form of a working demo that evaluates your entered expressions in Spencer-Brown's calculus of indications, and lets you watch the evaluation step-by-step. Hot from my Visual Basic compiler, you have only my word it's virus-free. You can run it in multiple copies to play with the arithmetic and the algebra (discover algebraic results of your own!). I'll email it to anyone on request. I have turned S-B's original notation upside- down and sideways for easier teaching and ease of handling in programming and computer entry. Your Windows/SYSTEM folder needs MsVBvm60.dll for it to run. I'll send that, too, on request.

Dave Zethmayr --- [e-mail]

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Summary: We Take the Form of Distinction for the Form
Review: I take the key sentences in Spencer-Brown's Laws of Form to be the first two sentences at the beginning of Chapter 1: "We take as given the idea of distinction and the idea of indication, and that one cannot make an indication without drawing a distinction. We take therefore the form of distinction for the form." This book is a carefully crafted and beautifully written account of how the act of imagining a distinction gives rise to worlds of multiplicity from a unity where no distinction is actually possible. The first mathematics that so arises is remarkably close to the boolean mathematics with which all logicians, engineers and philosphers are familiar. Once discovered it is easy to exhibit. Let < > stand for a typographical distinction between outside < inside > outside. Note that in imagining distinctions using linear typography, one must make extra cuts between right and left. Drawing circles in the plane is easier (and C. S. Peirce did this long before Spencer-Brown). Spencer-Brown uses a planar notation that is simple to write and less easy to type. In any case, we make a mathematics from the distinction < >. Think of < > as an "elementary particle" that can interact with itself in two ways. 1. It can interact with itself and produce itself, or it can produce two copies of itself from itself. < > ----- < > < > Read the dotted line in either direction. 2. It can interact with itself to cancel to nothing, or a pair of two copies of the particle can emerge from nothing. < < > > ----- Yes that's nothing on the right hand side, but maybe you would like a symbol for nothing. Ok. Let # stand for nothing. This means that you can erase # or put it in whenever you want to, and that means anywhere. Then we have < < > > ----- # With these modes of particle interaction we have an arithmetic of distinctions. For example < < > < > > -----< < > > ----- # The patterns of this arithmetic have their own algebra, and when one makes the critical distinction between < > as an operator, and < > as a value, this algebra gives rise to the patterns of boolean algebra. There is much more, but the key point is the simplicity of this approach. This simplicity can be applied to many complex systems to locate the key patterns that make them tick. The mark < > is itself an imaginary boolean value. At the outset the mark could be any imagined distinction at all, and the reader will have to ask how those distinctions managed to appear so solid and real. Two marks in a line do not create an inside and an outside. You the reader accomplished that trick. Then again, the mark was not boolean until the context became boolean, and operators separated from operands. This separation is a departure from the beginning. Later considerations in Chapter 11 of Laws of Form about imaginary values are related to this original imaginary state. The temporal interpretation of values i such that < i > = i calls the state of distinction into question, and either returns us to the imaginary source or propels us into temporality. Chapter 11 shows how digital circuitry has the structure of that apparently metaphysical discussion. And the theory of types? Well take a look at your Godel-Bernays set theory and realize that the usual resolution is to imagine sets and classes, with classes a bit more imaginary than sets. (A set is a member of a class. A class is never a member of anything.) The usual technical solution is to introduce imaginaries in the "right" place and to tell the users what they can say and what they cannot say. Spencer-Brown is rude enough and honest enough to admit this situation right in the beginning. There is no need for the theory of types because it is a matter of creativity just how you make your distinctions, and how you want to avoid inconsistency. How will you behave when the next new clever inconsistency in formal mathematics is discovered? A good reader of Laws of Form will be happy and ready to explore the anomaly.

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Summary: Laws of Form ( huh? )
Review: In a way, George Spencer-Brown's "Laws of Form" is an elaborate math puzzle. The author has given you the bare minimum of information to figure out what the heck he is talking about; your assignment ( should you choose to accept it ) is to investigate the fields of logic, symbolic logic, Boolean logic, and set theory, to attempt to reconstruct the mathematics behind the so-called Calculus of Indications presented in the book. In my own case, it took almost seven years of occasional attention to come up with the essential idea behind the math, namely the symmetry between AND-spaces and OR-spaces. It may not take you that long.
Contrary to what some other reviewers have written, Bertrand Russell did not praise this book--he seems to have been just as baffled by it as anyone else. He did praise the ideas presented in the book, but only after Spencer-Brown met with him and explained it to him.
It seems likely that the sections of the book were developed as lecture notes to be handed out in class. Presumably the professor would tell you what he was talking about, and the handouts would be supplemental reading. Unfortunately, all that we get in the book is the supplemental reading.
When you are looking for a tool, you don't want, or need, a math puzzle. This is why the notation and concepts presented in the book have never caught on with philosophers, mathematicians and engineers in spite of their clear superiority over the techniques of syllogism logic, symbolic logic, Boolean logic and set theory.
I have had a lot of fun with this book, but you shouldn't think you're going to get a lot out of it in your first reading.
...

Rating:
Summary: Obscure, flawed and self important little book
Review: Spencer Brown's book produced quite a stir when it was first published. Its nutty, zen like "definitions", its weird symbols, trivial proofs, Betrand Russell's weighty recommendation and overall odd presentation quickly gathered it an eager horde of fans from all disciplines notably system research. Never mind that what the book claimed to do was erroneous (it claimed to develop a one symbol algebra), never mind that the final result smacked of Boolean algebra, never mind that Pincava showed that it was isomorphic to certain well known algebraic systems... Along with other fads like the Q-Analysis of Atkins, Varela's autopoeisis nonsense, and other cybernetic balderdash, Spencer Brown's algebra died a natural death. Maybe not. Christopher Alexander's nonsense has been reborn as "Pattern Language Methodology" in Software "Engineering". So why not? Give a few years more, and it may yet find application in say, the "why" of salad dressings or some such useful human endeveaour.

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Summary: An outstanding intro to logic without Quantifiers
Review: This book is indeed mainly a new (but better) notation for Boolean algebra, a review of how Boolean algebra can be used to represent formal logic, all with New Age trappings derived from Wittgenstein, R D Laing, and from dubious etymology. To top it all off, Spencer Brown's claim that his formalism would be needed to prove the Four Color Theorem and Fermat's Last Theorem has been emphatically falsified.

Nevertheless, this is an astounding book. Boolean algebra is the formalism upon which all of information technology rests. Formal logic deserves a far greater place in educational practice than has been the case in recent decades. A number of Brown's more basic ideas should be incorporated into the junior high curriculum. Finally, some of Brown's advanced ideas such as the imaginary truth value, that memory precedes time, and so forth, deserve more academic attention than they have gotten to date. I emphatically believe that there is a lot here from which the professional mathematician and logician could benefit.

Rating:
Summary: An outstanding intro to logic without Quantifiers
Review: This book is indeed not much more than a very elegant re-exposition of Boolean algebra and the propositional calculus.
Furthermore, the essence of Brown's mathematical innovations were discovered by C S Peirce as early as 1885 (but published only after LoF was published). Nevertheless, LoF is no mean feat.
It radically simplifies sentential logic, switching circuit calculations, syllogisms. I use this book to solve logic problems arising in the computer programs I write.

Outside of electrical engineering, only a few mathematicians and logicians work with logic and Boolean algebra, which should be as commonly known as calculus and linear algebra.

I purchased this book in 1974, and have read many times since. EMail me at [e-mail] if you want a copy of my academic paper explaining the value of Spencer Brown's achievement.

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Summary: Definitely worth further investigation.
Review: This book is not easy to review. Of all the many comments made by others whether derogatory or favourable one impression does come to mind: and that is, this work `appears' to have remarkable potential. Throughout, Spencer-Brown lets the reader know how powerful his ideas are. Even if this seems a little egotistical and exagerated, given the actual results achieved in the book, the basis of the idea i.e. the human habit of making a distinction so the "world" can come into "existence" seems to have the potential to truly describe not only human perception but maybe the physical and conscious world as it stands. Consciousness is a natural part of the construction since the observer is in fact created by the fact he can draw a distinction between himself and the world. Similarly, physical attributes are describable since interaction and causation are fundamental aspects of the universe which the non-numerical mathematics of distinction can create through the fact that making distinctions twice means not making one and so in fact representing, quite by the way, the idea of wholeness; something which was pointed out by Henri Bortoft in his superb "The Wholeness of Nature". For a fuller development of Spencer-Brown's ideas see Edward R. Close and his "Transcendental Physics" where the concepts of dimension and time are constructed from such a mathematics.

Whether this is the be all and end all seems unlikely but its further development may outline some wonderful insights of the world. Definitely worth further investigation and not "content free" as some reviewers have said.

As for the writing style, that leaves something to be desired, Spencer-Brown writes in an obscure way and does not define all his terms along the way including his method of using certain terms to describe various aspects which have very different usages in related mathematics. It is understandable how, because of this obfuscation, his book has been panned by some. It must be remembered that even Bertrand Russell thought it worthwhile, certainly not an empty boast.

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Summary: An intriguing exposition of the foundations of logic
Review: Yes, Spencer-Brown probably got a lot from Peirce, and yes, his "system" is isomorphic to older systems, and yes, it's NOR gates. But his notation is as elegant as you can get to express zeroth order logic, and I think his claim is correct about developing a "natural" arithmetic for logic.

He manages to derive logic from something more primitive (2 rules about how to get the first glimmer of something out of nothingness), and then rightly points out that the derivation had to have logic in it implicitly, that you can't "prove" logic without already using it.

I have to admire the mind that can start from nothingness, and rigorously build up the world from it, even if only a bit of the world. And reminding us that "the universe is constructed in such a manner that it can see itself".

And while some intellectual fads (and a lot of impenetrably bad writing) followed in his wake, I would not be surprised to find some responsible thinkers making better use of this material. I found an unexpected reference to it in Bortoft's monograph "Goethe's Scientific Consciousness", for instance.