Customer Reviews

Laws of Form

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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 econ159@it.canterbury.ac.nz if you want a copy of my academic paper explaining the value of Spencer Brown's achievement.

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.
...

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.

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.

attempts to define a seemingly simple word: difference. In a pre-chaotic sea of jello where all worlds are possible, the emergent world that does follow is determined by the first distinction. Brown attempts to trace this initial severance and chart its after development. Eschewing the Aristotlean "A is not non-A" in favor of the more useful definition of distinction as "perfect continence," Brown goes on to enunciate two axioms 1) the value of a call made again is the value of the call, and 2) the value of a crossing made again is not the value of the crossing. From these, the book flows. Backwards it flows to pre-kindergarten conceptions of number, order, and eventually difference itself. On the way he solves the self-referential paradox by positing the existence of four distinct classes: true, false, meaningless and imaginary. The last of these has recently produced breakthroughs in software logic applied to virtual reality paradoxes. George Boole would have peed his pants over the motherlode of this book that finally offers the arithmetic of the algebra that still bears Boole's name. An incisive appendix "the Calculus applied to logic" renders university texts on sentential logic redundant. A similar achievement renders Bertrand Russell's theory of types (along with a chunk of Principia Mathematica) into a cocked hat...as Russell himself acknowledged in 1967. The idea paintakingly iterated in Laws of Form led to a significant patent for a device that counts entirely by logic, with switches only, and with no artificial time delays, an enormous energy gain in iteration-fraught engineering disciplines. (For validation of this, see Colin Johnson's website on analog computing). Dry wit and a delightsome British humor spice the endnotes, numerous appendices, and prescient forewords.

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.

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.

Philosophers, mathematicians, teachers, or anyone interested in logic or semiotics should acqaint themselves with this thin rigorous volume which advances a propositional form. The implications are the real reward as this singular exercise encourages investigation and invention in the representation of knowledge, a point of departure with real practical value. If you've read Bateson, Brand, and the second order cybernetics gang, this is a natural. If not, you will be intrigued into a rewarding realm of fundamental inquiry.

Spencer-Brown's articulation of the Laws of Form is of the same order of magnitude as Einstein's Theory of Relativity.

I have been reading and re-reading this book for more than 20 years. In full, at least 8 times. Countless "hits" to review specific ideas.

Reviewers here who have panned LOF might be right in thinking that my appreciation for this work is based on what I have read into it, not what it contains intrinsically.

On my first reading, I was perplexed and somewhat disappointed ("That's it?") But I felt that it deserved a second, more deliberate reading. I'm not a mathematician, but my fourth reading was assisted by conversations with a professional scholar of logic systems. And by my fourth reading I was better prepared to appreciate the subtlety and power of LOF.

The concepts regarding time as a function of memory (instead of the reverse) still awe me.

If I could only take three books with me (to Mars, to the future, to the past), LOF would be the *first* that I grabbed.

"The universe is constructed in such a manner that it can see itself." - GSB / LOF

A little tiny book that I have read scores of times in 20 years. For the mathematically inclined reader, creates a formal seed upon which new, clear concepts will crystalize for years. The high price will, unforturnately, discourage casual readers.