Dr. Randall Whitaker

  George Spencer Brown
Laws of Form (Calculus of Indications)

• Laws of Form

  The book presenting the calculus of indications

• George Spencer Brown

  Information about Spencer Brown himself

• Bibliography

  An archival presentation of the most comprehensive bibliography assembled on these topics. (On a separate Webpage)

• Links

  ...to Internet resources on Spencer Brown and/or his Work

As of 2001, many of the best Websites on Spencer Brown's work have disappeared.

This Focus File is an attempt to salvage and preserve some of the 'missing' resources, as well as provide a comprehensive set of pointers to those resources that survive.

In addition, many authors whose work explicitly intersects (or implicitly parallels) the issues and themes of autopoietic theory have made allusions to Spencer Brown and his calculus of indications.

These facts are the basis for considering Spencer Brown's work a relevant cognate area for people interested in the biology of cognition / autopoietic theory.

This Focus File is intended to serve as:

• An archival asset preserving materials otherwise endangered (or already disappeared from the Web)

• A reference aid in locating those materials currently available.

NOTE:  It is not my intention to establish or maintain a Spencer Brown Webspace per se. I am presenting these materials in the context of their relevance to the biology of cognition / autopoietic theory.

• More specific information on the book can be found in the Bibliography

• Access to the book's current edition can be pursued through the Book Shop

What's the relationship between Spencer Brown's work and autopoietic theory?

In his 1979 Principles of Biological Autonomy, Varela intensively analyzed and built upon Spencer Brown's calculus as an analytical instrument and a depictional schema. In addition, multiple papers Varela (co-)authored during the 1970's focused on the calculus of indications in the course of pursuing formal logic as a tool to be applied in autopoietic theory.

So how does Spencer Brown's work relate to the biology of cognition / autopoietic theory?

And what was so important about this relationship that it recommended the calculus of indications as the focus of so much attention by Varela?

"The theme of this book is that a universe comes into being when a space is severed or taken apart."  - (Laws of Form, 1972 edition, p. v)

Spencer Brown intersects Maturana and Varela by treating this 'severance' as involving an act of distinction.

Maturana and Varela identified distinction as the most fundamental element or act in an observer's cognitive operations. Through distinction, the observer "...specifies a unity as an entity distinct from a background and a background as the domain in which an entity is distinguished." (Maturana & Varela, Autopoiesis and Cognition, 1980, p. xxii)

"The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane."  - (Laws of Form, 1972 edition, p. v)

The notion that distinction is realized in (e.g.) the physical space by boundaries separating inside from outside has an obvious resonance with other key themes in Maturana and Varela's writings, including:

• The boundaries generated by, and definitive of, autopoietic unities
• The differentiation of domains of interactions for a system's (internal) components versus itself as a unity operating within a medium / environment
• The differentiation of phenomenal domains via which an observer can engage a system as either a simple or a composite unity

Moreover, Spencer Brown's allusion to the more abstract instantiation ('circle in a plane') opens up the door to depiction of distinctions generally and a schema for describing / explaining them.

"By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance."  - (Laws of Form, 1972 edition, p. v)

Spencer Brown set out in Laws of Form to address these 'basic forms' using a novel mode of representation (the 'mark'). This led to the creation of a calculus of indications - a formal logical framework complete with its own arithmetic.

Spencer Brown's calculus was therefore an obvious candidate framework for formalizing distinctions and hence providing a formal representational schema for addressing the operations of the observer. This explains why Varela dedicated much effort to analyzing (and eventually extending) the calculus of indications during the 1970's.

Beyond this, distinctions lie at the heart of Maturana's ontology of the observer. Every domain of explanations of an observer's praxis of living is constituted and/or contexualized by a 'domain of reality'. Such a domain of reality is interrelated with distinctions in two ways:

• "Each configuration of operations of distinctions that the observer performs specifies a domain of reality as a domain of operational coherences of his or her praxis of living in which he or she brings forth particular kinds of objects through their application..."

• "Although all domains of reality are different in terms of the operational coherences that constitute them, and, therefore, are not equal in the experience of the observer, they are all equally legitimate as domains of existence because they arise in the same manner as they are brought forth through the application of operations of distinction by the observer in his or her praxis of living."

  (Maturana, 1988a, "Reality...", p. 31)

The act of distinguishing specifies (even if only implicitly) both something referred to and the context in which it is manifest as such. Because a unity is brought forth only through distinction, "...each time we refer to a unity in our descriptions, we are implying the operation of distinction that defines it and makes it possible." (Maturana & Varela, The Tree of Knowledge, 1992 edition, p. 40)

As such, distinction serves as the fundamental element in the epistemologically constructivistic theories of Maturana and Varela. The relative subjectivity entailed in such constructivism is in turn implicit in Spencer Brown's book when he writes...

"The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please."  - (Laws of Form, 1972 edition, p. v)

How was Spencer Brown's work received?

A deep exploration of the debates that have swirled around Laws of Form is beyond the scope of this particular Web page. You can pursue that subject through the links provided below.

For now, I shall simply give you a flavor of the situation with the following excerpts drawn from issue number 6 of The Observer.

Haim Shaul wrote::

"Spencer-Brown's Laws of Form is a rather obscure book with perhaps an unusual viewpoint. Stafford Beer reviewed it in Nature when it was first published in the early 70s or late 60s. It has attracted the attention of people who have absorbed all the typical flotsam and jetsam of several disciplines and still find something left to be desired. What he does in this little book is so strange it has been left more or less in oblivion by academic humanity. The book is rather dense in the sense of intensely cogent and apparently a bit "mystical." He cites in the original hard cover a line from the Chinese text called Tao Te Ching in Chinese ... which is just the sort of thing which will turn many a head in another direction. If you like only what is familiar to you you may not like this little book. You must be willing to follow him out into his rather intense world to develop an appreciation of his efforts which could be called decent."

"... L.L. Whyte refers to this book with some frequency in his later writings. John Lilly, the dolphin fellow, also picked up on it. ... Spencer-Brown also wrote a book on Probability and Scientific Inference and contributed too to the literature of scientific parapsychology -- re statistical interpretation of parapsychological data as I recall."

"... Since Spencer-Brown arrives so much from left field as it were he has been ignored and not in any way assimilated by the basically conservative nature of academia."

"...When I first encountered Spencer-Brown's book I thought it highly interesting and provocative but I could hardly find a soul who shared my interest. That was in 1972 I think. I took the book to people I knew then in the Mathematics department here at Indiana University, and also to people in Physics, and Philosophy. Not a soul was familiar with the book. The typical and near universal response was: if this book has content I should already know of it, since I do not the book must lack content. I found out that what in my milieu was an atypical type could be made interested in this author or his book and that there were no atypical types in my milieu, or nearly none. I had a friend who was interested but he was most atypical I should think in any milieu, and I did eventually meet a few other people who had found merit and an interest in it. I mentioned Mr Eberhart [Steven Eberhard III, a mathematician at Washington State - ed.], who I met at a conference on multi-valued logic held here at Indiana University in 1975. But Mr Eberhart was also an anthroposophist, which was nothing in the least against him from my point of view but again in the pattern of an atypical personality. Well too John Lilly who had worked with dolphins with atypical experimental methodologies and atypical assumptions and weltanschauung had voiced his interest in Spencer-Brown's book but Mr Lilly too was an atypical personality. The same could be said of Lancelot Law Whyte who too had shown an interest. Stafford Beer, who reviewed Laws of Form in Nature seemed to me the most outside the pattern of odd-balls who took an interest in Spencer-Brown but he says in that review that he suspected that he was reviewing the work of a genius and saying such a thing seems to cut against some resentment and animus among all those who identify themselves with a certain idea of the establishment. New ideas, new points de depart, must crawl against a kind of ancestral miasma. ..."

Frances Heylighen wrote:

"I don't now much about the impact this book has had in philosophy, but it is referenced very frequently in cybernetics (especially the constructivist epistemologists of the "new" or "second order" cybernetics school, see e.g. Heylighen F., Rosseel E. & Demeyere F. (eds.) (1990): Self-Steering and Cognition in Complex Systems. Toward a New Cybernetics, (Gordon and Breach Science Publishers, New York).

I have myself been very impressed by Spencer-Brown's concept of "distinction", which I have elaborated in what I call a "distinction dynamics": a theory of how new distinctions arise out of combinations of existing ones."

Eric Oberle wrote:

"Brown has had an impact reaching far beyond cybernetics. Not only has he been invoked in recent debates about biological and psychological autopoiesis (as in the works of Varela and Maturana: Autopoiesis and Cognition: The realization of the living, Dordrecht 1980; Heinz von Foerster, Observing systems (Seaside California, 1981).

Brown has also been incorporated into Parsonian sociology in some interesting (and sometimes perverse) ways by Niklas Luhmann ..."

John D. Collier wrote:

"The Laws of Form are equivalent to propositional calculus. Spencer Brown showed LoF -> PC in his book. B. Banaschewski showed the opposite entailment in Notre Dame Journal of Formal Logic, 3, (1977): 507-509.

I always suspected that the Laws of Form and the Propositional Calculus (or Boolean algebra) were equivalent, and have treated them as such in my papers on distinction dynamics. It is nice to see that somebody actually proved this result, though.

As I guess people will start to wonder by now why we need Spencer-Brown then, if his algebra is merely equivalent to something which is very well-known, I want to remark that the axiom system of LoF is much simpler than the usual axiom systems for PC. Moreover, these axioms are derived from 2 extremely simple and intuitively understandable properties of the act of making a distinction."

As you can gather from the above comments, Laws of Form has received a mixed reception. Although in some circles it is treated with something approaching awe, there are detractors who dismiss Spencer Brown with extreme prejudice. Some dismiss him because of the formal equivalence between his logic and the Boolean / propositional logic it supposedly surpassed (cf. Collier's remarks above). Others are frank about having been put off by Spencer Brown's penchant for being cryptic -- both in his writings and in his behavior at meetings and lectures.

This has led to a polarization of opinion among the very few people who have stated opinions -- themselves a small subset of the few who have ever read Laws of Form. At one extreme are those who associate Spencer Brown's work with not only philosophy, but spirituality as well. The opposite extreme is (tastelessly) illustrated by the bibliographic entry for Laws of Form at the Santa Fe Institute Website whose stated basis for recommending the book is "Just for laughs!"

Photo from cover of:
Only Two Can Play This Game

(written under the pen name 'James Keys')

A Timeline of Spencer Brown's Life and Career

Source = Jorn Barger(?)

1923:  born 2 April, in Grimsby, Lincolnshire, England (father engineer, chessplayer, painter, writer)
1936-41:  Mill Hill; University of London
1941-43:  London Hospital Medical College
1943-47:  Royal Navy (telegraphist, radio mechanic)
1947-52:  Trinity College Cambridge (chess, gliding)
1949-50: Racing driving with Gavin Maxwell
1950-51:  Philosophy with Ludwig Wittgenstein
1951-52:  Perrott Student in Psychical Research (hypnosis)
1952-58:  Christ Church Oxford
1953.:  "Statistical significance in psychical research" published in Nature
1957:  Probability and Scientific Inference published
    [Paper citing this work]
1959-61:  Chief Logic Designer, Mullard Equipment Ltd
1963-64:  Adviser to British Rail
1964:  Soccer Correspondent to the Daily Express
1960ff:  Worked with Bertrand Russell in Foundations of Mathematics
1961:  Unpublished ur-LoF "An algebra for the natural numbers"
1963-68:  Senior Lecturer in Formal Mathematics, University of London Department of Extramural Studies
1968-69:  Professional psychotherapist
1969ff:  Department of Pure Mathematics and Mathematical Statistics in the University of Cambridge
1969:  Laws of Form is first published
1969:  UK seminar for Assoc of Teachers of Mathematics [eyewitness]
1970:  Founder of publishing imprint Cat Books
1970:  23 Degrees of Paradise published (a collection of poems as 'James Keys')
1971:  Only Two Can Play This Game (a comparison of western and eastern modes of thought and methods in the arts, philosophy, religion, and the sciences) as James Keys
1972:  Founder of publishing imprint Spencer Brown & Co
1973:  Esalen lectures on Laws of Form
1975:  Publishes melody-index The Directory of Tunes by Denys Parsons
1976:  Visiting Professor of Mathematics in the University of Western Australia
1977:  Married 'by special contract' Katherine Lynn Parker of Denver Colorado. Separated.
1977  Visiting Professor, Department of Computer Science, Stanford University
1977:  Consultant to Xerox PARC
1977-78:  Stanford and Palo Alto lectures:  The Four-color Map Theorem as a Problem in Formal Quaternions
1979:  Laws of Form republished
1980-81:  Visiting Professor, University of Maryland; lectures: What is Mathematics, Formal Arithmetics of the Second Order, and Cast and Formation Properties of Maps.
1982:  Founder of publishing imprint Universal College Press
1982:  "The Falkland Papers"
1985:  UNI-OPS-sponsored conference postponed indefinitely by "mutual agreement"
1980's?   Professional gambler (card counting); designing a board game
1991:  Jorn Barger visits GSB in London, turned off by megalomania
1994:  Laws of Form republished
1994:  A Lion's Teeth announced, 'fantastic tales'
1995:  Sentinel Training announced
1996:  EMCSR'96 Plenary Lecture (April 10) on graphs at University of Vienna [Announcement]

Laws of Form
(Richard Shoup / Jeffrey Mark James)

This is the latest location for the site I always considered the best on the Web. It contains the best-organized and most comprehensive collection of information of any of the Spencer Brown sites. Established in August 2000, it is somewhat 'hidden' by virtue of not showing up on any major search engine as of February 2001.

Primary URL:

http://www.lawsofform.org

Note: This resolves to the following URL (which you might find useful as a backup address):

http://www.rgshoup.com/lof/

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G. Spencer-Brown, The Laws of Form
John Miller

John provides his collection of notes on Spencer Brown's work, comments received about it, etc. Contains some interesting notes and gossip -- tidbits I'd never run across before.

http://sun.lclark.edu/~miller/spencer-brown.html

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G Spencer Brown resources on the Web
Jorn Barger

Includes a biographical timeline on Spencer Brown himself (the one reformatted and presented above).

http://www.robotwisdom.com/jorn/gsb.html

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Form follows distinction
Harry A. Pappo, Ph.D.

A brief introduction to Spencer Brown's calculus and a comparison of his work with traditional Boolean algebra. This presentation is a component of: Tutorials in: Simulation Technology For the Study, Design, and Training of Skills

http://home1.gte.net/simres/k1-lform.htm

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Brown, G. Spencer: Laws of Form Evolution of Consciousness
Book Notes by Bobby Matherne, 1999

Excerpted from A READER'S TREASURY, published by Julian Press/NY in 1972.

Provides a summary overview of Spencer Brown's ideas.

http://www.doyletics.com/arj/lofmart.htm

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The Form and Substance of Non-Numerical Arithmetics

A review of GSB's calculus embedded within a wider discussion. This site (formerly located at 'formal.com') presents this wider discussion in a manner which is every bit as cryptic as critics claim Spencer Brown to be. Considerable patience (and exploration of the internal links) is required to identify and obtain the information on Spencer Brown's calculus.

http://www.xenodochy.org/formal/fs_aritx.html

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Principia Cybernetica Entry: 'Distinction'

A short synopsis of the concept of 'distinction' and its relevance for cybernetics and systems theory.

http://pespmc1.vub.ac.be/DISTINCT.html

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Boundary Math
Department of Integrative Arts
Penn State University

A multi-part exploration of Spencer Brown's notions in the applied area of architecture.

http://wgn111.ce.psu.edu/semiotics/semiotics2.html

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Discussion Thread on Spencer Brown and his Calculus
The Math Forum (at WebCT.com)

This is a series of emails posted on Spencer Brown and his work, starting in 1995. Some of the messages include comments and factoids about the man and his calculus which cannot be found elsewhere.

Starting Point:

http://forum.swarthmore.edu/epigone/geometry-puzzles/71/71@forum.swarthmore.edu

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laws of form, the book, the author
Joao Leao

A single page site, with some fragments drawn from Jeff James' late Webspace on Spencer Brown.

http://gog.harvard.edu/~leao/Tangle/lof.html

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Boundary Mathematics & Architecture Theory

"Bringing the calculus of George Spencer Brown's "Laws of From" to bear on architectural conditions of intransitivity, concealment, consecution, and concinnity."

http://art3idea.ce.psu.edu/boundary/frames_site.html

Includes a brief tutorial entitled "Laws of Form" Basics. (Just click on the 'Boundary Math Step-by-Step' link).

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Glanville, Ranulph
The self and the other: The purpose of distinction, in Trappl, R. (ed.) Cybernetics and Systems '90. Singapore: World Scientific, 1990.

The nature of distinction drawing (Spencer Brown) is examined with special reference to the distinction between the self and the other. Glanville's critique of Spencer Brown and his calculus of indications.

http://www.univie.ac.at/constructivism/papers/glanville/glanville90-selfother.pdf

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Glanville, Ranulph
The cybernetics of value and the value of cybernetics. The art of invariance and the invariance of art, in Glanville, R. & de Zeeuw, G. (eds.) Problems of Values and (In)variants, Amsterdam: Thesis Publishers, 1995.

Abstract

In this paper, Spencer Brown's Logic of Distinctions is considered in the light of various amendments proposed by the author. These considerations, concerning (in the main) the notion of the separation of the mark and the value of the distinction, are examined in terms of value (meaning) and invariance (remaining, being) in art Objects. The contradiction between an older view of meaning as inherent in Objects and the more recent view of meanings being with the observer is elaborated; and the way that these attitudes can be seen as complementary is related to attitudes to and experiences of such art Objects.

http://www.univie.ac.at/constructivism/papers/glanville/glanville95-value.pdf

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G. Spencer Brown's Laws of Form & John Lilly's Take on It
Saul-Paul Sirag

http://www.stardrive.org/Sirag/

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Jokisch, Rodrigo
SUMMARIES OF CHAPTERS I. TO V. from:
Logic of Distinctions A Protologic for a Theory of Society.

English translation of: Kapitelzusammenfassungen I. bis V. (32-46), from: Logik der Distinktionen. Zur Protologik einer Theorie der Gesellschaft, Opladen:Westdeutscher Verlag, 1996.

Summary overview of sections of a German thesis. Contains some interesting remarks about Spencer Brown in the context of distinctions and differentiations as applied to the social.

http://www.geocities.com/Athens/Thebes/4211/jokischlog_i_v.html

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Brown, Mind and Logic
Acta Analytica Vol. 8, Issue 10, 1993
Table of Contents

Listing of abstracts for an issue apparently dedicated to analysis of Spencer Brown and his logic.

http://www.uni-mb.si/~actaana/toc10.htm

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