Laws of Form Bibliography
February 5, 1997 [Updated May 2003, but still contains some obsolete info -- rgs]
This bibliography lists references relating to the book Laws of
Form (1969) and to its author, George Spencer-Brown. The first
edition of this bibliography was assembled by Dave Keenan from an
inquiry posted to the USEnet newsgroups sci.logic and
sci.math in November, 1991, and the collated list was first
posted to those groups on December 7, 1991. It was first published on
the web in 1994.
This bibliography and more information about Laws of Form can be
found on the Laws of Form Web Site at
We would like to thank Dick Shoup, William Bricken, Tim Peters,
Randall Whitaker, Kevin Sahr, William Bricken, Cliff Joslyn, Lou
Kauffman, Jorn Barger, Nancy (Whole Earth Review), R Colin Johnson,
Ralph Gillmann, Ric Brier, Philip Meguire, Bruce Clark,
and the Sciences Citation Index for supplying these references.
Andrew, A. M. (1979). A comment on Varela's ``Calculus for self-reference''
(letter to the editor). International Journal of General Systems,
Asenjo, F. G. (1966). A calculus of antinomies. Notre Dame Journal
of Formal Logic, 7(1):103-105.
Baecker, D. (1993a). Kalkuel der Form. Suhrkamp, Frankfurt.
Baecker, D. (1993b). Probleme der Form. Suhrkamp, Frankfurt.
Baecker, D., ed., (1999). Problems of Form. Stanford: Stanford University
Press. Web site www.uni-wh.de/baecker.
Banaschewski, B. (1977). On G. Spencer Brown's Laws of Form. Notre
Dame Journal of Formal Logic, 18(3):507-509.
Bateson, G. (1988). Mind and Nature: A Necessary Unity. Bantam.
Bateson, G. (1991). A Sacred Unity. Harper, San Francisco.
Beer, S. (1969). Inventing Mathematics, Book review: Laws of Form. Nature,
Vol 223, Sept 27 1969, pp 1392-1393.
An extract appears on the cover of some editions of LoF, possibly with an incorrect reference.
Berkowitz et al., 1988
Berkowitz, G. C., Greenberg, D. R., and White, C. A. (1988).
An approach to a mathematics of phenomena: canonical aspects of reentrant
form eigenbehavior in the extended calculus of indications. Cybernetics
and Systems, 19(2):123-167.
Author's address: Gary C. Berkowitz, NeuroSoft Laboratories, 7777 Fay
Avenue, Suite K129, La Jolla CA 92037, USA. Also: 3284 Karok Avenue, San
Diego CA 92117, USA.
Bochvar, D.A. (1981). A 3-Valued Logical Calculus. History & Philosophy of Logic 2,
English translation. Original in Russian (1938) in Matematiceskij Sbornik 4 (46), 287-308.
Brand, S. (1981). Review: Laws of Form. In The Next Whole Earth Catalog,
page 31. Point Foundation, California, USA.
``This epochal book has spawned whole disciplines, vast and surly arguments,
and not a little mystical endeavor. Not bad for a rudimentary arithmetic
book. In the beginning God said, `Draw a distinction.' The primordial creative
act. You can take it from there.'' [publishing and availability info] [5
extracts of 2 to 4 paragraphs each from LoF]
Bricken, M., 1990a
Bricken, M. (1990a). A calculus of creation. Technical Report HITL-P-91-3,
Human Interface Technology Laboratory of the Washington Technology Center,
University of Washington, Seattle, WA.
Bricken, W. (1986a). A deductive mathematics for efficient reasoning. Technical
Report HITL-R-86-2, Human Interface Technology Laboratory of the Washington
Technology Center, University of Washington, Seattle, WA.
Bricken, W. (1986b). A simple space. In Proceedings of the Sign and
Space Conference, University of California at Santa Cruz. Also as Technical
Report HITL-R-86-3, Human Interface Technology Laboratory of the Washington
Technology Center, University of Washington.
Bricken, W. (1987a). Boundary numbers. Technical report, Advanced Decision
Bricken, W. (1987b). The efficiency of boundary mathematics for deduction.
Technical Report ADS-6824-1, Advanced Decision Systems.
Bricken, W. (1987c). Utilizing boundary mathematics for deduction. Technical
report, Advanced Decision Systems.
Bricken, W. (1990b). Boundary logic (boundary implementations). Technical
Report HITL-P-90-3, Human Interface Technology Laboratory of the Washington
Technology Center, University of Washington, Seattle, WA.
Bricken, W. (1990c). Extended abstract: A formal foundation for cyberspace.
Technical Report HITL-M-90-10, Human Interface Technology Laboratory of the
Washington Technology Center, University of Washington, Seattle, WA.
Bricken, W. (1992). Spatial representation of elementary algebra. In Proceedings
of the 1992 IEEE Workshop on Visual Languages, pages 56-62, Seattle,
Bricken et al., 1987
Bricken, W. et al. (1987). Implementation of the extended program
model for the intelligent program editor. Technical Report TR-1047-03, Advanced
Bricken and Gullichsen, 1989
Bricken, W. and Gullichsen, E. (1989). An introduction to boundary logic
with the Losp deductive engine. Future Computing Systems, 2(4). Also
as Technical Report HITL-R-89-1, Human Interface Technology Laboratory of
the Washington Technology Center, University of Washington.
Bricken and Nelson, 1986
Bricken, W. and Nelson, P. C. (1986). Pure lisp as a network of systems.
In Proceedings of the Second Kansas Conference: Knowledge-Based Software
Development, Kansas State University.
Briggs and Peat, 1989
Briggs, J. and Peat, F. D. (1989). Tubulent Mirror. Harper
See p.67. Mentions GSB and LoF in relation to 'time' and chaos.
Calvino, I. (1976). A sign in space. In Cosmicomics, pages 31-39.
Harcourt Brace Jovanovich.
Profound fiction. Does not cite GSB but is strongly related.
Campbell, H. (1987). Mathematical development of the Laws of Form and related
Brownian algebraic logics. In Artificial Intelligence for Advanced Concept
Aircraft, volume II. Aeronautical Systems Division, Wright-Patterson
Air Force Base.
A thorough search by a Wright-Patterson librarian did not turn up this
paper or the journal/proceedings. The Aeronautical Systems Division no longer
exists. Lou Kauffman, who wrote `Imaginary Values in Mathematical Logic'
when he consulted for Campbell in 1987, says he has not seen this paper
either, but at the time Campbell worked for Systran Corporation, Dayton,
Ohio, 45432, doing classified work for the Air Force.
Comfort, A. (1981). Comments on Kauffman and Varela's `Form Dynamics'.
Journal of Social and Biological Structures, 4:91-96.
Includes a rejoinder from Kauffman and Varela.
Comfort, A. (1989). A bridge to twenty-first century science. The Lancet,
Mostly a review of Penrose, R. (1989) The Emperor's New Mind.
Only cites LoF in passing: ``True, but it [the anthropic principle or Pangloss's
theorem] fails to answer Spencer-Brown's fundamental question: how is it
that the universe has aquired the capacity to be aware of itself?''
Cronen et al., 1988
Cronen, V. E., Chen, V., and Pearce, W. B. (1988). Coordinated
management of meaning: A critical theory. In Kim and Gudykunst, editors, Theories
in Intercultural Communication, pages 66-98.
Cronen et al., 1982a
Cronen, V. E., Johnson, K. M., and Lannamann, J. W. (1982a).
Paradoxes, double binds, and reflexive loops: An alternative theoretical perspective.
Family Process, 20:91-112.
Cronen and Pearce, 1981
Cronen, V. E. and Pearce, W. B. (1981). Logical force in interpersonal
communication. Communication, 6:5-67.
`The hierarchical operator, [cross], is taken from Brown's Laws of
Form, and is read ``in the context of.'' For example, [X over Y-cross],
means ``Y in the context of X'' and implies that whatever is asserted about
Y is limited to this context.' p.24
Cronen et al., 1982b
Cronen, V. E., Pearce, W. B., and Harris, L. M. (1982b).
The coordinated management of meaning: A theory of communication. In Dance,
E. X., editor, Human Communication Theory; Comparative Essays,
Croskin, C., (1978). Ways of Knowing. Cybernetica, 21: 185-92.
Cull and Frank, 1979
Cull, P. and Frank, W. (1979). Flaws of form. International Journal
of General Systems, 5(4):201-211.
Doyle et al., 1987
Doyle, B., Friedman, M., and York, B. (1987). An introduction to forms
and logic. Technical Report BUCS-87-008, Boston University.
Edmonds, Jr, J. D. (1978). Hypercomplex number approach to Laws of
Form and logic. Speculations in Science and Technology, 1(3):245-251.
Engstrom, J. S. (1994). Natural numbers and finite sets derived from
G. Spencer-Brown's laws of form. Thesis, Master of Science in Mathematics,
Maharishi International University, Fairfield, IA.
Author's address: ``Jack'' John S. Engstrom, c/o Department of Mathematics,
Maharishi International University, Fairfield, Iowa,
Etter, T. (1991). Are the Laws of Form non-boolean? ANPA West (Journal
of the Western Chapter of the Alternative Natural Philosophy Association),
Author's address (also ANPA West address): 112 Blackburn Avenue, Menlo
Park CA 94025. Ph: (415) 324 4235. Email: firstname.lastname@example.org.
Flemons, D. G. (1989). Completing Distinctions. Shambhala.
Family Therapy, Taoism, Gregory Batson, and GSB.
Fox, J. (1994). Forth meets laws of form.
Available on the WWW at http://www.ultratechnology.com/laws.html.
Fry, R. (1994a). Neural processing of information. In Proceedings of
the IEEE International Symposium on Information Theory.
Fry, R. (1994b). Observer-participant models of neural processing. IEEE
Transactions on Neural Networks. In press.
Gardner, M. (197?). Review. Scientific American, 1980 (2): 14.
The comment ``beautifully written but content free'' is due to John Horton
Conway (of `game of life' fame). It appears in a Martin Gardner column where
GSB is referred to as the ``maverick British mathematician ...''. ALL of
mathematics (pure mathematics!) is content free. SOME of mathematics is
beautifully written. Conway's remark is a high compliment; backhanded, but
high! Unfortunately, GSB took umbrage at the Conway remark, or so he often
Glanville, R. (1979). Beyond the boundaries. In Ericson, R., editor, Improving
the Human Condition: Quality and Stability in Social Systems. Springer,
Glanville and Varela, 1980
Glanville, R. and Varela, F. J. (1980). Your inside is out and your
outside is in. In Lasker, G., editor, Applied Systems and Cybernetics,
volume II. Pergamon, Oxford.
Goff and Lehmann, 1994
Goff, A. and Lehmann, D. (1994). Non-linear logic: A formal system of mathematical
logic that can indicate and resolve self-referential forms. Unpublished book
A 4 valued self-referential system of logic (true, false, paradox, indeterminate)
with standard references to Spencer-Brown (p.11), Gödel, and Cantor.
They appear ignorant of Kauffman's mathematical treatment, but then they
focus on physical interpretations as feedback circuits. Incomplete document
without a bibliography.
Goguen and Varela, 1979
Goguen, J. A. and Varela, F. J. (1979). Systems and distinctions;
duality and complementary. International Journal of General Systems,
Goldman and Kauffman, 1993
Goldman, J. and Kauffman, L. H. (1993). Knots, tangles and electrical
networks. Advances in Applied Mathematics, 14:267-306.
Gould, W. E. (1977). Review: Laws of Form. Journal of Symbolic
Grattan-Guinness, I. (1982). Psychology in the foundations of logic and
mathematics: the cases of Boole, Cantor and Brouwer. History and Philosophy
of Logic, 3(1):33-53. "It is a very beautiful version of the propositional calculus,
and I cannot understand why it has not become a standard method in logic textbooks."
Grattan-Guinness, I. (2001). The Search for Mathematical Roots, Princeton Univ. Press, 2001.
Pg 527 contains a flawed discussion of LoF and Bertrand Russell, equates the Mark with Sheffer Stroke.
Gunji, Y. (1990a). The algebraic properties of finite cellular automata.
Physica D, 41:282-294.
Gunji, Y. (1990b). Pigment color patterns of mollusks as an autonomous
process generated by asynchronous automata. Biosystems, 23(4):317-334.
Gunji and Nakamura, 1991
Gunji, Y. and Nakamura, T. (1991). Time reverse automata patterns generated
by Spencer-Brown modulator - invertibility based on autopoiesis. Biosystems,
Gunji and Norio, 1991
Gunji, Y. and Norio, K. (1991). Artificial life with autonomously emerging
boundaries. Applied Mathematics and Computation, 43(3):271-298.
Haddawy et al., 1987
Haddawy, P., Bricken, W., and Fung, R. (1987). The problem of robustness:
A multi-valued logic approach. Technical Report TR-5928-01, Advanced Decision
Systems, Mountain View, CA.
Handler et al., 1989
Handler, I., Kauffman, L. H., and Sandin, D. (1989). On crossing the
boundary of the Mandelbrot set. In Computers in geometry and topology,
number 114 in Lecture Notes in Pure and Applied Mathematics, pages 151-177.
Harris and Cronen, 1979
Harris, L. M. and Cronen, V. E. (1979). A rules-based model for
the analysis and evaluation of organizational communication. Communication
Hellerstein, N. S. (1984). Diamond: A Four-valued approach to
the problem of paradox. PhD thesis, University of California, Berkeley.
Author's address: Mathematics Department, University of California at
Davis. Also: 306 1/2 Montcalm Street, San Francisco, CA 94110, USA. Ph:
(415) 641 6135.
Hellerstein, N. S. (1985). Diamond: A logic of paradox. Cybernetic,
Hellerstein, N. S. (1994a). Contra Cantor. Unpublished.
Hellerstein, N. S. (1994b). N-fold logic - or - paradox island. Unpublished.
Hellerstein, N. S. (1997). Diamond: A Paradox Logic, World Scientific,
Series on Knots and Everything - Vol. 14.
Hellerstein, N. S. (1998). Delta: A Paradox Logic, World Scientific,
Series on Knots and Everything - Vol. 16.
Heylighen, F. (1989). Causality as distinction conservation: a theory of
predictability, reversibility and time order. Cybernetics and Systems,
Heylighen, F. (1990a). Classical and non-classical representations in physics
I. Cybernetics and Systems, 21:423-444.
Heylighen, F. (1990b). Classical and non-classical representations in physics
II: Quantum mechanics. Cybernetics and Systems, 21:477-502.
Heylighen, F. (1990c). Relational closure. In Trappl, R., editor, Cybernetics
and Systems '90, volume 21, pages 335-342. World Science.
Mathematical concept for distinction-making and complexity analysis.
Heylighen, F. (1990d). Representation and Change. A Metarepresentational
Framework for the Foundations of Physical and Cognitive Science, Communication
and Cognition. PhD thesis, Gent, Belgium.
Heylighen, F. (1992). Non-rational cognitive processes as changes of distinctions.
In van de Vijver, G., editor, New Perspectives on Cybernetics. Self-Organization,
Autonomy and Connectionism, Synthese Library v.220, pages 77-220. Kluwer
Holt, C. M. (1994). An algebra of lines and boxes. In Proceedings
of the 1994 IEEE Symposium on Visual Languages.
Author's WWW page http://email@example.com.
Hopkins, M. (1994). Laws of Form as a functional calculus. Posted to the
USEnet newsgroup sci.math Mar 13, 1994 and updated to sci.logic
An attempt to translate LoF, the whole book, including the meta-level
canons, principles and rules, into a functional language and categorical
combinator system in the spirit of CAML.
Howe and von Foerster, 1975
Howe, R. and von Foerster, H. (1975). Introductory comments to Francisco
Varela's calculus for self-reference. International Journal of General
Jacobs et al., 1988
Jacobs, W. J., Blackburn, J. R., Buttrick, M., and Harpur, T. J.
(1988). Observations. Psychobiology, 16(1):3-19.
Only cites LoF in passing: ``More bluntly, the way that we describe and
classify events in the natural world determines the theoretical description
that can be derived from it (...; Spencer-Brown, 1979; ...).''
James, J. M. (1993). A calculus of number based on spatial forms.
Thesis, Master of Science in Engineering, University of Washington.
Available on this site at http://www.lawsofform.org/collection.html.
James and Bricken, 1992
James, J. M. and Bricken, W. (1992). A boundary notation for visual
mathematics. In Proceedings of the 1992 IEEE Workshop on Visual Languages,
pages 267-269, Seattle, Washington.
Johnson, R. C. (1994a). Breaking the laws of logic. Electronic
Engineering Times, pages 1,31,34,35. Part 1 of 4.
CMP Publications, 600 Community Drive, Manhasset, NY 11030.
Johnson, R. C. (1994b). Laws of form are gaining support. Electronic
Engineering Times, pages 31-33. Part 2 of 4.
Johnson, R. C. (1994c). Building the laws of form. Electronic
Engineering Times, pages 43-62. Part 3 of 4.
Johnson, R. C. (1994d). Laws of form exceed bounds. Electronic
Engineering Times, pages 37-39. Part 4 of 4.
Jokisch, R. (19xx). Logik der Distinktionen. Zur Protologik einer Theorie
der Gesellschaft. Unknown.
In German and English. The title translates to ``Logic of Distinctions. A Protologic
Towards a Theory of Society.''
Kauffman, L. H. (1977). Review #4892: Laws of Form. Mathematical
Kauffman, L. H. (1978a). DeMorgan algebras - completeness and recursion.
In Proceedings of the Eighth International Symposium on Multiple-Valued
Logic, pages 82-86. IEEE Computer Society Press.
Kauffman, L. H. (1978b). Network synthesis and Varela's calculus.
International Journal of General Systems, 4:179-187.
Kauffman, L. H. (1980). Complex numbers and algebraic logic. In Proceedings
of the 10th International Symposium on Multiple-Valued Logic, pages 209-213.
IEEE Computer Society Press.
Kauffman, L. H. (1983). Multiple viewpoint-a review essay. Journal
of Social and Biological Structures, 6:265-271.
Kauffman, L. H. (1984). On the form of self-reference. In Proceedings
of the 1984 General Systems Society Meeting, Los Angeles.
Kauffman, L. H. (1985a). Sign and space. In Religious Experience
and Scientific Paradigms, Proceedings of the IASWR Conference, pages
118-164, Stony Brook, NY. Institute for Advanced Study of World Religions.
Kauffman, L. H. (1985b). Transformations in special relativity. International
Journal of Theoretical Physics, 24:223-236.
Kauffman, L. H. (1986a). Map Reformulation. Princelet Edition
#30, 25 Princelet Street, London, U.K.
An account of a seminar with the author and a group of cybernetic musicians
(Herbert Brun's group) and Annetta Pedretti (the publisher), in which they
explored GSB's approach to the 4-color map theorem. The book is incomplete,
but the author is willing to supply copies of it plus an appendix that brings
it partially up to date. Author's address: Dept Math Stats & Comp Sci,
University of Illinois at Chicago (M/C 249), PO Box 4348, Chicago, IL 60680.
Kauffman, L. H. (1986b). Arithmetic in the form. Technical report,
University of Illinois at Chicago, Dept Math Stats & Comp Sci.
Kauffman, L. H. (1986c). String arithmetic. Technical report,
University of Illinois at Chicago, Dept Math Stats & Comp Sci.
Kauffman, L. H. (1987a). Imaginary values in mathematical logic. In
Proceedings of the 17th International Symposium on Multiple-Valued Logic,
pages 282-289, Boston, MA. IEEE Computer Society Press.
Kauffman, L. H. (1987b). On Knots. Princeton University Press.
See chapter VI, sections 4, 5, 8.
Kauffman, L. H. (1987c). Self-reference and recursive forms. Journal
of Social and Biological Structures, 10:53-72.
Kauffman, L. H. (1987d). Sign and space - an introduction. Handwritten
notes toward a book of the same title. 68 pages.
Kauffman, L. H. (1988a). The form of arithmetic. In Proceedings
of the 18th International Symposium on Multiple-Valued Logic. IEEE Computer
Kauffman, L. H. (1988b). Notes on the arrow. Technical report, University
of Illinois at Chicago, Dept Math Stats & Comp Sci.
Kauffman, L. H. (1988c). Special relativity and a calculus of distinctions.
Technical report, University of Illinois at Chicago, Dept Math Stats &
Kauffman, L. H. (1989). FormCreation. 30 pages. Poetry regarding 0,
1, 2 and 3 dimensions, boundaries, knots and self-reference.
Kauffman, L. H. (1990a). An integral heuristic. International
Journal of Modern Physics A, 5(7):1363-1367.
Kauffman, L. H. (1990b). Map coloring and the vector cross product.
Journal of Combinatorial Theory, Series B, 48(2):145-154.
Kauffman, L. H. (1990c). Robbins algebra. In Proceedings of the
20th International Symposium on Multiple-Valued Logic, pages 54-60, Charlotte,
NC. IEEE Computer Society Press.
Kauffman, L. H. (1990d). Schrodinger's cat and the cheshire cat: Quantum
mechanics and Laws of Form. ANPA West (Journal of the Western Chapter
of the Alternative Natural Philosophy Association), 2(2):9-33.
Kauffman, L. H. (1991a). Knots and physics. World Scientific,
Kauffman, L. H. (1991b). The primary algebra of Spencer-Brown is non-boolean.
ANPA West (Journal of the Western Chapter of the Alternative Natural Philosophy
Kauffman, L. H. (1992a). Link polynomials and a graphical calculus.
Journal of Knot Theory and its Ramifications, 1(1):59-104.
Kauffman, L. H. (1992?b). Reflections on reflexivity - the form of
self-reference. Possibly an update of Kauffman, L. H. (1984) `On the Form
Kauffman, L. H. (1993b). Circuit automata. Unpublished.
Kauffman, L. H. (1994a). Knot automata. In The Twenty-Fourth International
Symposium on Multiple Valued Logic, pages 328-333, Boston, MA. IEEE Computer
This paper is a condensed version of Kauffman (1993) `Circuit Automata'.
Kauffman, L. H. (1994b). Knot logic. In Kauffman, L. H., editor,
Knots and Applications. World Scientific Press. In press.
Kauffman, L. H. (1994c). Ways of the game - play and position play.
In Brier, S., editor, Cybernetics and Human Knowing. The Royal School
of Librarianship, Aalborg Branch, Langagervej 4, DK-9220 Aalborg 0st, Denmark.
Kauffman, L. H. (2001). On the map theorem.
Discrete Math, 229 (2001), no. 1-3, 171-184.
Kauffman, L H (2001) The Mathematics of Charles Sanders Peirce,
Cybernetics and Human Knowing 8, 79-110.
Shows how Peirce (1839-1914) discovered the notation of LoF in 1886, and later developed
his existential graphs, which can now be seen as a clear anticipation of the primary algebra of LoF.
Kauffman and Baadhio, 1993
Kauffman, L. H. and Baadhio, R., editors (1993). Quantum Topology.
Knots and Everything. World Scientific.
Contains an introduction to quantum topology by L. H. Kauffman that
is an example of the interaction of formal mathematics with physics and topology.
Kauffman and Brier, 2001
Kauffman, L. H. and Brier, S., editors (2001). Cybernetics and Human Knowing 8, No. 1-2.
Special Issue: Peirce and Spencer-Brown: History and Synergies in Cybersemiotics.
"In this issue of Cybernetics and Human Knowing we have a collection of papers devoted to
the cybernetics and mathematics of Charles Sanders Peirce with a special focus on its
synergies with Spencer Brownís thinking."
Kauffman and Solzman, 1981
Kauffman, L. H. and Solzman, D. M. (1981). Letter to the editor.
International Journal of General Systems, 7:253-256.
A response to Cull and Frank (1979) and Kohout and Pinkava (1980).
Kauffman and Varela, 1980
Kauffman, L. H. and Varela, F. J. (1980). Form dynamics. Journal
of Social and Biological Structures, 3:171-206.
Keenan, D. C. (1991). To dissect a mockingbird - a graphical notation
for the lambda calculus with animated reduction. Unpublished.
Available at the author's WWW page http://users.bigpond.net.au/d.keenan/.
Keenan, D. C. (1992). Learning logic functions explicitly by back-propagation
in nor-nets. In Proceedings of the International Workshop on Emerging
Technologies and Factory Automation, pages 435-440, Melbourne, Australia.
Keys, J. (1970). Twenty-three Degrees of Paradise. Cat Books,
17 Halifax Rd, Cambridge, UK.
Poetry. ``James Keys'' is a pseudonym of George Spencer-Brown.
Keys, J. (1971). Only Two Can Play This Game. Cat Books, 17 Halifax
Rd, Cambridge, UK.
Love poetry plus scientific mysticism.
Keys, J. (1972). Only Two Can Play This Game. Julian Press, New
First US edition.
Keys, J. (1974). Only Two Can Play This Game. Bantam (Ballantine?),
Kohout and Pinkava, 1980
Kohout, L. J. and Pinkava, V. (1980). The algebraic structure of the
Spencer Brown and Varela calculi. International Journal of General Systems,
Krippendorf, Klaus (1984). "Paradox and Information" in
Dervin, Brenda, and Boigt, M. J., Progress in Communication Science,
Vol. 5, pp 45-72. Ablex Publishing, Norwood, NJ.
Laycock, S. W. (1994). Mind as Mirror and the Mirroring of Mind:
Buddhist Reflections on Western Phenomenology. State University of New
York (SUNY) Press.
He refers to Brown, G. Spencer on pages 31, 40-44 in the context of a
discussion of distinction which refers to Buddhist dialectics, Kant, Heidegger,
Derrida, etc. I wouldn't say he deals directly with LoF but rather that
he uses quotes from LoF as a springboard for his philosophical discussion
of distinction (which is insightful and interesting).
LeTourneau, J. (1993). Prime arithmetics. Overhead projector slides which
accompanied a lecture.
A boundary arithmetic system based on prime factors. Address: Prime Arithmetics,
3410 Geary Street, Suite 311, San Francisco, CA 94118, USA.
Lilly, J. C. (1972). The Centre of the Cyclone. Granada Publishing
Limited. Published in Great Britain in 1973 by Paladin Frogmore, St Albans,
Herts AL2 2NF. Reprinted 1973,74,77.
Only contains, as a preface, a quote from LoF pp109-110 ``Discoveries...
furtively.'', without comment.
Long, J. (1995). A study of notation: The structure of perceptual revolutions.
Available on the WWW at http://www.cs.vu.nl/~mmc/tbr/content_pages/repository/nel/ason.html.
``...a lengthy study of notation and of Ultra-Structure''
Niklas Luhmann, "Frauen, Maenner und George Spencer Brown". In:
Zeitschrift fuer Soziologie 17 (1988), pp. 47-71. [in German]
Maturana and Varela, 1980
Maturana, H. R. and Varela, F. J. (1980). Autopoiesis and
Cognition: The Realization of the Living. D. Reidel, Boston.
Maturana and Varela, 1987
Maturana, H. R. and Varela, F. J. (1987). The Tree of Knowledge:
The Biological Roots of Human Understanding. New Science Library, Boston.
McFarlane, T. (1994). The play of distinctions. Available at
An essay exploring the philosophical significance of distinction and its
essential role in the process of creation, sprinkled with suggestive quotes
Meguire, Philip (2003). Discovering Boundary Algebra: A Simple Notation for Boolean Algebra
and the Truth Functors, International Journal of General Systems 32: 25-87.
According to the author, this publication contains many errors. Contact him at
firstname.lastname@example.org for a corrected PDF version, or just read the following.
Meguire, Philip (2007). Boundary Algebra:
A Simple Notation for Boolean Algebra and the Truth Functors.
Merrell, Floyd (1982). Semiotic Foundations: Steps toward an Epistemology
of Written Texts. Indiana University Press, Bloomington, IN.
In a note (p.153) he says ``I cannot overestimate the influence of Spencer-Brown's
`Laws of Form' on this book.''
Merrell, Floyd (1995). Semiosis in the Postmodern Age. Purdue University Press.
The index has 14 entries under Spencer-Brown and several for Varela, Maturana, and
Kauffman. The author's real passion is for C. S. Peirce.
Miller and Spencer-Brown, 1966
Miller, J. C. P. and Spencer-Brown, D. J. (1966). An algorithm
for evaluation of remote terms in a linear recurrence sequence. The Computer
D. J. Spencer-Brown is G. Spencer-Brown's brother who died in 1976.
Mingers, J. (1995). Self-Producing Systems: Implications and Applications
of Autopoiesis. Plenum.
Chapter 4, entitled ``Mathematics and Models for Autopoiesis,'' examines
LoF and Varela's extension. It concludes, ``Mathematically we have seen
attempts to model self-reference by using the laws of form, but it is not
really clear at this point how valid or usefuul this has been.''
Nicod, J. (1917). A reduction in the number of the primitive propositions
of logic. Proceedings of the Cambridge Philosophical Society, XIX:32-41.
Here Nicod marries the Sheffer stroke (NAND, dually NOR) to Principia Mathematica (PM),
a marriage the second edition of PM cited warmly and took on board. Nicodís notation
curiously anticipates LoF. Nicod denotes negation by the overbar, and material implication
by connecting the overbar to the Sheffer stroke, which effectively results in the LoF "mark".
Let concatenation express the Sheffer stroke and (p) the negation of p; Nicod then writes
"if p then q" by p(q) and sometimes even by (q)p. The latter notation is, in effect, the
LoF "mark". Nicodís article would have been readily available in any British research library
during the 1960s. Appendix 1 of LoF demonstrates that Sheffer's (1913) postulates are primary
algebra consequences. -- Philip Meguire
Odegard, S. L. (1989). Chapter 4a: The extension. Posted to Usenet
newsgroup sci.math 29-Nov-89.
Olson and Sawada, 1988
Olson, A. T. and Sawada, D. (1988). Computer, quantized time and human
duration. Cybernetica, 31(2):65-76.
Orchard, R. A. (197?). Title unknown. PhD thesis, College
of Staten Island?
Orchard, R. A. (1975). On the Laws of Form. International Journal
of General Systems, 2:99-106.
Oshins, E. (1990). Why Brown's `Laws of Form' and Pribram's ``Hologram
Hypothesis'' are ``...just what is not relevant in a `quantum context' ''.
In Young, F., editor, Causal Structure and the Quantum, Instant Proceedings
of the 6th Annual Meeting of ANPA West, Stanford University. Alternative
Natural Philosophy Association.
Author's address: Department of Physics, Stanford University, Stanford
CA 94305-2196, USA.
Oshins, E. (1991). About models and muddles, part I: Why Brown's `Laws
of Form' and Pribram's ``Hologram Hypothesis'' are not relevant to quantum
physics and quantum psychology. In Manthey, M., editor, Alternatives in
Physics and Biology, Proceedings of the XII Annual International Meeting of
the Alternative Natural Philosophy Association (ANPA), Cambridge, UK.
Largely a correction and extension of Oshins (1990). Oshins describes
this work as ``an alternative to LoF in which the transposition axiom is
violated thereby giving a nondistributive lattice for linguistic (like Epimenedes)
and psychological propositions, similar to von Neumann's formulation of
quantum logic for physical propositions.''
Parsons, D. (1975). The Directory of Tunes and Musical Themes.
Spencer-Brown & Co., Cambridge, England.
Not related in any way to LoF but contains a foreword by GSB as the publisher.
Each tune is identified by a string of U, D, R, representing movement Up,
Down, or Repeat from the last note. Amazingly, this works pretty well in 15 and
often fewer notes. The book is divided into two parts, classical themes
and popular melodies.
Pearce and Cronen, 1980
Pearce, W. B. and Cronen, V. E. (1980). Communication, Action,
and Meaning: The Creation of Social Realities. Praeger, New York.
Peat, F. D. (1987). Synchronicity: The Bridge Between Matter and
Mind. Bantam Books.
See p.198-206. A fairly basic description plagued by the all too common
typos where blanks, and in this case their preceding equals signs, get lost.
Pedretti, A. (1980). Epistemology, semantics, and self-reference. Journal
of Cybernetics, 10(4):313-339.
Peirce, C. S. (1931). Collected Papers of Charles Sanders Peirce.
Harvard University Press, Cambridge.
"The semantics and proof procedure of Charles S. Peirce's graphical logic,
to which he devoted much effort over the period 1895-1910, in some ways resemble
those of LoF's primary algebra. However all work to which GSB would have had ready
access in the 1960s UK dismissed that graphical logic as hopelessly eccentric (Meguire 2003).
Peirce fully anticipated LoF's 'mark' in two papers written in 1886, but not published
until 1993 (Kauffman 2001). But the primary arithmetic, the self-duality of the primary
algebra, and LoF's metatheory all find no echo in Peirce." -- Philip Meguire
Peirce, Charles S. (1993). Writings of C.S. Peirce: A Chronological Edition, Vol. 5.
Kloesel, C. et al, eds., Indiana University Press.
A very rare instance of scholarly work of the first rank citing LoF, and favorably at that.
Pountain, D. (1982). Namer of the nameless. Australian Personal Computer,
A non-technical review of LoF. It probably appeared in Byte magazine at
about this time also.
Roberts, D. D. (1973). The Existential Graphs of Charles S. Peirce.
Mouton, The Hague.
One of two books (the other being Shin, 2002) on C. S. Peirce's graphical logic,
a logic quite similar in spirit to the primary algebra of LoF.
Roberts, D. D. (1992). The existential graphs. In Lehmann, F., editor,
Semantic Networks in Artificial Intelligence. Pergamon Press. Also
in Computers and Math. with Applications, 23(2-9).
Rolt, C. E. (1920). Dionysius the Areopagite on the Divine Names
and the Mystical Theology.
GSB appears to have been influenced by this book based on a translation
from 5th century Greek.
Russell, B. (1967). The Autobiography of Bertrand Russell 1944-1967,
volume III. George Allen and Unwin Ltd, London.
In one paragraph on page 166, Russell describes the time in 1965 that
GSB had him go over GSB's work. This passage is quoted on the cover of some
editions of LoF.
Schwartz, D. G. (1981). Isomorphisms of Spencer Brown's Laws of Form
and Varela's calculus for self-reference. International Journal of General
An analysis which purports to equate Varela's calculus with Kleene's `total
Shimogawa and Takahara, 1995
Shimogawa, T. and Takahara, Y. (1995). Reconstruction of G. Spencer Brown's
theme. Available on the WWW at http://musashi.syllabus.ne.jp/syllabus/tT1M94001.html.
Rehashes LoF with a proof of soundness and a comparison to propositional
Shin, S-J (2002). The Iconic Logic of Peirce's Graphs. MIT Press.
Shoup, R. G. (1993). A complex logic for computation with simple interpretations
for physics. In PhysComp '92, Workshop on Physics and Computation.
Shoup, R. G. (1995). Space, Time, Logic, and Things. In PhysComp
'94, Workshop on Physics and Computation. IEEE Press.
Soto-Andrade and Varela, 1984
Soto-Andrade, J. and Varela, F. J. (1984). Self-reference and fixed
points-a discussion and extension of Lawvere's theorem. Acta Applicandae
Spencer-Brown, G. (1953). Statistical significance in psychical research.
Spencer-Brown, G. (1957). Probability and Scientific Inference.
Spencer-Brown, G. (1961). An algebra for the natural numbers. Unpublished.
Spencer-Brown, G. (1965). British patent specifications
1006018 Lift Control System and 1006019 Lift Control Circuit Arrangement.
Spencer-Brown, G. (1969). Laws of Form. Allen and Unwin, London.
Spencer-Brown, G. (1972). Laws of Form. Julian Press, New York,
First American edition. A second preface was added relating to ordinary
imaginary numbers and the elimination of the Whitehead Russell Theory of
Types. 4 pages.
Spencer-Brown, G. (1979). Laws of Form. E. P. Dutton, New York,
A third preface was added relating to the four colour map theorem. 2 pages.
Spencer-Brown, G. (1982). The Falkland papers.
Spencer-Brown, G. (1994a). Laws of Form. Cognizer Connection,
A fourth preface was added in which he talks about ``triunions'' or ``triple
identities'' such as of reality, appearance and awareness, or imaginability,
possibility and actuality, or what a thing is, what it isn't and the boundary
between them. He claims/acknowledges that Sakyamuni (the Buddha) is ``the
only other author who evidently discovered these laws.'' He invites the
reader to join a siblinghood and help found a school of his methods for
intuitively feeling and naturally acting upon the consequences of there
being nothing. He calls the method and the siblinghood ``tarati.'' 4 pages.
An extraduction was also added in which he claims to be, since before he
could talk, a liberated being with unlimited thinking, superior to Bach,
Mozart, Newton and Gauss, and asks for money and volunteers to help him
found schools for superintelligent children such as he was. 5 pages. May
be available from Bookmasters (800) 247-6553.
Spencer-Brown, G. (1994b). A Lion's Teeth. Bohmeier Verlag, Soltendieck,
This book is announced at the end of the 1994 Limited Edition of LoF.
It is to contain ``The most fantastic tales ever told.'' Publication is
stated to occur in August/September 1994. Inquiries should be directed to:
Bohmeier Verlag, Im Grund 1, D-29594 Soltendieck, Germany Voice: +49 5874
801, Fax: +49 5874 1682.
Stewart, I. (1993). A partly true story; fuzzy logic; includes related
article; Mathematical recreations. Scientific American, 268(2):110.
Humorous. Paradoxes as dynamic systems. Does not cite GSB but is related.
Swiggart, P. (1974). Self-reference in formal languages. Notre Dame
Journal of Formal Logic, 15(4).
Thomas, I. (1971). Review of Spencer Brown's The Laws of Form,
Zentralblatt fur Mathematik CCVII, 7.
Thompson, W. I. (1989). Imaginary Landscape. St. Martin's
See p.114. Mentions GSB lectures at Esalen in discussion of Varela.
Trudeau, R. J. (1989). Introduction to Graph Theory. Dover.
See First Chapter.
Turney, P. (1986). Laws of form and finite automata. International
Journal of General Systems, 12(4):307-318.
Unknown (1958?). Book review: Probability and scientific inference. Scientific
American. Month and pages unknown.
Varela, F. J. (1975). A calculus for self-reference. International
Journal of General Systems, 2:5-24.
Varela, F. J. (1979a). ??? International Journal of General Systems,
Varela, F. J. (1979b). The extended calculus of indications interpreted
as a three-valued logic. Notre Dame Journal of Formal Logic, 20(1):141-146.
Varela, F. J. (1979c). Principles of Biological Autonomy.
Elsevier, North Holland, New York.
In case you can't get hold of LoF, chapter 11 and appendix B of this book
are the next best thing. Chapters 12 and 13 extend GSB's work. This book
includes material from the following papers: Varela (1975), Varela and Goguen
(1978), Varela (1979b), Kauffman and Varela (1980).
Varela and Goguen, 1978a
Varela, F. J. and Goguen, J. A. (1978a). The arithmetic of closure.
Journal of Cybernetics, 8:48-64.
Varela and Goguen, 1978b
Varela, F. J. and Goguen, J. A. (1978b). The arithmetic of closure.
Progress in Cybernetics and Systems Research, 3:38-64.
von Foerster, 1972
von Foerster, H. (1972). Review: Laws of Form. In The Whole Earth Catalog.
Random House, New York.
An extract appears on the cover of some editions of LoF.
von Kibéd, 1989
Matthias Varga von Kibéd, "Wittgenstein und Spencer Brown".
In: Paul Weingartner and Gerhard Schurz (eds.), Philosophie der Naturwissenschaften.
Akten des 13. Internationalen Wittgenstein Symposiums, Wien: Hoelder, Pichler
& Tempsky, 1989, pp. 402-406. [in German]
von Meier, 1973
von Meier, K., editor (1973). Spencer-Brown Transcript, Art 269.
Unpublished transcript of discussions with GSB at the Aum conference at Esalen
Institute, Big Sur, CA (March 19-20, 1973).
122 pages. Other participants: John Lilly (Chairperson), Kurt von Meier,
Alan Watts, Douglas Kelley, Heinz von Foerster, Gregory Bateson, Walter
Barney, Karl Pribram, Lu Ann King, Ram Dass, Brendan O'Regan.
Vuillemin, J. (1993). On circuits and numbers. Research Report 25, Digital
Does not cite Laws of Form but is strongly related to the reentrant form
dynamics thread, particularly Berkowitz et al (1988).
Waldo, I. (1978). Nagarjuna and analytic philosophy, II. Philosophy East
and West, 28(3):287-298.
Walgate, R. (1977). A colourful character. New Scientist, 6-Jan:6.
A biographical and non-technical mention of claimed four-color proof.
Watts, A. (1975). Tao: The Watercourse Way. Penguin, London.
Only cites LoF in passing on page 45: ``The `chaos' of hsu:an is the nature
of the world before any distinctions have been marked out and named, the
wiggly Rorschach blot of nature. But as soon as even one distinction has
been made, as between yin and yang or 0 and 1, all that
we call the laws or principles of mathematics, physics and biology follow
of necessity, as has recently been demonstrated in the calculus system of
G. Spencer Brown.''
Watts, A. (1980). Om: Creative Meditations. Celestial Arts.
See p.66-91. A collection of transcribed talks by Allan Watts. The piece
called ``Intellectual Yoga'' compared pure math to yoga and brings in disscussions
of Wittgenstein and ``Tractatus'', and GSB and LoF.
Whyte, L. L. (1972). Review: Laws of Form. British Journal of
Philosophy of Science, 23:291-292.
Copyright © 2000-2007
all rights reserved.