Laws of Form

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Laws of Form (in the original English Laws of Form, short: LoF ) is the title of a work by George Spencer-Brown from 1969 that touches on the philosophy of logic , basic mathematical research, cybernetics and epistemology . The book presents three graphical calculi that build on each other , which Spencer-Brown developed with the aim of a universal algebra:

  1. the primary arithmetic (Chapter 4)
  2. the primary algebra (Chapter 6)
  3. Second order equations (Chapter 11)

All calculi are based on the only fundamental operation of discrimination . Occasionally only primary algebra is referred to as Laws of Form .


Spencer-Brown's starting point is the logical form of distinction. The basic act described as “ draw a distinction !” Is then combined with itself and in this way creates a variety of new forms that are used as basic concepts ( true and false , symbol , signals , names , processes, themselves self-changing forms, operators , etc.) can be viewed. With these terms, formal calculations of logic and mathematics can be represented.

In the course of his investigation, Spencer-Brown questions classical logic as the basis of mathematics and reinterprets it. The epistemological concepts of the Laws of Form are important . “Nothing at all can be known through storytelling” is one of the core statements of the LoF. Accordingly, knowledge can only be obtained through experience from the results of practical action. In contrast to the classical calculi of logic, which are supposed to map the logical structure of facts and statements, George Spencer-Brown understands the logical form as something that corresponds to knowledge as a process and action. While it is arbitrary for a formal language what its non-logical constants denote, in the Laws of Form distinguishing and denoting itself - as a simultaneous act - becomes the starting point for formal operations. Spencer-Brown's calculus provides not only a formal syntax and semantics, but also a formal semiotics . In addition, the concept of re-entry of the form (into the form) offers the possibility of formulating a formal idea of ​​change of state and memory.

The Laws of Form are regarded by their followers as the minimal calculus for mathematical truths. Representatives of several branches of science therefore explicitly refer to the Laws of Form, such as Niklas Luhmann in his sociological systems theory or Humberto Maturana in the theory of radical constructivism .


Spencer-Brown's formal calculus was first published in 1969 under the title Laws of Form . The book has been reprinted several times since then and translated into various languages. Spencer-Brown developed the idea for the book while working as an engineer at British Railways , in which he was commissioned to develop electrical circuits for counting wagons in tunnels in the late 1950s. The technical problem, which was fundamental at the time, was to let the counters count up and down and to save wagons that had already been counted. Spencer-Brown solved the problem by using previously unknown imaginary Boolean values and "a new problem arose at the same time: his idea worked, but there was no mathematical theory that could justify this approach". The elaboration of the calculus that allowed these new imaginary Boolean values ​​was the trigger for the Laws of Form .

The book has 141 pages, 55 of which cover the mathematical calculation, and is considered difficult to understand for laypeople. There are a number of "explanatory books" for the LoF.

Basic concepts of the Laws of Form

The first chapter of the Laws of Form is preceded by six Chinese characters, which can be translated as follows: "The beginning of heaven and earth is nameless". Without labels, the world is empty and indefinite. The designation of something, however, presupposes a distinction: what is designated must be distinguished from the rest. For Spencer-Brown, the idea of ​​designation and differentiation form the starting point, whereby he gives the distinction logical priority: “We take as given the idea of ​​distinction and the idea of ​​indication, and that we cannot make an indication without drawing a distinction. "

The distinction divides the initial uncertainty into areas. The distinction is clear when it completely separates the areas from one another, so that a “point only gets from one side to the other by crossing the common border”. Because the distinction draws a closed border, it constitutes what it encloses as an object that can be designated and thus a difference between inside and outside. The object is exactly and completely enclosed by the distinction. ("Distinction is the perfect continence"). It is therefore sufficient to introduce the form of distinction as the only symbol (“We take therefore the form of distinction for the form”). The distinction made so, by an icon on the inner or outer side (token) marks are.

In the literature, a circle on a white sheet of paper is occasionally cited as an example: The circle clearly separates the outside from the inside in the sense that you can only get from the outside to the inside or vice versa if you “cross” the circular line . The completely enclosed circular area is clearly differentiated from the surrounding space as the inside ("displayed" side) - unmarked space (literally "unmarked space").

Differentiation by crossing

Spencer-Brown uses the English word "cross" to mark a distinction, which can and should be read as a noun (mark), but also as a request (cross!). This is important in that the term “marking” is introduced at a later date in the Laws of Form . Making a distinction is facilitated by the above. Definition equated with the crossing of a border and the different values ​​of the sides of a distinction. Two basic operations are carried out through a distinction: Either one changes from an unmarked state to a marked state (e.g. we start from a blank sheet of paper, mark a "circle" and thus go from "non-circle" to " Circle ") or one changes from a marked state to an unmarked state (eg: we start from a" circle "and by making a distinction go to" non-circle ").

Differentiation by naming

Then - in an epistemological context - the Laws of Form describe the connection between distinction, motive, value and naming a name: A distinction presupposes a motive (of the distinctive), and there can be no motive unless the content is viewed differently in value become. A distinction therefore presupposes that there is someone who makes the distinction, and that this actor sees a difference in value that prompts him to make the distinction. Since a distinction denotes a content that has a value, this value can also be named, and the name can be identified with the value of the content ("Thus the calling of the name can be identified with the value of the content").

According to the Laws of Form you have two ways to make a distinction: that of crossing, ie of making a distinction by exceeding a limit, and the Nennens, so the use of a name representative of the distinction.

In the original of the Laws of Form Explanation
Axiom 1 The value of a call made again is the value of the call. A renewed mention (= differentiation) is the value of the (original) mention.
In a mathematical sense, the value of a distinction does not change if you name it again.
Axiom 2 The value of crossing made again is NOT the value of the crossing. A distinction by crossing does not lead back to the same state.
In the example: the first distinction leads to "circle", the second distinction leads to "non-circle".

Symbolic illustration

The Laws of Form then introduce a symbol for the demarcation of a distinction, represented by the cross:   . What is written under the "angle" at the bottom left is separated from everything else (you can imagine the cross as a closed rectangle). cross and blank page (a blank page with no characters) are the basic terms for passing or failing a Spencer-Brown form . In text representations, the closed demarcation is also represented by brackets: for example with [] or <>, the empty page with a point “.” Or “{}”. The general shape indicated by the cross corresponds to a demarcation that separates one area from another. It says as much as here-so! and there - across the border - definitely not-so! In addition to the angle, other symbols are therefore also possible, such as a circle.

In this symbolic language, the above two basic axioms can be formalized as follows:

From axiom 1: "The value of a call made again is the value of the call": the form of condensation:       
From axiom 2: "The value of crossing made again is NOT the value of the crossing": the form of cancellation:   

Then Spencer-Brown introduces the concept of depth of space , which allows symbols to be nested in one another and therefore leads to structurally rich forms. In addition, four fundamental canons are presented that deal with the rules for handling such forms. The forms can be gradually substituted ("shortened") according to the above two basic axioms (see in particular the 3rd canon of substitution: "In any expression, let any arrangement be changed for an equivalent arrangement") as in the following example:

by condensation by repeal

Note on the process: First the two lower left 'crosses' become a 'cross' by condensation according to axiom 1. Then the resulting nested 'cross' and the already existing nested 'cross' are canceled according to Axiom 2. A 'cross' remains.

At the end of Chapter 3 of the Laws of Form , GSB clarifies what he means by the preceding so-called indication calculus - namely the calculus that is determined by taking the two above basic forms as a starting point (“Call the calculus determined by taking the two primitive equations as initials the calculus of indication ") - and it leads over to primary arithmetic, which should include all statements and be limited to all statements that result from the indication calculus (" Call the calculus limited to the forms generated from direct consequences of these initials the primary arithmetic ").

Primary arithmetic

In Chapter 4 of the LoF , GSB carries out the above Calculus of indications into what is known as “primary arithmetic”. The starting point are the instructions (“initials”) obtained from the form of condensation and the form of crossing.

Initial 1 (number):        (allows changes in the number of marks)
Initial 2 (order):    (allows deletion or addition of marks)

and develops nine theorems on primary algebra from these two initials:

theorem Explanation
1 The shape of any finite number of crosses can be viewed as the shape of an expression. Theorem of “shortening” (see above).
2 If any space penetrates an empty cross, then the value that is denoted in that space is the marked state. If an arithmetic expression ("c") consists of any partial expression (here: "b") and there is a single cross next to it, then the value of the expression is the highlighted state.
3 The simplification of an expression is clear. If an expression according to Theorem 1 can be simplified in different ways, then in all possible ways the result will clearly be the marked or the unmarked state.
4th The value of any expression constructed by taking steps from a given simple expression is different from the value of any expression constructed by taking steps from a different simple expression. Theorem 3 is reversed: expressions can be "nested" by expanding them. Different output expressions have different results.
5 Identical expressions express the same value. N / A
6th Expressions of the same value can be identified with each other. N / A
7th Expressions that are equivalent to the same expression are also equivalent to each other. N / A
At this point, variable partial expressions are introduced into the LoF . The argument is made by substituting both the marked and the unmarked state for p and both cases lead to the same result (the unmarked state).
9 rprqpq Also called the variance condition. If r is the unmarked state, the equations are immediately identical. If r is the marked state, then on the left side the inner parts are marked after the application of Theorem 2 twice and the inner crosses cancel after initial 2, so that only the marked state remains on the left side. In this case, the right side of the equation is also the marked state according to Theorem 2.

Theorems 8 and 9 also serve as the initials of Brownian primary algebra.

Primary algebra

In Chapter 6 of the LoF , GSB defines theorems 8 and 9 of primary arithmetic themselves as the initials of primary algebra. On the basis of this initial, GSB demonstrates nine so-called "consequences" (development of forms through the consistent application of the permitted calculation steps): "We shall proceed to distinguish particular patterns, called consequences, which can be found in sequential manipulations of these initials."

In Chapter 8 of the LoF , GSB tries to show that every consequence in algebra must point to a provable theorem about arithmetic. This is followed by the proof that every theorem can actually be demonstrated about arithmetic in algebra (Chapter 9) as well as the independence of the two initial algebraic equations (Chapter 10).

With regard to Gödel's theorem of incompleteness , the postulated simultaneous completeness and consistency of primary algebra in particular caused discussion in the literature. “An obvious assumption is that Kurt Gödel's sentences do not apply here, because the Laws of Form allow the imaginary to be represented […] If the imaginary is inherent in a formal system, Gödel's sentences can no longer be applied to this system apply. "

2nd degree equations

Second degree equations in the meaning of LoF are obtained by representing infinite algebraic expressions as finite equations through self-referentiality. The term imaginary value is introduced, which is supposed to express the oscillation between marked and unmarked state, and which is used in the later sequence for the application of complex values in algebra, which in turn are called “analogies to the complex numbers in ordinary (numerical) algebra “Can be used. For the mathematical form of the representation of the imaginary value, GSB coined the term re-entry (" re-entry ") of the form into the form.

The starting point is the demonstration that the u. a. mathematical form can be transformed into an infinite term of the same recurrence by consequent transformation according to the rules of the LoF . This infinite repetition can be converted into a finite formalism (expressed here by ) which is identical to the whole expression in every integer depth. Since the form reappears in its own space, it was given the name re-entry .

a b ... abab a b

On this basis, GSB develops two such recursive functions: The function (“memory function ”), which is used both for  as well as for empty space is { }fulfilled, as well as ("oscillation function"), the solution of which is not a fixed expression, but extends infinitely.


is used in the formalism of LoF to express a mathematical form of time. is only solvable if it is equated with infinitely nested , and if the equation is to solve, it must be extended infinitely . This equation leads - although it cannot be solved in space (i.e. with the means of primary arithmetic ) - to an idea of ​​time by dissolving the “one after the other” of the states and only looking at the imaginary state of the form .

Meaning and reception


While the formalism of LoF can largely be transferred to that of Boolean algebra , there is a fundamental difference between the two: While Boolean algebra uses the laws of logic, in particular the theorem of the excluded third party, as an axiomatic basis, this assumption does not apply in Brownian algebra. It is believed that the LoF represent the "undiscovered" arithmetic of Boolean algebra .

In the eleventh chapter of the Laws of Form , oscillating values ​​are introduced for forms that are based on self- reference . A certain form can be called up again within itself through a reentry . The oscillating values ​​("<>" or ".") Are not interpreted as a contradiction or syntax error that should be forbidden by a type theory, for example. Spencer-Brown interprets the oscillation between two values ​​rather as "mathematical time". In the note to Chapter 11, reference is made to the parallel to the root of −1, which can also be represented as an imaginary number as an oscillation between 1 and −1 (cf. Louis H. Kauffman ). Based on the traditional representation of imaginary numbers as the points of the y-axis in the complex plane , the y-axis becomes the notional placeholder for the oscillation. This approach is significant for physics insofar as it relies on complex numbers to describe natural processes .

In 1975, the Chilean biologist and systems scientist Francisco Varela presented an extension of Brown's indication calculus to a trivalent calculus.

As a follow-up to the LoF, Spencer-Brown himself proposed nine mathematical and philosophical applications, including for the four-color problem , the Riemann hypothesis , the Goldbach hypothesis and the Fermat hypothesis .

Unmarked space

Outside of mathematics, the Laws of Form give one a special meaning, the observer's dilemma: every observation ( distinction ) made by an observer implies a second distinction. The first is the (possibly also multi-valued) distinction of the observed object (“The number of people wearing glasses is increasing”), the second is the implicitly underlying distinction between what was observed and what was not (here, for example, the number of blind people, hearing aid wearers , cell phone owners, total population, etc.).

This recessed at each observation room of the Non-Observed are Spencer-Brown is now the name space unmarked . This space arises with every observation - scientific, epistemological, phenomenological. Conversely, when comparing a phenomenon and its description, the unmarked space is always involved .

Such an observation of observation is also called “ re-entry ” and can be used universally as a theoretical figure, i.e. also beyond mathematics. The sociologist Niklas Luhmann , for example, translates it as a re-entry into the distinction and becomes a central figure in Luhmann's systems theory .

Systems theory

In systems theory in particular, the LoF received attention beyond mathematics. Parallels between the LoF and fundamental concepts of systems theory (e.g. differentiation, observation as the separation of object and environment, knowledge as construction and recursion, etc.) were drawn again and again. Niklas Luhmann pointed out that he took his fundamental difference theory approach from the LoF . Parallels were also made with the concepts of radical constructivism and sociology. The German sociologist Dirk Baecker has compiled applications and interpretations of the LoF for sociology in two collections of essays .


Critics of the Laws of Form point to the synonym for Boolean algebra and contradict Spencer-Brown's claims about self-reference. In the abstract of a publication from 1979 entitled Flaws of Form (English for. Errors of form ) the authors write:

G. Spencer Brown's Laws of Form is popular with social and life science scholars. Proponents claim that the book introduces a new logic that is ideally suited to their research areas, and that the new logic solves the problems of self-referentiality. These claims are wrong. We show that Brown's system is a Boolean algebra in an obscure notation and that his "solutions" to the problems of self-reference are based on a misunderstanding of Russell's paradox. "

See also


Primary text
  • George Spencer-Brown: Laws of Form. Allen & Unwin, London 1969 (first edition).
    • George Spencer-Brown: Laws of Form. 1994. Portland OR: Cognizer Company, ISBN 0-9639899-0-1 (latest edition).
    • German translation: Laws of Form. Bohmeier Verlag, Leipzig 2008, ISBN 978-3-89094-580-4 .
Secondary literature
  • Dirk Baecker (ed.): Calculus of form. Suhrkamp, ​​Frankfurt / Main 1993, ISBN 3-518-28668-4 .
  • Dirk Baecker (Ed.): Problems of Form. 1988, ISBN 0-8047-3424-0 .
  • Louis H. Kauffman: The Mathematics of CS Peirce. (PDF; 171 kB). In: Cybernetics and Human Knowing.  8 (2001), pp. 79-110.
  • Holm von Egidy: observing reality. Difference Theory and the Two Truths in Madhyamika Buddhist Philosophy. Carl Auer Systems Verlag, Heidelberg 2007 (e-book). ISBN 3-89670-328-5 .
  • Felix Lau: The Form of Paradox. An introduction to the mathematics and philosophy of the Laws of Form by George Spencer Brown. Publishing house for systemic research in Carl-Auer-Verlag, Heidelberg 2005/2008, ISBN 3-89670-352-8 .
  • Niklas Luhmann: The science of society. Suhrkamp, ​​Frankfurt / Main, 1994, ISBN 978-3-518-28601-2 .
  • Tatjana Schönwalder-Kuntze, Katrin Wille, Thomas Hölscher: George Spencer Brown. An introduction to the 'Laws of Form'. VS Verlag für Sozialwissenschaften, Wiesbaden 2004/2009, ISBN 3-531-14082-5 .
  • Francisco Varela: A calculus for self-reference. In: International Journal of General Systems. 2, pp. 5-24.
  • Matthias Varga von Kibéd, Achim Ferrari: George Spencer Brown. The Spencer Brown distinction theory and the distinction form constellation. Aachen 2008, ISBN 978-3-942131-04-9 (DVD box).

Web links

Individual evidence

  1. "A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics." In: LoF. 1969, p. 11 of the introduction.
  2. ^ Laws of Form. P. 12 of the introduction.
  3. a b Lau, 2008, p. 9.
  4. See list of secondary literature .
  5. Schönwälder-Kuntze / Wille / Hölscher, p. 64 f.
  6. a b c d e George Spencer-Brown: Laws of Form. Allen & Unwin, London 1969, p. 1.
  7. Lau, p. 40 ff.
  8. To mark both sides with symbols, see Axiom 1.
  9. Lau, 2008, p. 46 ff.
  10. ^ Laws of Form, p. 4
  11. ^ Laws of Form, p. 5
  12. ^ Laws of Form. P. 7.
  13. ^ Laws of Form. P. 11.
  14. according to axiom 1; Laws of Form, p. 4
  15. according to axiom 2; Laws of Form, p. 5
  16. LoF. P. 28.
  17. Schönwälder-Kuntze / Wille / Hölscher, pp. 140 ff.
  18. Lau, p. 83.
  19. LoF. P. 11 of the introduction.
  20. ^ "Nevertheless [...] it is real in relation with time and can, in relation with itself, become determinate in space, and thus real in the form." In: LoF. P. 61.
  21. Lau, p. 119 ff.
  22. ^ Louis H. Kauffman : Time, Imaginary Value, Paradox, Sign and Space. (PDF; 180 kB).
  23. Varela, 1975.
  24. Appendices 1 to 9 of the English edition of LoF, Bohmeier Verlag, Heidelberg, 2008.
  25. ^ Laws of Form. EP Dutton, New York, 1979, pp. 19, 111, 125.
  26. Lau, p. 21; Luhmann 1994.
  27. ^ Paul Cull and William Frank: Flaws of Form In: Int. J. General Systems , 1979, Vol. 5, pp. 201-211, doi: 10.1080 / 03081077908547450