Laws of Form

Laws of the form ( in the original English Laws of Form, in short: LoF ) is the title of a work by George Spencer -Brown in 1969, the philosophy of logic, mathematical basic research, cybernetics, and epistemology touched. This book presents three inter- graphical calculi, the Spencer -Brown developed with the aim of universal algebra:

All calculations are based on the single basic operation of distinguishing. Only occasionally is referred to as the primary algebra rules of form.

• 2.1 distinction by crossing
• 2.2 distinction by Name
• 2.3 Symbolic representation

• 6.1 Mathematics
• 6.2 Unmarked Space
• 6.3 System Theory

General

The starting point of Spencer - Brown is the logical form of discrimination. The as "Meet a distinction " ( draw a distinction ) circumscribed basal Akt is combined in addition to itself and generates in this way a variety of new forms, as the basic concepts ( true and false, symbol, signals, names, processes, to self-changing forms, operator, etc.) may be considered. With these concepts to formal calculi of logic and mathematics can be presented.

Spencer -Brown is in progress of his investigation classical logic as the basis of mathematics in question and re-interprets them. Important are the epistemological concepts of the Laws of Form. " Nothing at all can be known by telling " is a key message of the LoF. Knowledge one attains therefore only in the experience of the results of practical action. In contrast to classical calculus of logic that is intended to reflect the logical structure of facts and statements, of course George Spencer -Brown logical form as something that corresponds to the knowledge as a process and action. While it is optional for a formal language, what their non-logical constants denote, in the Laws of Form is the distinguishing and designating himself - as a simultaneous act - the starting point for formal operations. So Spencer - Brown's calculus not only provides a formal syntax and semantics, but also a formal semiotics. Moreover, the concept of re -entry of the form (into the form) offers the possibility to formulate a formal notion of state change and memory.

The Laws of Form to be considered by its adherents as a minimal calculus for mathematical truths. Several branches of science, therefore, rely explicitly on the Laws of Form, as Niklas Luhmann in his sociological system theory and Humberto Maturana in the theory of radical constructivism.

Publication

Spencer - Brown's formal calculus in 1969 published form for the first time under the title of Laws. The book has since been reprinted several times and translated into several languages. The idea for the book developed Spencer -Brown already during his work as an engineer for British Railways, in which he was commissioned in the late 50 years trying to develop electrical circuits for the counting of cars in tunnels. It was the - at that time fundamental - the technical problem to be counting down the counter and already counted to save cars. Spencer -Brown solved the problem by the use of so far unknown imaginary Boolean values ​​and there " at the same time created a new problem: His idea worked, but there was no mathematical theory that could justify this approach ." The development of the calculus, which allowed this new imaginary Boolean values ​​, was the trigger for the Laws of Form.

The book contains 141 pages, of which 55 comprise the mathematical calculus, and is considered difficult to understand for the layman. There are a number of " books explaining " to the LoF.

Basic Concepts of Laws of Form

The first chapter of the Laws of Form are six Chinese characters prefixed which may be translated as follows: "The beginning of heaven and earth is nameless ." Without names the world is empty and indeterminate. However, the designating something puts forward a distinction: the signified must be distinguished from the rest. The presentation of the name and of distinguishing form for Spencer -Brown the starting point, though he admits the distinction logical priority: " We take as givenName the idea of distinction and the idea of indication, and did we can not make an indication without drawing a distinction"

The distinction divides the initial uncertainty in areas. The difference ( " distinction" ) is then clearly if the regions separated completely from each other so that a " point from one side only reaches to the other by intersecting the common boundary " The fact that the differentiation takes a closed border, it constitutes what it encloses, as bezeichenbares object and thus a difference between inside and outside. The property is enclosed accurate and complete to the distinction. ( " Distinction is the perfect continence " ) It is therefore sufficient to establish the shape of the distinction as a single symbol ( " We take THEREFORE the form of distinction for the form" ). The distinction made so can be labeled by a symbol on the inside or outside ( "token ").

In the literature, a circle on a white sheet of paper is cited as an example occasionally: The circle separates clearly the outside from the inside, in the sense that one can only get from outside to inside or vice versa, if you cross the circle line ( "cross "). The fully enclosed circular area is ( " displayed " page ) distinguished clearly from the surrounding space as inside - unmarked space ( often written as unlabeled space ).

Distinction by crossing

Spencer -Brown used for marking a distinction the English word " cross", which can be read as a noun (mark ), but also as an invitation ( kreuze! ), and should be. This is significant, as is introduced at a later stage in the Laws of Form, the term " marker ". Process of making a distinction is equated by the o a definition with the crossing of a border and the different values ​​of the sides of a distinction. By distinguishing two basic operations: Either you switch from an unmarked state to a selected state (eg: We start from a blank sheet of paper, mark a "circle " and thus go from "non- circular " to " circle " ) or you can switch from a marked state in an unmarked state (for example: We start from a " circle "and walk through differentiation over to" non- group " ).

Distinction by Name

Then - in an epistemological terms - describe the Laws of Form, the relationship of distinction, design, value and specifying a name: A distinction is a motif ( the Discriminating ) ahead, and there can be no motive, if not seen content varies in value be. So a distinction presupposes that there is someone who makes the distinction, and that this actor has a value difference does that causes him to 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, one has two ways to make a distinction: the cruising, ie of making a distinction by exceeding a limit, and of calling, so the use of a name representative of the distinction.

Symbolic representation

The Laws of Form then introduce a symbol for the demarcation of a distinction, represented by the cross. Here is what is written on the bottom left under the " angle " of all other accrued ( the cross one can imagine as a closed rectangle ). cross and blank page ( a blank page with no characters) are the basic expressions for the existence or non- existence of a Spencer -Brown -shape. In text displays in closed boundaries is also represented by brackets: for example, by [ ] or < >, the empty side by a point or by "{}" ". ". The general form 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, therefore, other symbols are possible, such an encirclement.

In this symbolic language can the above two basic axioms formalize as follows:

Then Spencer -Brown introduces the concept of depth of field a, the nested symbols allowed, and therefore leads to structurally rich forms. Furthermore, presented four fundamental canons, treat the rules for the treatment of such forms. The forms can gradually substituted ( " cut " ) according to the above two basic axioms (cf. in particular the third canon of substitution: "In any expression, let any arrangement be changed for at equivalent arrangement" ); as in the following example:

At the end of Chapter 3 of the Laws of Form clarifies GSB what he meant by the previous so-called indication calculus - namely the calculus, which is determined by taking the above two basic forms as a starting point ( "Call the calculus deterministic mined by taking the two primitive equations as initials the calculus of indication " ) - and he leads us to the primary arithmetic, which include all statements and should be limited to any statements that arise from the indication calculus ( " Call the calculus limited to the forms generated from direct synthesis initials Consequences of the primary arithmetic " ).

Primary arithmetic

In Chapter 4, the LoF GSB leads the indication o a calculus in a so-called "primary arithmetic " on. The starting point are the lessons learned from the shape of the condensation and the form of cruising instructions ( " Initial" ).

And developed from these two initials nine theorems for primary algebra:

Theorems 8 and 9 serve as initial Brownian primary algebra.

Primary Algebra

In Chapter 6, the LoF GSB defines the theorems 8 and 9 of Primary Arithmetic turn as the primary initial algebra. Based on these initial GSB demonstrates nine so-called " consequences" ( development of forms through consistent application of the allowed computational steps ): . " We shall proceed to distinguish Particular patterns, called Consequences, Which can be found in sequential manipulation of synthesis initials "

In Chapter 8 of the LoF GSB seeks to show that each consequence must point to a provable theorem on arithmetic in algebra. Following this is the proof that indeed every theorem can be demonstrated on the arithmetic in algebra (Chapter 9) and the independence of the two algebraic initial equations ( Chapter 10).

In terms of Gödel's incompleteness theorems, in particular the postulated simultaneous completeness and consistency of the primary algebra for discussion in the literature have provided. " An obvious guess is that the sets of Kurt Godel not apply here because the Laws of Form to represent the imaginary allow [ ... ] If the imaginary a formal system is inherently, the sets of Gödel can no longer this system apply. "

2nd degree equations

2nd degree equations in the importance of LoF is obtained by infinite algebraic expressions are represented by self-reference as a finite equations. There the concept of imaginary value is introduced, which is intended to express the oscillation between labeled and unlabeled state, and is used in a later episode to the use of complex values ​​in the algebra, in turn, as " analogies to the complex numbers in the usual ( numerical ) algebra " can be used. For the mathematical form of the representation of the imaginary value GSB coins the term of re-entry ( " reentry " ) of the mold into the mold.

The starting point is the demonstration that the mathematical form of, inter alia, can be transformed by logical transformation according to the rules of the LoF in an infinite term of the same recurrence. This infinite repetition can be divided into a finite formalism ( here expressed by the f) transfer, which is identical with the whole expression in each integer depth. Since the form occurs again in her own room, she was named the re-entry.

On this basis, GSB developed two such recursive functions: The function G ( "memory function "), which is fulfilled both for and for the empty space { }, and O ( " oscillation function "), whose solution is not a fixed expression, but rather infinitely extended.

O is used in the formalism of the SOF in order to express a mathematical form of the time. O: f = ( f ) is only solvable if she with infinitely nested f: ... (()) ... is equated, and if f is to solve the equation, f must be infinitely extended. This equation leads - although in space ( ie the means of primary arithmetic) can not be resolved - thereby a notion of time by dissolving the " succession " of states and considered only the imaginary state of the form.

Significance and reception

Mathematics

While the formalism of LoF in the Boolean algebra can be converted to a large extent, exists between the two is a fundamental difference: while the Boolean algebra the laws of logic, especially used here the law of excluded middle as an axiomatic basis, applies this assumption is not in Brownian algebra. It is believed that the LOF of representing " undiscovered " arithmetic to Boolean algebra.

In the eleventh chapter of the laws of the form oscillating values ​​for the forms to be introduced, which are based on self-reference. It can be called from within its own back by a reentry a certain shape. The oscillating values ​​( "<>" or "." ) Are interpreted not as a contradiction or syntax error it would apply to prohibit about by a type theory. Spencer -Brown suggests the oscillation between two values ​​rather than " mathematical time ". In the note to Chapter 11 is referred to the parallels with the square root of -1, which can be regarded as imaginary number as oscillation between 1 and -1 (see Louis H. Kauffman ). Taking the traditional representation of the imaginary number as the points of the y-axis in the complex plane based on the y-axis so that the conceptual placeholders for the oscillation. This approach is important for the physics, insofar as it relies on complex numbers to describe natural processes.

The Chilean biologist and systems scientist Francisco Varela in 1975 presented an extension of the Brownian indication calculus to a trivalent calculus.

GSB has itself proposed nine mathematical and philosophical applications in the wake of the LoF, inter alia, for the four-color problem and the Riemann Hypothesis. Goldbach's conjecture and Fermat's Last Theorem.

Unmarked Space

Outside mathematics is from the Laws of Form a special meaning given to the observer dilemma: Any action taken by an observer watching ( distinction ) therefore implies a second distinction. The first is the (possibly multivalued ) distinction of each observed object ( "The number of eyeglass wearers increases " ), the second is the implicit underlying distinction what is observed and what is not (here about the number of the blind, the hearing aid wearer, of mobile phone owners, of the total population, etc.).

This recessed at each observation space of non - Observed Spencer -Brown is now the name unmarked space. For each - scientific, epistemological, phenomenological - observation arises this space. Conversely, suppose in the comparison as between a phenomenon and its description is always the unmarked space in the game.

Such observation of the observation is also called " re-entry" and is universally recognized as Figure theory, so also on the mathematics beyond use. It is at about the sociologist Niklas Luhmann translated as re-entry into the differentiation and a central figure in the theory luhmann between systems theory.

Systems theory

In particular, in systems theory, the LoF found one on the mathematics beyond compliance. So parallels the LoF on basic concepts of systems theory were drawn (eg differentiation, observation as a separate object and environment, knowledge as a construction and recursion, etc.) again. Niklas Luhmann has pointed out that he has taken his basic theoretical difference approach the LoF .. also parallels were made to 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.