"A new calculus, of great beauty and depth" (Bertrand Russell).
The Theory
Concept of form
A shape is the surface appearance of something. At the language level, a form is pure syntax, where the semantics is open, i.e., it can be interpreted in multiple ways. For example, a formal axiomatic system admits different interpretations (so-called "models"). Different models may share the same axioms, with the same form, which reveals structural (formal) analogies between apparently different systems.
Laws of form
Published by George Spencer-Brown (GSB) in 1969, the laws of form constitute an attempt at a unifying paradigm based on the essential forms underlying different domains (mathematics, logic, physics, biology, linguistics, etc.). They are based on the concept of "distinction" (symbolized by the angular symbol "⌉", also called "mark"), which represents the creation of a space (from the void) and its division or separation into two subspaces: one internal (content) and the other external (context). The mark establishes the boundary between the two created spaces. The mark of the laws of form symbolizes or represents consciousness, which connects the opposites: inner space and outer space. For example, the tracing of a circumference in a flat space creates a distinction, by dividing the space into two subspaces. The absence of distinction is emptiness.
Space is an experience associated with consciousness. The mark is associated with the conscious, the differentiated. Emptiness is associated with the unconscious, the undifferentiated. Consciousness emerges from the process of making or perceiving distinctions: something emerges from the unconscious (the void) to become conscious. The mark is the common boundary between the internal and external worlds.
Showing a preference for one of the two subspaces is an "indication." An indication is a type of distinction: a value distinction. To make an indication it is necessary to first make a distinction. Knowledge is constructed by distinctions and indications in reality.
An alternative symbol for the mark, more convenient to use, is a pair of parentheses () to delimit (linearly) the inner and outer spaces. We could also use a circle as a two-dimensional notation. In a circle there are 3 zones: the inner, the outer and the circumference, which joins the inner and the outer.
GSB uses only two forms of mark composition:
The sequence: ⌉⌉()()
The hierarchy: ⌉⌉ (())
With the mark and the two composition forms you can create expressions (forms) as complex as you want, for example:
()((()()()(()()()())()
Once the mark symbol and its absence (the void) are defined, two very simple laws are defined:
Calling law: ()() = ()
Crossing law: ((()) = (the void)
As can be seen, there are no operators. The mark acts as both operator and operand and represents the foundation of consciousness and the basis of the language used to construct or perceive reality.
By means of these two laws one can perform calculations (calculus of distinctions). The result of a calculation is the mark or void. For example, the expression in the above example, calculated, produces ().
The two shape laws can be interpreted in both senses:
To the right: Condensation. To the left: Confirmation.
Towards right: Cancellation. To the left: Compensation.
Algebra of the laws of form
Using variables (representing the mark or the void) one can construct algebraic expressions like
(((x)y)z(x))
From the two laws of form (the axioms) numerous theorems can be deduced. For example:
xx = x
((x)) = x
((x)x) =
(x)x = ()
()x = ()
((x)y)x = x
(xy)y = (x)y
((x)(y))z = ((xz)(yz))
(((x)y)z) = (xz)((y)z)
((x)(y))((x)y) = x
The first two expressions are the two form laws expressed algebraically (variables are used). The original form laws are expressed arithmetically (i.e., only constants are used).
The advantages of notation with variables are:
It allows expressions to be "calculated" by applying the two rules of the laws of form. For example, the expression (x)((x)(x)((x)(x))(x) reduces to (x)x(x).
There are no operators. There is only a delimiter (the pair of parentheses) as the only differentiator. This type of mathematics is called "Boundary Mathematics" (Boundary Mathematics).
You can create recursive expressions, i.e., make self-references to an expression in the form of re-entry (re-entry). Examples:
f = (a(bf)) yields the infinite expression (a(b(a(b(a(b(...)))))))
x = (x) produces the oscillator x (x) x (x) x ...
In this case what you get is an expression that oscillates between two values or states. GSB interprets this as the creation of time, in the same way that the distinction creates space.
In the case of interpretation in the domain of logic, this leads to "imaginary logic", just as in arithmetic the imaginary number i = √(−1) can be interpreted as an oscillator between the values 1 and −1. The latter GSB justifies this because i = −1/i and because the only possible values for i are 1 and −1, so it follows that 1 = −1.
Interpretations of the laws of form
The laws of form admit many interpretations. For example, they can be interpreted as:
Boolean logic. This type of logic is called "Boundary Logic" (Boundary Logic).
Classic notation
New notation
F (false)
the vacuum
T (true)
()
a' (not a)
(a)
a∨b (a or b)
ab
a∧b (a and b)
((a)(b))
a→b (a implies b)
(a)b
a↔b (a equivalent to b)
(((a)b)((b)a))
If a Then b Else c
((((a)b)(ac)))
Algebra of sets:
Classic notation
New notation
∅ (empty set)
the vacuum
U (universal set)
()
a' (complementary to a)
(a)
a∪b (union)
ab
a∩b (intersection)
((a)(b))
a−b (difference)
(a)b
Circuit algebra (current switches):
Classic notation
New notation
0 (no current)
(vacuum)
1 (there is current)
()
a' (reverse switch)
(a)
a∪b (union)
ab
a∩b (intersección)
((a)(b))
Applications of the Laws of Form
The laws of form have inspired numerous authors, who have applied them (sometimes with variants) to different domains: philosophy, cybernetics, art, computation, logic, sets, arithmetic, topology, circuits, semiotics, neural networks, etc.
Virtual or imaginary logic
Louis H. Kauffman has developed applications of virtual logic.
Charles H. Moore, inventor of the Forth language, has applied imaginary logic to simplify circuit design.
Jeff Fox has studied the parallel between the laws of form and the Forth language.
Bounding Mathematics
William Bricken is the creator of Boundary Mathematics (Boundary Mathematics), a formalism based on the use of delimiters (parentheses), instead of operators, which has the advantage of greater expressiveness, as well as simplifying computation. He has also developed a language called LOSP, which uses this type of mathematics, and has implemented it on a chip as an engine for virtual reality applications.
Transcendental physics
Edward R. Close has developed "Transcendental Physics", a discipline that integrates (by calculating distinctions) physical reality (in its aspects of space, time, matter and energy) and the consciousness of the observer. It is based on the following principles:
The underlying reality is continuous and infinite, but our perception is discrete. The laws of physics in relativity and quantum mechanics are apparently different (relativity is presented to us as continuous and quantum mechanics is presented to us as discrete), but this is due to our perception.
The vacuum is infinite and continuous. Consciousness, itself, is the substance of emptiness. The act of distinction requires the existence of consciousness. Distinction indicates the existence of "something", which is necessarily perceived as discrete, since it is associated with consciousness.
The calculus of distinctions makes it possible to represent all aspects of reality. The whole universe can be described in terms of the consequences of making distinctions.
There are several kinds of distinctions. Matter is condensed energy. Energy is expanded matter. Time and space are interrelated through the transformations of matter into energy and vice versa.
Calculation of distinctions in biological systems
Francisco Varela has applied the calculus of distinctions to biological systems. In his book "Principles of Biological Autonomy" he extends the GSB system to a logical system of three values, adding to distinction and emptiness a third value: self-reference, symbolized by the mythical Ouroboros, which represents self-regeneration, renewal and endless identity. This third symbol is also a distinction and constitutes a synthesis of the opposites (distinction and emptiness), a unity of a higher order. These three primary entities, according to Varela, constitute reality.
For Varela, self-reference, acting at different levels, in a biological system is what allows it to operate autonomously and be capable of self-regeneration, where the system is structurally open but functionally closed [see Addendum - Autopoiesis].
Numerical computation system
Jeffrey M. James has devised a numerical calculation system based on Delimiting Mathematics. It uses three types of delimiters:
[] for ln (neperian logarithm)
<> for generalized inverse (valid for all operations).
() for powers of base e (Euler's constant).
Concatenation is interpreted as addition (there is no specific delimiter for this operation).
Classic notation
New notation
a
a
−a
<a>
e^a
(a)
a+b
ab
a−b
a<b>
a*b
([a][b])
a*b*c
([a][b][c])
1/a
(<[a]>)
a/b
([a]<[b]>)
a^b
(([[a]][b]))
logab
(([[a]]<[b]>))
3√a
(([[<a>]]<[b]>))
The product and exponentiation expressions are justified by the following expressions:
Since curved parentheses already have a meaning in MENTAL, we can use for the distinction (mark) the same parenthesis, but with some differentiating attribute (bold, italic, color, etc.) or use another type of parenthesis. In the latter case, we could use, for example, the parentheses ⌈ and ⌉. In this case, the laws of form, in their arithmetic version, would be:
Calling law:
( ⌈⌉ ⌈⌉ = ⌈⌉ )
Crossing law (crossing):
( ⌈⌈⌉⌉ = θ )
The laws of form, in their algebraic form, would be:
⟨( ⌈x⌉ ⌈x⌉ = ⌈x⌉ )⟩
⟨( ⌈⌈x⌉⌉ = x )⟩
where x is a null expression (θ) or the mark ⌈⌉. In the case of x = θ, you have the arithmetic version of the laws of the form.
Bounding mathematics
In this aspect, it should be emphasized that MENTAL uses 6 delimiters to express shapes:
Form
Delimiters
Generic expression
⟨...⟩
Sequence
(...)
Set
{...}
Normal distribution
[...[...]...]
Linear distribution
[...⌊...⌋...]
If you want to use more delimiters, instead of operators, you can use these delimiters, but with a differentiating attribute (bold, italic, underline, color, etc.).
Self-referential expressions
We use potential substitution here. The above examples would be expressed as follows:
( f =: ⌈a ⌈b f⌉⌉ ) // represents the infinite expression
⌈a ⌈b ⌈a ⌈b ⌈a ⌈b ⌈...⌉⌉⌉⌉⌉⌉⌉⌉
( x =: ⌈x⌉ ) // represents the oscillator x ⌈x⌉ x ⌈x⌉ x ...
Addenda
Autopoiesis
It is a neologism created in 1971 by Humberto Maturana and Francisco Varela to explain the organization of biological systems. It comes from the Greek "poiesis", production. It refers to the distinctive characteristic of living beings to self-produce, to self-regenerate by means of a circular, self-referential or reentrant organization. An autopoietic system continuously produces itself using resources from the environment, so that producer and product, doing and being, subject and object, are the same thing.
Living beings are self-referent autonomous beings. Not every autonomous entity is a living entity. Self-reference is a type of autonomy and is what characterizes living beings.
Living systems are simultaneously autonomous systems and dependent on the environment. It is a paradox, not conceivable with traditional dichotomous thinking (true/false, yes/no, etc.), but with a model of thinking that harmonizes analytical and synthetic thinking, linear and circular. In short, with the union of opposites.
The theory of autopoiesis is based on cybernetic theory (Wiener, Ashby, von Foerster, etc.) but with two important conceptual contributions:
Structural coupling.
It refers to the capacity of a living being to constantly change its structure in a flexible and congruent way with the modifications of the environment. This circular structural coupling, of constant being-environment dialogue, occurs at multiple levels.
Operational closure.
For life to be possible, it is necessary for the living being to close itself, to close itself to the environment, in such a way that, in the face of the dynamics of the environment, the organization of the living being (its functionality, its identity, its globality) remains invariant. Operational closure is due precisely to the self-referential quality.
That is, living beings are structurally open and functionally closed.
The concept of autopoiesis has overflowed the limits of biology to be applied in other domains such as sociology, anthropology, psychotherapy, etc., having become an important concept for the investigation of reality.
Bibliography
Laws of Form
Bricken, William. What´s the Difference? Contrasting Boundary and Boolean Algebras. October 2005. Internet.
Close, Edward, R. Transcendental Physics. toExcel Press, 1997.
Forth Meets Laws of Form. Internet.
James, Jeff. A Calculus of Number Based on Spatial Forms. Thesis, Master of Science in Engineering. University of Washington. http://www.lawsofform.org/ collection.html
Las Leyes de la Forma. http://www.lawsofform.org/lof.html
Spencer-Brown, George. Laws of Form. Allen and Unwin, London, 1969. 2nd edition: Julian Press, New York, 1972.
Autopoiesis
Maturana, Humberto. La realidad: ¿objetiva o construída?. Anthropos – Universal Iberoamericana – Steiso, Barcelona. Tomo I: Fundamentos biológicos de la realidad, 1995. Tomo II: Fundamentos biológicos del conocimiento, 1996.
Maturana, Humberto y Varela, Francisco. De máquinas y seres vivos. Editorial Universitaria, Santiago de Chile, 1972. Existe una edición de 1974 con el subtítulo “Autopoiesis: la organización de lo vivo”.
Maturana, Humberto y Varela, Francisco. El árbol del conocimiento. OEA, Santiago de Chile., 1984. Editorial Debate, 1996.
Maturana, Humberto y Varela, Francisco. Autopoiesis and cognition: the organization of the living. Boston: Reidel, 1980.
