George Spencer-Brown’s Vita• George Spencer-Brown worked with Bertrand Russell in Foundations of

Mathematics from 1960 onwards. He is the part-editor of material for Russell’s Autobiography.

• He worked with Ludwig Wittgenstein in the Foundations of Philosophy in 1950-51.• He worked with J D Boyd in experiments in the special senses, and with D T Harris

in the medical uses of hypnosis. With the Royal Navy, he undertook successful trials of hypnosis for dentistry and the retraining of wounded personnel. Laurelled chess master and chemist, race car driver (member of the Institute of Advanced Motorists since 1962), and expert pilot with the Royal Air-force. Published in statistics, probability, psychical research, logic, and poetry. Worked with Winston Churchill’s advisor Lord Cherwell on Goldbach’s Conjecture and other advanced mathematical problems from 1954 until his death in 1957. He is custodian or Lord Cherwell’s unpublished mathematical papers and correspondence.

• Chief Logic Designer for Mullard Equipment Ltd., 1959-61, consultant from 61 on. Advisor to British Rail, 1963-64. Inventor of the first modular lift and elevator control units, British Patent Specifications 1006018 and 1006019 (1965), now in general use in simplex and multiplex elevator systems constructed by Marryat and Scott. Joint inventor (with D J Spencer-Brown) of the first transistor counting-module to employ complex formal arithmetic signals, now in use by British Rail.

George Spencer-Brown’s Vita• Senior Lecturer in Formal Mathematics with the University of London Department of Extramural Studies,

1963-1968. Work in military Communications for U K Government 1965-66. Manager, Development Division International Publishing Corporation Ltd, 1966-68. Worked in Computer Science and Number Theory with J C P Miller 1963 onwards. In collaboration with him and with D J Spencer-Brown discovered new primality tests and factoring methods. Member of the Department of Pure Mathematics and Mathematical Statistics in the University of Cambridge since 1969. Life Fellow of the Cambridge Philosophical Society.

• Visiting Professor of Mathematics in the University of Western Australia 1976. Visiting Professor, Department of Computer Science, Stanford University, 1977. Consultant to Xerox Palo Alto Research Corporation 1977-78. Stanford and Palo Alto lectures The Four-color Map Theorem as a Problem in Formal Quaternions 1977-78. Visiting Professor of Pure Mathematics in the Department of Computer Science, University of Maryland, 1980-81. Seconded to Federal Naval Research Laboratory, Washington D C, as adviser in Military Communications. Contributions include new discoveries in optics, and in coding and code-breaking.

• Washington lectures 1980-81: What is Mathematics, Formal Arithmetics of the Second Order, and Cast and Formation Properties of Maps.

• Practiced full-time as a professional psychotherapist 1968-69. Successfully implemented hypnosis and sleep-learning techniques to enhance performance in sporting and other competitive activities. Specialist in the training and education of gifted and super-intelligent children. Continues to practice world-wide on a part-time basis.

• Publications: Probability and Scientific Inference, London 1957, 1058; Laws of Form, London 1969, 1971, New York 1972, 1973, 1977, 1979 (writing as G Spencer-Brown); Twenty-Three Degrees of Paradise (verse), Cambridge, 1970; Only Two Can Play This Game (a comparison of western and eastern modes of thought and methods in the arts, philosophy, religion, and the sciences), Cambridge 1971, New York 1972, 1973, 1974 (writing as James Keys).

• His latest proof of Riemann’s hypothesis via Denjoy’s equivalent can be found at laws-of-form.net.

EXCERPTSFrom Laws of Form

A NOTE ON THE MATHEMATICAL APPROACHFirst Paragraph

The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can being to see how the familiar laws of our own experience follow inexorably from the original act of severance. The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it, and the world may seem like shifting sand beneath our feet.

NOTE: Paragraph 2

Although all forms, and thus all universes, are possible, and any particular form is mutable, it becomes evident that the laws relating such forms are the same in any universe. It is this sameness, the idea that we can find a reality independent of how the universe actually appears, that lends such fascination to the study of mathematics.

NOTE: Paragraph 2

That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together , is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the threads as best he can. Here the story is traced from the beginning.

1

THE FORM

We take as given the idea of distinction and the idea of indication, and that we cannot make an indication without drawing a distinction. We take, therefore, the form of distinction for the form.

Definition

Distinction is perfect continence.

That is to say, a distinction is drawn by arranging a boundary with separate sides so that a point on one side cannot reach the other side without crossing the boundary. For example, in a plane space a circle draws a distinction.

Once a distinction is drawn, the spaces, states, or contents on each side of the boundary, being distinct, can be indicated.

There can be no distinction without motive, and there can be no motive unless contents are seen to differ in value.

If a content is of value, a name can be taken to indicate this value.

Thus the calling of the name can be identified with the value of the content.

Axiom 1. The law of calling

The value of a call made again is the value of the call.

That is to say, if a name is called and then is called again, the value indicated by the two calls taken together is the value indicated by one of them.

That is to say, for any name, to recall is to call.Equally, if the content is of value, a motive or an

intention or instruction to cross the boundary can be identified with the value of the content.

Axiom 2. The law of crossing

The value of a crossing made again is not the value of the crossing.

That is to say, if it is intended to cross a boundary and then it is intended to cross it again, the value indicated by the two intentions taken together is the value indicated by none of them.

That is to say, for any boundary, to recross is not to cross.

2FORMS TAKEN OUT OF THE FORM

Construction

Draw a distinction.

Content

Call it the first distinction.Call the space in which it is drawn the space severed or cloven by the distinction.Call the parts of the spaces shaped by the severance or cleft the sides of the distinction or, alternatively, the spaces, states, or contents distinguished by the distinction.

IntentLet any mark, token, or sign be taken in any way with or with regard to the distinction as a signal.Call the use of any signal its intent.

First canon. Convention of intention

Let the intent of a signal be limited to the use allowed to it.Call this the convention of intention. In general, what is not allowed is forbidden.

Knowledge

Let a state distinguished by the distinction be marked with a mark

of distinction.

Let the state be known by the mark.

Call the state the marked state.

Form

Call the space cloven by any distinction, together with the entire content of the space, the form of the distinction.

Call the form of the first distinction the form.

Name

Let there be a form distinct from the form. Let the mark of distinction be copied out of the form

into such another form.Call any such copy of the mark a token of the mark.Let any token of the mark be called as a name of the

marked state.Let the name indicate the state.

Arrangement

Call the form of a number of tokens considered with regard to one another (that is to say, considered in the same form) an arrangement.

Expression

Call any arrangement intended as an indicator an expression.

Value

Call a state indicated by an expression the value of the expression.

Equivalence

Call the expressions of the same value equivalent.Let a sign

of equivalence be written between equivalent expressions.

Now, by axiom 1,

.

Call this the form of condensation.

Instruction

Call the state not marked with the mark the unmarked state.Let each token of the mark be seen to cleave the space into which it is copied. That is to say, let each token be a distinction in its own form.Call the concave side of a taken its inside.Let any token be intended as an instruction to cross the boundary of the first distinction.Let the crossing be from the state indicated on the inside of the token.Let the crossing be to the state indicated by the token.Let a space with no token indicate the unmarked state.Now, by axiom 2,

Call this the form of cancellation.

The Primary Arithmetic (Ch. 4)George Spencer-Brown continues in successive chapters of Laws of Form to use the notion of distinction and form developed in chapters 1 and 2 given in these excerpts, to develop the primitive notion of calculation needed to develop the primary arithmetic, which is non-numerical, called the calculus of indications, dealing with the two initials of number (Axiom 1: the law of calling; condensation and confirmation) and order (Axiom 2: the law of crossing; cancellation and confirmation*).

“Call the calculus limited to the forms generated from direct consequences of these initials the primary arithmetic.”

* The respective operations of condensation and confirmation are complimentary and inverse operations with regard to the direction of the law of calling, as cancellation and compensation are to the law of crossing. See this beautiful economy in the Fifth canon, Expansion of reference, from Chapter 3, “The Conception of Calculation”.

The Primary Algebra (Ch. 6)At the beginning of the fourth chapter “The Primary Arithmetic”, after an image of the initials of the calculus of indications, he explains that the general patterns distinguished therein are called Theorems. After a number of such theorems and some more canons are given expression in that chapter, we are able to see the generalities of it such that we can take tokens of variable form to indicate expressions in the primary arithmetic, which leads to a new calculus, taken out of the first, called the primary algebra, in the fifth chapter “A Calculus Taken Out of the Calculus”. Tokens of constant form are in the Laws of Form distinctions, which indicate the first distinction.

There are two initials of the primary algebra, given at the beginning of the sixth chapter, “The Primary Algebra”. They are position and transposition. And just as in the Primary Arithmetic, there is a beautiful economy of operations pertaining to the initials: Position means “take out” one way, and “put in” the other. Transposition means collect and distribute, again, in reverse directions. He then states,

“We shall proceed to distinguish particular patterns, called consequences, which can be found in sequential manipulations of these initials.”

Theorems of the Second Order (Ch. 7)

A number of Theorems (10-15) are developed, building on pervious forms (definition, axioms, canons, arithmetic initials, algebraic initials, theorems, rules, consequences, and expressions of completeness).

Re-uniting the Two Orders (Ch. 8)

Here he introduces Theorem 16, “The bridge”, and the eighth Canon, “Principle of transmission”.

Equations of the Second Degree (Ch. 11)

Here he introduces “Re-entry” (i.e., of the form into itself). Re-entry is a very popular notion in the interpretation of Laws of Form. Here is the explanation of Re-entry:

“The key is to see that the crossed part of the expression at every even depth is identical with the whole expression, which can thus be regarded as re-entering its own inner space at any even depth.”

He then introduces the Imaginary state, involving an infinite expression and the fact of the ninth Canon, the Rule of Demonstration (A demonstration rests in a finite number of steps), it seems to me, although he makes no explicit reference to that Canon under the introduction of the Imaginary state, it is implied by the order of the work.

The next section is called Time, and has a graphic of a circle and a shaded region outside the circle to indicate a higher-dimensional (second-degree) frame of reference (indicational space).

TimeSince we do not wish, if we can avoid it, to leave the form, the state we envisage is not in space, but in time. (It being possible to enter a state of time without leaving the state of space in which one is already lodged.

FrequencyIf we consider the speed at which the representation of value travels through the space of the expression to be constant, then the frequency of its oscillation is determined by the length of the tunnel. Alternatively, if we consider this length to be constant, the frequency of the oscillation is determined by the speed of its transmission through space.

VelocityWe see that once we give the transmission of an indication of value a speed, we must also give it a direction, so that it becomes a velocity. For if we did not, there would be nothing to stop the propagation proceeding as represented to t4 (say) and then continuing towards the representation shown in t3 in stead of that shown in t5.

FunctionWe shall call an expression containing a variable v alternatively a function of v. We thus see expressions of value or functions of variables, according to from which point of view we regard them.

Oscillator functionIn considering the indications of value at the point p in Figure 1 [Illustration of six states of the tunnel diagram], we have, in time, a succession of square waves of a given frequency.

RELEVANCE TO THIS CLASS???

From Note 2 in the Notes section of the book:

Where Wittgenstein says

whereof one cannot speak,thereof one must remain silent

he seems to be considering descriptive speech only. He notes elsewhere that the mathematician, descriptively speaking, says nothing. The same may be said of the composer, who, if he were to attempt a description (i.e., a limitation), of the set of ecstasies apparent through (i.e., unlimited by) his composition, would fail miserably and necessarily. But neither the composer nor the mathematician must , for this reason, be silent.

He goes on to quote Russell from the introduction of the Tractatus, and suggests Russell’s suggested loophole out of the hierarchy of language, is the injunctive faculty of language (from the primal injunction of construction: Draw a distinction.)

“We see now that the first distinction, the mark, and the observer are not only inter-changable, but, in the form, identical.”

In a fourth preface to Laws of Form, Spencer-Brown continues on this point, stating that the thirdness is essential. He calls it triplicity. In his book Only Two Can Play This Game, he explains that these three are none other than the Holy Trinity! They are:

The Marked State, The Mark, and the Unmarked State

see: http://uboeschenstein.ch/texte/spencer-brown-LoFVII.html

In Appendix 2, “The calculus interpreted for logic”, he says:

“…All forms of primitive implication become redundant, since both they and their derivations are easily constructed from, or tested by reduction to, a single cross. For example, everything in pp 98-126 or Principia mathematica can be rewritten without formal loss in one symbol

provided, at this stage, the formalities of calculation and interpretation are implicitly understood, as indeed they are in Principa. Although some 1500 symbols to a page, this represents a reduction of the mathematical noise-level by a factor of more than 40000.

George Spencer-Brown

01144-1985-844-855

Dr. John Cunningham Lilly (1915-2001)

George Spencer-Brown and Second-Order Cybernetics

Dr. Heinz von Foerster (1911-2002)

Louis KauffmanWilliam BrickenRichard Shoup

Humberto MaturanaFrancisco Varela

Randall Wittaker

Niklas Luhmann

Alan Watts

Francis Heylighen

Thomas McFarlane, CIIS

Dirk Baecker

Wittgenstein and Russell

Laws of Form

"To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity. Not reasoning. Not calculating. Not busy behavior of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set about it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are being continually thrust upon them."

-G. Spencer Brown, from Laws of Form