Thursday, May 24, 2012

Tao of form

The text of philosophical logic that historically was the foundation of the cybernetic epistemology is Laws of Form by G. Spencer-Brown, first published in 1969. Brown is a figure always surrounded by a shroud of mystery, he is considered an eclectic expert in various disciplines - a polymath - he associated and studied with figures like Russell, Wittgenstein and Laing and has strongly influenced with his work many authors of the systemic-cybernetic movement such Von Foerster, Maturana, Varela, Kauffman and Bateson himself. For example he is also novelist James Keys with his mystic vision of the "five levels of eternity":

A story by Bateson on a meeting with Spencer-Brown together with Heinz Von Foerster, from Keeney (1977), shows how Brown manages to keep his territory obscure: 

I talked to Von Foerster the morning before I met Brown to see if I was getting it right. I said these upside-down L-shaped symbols of this fellow are some sort of negative ... He said, "Yes, you've got it Gregory". At that moment Brown came into the room and Heinz turned to Brown and said, "Gregory has got it - those things are sort of negative". And Brown said, "They are not!".

G. Bateson, quoted by Keeney, "Aesthetics of Change", 1983

Von Foerster saw Spencer-Brown as similar to Wittgenstein (of which he was the nephew) and Don Juan, Carlos Castaneda's teacher, in that all three shared "a state of melancholy that befalls those who know that they know".

The historical-logical base of the Brown text is the overcoming of the classical Logic (or Aristotelian) that the monumental work by Russell and Whitehead Principia Mathematica of 1910 sought to preserve against mathematical paradoxes and demonstrated impossible in 1931 by the Gödel's incompleteness theorems. Since 1931 any type of formal logic should explicitly take into account the existence of logical paradoxes since, as demonstrated by Gödel, in any formal system "powerful" enough - such arithmetic - one can derive undecidable propositions, which are neither true nor false but paradoxical: if they are true then they are false and back.
As stated by Brown in the preface: 

Recalling Russell's connexion with the Theory of Types, it was with some trepidation that I approached him in 1967 with the proof that it was unnecessary. To my relief he was delighted. The Theory was, he said, the most arbitrary thing he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved.

The theory of logical types, though refuted in formal logic by the work of Gödel, has defined the fundamental idea of logical levels of meta- type, applied by Bateson on human and animal interaction and communication models, and fundamental to define meta-classes, meta-terms, meta-descriptions and meta-explanations in the theory of complexity.

The subject is introduced by Brown in his preface:


Apart from the standard university logic problems, which the calculus published in this text renders so easy that we need not trouble ourselves further with them, perhaps the most significant thing, from the mathematical angle, that it enables us to do is to use complex values in the algebra of logic. They are the analogs, in ordinary algebra, to complex numbers a + b √- 1 . My brother and I (*) had been using their Boolean counterparts in practical engineering for several years before realizing what they were. Of course, being what they are, they work perfectly well, but understandably we felt a bit guilty  about using them, just as the first mathematicians to use 'square roots of negative numbers' had felt guilty, because they too could see no plausible way of giving them a respectable academic meaning. All the same, we were quite sure there was a perfectly good theory that would support them, if only we could think of it.

The position is simply this. In ordinary algebra, complex values are accepted as a matter of course, and the more advanced techniques would be impossible without them. In Boolean algebra (and thus, for example, in all our reasoning processes) we disallow them. Whitehead and Russell introduced a special rule, which they called the Theory of Types, expressly to do so. Mistakenly, as it now turns out. So, in this field, the more advanced techniques, although not impossible, simply don't yet exist. At the present moment we are constrained, in our reasoning processes, to do it the way it was done in Aristotle's day. The poet Blake might have had some insight into this, for in 1788 he wrote that 'reason, or the ratio of all we have already known, is not the same that it shall be when we know more.'

Recalling Russell's connexion with the Theory of Types, it was with some trepidation that I approached him in 1967 with the proof that it was unnecessary. To my relief he was delighted. The Theory was, he said, the most arbitrary thing he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved.

Put as simply as I can make it, the resolution is as follows. All we have to show is that the self-referential paradoxes, discarded with the Theory of Types, are no worse than similar self-referential paradoxes, which are considered quite acceptable, in the ordinary theory of equations.

The most famous such paradox in logic is in the statement, 'This statement is false.'

Suppose we assume that a statement falls into one of three categories, true, false, or meaningless, and that a meaningful statement that is not true must be false, and one that is not false must be true. The statement under consideration does not appear to be meaningless (some philosophers have claimed that it is, but it is easy to refute this), so it must be true or false. If it is true, it must be, as it says, false. But if it is false, since this is what it says, it must be true.

It has not hitherto been noticed that we have an equally vicious paradox in ordinary equation theory, because we have carefully guarded ourselves against expressing it this way. Let us now do so.
We will make assumptions analogous to those above. We assume that a number can be either positive, negative, or zero. We assume further that a nonzero number that is not positive must be negative, and one that is not negative must be positive.
We now consider the equation


Transposing, we have 


and dividing both sides by x gives ,


We can see that this (like the analogous statement in logic) is self-referential: the root-value of x that we seek must be put back into the expression from which we seek it.

Mere inspection shows us that x must be a form of unity, or the equation would not balance numerically. We have assumed only two forms of unity, +1 and — 1, so we may now try them each in turn. Set x = +1. This gives

+ 1= -1/+1= - 1

which is clearly paradoxical. So set x= -1. This time we have

- 1= -1/-1= + 1

and it is equally paradoxical.
Of course, as everybody knows, the paradox in this case is resolved by introducing a fourth class of number, called imaginary, so that we can say the roots of the equation above are ± i, where i is a new kind of unity that consists of a square root of minus one.


Cambridge, England
Maundy Thursday 1972

(*): there are reasons to believe Spencer-Brown is only child.

Brown's epistemology is clearly stated in the beginning of text:


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 begin 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.

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 which is independent of how the universe actually appears, that lends such fascination to the study of mathematics. 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 thread as best he can. Here the story is traced from the beginning.

Unlike more superficial forms of expertise, mathematics is a way of saying less and less about more and more. A mathematical text is thus not an end in itself, but a key to a world beyond the compass of ordinary description.

An initial exploration of such a world is usually undertaken in the company of an experienced guide. To undertake it alone, although possible, is perhaps as difficult as to enter the world of music by attempting, without personal guidance, to read the score-sheets of a master composer, or to set out on a first solo flight in an aeroplane with no other preparation than a study of the pilots' manual.

In the first section Brown outlines what a form is:

T H E  F O R M

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.

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 into the content can be taken to indicate this value.
Thus, also, the crossing of 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.

In the following he defines how forms are taken out of forms:

F O R M S  T A K E N  O U T  O F  T H E  F O RM

Draw a distinction.

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 space shaped by the severance or cleft the sides of the distinction or, alternatively, the spaces, states,or contents distinguished by the distinction.

Let 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.

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.

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.

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.

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.

Call any arrangement intended as an indicator an expression.

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

The mark or cross operator:

is the main symbol used by Brown.
The symbol represents the distinction between its inside and outside:

The inside state defined by the symbol is called marked state, the outside state unmarked state, meaning by state the two sides of a distinction, where the symbol itself represents the distinction between the two states:

with such definitions and axioms Brown derives a formal logical system for the description of primary arithmetic, primary algebra and second order equations.