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Laws of Form

Form follows distinction.

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Index of Page Topics

I'll Bite -- What are we talking about?

G. Spencer Brown

Why should we bother with this stuff?

Boolean Algebra

Solving Boolean Equations

Ordinary Algebra

The Beginnings

Complex Numbers

The Primary Arithmetic

Quadratic Equations

The Primary Algebra

Math Functions

Second Degree Equations

Math Models

Ordinary Algebra

Symbolic Logic

Logic

Math

Godel Theorem

 

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I'll Bite -- What are we talking about?

What we're talking about is Boolean algebra.

Oh, boy -- out of the frying pan into the fire! What in the world is Boolean algebra?

Boolean algebra is a form of algebra that's commonly associated with ordinary, or 2-valued, logic; it's an algebra of statements, just as ordinary algebra is an algebra of numbers. So think in terms of language forms in place of equations.

Boolean algebra was designed to formalize logic -- which makes it part of logic, or so it would seem. However, as algebra it belongs in math (as math modeling), particularly since it can be interpreted in more than one way, including something other than a system of 2-valued logic. It is a theory and is more general than simple logic, so it naturally goes into math.

For the same reason, the subject identified as the laws of form -- developed by G. Spencer Brown -- is also a subject of math. It, too, was designed to fit ordinary logic, but it was also meant to go beyond 2-valued logic. In Laws of Form Brown wants to separate algebras of logic from the subject of logic itself, and re-align them with math.

If that's too muddy for you, please bear with me a bit.

 

A Difference

An important difference between traditional Boolean algebra and the algebra of Brown is that Brown's system is intended to generalize traditional Boolean algebra to handle multi-valued logic (where you get more than just true or false values for statements). Indeed, Brown says math itself not only aims to "provide a shorthand" for what is actually spoken, but also to "extend consideration of a subject to what is common with other subjects." So reasoning and computing become secondary functions. In Brown's words:

The discipline of mathematics is seen to be a way, powerful in comparison with others, of revealing our internal knowledge of the structure of the world, and only by the way associated with our common ability to reason and compute.

(How we happen to have such an amazing capability is another matter -- a mystery -- but it goes along with our creative ability to read the world around us. We don't just look at the world -- even babies (or pets) can look. There's a difference between looking and seeing. You could look but not see. Seeing involves "insight." A fruitful point of view. You need understanding.

One of the big jobs of math, in other words, is to perform abstractions -- to create models of more general validity from more empirical models.

[The] subject matter of logic, however symbolically treated, is not, in as far as it confines itself to the ground of logic, a mathematical study. It becomes so only when we are able to perceive its ground as a part of a more general form, in a process without end [Like Zen?]. Its mathematical treatment is a treatment of a form in which our way of talking about our ordinary living experiences can be seen to be cradled. It is the laws of this form, rather than those of logic, that I have attempted to record.

The subject of Brown's concern is thus the more general form so perceived:

The problem is simply this. In ordinary algebra, [complex values] are accepted as a matter of course, and the more advanced techniques [i.e., those for solving equations] would be impossible without them. In Boolean algebra (and thus for example in all our reasoning processes) we disallow them [i.e., imaginary numbers]. ... At the present moment we are constrained, in our reasoning processes, to do it the way it was done in Aristotle's day [using only true/false statements].

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Put as simply as I can make it, the resolution is as follows. All we have to do is show that the [self-referential paradoxes], discarded with [Bertrand Russell's Theory of Types], are no worse than similar self-referential paradoxes, which are considered quite acceptable, in the ordinary theory of equations [,which paradoxes are solved by introducing the [class of numbers called imaginary]].

(In quantum mechanics, imaginary numbers are used to characterize properties of probability states, which are interpreted to be "potential" states.)

 

Another Difference

Another important difference between traditional Boolean algebra and Brown's algebra is that his system has an arithmetic, whereas ordinary Boolean algebra does not have an arithmetic.

This emphasizes the distinction to be made between math and logic -- in effect treating logic as applied math, like physics or engineering. On a parallel with ordinary (numerical) algebra, which provides solutions to ordinary algebraic equations of degree greater than one, Brown's algebra is intended to provide solutions to (non-numerical) Boolean equations of degree greater than one -- something that ordinary Boolean algebra does not provide.

 

Math as Injunction

It will serve you well to understand that Brown sees the primary form of math communication to be injunction rather than description -- collections of commands or directives or pointers that lead to the experience, rather than words about the experience. This will help you grasp what he's doing. As he himself says:

... In this respect [math] is comparable with practical art forms like cookery, in which the taste of a cake, although literally indescribable, can be conveyed to a reader in the form of a set of injunctions called a recipe. Music is a similar art form, the composer does not even attempt to describe the set of feelings occasioned through them, but writes down a set of commands which, if they are obeyed by the reader, can result in a reproduction, to the reader, of the composer's original experience.

(Except that it can't exactly, unless you have the producer's brain; our brains are similar but different creative echanisms.)

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Why should we bother with this stuff?

There are several reasons for studying Brown's work, aside from its intrinsic value and very powerful ideas. These reasons are:

  1. To help develop the relationship between simulation and the various multi-valued information systems -- including statistics and fuzzy theory.
  2. To understand how the meaning of meaning is affected by the relationship, since all of the systems have to do with representing and thus interpreting the real world.
  3. To apply the modeling ideas meaningfully to simulation.

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Solving Boolean Equations

Ordinary Boolean algebra deals with ordinary statements and statement equations (statement forms) expressed with them; these are non-numerical equations, which define our reasoning processes. But it is so formulated that it admits only Boolean equations of degree 1, or linear equations. (In the algebra of numbers such a policy would exclude equations having variables with exponents equal to 2 or greater.) Therefore logic is in the position that ordinary algebra was in prior to the invention of complex numbers, when only real numbers provided possible solutions to equations. (Now we might use forms with an imaginary truth value component.)

Math guys had gotten it into their heads that all (numerical) equations of degree 2 should be solvable. But as it happened there were some equations that could not be solved. That's when George Cantor extended the range by admitting complex numbers as potential solutions.

Brown posed a similar task for himself in trying to extend the proofs of statement equations to those higher than degree 1. He feels that more statements are provable than those that can be proved when only statements of the first degree are allowed in the proofs.

It is generally agreed that contradictory statements can be used to prove any assertion, including false ones. Brown's attention was drawn to self-referencing statements, which are considered to be self-contradictory, but Brown accepts them into his system as full-fledged members. (Very bold!) The problem as I see it is how to do it without creating contradictions. Compare this with the use of 'i' in electrical engineering.

 

To be able to solve Boolean equations of degree greater than 1, Brown extends the Boolean algebra to include evaluations true, false, meaningless, and imaginary for statements. He does with the algebra of statements what Cantor did with the algebra of numbers -- adding imaginary statements.

The fact that imaginary values can be used to reason towards a real and certain answer, coupled with the fact that they are not so used in mathematical reasoning today, and also coupled with the fact that certain equations plainly cannot be solved without the use of imaginary values, means that there must be mathematical statements (whose truth or untruth is in fact perfectly decidable) which cannot be decided by the methods of reasoning to which we have hitherto restricted ourselves.

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The Beginnings

The underlying principle of Brown's Laws of Form is the notion of distinction. He begins the work as follows:

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. (My emphases)

You can think of the notion of distinction in different ways, but the core idea is distinguishing one thing from another in nature or properties, or, what to me is recognizing patterns. The indication expresses the distinction. This is the fundamental task of the teacher, to bring you to the point where you can see (read) what's happening. The indications are your patterns of experience. For example, you must already have made a distinction between yourself and the rest of the world. Having done so, having made that first distinction, you have put a boundary between yourself and the rest of the world and thus have created a world out of the void, which becomes the form. ('Void', because there can be no "things" prior to the first distinction.) The two units -- you and the other -- have come from no thing.

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 [flat] space a circle draws a distinction.

Similarly, in a three space a distinction is drawn by a spherical surface. On a doughnut you can draw many circles that make a distinction between points on one side and points on the other, but also many circles that don't make such a distinction. You also make distinctions between sounds, or between colors. Or between objects or ideas. Between higher or lower values, or between rules of different kinds. And so on.

Given the distinction, the states or contents on each side of the line of distinction can be indicated. And of course you cannot indicate differences between things that are not distinct, which is to say you can't talk about things that aren't yet different things. Brown develops the implications of distinctions using an abstract form that represents any such distinctions.

There are two basic axioms to Brown's system: The law of calling, and the law of crossing:

Axiom 1. The law of calling

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

This axiom says you can repeat an indication -- you can recall a name -- without affecting the call. It's still the same call. You don't change anything with a second call.

Axiom 2. The law of crossing

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

This axiom says that two successive crossings of a boundary bring you back to the original side of the boundary -- your starting point. The first crossing takes you to the other side of the line, the second brings you back. There is no third space that can be crossed to. The second crossing only nullifies the first crossing.

Brown allows for the marking of states that are distinguished by distinctions. The mark that he uses is one I can't reproduce here, so I take the liberty of using my own mark, which is simply the pair of parentheses:

(..)

Brown also allows for equality, and so by Axiom 1 he identifies the form of condensation:

(..)(..) = (..)

The idea is that concatenation of the mark with itself is the same as the mark.

States formed by distinctions can also be left unmarked and are indicated by blanks. By Axiom 2, then, he can say:

((..)) = ...

This is called the form of cancellation, and simply says that a double marking is the same as no marking.

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The Primary Arithmetic

In Brown's system the primary arithmetic is a first order arithmetic, so-called, and is to be compared to the ordinary arithmetic of numbers: 3 + 5 = 8, 45 - 20 = 25, 2 x 7 = 14, and so on. It's on the level of "numbers" rather than on the level of the general representations of numbers, like a, b, c, ... x, y, z -- the things you find in ordinary algebra. There is only one number system, but more than one representation of it. The same is true for the non-numerical arithmetic that Brown develops. Remember, he is dealing with statements -- not numbers.

Whether dealing with numbers or statements, though, a basic characteristic of any arithmetic is that it admits of calculations. You make calculations -- or more simply, you calculate -- by taking any of the appropriate steps in the arithmetic, which is to say that you allow certain changes to be made in the arrangements of the elements of the system (valid expressions in the system) to yield equivalent arrangements. Brown allows the following steps, or changes, to be made in the arrangements:

(..) ® (..)(..)

(..) ® (((..)))

These steps can be taken in either direction, left to right or right to left. Keep in mind, too, that the arrangements taking these forms can be of any level of complexity.

The calculus of indications can now be defined as the calculus determined by starting with the two primitive equations:

(..)(..) = (..)

((..)) = ...

From these equations Brown proves nine theorems. See Chapter 4 of his book. 

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The Primary Algebra

From the calculus of indications, identified above, Brown now takes us to his primary algebra. He first allows tokens of variable form, like

a, b, ...,

that indicate expressions in the primary arithmetic. They are like the variables in ordinary algebra, except they refer to statements. Together with the variables, he provides constants by means of the tokens:

(..)

which indicate instructions to cross the boundary of the first distinction.

Now he draws on one of the theorems of the primary arithmetic (T. 8) to provide what he calls the form of position:

((p)p) = ...

From another theorem (T. 9) he offers the form of transposition:

((pr)(qr)) = ((p)(q))r

These two forms, together with two rules of calculation -- substitution and replacement -- then provide the starting point for the algebra. They become the postulates from which he now draws consequences by demonstration. You can find the various demonstrations in Chapter 6 of Laws of Form.

Brown carefully distinguishes between proofs of theorems from axioms in the arithmetic, on the one hand, and demonstrations of consequences from postulates in the algebra, on the other hand.

 

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Second Degree Equations

Hitherto we have obeyed a rule (theorem 1) which requires that any given expression, in either the arithmetic or the algebra, shall be finite. Otherwise, by the canons so far called, we should have no means of finding its [truth] value.

It follows that any given expression can be reached from any other given equivalent expression in a finite number of steps. We shall find it convenient to extract this principle as a rule to characterize the process of demonstration. (My emphasis)

That's how Brown starts his chapter on second degree equations. He then generates a step-sequence that can be carried on without limit. Each step in the sequence is legitimate and yields an equivalent expression. However, inasmuch as the procedure is endless, there is no reason to expect that it should continue to have the same value it had at the start -- keeping in mind that only a procedure having a finite number of steps assures an equivalent value.

In fact, he shows that, by an unlimited number of steps from an expression, e, we can indeed reach another expression, e', which isn't equivalent to e. This introduces indeterminacy in the expression, e'.

Because of the unlimited nature of the step-sequence, part of the whole expression is identical to the whole expression! This is a form of self-representation, which you also find in fractals. The original expression is:

((a)b)

and leads to the expression:

((((...a)b)a)b).

Therefore,

f = ((((...a)b)a)b)

= ((fa)b),

which is the self-referencing function, an implicit function of itself.

This is compared to the form obtained in ordinary (numerical) algebra using an expression like:

x2 + ax + b = 0.

Dividing through by x re-expresses it as a self-referencing form:

x = -a -b/x

And an unlimited repetition of the substitution then leads to:

x = -a -b/(-a -b/(-a -b/(...))),

which is no longer self-referential.

You might note the similarity with Fractals, in which the whole of a pattern is reflected as part of the pattern

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