Boundary Logic

Replacing truth with existence, and deduction with deletion

If you could design math and computing from scratch and without assumptions, what would it look like?

Here’s one recipe: move away from abstract symbols and use one relation, that of containment. That’s what boundary math, logic, and unary computing based on them do.

This is the third post in the Containment series. Containment, besides being the very architecture of our material world, is a fundamental organizing principle of life, language, and thought. The first essay reviews all kinds of containers, from the biological cell to the physical and metaphorical containers we create. It ends with a review of containment in philosophy and logic. The final part is expanded in the second essay, delving into the Calculus of Indications by George Spencer-Brown, as described in his book Laws of Form. That book has inspired people across various fields: mathematics, biology, psychology, cybernetics, philosophy, music, cognitive sciences, computer science, and social sciences. One prominent non-symbolic system for rigorous thinking, heavily influenced by Laws of Form, is the Boundary Math. This current post, third in the series, provides a glimpse at that system. The Boundary Math (aka Iconic Math) was developed by William Bricken with contributions from Jeffrey James and Louis Kauffman.

What is Boundary Math (aka Iconic Math)? William Bricken explains:

Iconic math is rigorous thinking that looks and feels like what it is intended to mean. Postsymbolic thought is embodied experience. Our topic for the moment then is the deconstruction of common arithmetic based on the formal principles first developed by Spencer Brown, with the American philosopher Charles S. Peirce laying the groundwork at the turn of the twentieth century, and with our nomadic ancestors over 30,000 years ago providing tallies as the original substance from which numbers sprang.

The remaining part of the post presents some features of this system.

Arithmetic

When addition is thought of in embodied terms, then it is simply about putting things together. Or, in boundary terms, that is the action of removing a shared boundary.

⟮●●●⟯⟮●●●●⟯ ⇒ ⟮●●●●●●●⟯

It can be done with any number of ensembles.

⟮●●●⟯⟮●●●●●⟯⟮●⟯ ⇒ ⟮●●●●●●●●●⟯

Symbolic addition imposes sequence. For the example above, it will be either first 3 + 5 and then add 1, or 3 + 1 and then add 5, or 5 + 1 and then add 3. With boundary arithmetic, the removal of all boundaries can be done in parallel.

This boundary deletion can be expressed as a void-based transformation:

⟯⟮ ⇒

When ⟯⟮ is transformed into nothing, ensembles get fused.

The polarity of an ensemble can be changed by putting it in angle brackets. On such a basis, subtracting two ensembles is done by matching the magnitude of positive and negative forms.

⟮●●●● <●●●>⟯ ⇒ ⟮●⟯

The next important operation in boundary arithmetic is substitution. Substitution is powerful. It is used, for example, for what in conventional arithmetic we call multiplication and division. I experienced an aha-moment when I saw how multiplication and division are just specific cases of substitution.

Substitution is expressed with a special container like this one:

⟬A C E⟭

which is the instruction to substitute A for C in E. This instruction can be represented in the following way:

⟬PUT FOR INTO⟭

⟬A C E⟭ can be translated as (A×E)/C. This makes it easy to see that multiplication is the special case in which C is one, so we have:

⟬⟮●●●⟯ ● ⟮●●⟯⟭ = ⟮●●●●●●⟯

The multiplication principle is that every unit from one ensemble interacts with every unit from another ensemble. Once represented visually, we can realize how intuitive it is.

Division is a special case of substitution, where

⟬● C E⟭ ⇒ substitute ● for every C in E

Substitution includes structural identities such as global substitution:

⟬A E E ⟭ = A,

self-substitution:

⟬A E E ⟭ = A, and two void-substitions:

⟬ C E ⟭ = A ⇒ delete C from E

⟬A E ⟭ = A ⇒ construct A within E,

where in the first case there is nothing in the “PUT” position and in the second in the “FOR” position.

The natural question now is how to deal with large numbers.

The solution is depth-value. Crossing the boundary outward changes the value of one order of magnitude. The magnitude will depend on the base. In the case of 10-base, that crossing will increase the value by 10. Now all digits can be reused except 0. Then, 2025 can be drawn like this

The calculation of depth value forms involves natural operations such as put, merge and cancel. See chapter 3 of Iconic Arithmetic for details.

Logic

What is Boundary Logic? William Bricken explains:

Boundary logic replaces truth by existence, deduction by deletion, logical connectives by patterns of containment, and sequential inference by concurrent substitutions, without losing any of the capabilities of symbolic logic.

Boundary logic is based on the concept of physical containment, defined as the gestalt between inside, outside and boundary. Unlike Western conventional logic, there is no duality, no true and false, just a single concept for existence.

There are other differences. Boundary logic is iconic, rather than symbolic. Icons maintain structural resemblance to their meaning. Boundary logic is based on information, which is contextual, rather than on truth, which is an absolute value.

Boundary logic reuses the law of crossing:

(()) =

However, the law of calling does not show where the marks are contained. In such a way, it sort of introduces another relation that can be called “taken together.” When the unwritten cross is added, the result is again nothing:

(()()) =

That’s how the boundary logic allows only containment and is fully void-based.

This last form may seem strange. It will become clearer when the concept of pervasion is introduced.

Boundary logic has three axioms, for which only one variable is used. Unlike Laws of Form, it doesn’t include transposition, which can be looked at as another operation, a rearrangement, which is not containment. Here are the three axioms of Boundary Logic:

Occlusion: (() A ) =

Involution: ((A)) = A

Pervasion (A ⋮ A ⋮ ) = (A ⋮ ⋮ ),

Where ⋮ stands for any boundaries, including none.

Pervasion treats boundaries as semi-permeable. Replicas are created inwards and deleted outwards.

Now we can see how pervasion is replacing the law of calling with void-based equivalence

So, if we have this initial form:

(()(()))

then the deepest mark can be deleted, so we get: (()()), but since ⋮ also works in case of no boundaries, we get

(()()) = (()) =

If you want to learn more, here’s a video that provides a detailed explanation and demonstration of the axioms of Boundary logic

Computing

If you could rebuild computation from scratch without any concern for backward compatibility, what would you do? That’s the question the Natural Computing project started with and achieved some remarkable results. Watch here, William Bricken, tell the story