Laws of Form is something more fundamental.  In his introduction, George Spencer-Brown writes: "A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics."  Later on in the Introduction, GSB writes: "One of the motives prompting the furtherance of the present work was the hope of bringing together the investigations of the inner structure of our knowledge of the universe, as expressed in the mathematical sciences, and the investigations of its outer structure, as expressed in the physical sciences. Here the work of Einstein, Schrodinger, and others seems to have led to the realization of an ultimate boundary of physical knowledge
in the form of the media through which we perceive it. It becomes apparent that if certain facts about our common experience of perception, or what we might call the inside world, can be revealed by an extended study of what we call, in contrast, the outside world, then an equally extended study
of this inside world will reveal, in turn, the facts first met with in the world outside : for what we approach, in either case, from one side or the other, is the common boundary between them."On the nature of mathematics, GSB writes: "A major aspect of the language of mathematics is the degree of its formality. Although it is true that we are concerned, in mathematics, to provide a shorthand for what is actually said, this is only half the story. What we aim to do, in addition, is to provide a more general form in which the ordinary language of experience is seen to rest. As long as we confine ourselves to the subject at hand, without extending our consideration to what it has in common with other subjects, we are not availing ourselves of a truly mathematical mode of presentation.
"What is encompassed, in mathematics, is a transcedence from a given state of vision to a new, and hitherto unapparent, vision beyond it. When the present existence has ceased to make sense, it can still come to sense again through the realization of its form."

Louis Kauffman's work, as he says, is "A presentation of the topology of curves in the plane and how moves on these curves embody the themes of the Calculus of Indications of Laws of Form." He also has a section on "A presentation and discussion about idemposition (the principle that common boundaries cancel). How idemposition of curves in the plane reproduces the calculus of indications and how the calculus of idemposition is related to the Four Color Theorem".  Now this process of common boundaries cancelling appears, to me, to have application in the chemistry of cells and cell membranes, but that is many levels of abstraction away from the mathematical and topological roots.

Knot theory is a part of Topology, so it is covered as well by Laws of Form.

As for its applicability to computing, in Laws of Form, GSB develops everything that is needed to build Finite State Machines and Turing Machines.  In living creatures, the DNA and RNA form the Turing Tapes, which are Universal Turing Tapes as they contain the program as well as data, and the various enzymes are the Finite State Machines that operate on the tapes.  Again this is many levels of abstraction away from the roots in the Laws of Form.

Your bring up quaternions is interesting.  As written, Laws of Form covers only scalar forms.  Forms that have one inside.  This leads to all of the possible scalar numbers.  It leads to numbers as measurement of length and angles.  I am currently exploring what it takes to add vectors, and matrices to the scalar Laws of Form.  Forms of Distinctions that have one  or more inside boundaries. That is where spinors, and quaternions will come in.

Chapter 11, of Laws of Form is devoted to Equations of the Second Degree that develops imaginary states and circuitry.   

As to what is wrong with Frege, I will quote Louis Kauffman: "We urge the reader to read or reread Laws of Form. In our view Spencer-Brown’s book is the most wonderful addition to philosophy, foundations of mathematics, logic and epistemology since the advent of symbolic logic with Boole, Frege, Peirce, Russell-Whitehead and Wittgenstein. Laws of Form begins with the injunction “make a distinction!” This leads directly to the epistemology of things (another name for distinctions) and the understanding that “A thing is identical with what it is not.” A distinction occurs when a space or whole is apparently taken apart (into an inside and an outside) and these two parts cohere into the whole. By choice we designate one side and call it the thing, but it could just as well have been the other side that was the thing. You, indeed, are what you are not. And if you carry this further and ask that the whole Universe be a thing, why then what the Universe is not is Nothing and so the Universe is identical with Nothing. This is paradoxical. Form is emptiness, emptiness is form. The form we take to exist arises from framing nothing. “We take as given the idea of a distinction, and that one cannot make an indication without drawing a distinction. We take therefore the form of distinction for the form” (Spencer-Brown, 1969, p.1)."

By ignoring the "imaginary" states in logic, all of the symbolic logicians have limited themselves to an incomplete subject.

Hope this helps.  Best regards,
Lyle Anderson
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