15) Philosophical aspects of Multiplicity in Multiple Form Logic^™

George Spencer Brown in "Laws of Form" had adopted a monotheistic interpretation of Forms: He identified "God the Father" with the First Distinction itself, "God the Son" with the Outer Space which resides outside the First Distinction, and "God the Holy Ghost" with the Inner Space inside the First Distinction. These ideas were also expounded in "Only Two Can Play This Game" and led to Spencer Brown's strange theological doctrine that "The Holy Spirit is Female" (identified with "Nirvana", or the Clear Light of the Void).

In contrast, when Multiple Form Logic was created, the philosophical inspiration behind it was Polytheistic, rather than Monotheistic: created with a different metaphysical vision, it adopted a polytheistic interpretation of Forms: Only the "Formless Void", or the Space where no distinction has yet been created, was regarded as "unique in the Universe" and Monotheistic; just like the ancient Greek religion identified Chaos, or "the Void", as the "Mother of all the Gods", but the Gods and Goddesses were multiple. Thus the "One God", which is the "All", or "All the Forms in the Universe", became a "construction" generating Axiom 1 of Multiple Form Logic.

Nevertheless, axiom 1 is not essential for deriving some interesting consequences from the other two axioms. On the contrary, I strongly suspect that if we do not take Axiom 1 for granted, we end up with a  type of "Alternative Logic" or "Quantum Logic", closer to the work of Eddie Oshins and his "Quantum Psychology". (It is interesting to see what happens to the "generalised distributive law" in such an alternative boundary logic).

However, In private communications with another member of the "Laws of Form" forum (Mr. Philip Meguire), he pointed out that one of George Spencer Brown's achievements was the reduction of Propositional Logic to arithmetic axioms. I.e. the "initials" of Spencer Brown's "Primary Algebra" were not axioms "as such", but were proved as consequences from Spencer Brown's "axioms of the primary arithmetic" (I1 and I2).

So I began to think more about this issue, and concluded that there is no  reason why Multiple Form Logic cannot be defined in a similar way (if we desire this), derived from a kind of "Primary Arithmetic".  Except that (in this case) the "new arithmetic axioms" are not going to be the same as George Spencer Brown's.

First of all, the existing axioms of the "primary arithmetic" need to be interpreted differently, within "Laws of Form" itself: Instead of  "1 or 1 = 1"  and  "not(1) = 0"  (as Brown assumed) they must be re-interpreted as  "1 XOR 1 = 0"  and  "1 OR 1 = 1"  (as explained in the section "All  we need is OR and XOR"). This is the only correct and consistent way of interpreting the "Primary Arithmetic". Once we adopt this interpretation, the remaining material in "Laws of Form" follows naturally as before, except that the interpretations of his logic become absolutely consistent with Boolean Algebra. For example, Spencer Brown's "imaginary operator" is now seen as "XOR", and it becomes consistent with conventional work in the construction of counters and pseudo-random sequence generators (etc), all of which are based in Exclusive-OR gates, for their operation.

Secondly, I sometimes suspect that there is a deeper foundation, than the Three Axioms of Multiple Form Logic, which constitutes an "alternative primary arithmetic". I can not sustain dogmatically the romantic polytheism of my early twenties, where Distinctions were seen as "inherently multiple". E.g. How did they become Multiple, in the first place? There could be a kind of deeper process, something like a "primary construction", which distinguishes between Forms at a more fundamental level than the Three Axioms: Such a "primary construction" would then become the origin of the "distinction between distinctions", a kind of  "meta-distinction", like a "passage-way" from the "One God Universe" to the Polymorphic and Polytheistic "Universe" of Multiple Forms. - Well, why not?

Suppose, for example, that (in a plane space of representation) we draw two distinctions, as circles within this space (ala George Spencer Brown):



Initially, there is no distinction between these two distinctions, which means that: Unless we perform within this space some kind of "additional distinction", these two distinctions (which we have drawn) collapse into only one, in accordance with George Spencer Brown's Axiom 1 of the "primary arithmetic":



Suppose now, that we require these two distinctions to "co-exist", without collapsing into only one. Then, there must be some kind of "additional distinction" that we can make, and which distinguishes between these two distinctions, most probably at a kind of "meta-level" to these distinctions themselves.

I.e. if they cannot be distinguished between them, they have to collapse into one. On the other hand, if they can be distinguished between them, then there must be a "new type of distinction" at a kind of "metalevel", which has the ability to "distinguish between distinctions". However, if our "vocabulary" (of distinctions) does not contain any such "meta-distinction", or any other distinction than "the One" (the "Marked State" of George Spencer Brown), it becomes by definition impossible to distinguish between (any) two distinctions, so that all our attempts to distinguish between (any) two distinctions (such as the ones we drew) are bound to fail, from the beginning! (These considerations should be made side by side with Primordial Theorem 1, of section 2).

Multiple Form Logic was created (partly) by realising this problem, and by attempting to solve it in a different way than "accumulations" of Brown's "marked state": The problem was solved by assuming all distinctions to be inherently multiple, rather than one. Thus, it becomes possible for different distinctions to co-exist naturally at all levels. Only when they are the same, do they collapse into One (by Theorem T2), or "cancel out each other" (by Axiom 2). Nevertheless, it is still possible to construct a special distinction  ("1"), defined as "the union of all distinctions in the universe", which generates Axiom 1 if we wish (and so on).

So, to prevent any two distinctions from collapsing into one, all we need is to assume that they are different distinctions:



However, the new question that arose naturally, from Mr. Meguire's remark about the primacy of arithmetic to algebra, is this: Is there another way to define distinctions, e.g. using only two distinctions, one of which is the "basic distinction", and the other one a kind of "meta-distinction" (or "distinction between distinctions"),  responsible for generating arithmetically all the different distinctions in the "universe" of Multiple Forms?

Well, other systems of Boundary Algebra indicate that there indeed exist many possible ways to define a mechanism that generates the natural numbers, within a Logic that can be made similar to Multiple Forms. And once we have the natural numbers, we then can associate each natural number with a unique single distinction, so as to get different distinctions, without having to assume that they are different axiomatically. Then the universe of Multiple Form Logic can become a construction, arising out of a very small number of other, more fundamental distinctions, and also some modified axioms capable of accomodating a suitable construction mechanism.

Moreover, my intuition compels me to speculate that only axiom 3 of Multiple Form Logic is fundamental in this "universe". I.e. we can play around with the other two axioms as much as we like, generating all kinds of logic systems with different properties, but it seems unlikely that we can play around too much with axiom 3.

An interesting example of creating a logic system which is a superset of Multiple Form Logic, and yet also obeys the existing three axioms, is the following: -A very slight modification of Axiom 2 is to assume that identical distinctions collapse into different kinds of Void. (Lou Kauffman has discussed extensively some interesting consequences of assuming zero -or "voidness"- to be multiple rather than one). The natural numbers can then be generated by assuming a succession of "multiple voids", where the first void is the outcome of just one pair of identical distinctions cancelling out each other, the second void is the outcome of two such pairs, and so on. The advantage of this approach is that axiom 2 of Multiple Form Logic is still valid, as it stands, for the particular case of two distinctions collapsing into void, and it remains valid in other cases too, if we assume that the different kinds of void do not affect logical operations within Multiple Form Logic itself, but only affect operations in a kind of "parallel universe", co-existing peacefully with Multiple Forms. In this universe, different voids may follow other laws, that do not affect Boolean operations, but only arithmetical ones, as well as certain kinds of "meta-operations" that generate different distinctions. The "different distinctions" can then be constructed within this "parallel universe", but remain invisible within Multiple Form Logic, exactly like quarks are totally invisible within "classic" Physics, but concern Quantum Physics. I.e. whenever we enter this "parallel universe" to examine more closely what is going on, what appears (to Multiple Form Logic) as inherently different distinctions, is the result of certain more fundamental "hidden operations", taking place in a parallel universe which contains mechanisms of arithmetic.

This is just one possible path of further research about Multiple Form Logic, that I'm currently contemplating. The philosophical disadvantage of this path, is of course the existence of multiple voids: It seems to rely on shifting the problem elsewhere, rather than solving it. Then again, perhaps the problem is partly unsolvable, anyway: The problem of "Multiplicity versus Oneness" in metaphysics, is considered unsolvable e.g. within Buddhism: In Buddhism, it is considered useless to speculate about the Origins of the World, while it is said that "the World has always existed" with many beings and gods who have existed since uncountable time, because there was no beginning of Time (or: even if there was, it is useless to think about it, since our minds are "not capable of resolving such issues"; e.g. even if physical time can be traced back to the "Big Bang", there is nothing to stop us from assuming a "previous phase", before the Big Bang took place, about which we are never going to get any useful information, and so on... ) Moreover, if we assume that this universe of multiple beings and Multiple Forms can be "explained" by another "parallel universe", where the forms are arithmetically constructed, we may be acquiring a certain formal advantage that creates other kinds of new problems, which we cannot imagine yet. - Anyway, if you have new ideas about all this (wild speculation) let me know! ;)

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