This is a blog about George Spencer-Brown’s ideas (expounded in “Laws of Form“), as well as about his numerous philosophical disciples, a rather big crowd that includes extraordinary individuals, as well as ordinary people (who have been influenced by George Spencer-Brown‘s ideas): E.g. Louis Kaoufman, Richard ShoupArt Collings, Dave KeenanWilliam Bricken, Tom McFarlaneBen Goertzel, Eddie Oshins, Francisko Varela, Natalia Petrova, Jeff James, etc. and… myself – through Multiple Form Logic.
This blog’s header depicts the two fundamental axioms or “initials” of George Spencer-Brown’s “Primary Arithmetic“: The image on the left is the “Law of Calling” and the one on the right is the “Law of Crossing“, i.e.
Law of Calling
Law of Crossing
There is a certain revival of George Spencer-Brown’s ideas, taking place nowadays, and this blog will -hopefully- contribute to creative public discussions about GSB’s ideas.
George Spencer Brown
There is already a Yahoo group dedicated to “Laws of Form” (started many years ago by Mr. ) called the “Laws of Form forum”. However, all discussions in that Yahoo group are private, i.e. not visible to non-members (and to Search Engines).
An important goal of this blog is to gather here as much information as possible, about “Laws of Form”, George Spencer-Brown. and any related research, or references to all this by others (including critics)! As an appetizer, my “del.icio.us” collection of (more than 5600) links (http://delicious.com/omadeon) also contains:
- http://delicious.com/omadeon/LoF (about “Laws of Form”)
- http://delicious.com/omadeon/GSB (about George Spencer-Brown)
Another goal of this blog is to provide information about “Multiple Form Logic“, my own formal / philosophical extension of GSB and “Laws of Form”. This work has received some recognition in recent years, e.g.
- The research of Mr. Ralf Barkow, a Computer Scientist who published some interesting unifications of Multiple Form Logic with the Pile System (invented by Erez Elul).
- Wikipedia references here (“related work to Laws of Form”) and here (“external links about Laws of Form”); written by some (unknown) nice people (most probably a group of researchers from the official “Laws of Form Forum) – who reviewed my work. Here are these references:
The Multiple Form Logic, by G.A. Stathis, “generalises [the primary algebra] into Multiple Truth Values” so as to be “more consistent with Experience.” Multiple Form Logic, which is not a boundary formalism, employs two primitive binary operations: concatenation, read as Boolean OR, and infix “#”, read as XOR. The primitive values are 0 and 1, and the corresponding arithmetic is 11=1 and 1#1=0. The axioms are 1A=1, A#X#X = A, and A(X#(AB)) = A(X#B).
Τhe Multiple Form Logic, by G.A. Stathis, owes much to the primary algebra.
BTW, in case you got the… wrong idea(!) I did not write these references! Oh no! 🙂 Besides, although I feel greatful towards these… nice people, I do not agree with… everything they wrote! 🙂
E.g. as regards their reference to “Multiple Form Logic, which is not a boundary formalism”… I believe that -on the contrary– Multiple Form Logic IS a “boundary formalism”; a very fundamental and radical one, in fact! My view is also that Multiple Form Logic changes the way we think of boundaries, as such; enhancing the ontological nature or (if you prefer) the existential fabric of (Reality consisting of -) boundaries, in (at least) 2 ways:
- 1) To start with, boundaries are multiple (rather than the Unique, one-and-only Form in “Laws of Form”). Other multi-valued extensions to “Laws of Form” have been proposed by others, e.g. Ben Goertzel’s “Ons Algebra”.
- 2) Secondly, to our amazement -perhaps- boundaries are also (or can also be) mathematical “first class citizens“; i.e. boundaries are entire expressions (inside their own system) themselves! (This entails a conceptual difficulty of a need for a paradigm shift, as well as a practical difficulty of visual representation; a challenge for Visualisation software; e.g. “DreamProver” -my own visual theorem prover, totally un-funded, delayed, long overdue; more about it… soon! – hehe)
Furthermore, leaving aside my Theorem Proofs in Multiple Form Logic and More Theorem Proofs (about William Bricken’s system, etc)…
- Some cool alternative proofs of Multiple Form Logic theorems have been produced by Mr. Art Collings, a professional mathematician.
Now, although Mr. Art Collings is an uncompromising critic of my work (!) I think that his criticisms have been of immense benefit! His professional mathematical expertise has helped me clarify some very important issues, while his objections to some of my (occasionally… wild) philosophical claims (hehe) have been extremely valuable and thought-provoking, despite the fact we (usually) don’t agree. E.g. Some time ago he had expressed strong doubts (HERE) about the “completeness of Multiple Form Logic, saying that it would be (probably) impossible to prove. However, soon afterwards, I produced a verifiable formal proof of the contrary, which he then verified (which… was nice of him). He subsequently produced his own (different) proofs, after doing some research. Most notably, these communications took place (more-or-less) in public: -In the (semi-)public “Laws of Form Forum“, a Yahoo Group where William Bricken also participates. Some interesting logical problems have been solved (with mathematically sound answers) in those public communications. However, there is still, a… tiny open problem:
- My proof of Theorem T12 (here), stating that William Bricken’s “Boundary Algebra” is “a special instance of Multiple Form Logic” is still not accepted by William Bricken himself, on (more-or-less) philosophical grounds!
- Nevertheless, this proof has been accepted by other researchers, e.g. Ralf Barkow. Also, Mr. Tasos Patronis (Ph.D), a Greek mathematician who -I must admit- is also a good friend of mine (OK, so maybe he’s a bit biased -hehe)
- The real problem in this case, is not the proof itself (which is undoubtedly consistent) but whether or not certain philosophical and mathematical (meta-)criteria are also satisfied, validating my… wild claim that Multiple Form Logic is more generalised and more fundamental, as a “theory of boundaries”, than Bricken’s “Boundary Logic”.
William Bricken’s main objection is that Multiple Form Logic is a “higher abstraction” than his “boundary logic”. So, philosophically (he argues) it would be wrong to regard it as “more fundamental” than his system. However, my contentions are:
1) that this is not a drawback, but a formal advantage (since Bricken’s logic follows, as a “special instance” –provably– precisely by assuming only one type of Form in M.F. Logic) and
2) that all Forms are multiple – from the very beginning – i.e. that Multiplicity is fundamental, in this Universe! Go(d) figure… 🙂
Ah well – all this is a loooong story, beginning here:
(and the rest is in the “Laws of Form Forum” archives).
P.S. Art Collings (much to the delight of William Bricken) made use of the (well-known) fact that the “XOR” relation can be re-expressed as a composite expression containing only ORs and NOTs, to prove that one can -indeed- construct (without adding any new axioms or unproved assumptions) a Multiple Form Logic system by using the XOR relation as a new abstraction, inside (Bricken‘s) system. However, it appears that Bricken’s system itself follows (provably) as a special instance of Multiple Form Logic, if (and only if) all different forms are fused into one .
-So… which is the “chicken” and which is the “egg”?
- Well, devotees of the “Simplest Egg” in the Universe (that can probably make the smallest… omelet) say “boundary logic” is “more fundamental”.
- Proponents of Simplicity… NOT necessarily being associated with “the One” but probably (a) being of a strange new Fundamental Quality (that can be) called “A Priori Multiplicity”… can say that (omelets being in need of many eggs, anyway) …Multiple Form Logic is “more fundamental”.
Still others, may blatantly theologize: –Is there one God, or many Gods?
- Hm… does it… matter, how many Gods you imagine? 🙂
– Ah Well, …It may turn out, actually… . . . . that it doesn’t matter!!! 🙂
- What does appear (to matter) however, is not “the number of Gods” but the number of… sacrifices(!):
- Worshipping the Logic of Only One God (=Truth Value) …it turns out that you need an exponentially larger amount of Deductions (Proof steps, i.e. logic computation) than for the Logic of Multiple Gods ….er… Forms. More about this astonishing fact will be explained further, during the course of events to come. (Appetizers are here).