Saturday, May 24, 2014

LoF: Recalling and Crossing from the Unmarked Binaric

LoF symbolizes one binaric (one of the two binaric entities) by no-mark.

Such symbolization is seemingly not only for elegance in calculation but expresses an asserted degenerate identity in LoF, if not binarics itself, of calculation and calculated, operation and operand, instruction and name.

But one consequence of such symbolization is that the law of calling as symbolized in LoF (here in parenthetic form)

( ) ( ) = ( )

appears to imply that that law applies only to the marked binaric, whereas it must apply to both binarics, as can be seen by symbolizing the marked and unmarked binarics as _m_ and _n_ respectively (as is done in LoF occasionally)

m = ( )

n =

and the law of calling with regard to each binaric as

m m = m

n n = n

The law of calling is more correctly and commodiously expressed in LoF's own algebraic Consequence 5 as

a a = a

Similarly, LoF's symbolization of the law of crossing

( ( ) ) =

again appears to imply that that law applies only to the marked binaric, whereas it must also apply to the unmarked binaric, symbolizable as

( ) = ( )

These last two symbolizations and points are perhaps better understood by again using the labels _m_ and _n_ (in part)

( m ) = n


( n ) = m

Note that last parenthetic

( ) = ( )

appears to correspond to the usual law of identity, more correctly and commodiously expressed in binarics as

m = m

n = n

and generally as

a = a

and which does indeed suggest an identity of crossing and identity.

Tuesday, February 25, 2014

Progress (Sort Of) Report

Progress has been slow in my study of ternarics and higher enarics (how complex simplicity can be!), but I hope to finish a small (about 20 pp.) paper I've been working sporadically on over the last couple of years on this subject called "Toward Ternarics" and put it in this year here and in the Yahoo group lawsofform. This paper as its title indicates will be very far from complete even so (more on this below). The present post is more a recitation of mea-culpas, if that's the phrase, if anything.

One recent realization, and this by way of at best qualified mea culpa, has to do with my invocation on various occasions in the Yahoo group of "groundedness" in enarics, which in fact arose from a misunderstanding on my part of the direction of indication in the calculus of form as presented in LoF: Indication is outward, in the opposite direction from depth. Although (the qualification referred to) grounded form may well prove to be in the end another variety of form.

Another and much more important realization has been that The Enary Plenary Parenthetics (TEPP) that I've put forth here and in the Yahoo group represent only a small subset of the varieties of form at each level above the binaric, and a logarithmically smaller subset at that with each increase in order, from ternaric to quaternaric, quaternaric to quinaric, and so on. There are explosions combinatorial and otherwise of properties absent or degenerate in Lof and the binaric as form and analysis increase in order above the binaric; I can barely do more at present than point to the fact (this is another excuse for my slow progress). Essentially, the enary plenaries treated in TEPP are "binaric-like" (I cannot describe it any better here). I strongly doubt any of us will live to see the complete explication of the properties of even the orders treated in TEPP, up to the undecenaric: The ternaric volume corresponding to LoF must necessarily be a larger and likely a much larger volume, and that corresponding to the quaternaric a multi-volume set. (I console myself with the reflection that this is not complexity but richness.)

"Toward Ternarics" will need to be made mutatis mutandis into a new but necessarily still provisional introduction to TEPP. I also need to fix the missing-forms problem above the nonaric in those tables; to generate and release the corresponding tables of what I've been calling "eversions" and now call "perspects" (the internal views in each form, taken as a series of interconnecting cells with the appropriate visibility) as well; to simplify and release the software tools involved (see the next paragraph); and to generate the numeric sequences corresponding to the enary plenaries and perspects in various ways, if only to help people realize when they're dealing with phenomena best modeled by enaric analysis. ("This is not complexity but richness" indeed.)

With the regard to the software tools involved, which are unavoidably going to eventually become very numerous and complex indeed, I tend to write portmanteau warez, which is a fundamental programming error, other than in software intended for nonprogrammer use: The correct "software tools philosophy" is that warez should consist of tightly-defined utilities each doing one thing well, which can then be cobbled together in various pipelines as needed, as in shell languages such as bash and the multimedia language Gstreamer. Jim Snyder-Grant of the Yahoo group wrote a program generating the initial forms at each order (before checking for duplicates and equiforms, eg "(())()" and "()(())") that is one such, as is my "eversions.c" (soon to become "perspects.c")  generating perspects from a supplied form (see the "Software" directory in the Yahoo group), although both should be rewritten to take their inputs from standard input (keyboard or redirected output), for the sake of pipelining. Such arrangement allows software to be used as flexibly as possible, and in ways unimagined by the original programmers. (I might as well add here that Jim's program generates parenthetics deepest on the left, and mine on the right, and while I've come around to believing that left-weighted is best, for ease of taxonomic and dictionary use—another revision of TEPP!—both and related warez should probably be written to work either way.)

One result of reflection upon the ternaric and higher enaric properties and explosions referred to above has been a correction of another foolish error of mine with regard to LoF and the binaric.

GSB says in LoF, "Notes", "Chapter 2", that "In conceiving the calculus of indications, we begin at a point of such degeneracy as to find that the ideas of description, indication, name, and instruction can amount to the same thing."

I misremembered this as embodying a claim that form and symbolization are identical or degenerate in LoF and the binaric, with which I casually if not sloppily (even mystically) did not disagree.

Call an expression combining several subexpressions such as "()()" a "symplex", or "symplexed" (and an involved expression such as "(())" an "implex" or "implexed"—this terminology is from "Toward Ternarics").

The symbolization of symplexes in LoF is (borrowing a term from arithmetic theory) "commutative", eg

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

but the form of course is not, the form represented on both sides of the equation being exactly the same (which is, after all, exactly what that equation symbolizes):

Form and symbol are therefore not identical or degenerate even in LoF and the binaric.

Which is one explosion one can point to resulting from the increase in enaric and analytical order from the unaric to the binaric.

Finally, I am troubled by what I call the "low-hanging fruit" problem with regard to LoF and binarics themselves: Generally, whenever a new form of analysis is introduced, there will be a quick harvesting of problems the solutions of which are then rendered easier or even possible by the new analysis. We have not seen this with regard to LoF and binarics in the half-century since LoF was published. It is true that many varieties of binaric analysis were already in use long before LoF, and indeed LoF represents a pushing downward hierarchy-of-analysis-wise from one such, the algebraic treatment of propositional-logical truth-values, and the application of the algebra of form to the solution of propositional logical problems in Appendix 2 of LoF represents only a beautiful simplification of what was already used and known as "binary resolution". But what about the use of binaric imaginaries, in logic and elsewhere? GSB used one such in a switching-problem. Where are the others? Even GSB has not produced such (and the last I read was totting up seemingly metaphysical ternaries like a Welsh bard, although it's true that taxonomy comes first, hence TEPP). One can point to Maxwell's originally-controversial use of arithmetic imaginaries in his electromagnetic field equations which relatively quickly led to acceptance and widespread use thereof.

Where are the applications of binarics and (more importantly) binaric and logical imaginaries likewise?