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?
Tuesday, February 25, 2014
Monday, February 4, 2013
Between Unity and LoF
The simplest of all cases would be undifferentiable or at least undifferentiated unity.
Unity was primitively recognized and denominated as "abyss", "void", "nameless" and "chaos".
And the simplest of all analyses would be such what might be called "unarics", limited to such recognition and denomination of unity, with no calculation possible.
The simplest of all cases which permit calculation are those analyzable in terms of two indexes, which analyses, calculuses, arithmetics and algebras are called binarics here, and of which George Spencer Brown's Laws of Form (LoF) of course presents a most elegant and thorough description and development.
Note, however that LoF's law of calling
[ ] = [ ] [ ]
can be, but law of crossing
[ [ ] ] =
cannot be, represented on a line, by segments of different colors.
This proves that there exist still simpler cases and corresponding binarics than that presented in LoF, binarics in which the law of calling but not the law of crossing is valid, but which cases and binarics like that of unity and "unarics" are incalculable.
And it also suggests some correspondence of form and representation on the one hand and dimensionality on the other.
But note too that calling involving more than two indexes and therefore similarly simplified what are called here enarics of number of indexes N > 2 can be represented on a line, by use of line-segments of more than two colors, which proves that there exist cases and such enarics in which the law of calling but not that of crossing is valid.
And could there not equally be cases and enarics where the law of crossing but not calling is valid?
Keywords: abyss, analysis, binarics, calculation, calling, chaos, crossing, dimension, enarics, George Spencer Brown, Laws of Form, LoF, nameless, unity, void
Unity was primitively recognized and denominated as "abyss", "void", "nameless" and "chaos".
And the simplest of all analyses would be such what might be called "unarics", limited to such recognition and denomination of unity, with no calculation possible.
The simplest of all cases which permit calculation are those analyzable in terms of two indexes, which analyses, calculuses, arithmetics and algebras are called binarics here, and of which George Spencer Brown's Laws of Form (LoF) of course presents a most elegant and thorough description and development.
Note, however that LoF's law of calling
[ ] = [ ] [ ]
can be, but law of crossing
[ [ ] ] =
cannot be, represented on a line, by segments of different colors.
This proves that there exist still simpler cases and corresponding binarics than that presented in LoF, binarics in which the law of calling but not the law of crossing is valid, but which cases and binarics like that of unity and "unarics" are incalculable.
And it also suggests some correspondence of form and representation on the one hand and dimensionality on the other.
But note too that calling involving more than two indexes and therefore similarly simplified what are called here enarics of number of indexes N > 2 can be represented on a line, by use of line-segments of more than two colors, which proves that there exist cases and such enarics in which the law of calling but not that of crossing is valid.
And could there not equally be cases and enarics where the law of crossing but not calling is valid?
Keywords: abyss, analysis, binarics, calculation, calling, chaos, crossing, dimension, enarics, George Spencer Brown, Laws of Form, LoF, nameless, unity, void
Saturday, June 30, 2012
A Possibly New Primary Arithmetic Theorem:"Volution"
The primary arithmetic theorem "Variance" (T9) of Laws of Form (hereafter "LoF") can be written in square-bracket notation as
[[pr][qr]] = [[p][q]]r
(square-bracket is preferable to paren notation in text, since it doesn't conflict with the literary use of parentheses).
(The theorem interpreted as logic—see Appendix 2 of LoF—might read " '( p OR r ) AND ( q OR r )' is equivalent to '( p AND q ) OR r'.)
The theorem itself is proven in LoF by application of Theorem 16, "The bridge" (see the final remark below), showing it holds in either case or value of r, reducing to
[] = []
when r is dominant ( r = [] ) and
[[p][q]] = [[p][q]]
when r is recessive ( r = ).
Let p be dominant instead, and the theorem reduces to
qr = qr
while letting q be dominant yields
pr = pr
But let p be recessive, and the theorem reduces to
[[r][qr]] = r
while letting q be recessive yields
[[pr][r]] = r
The left-hand expressions of these equations are of course equiformal, since
[[r][qr]] = [[qr][r]]
and the equations therefore represent a single form of equation.
This equation can be rewritten as
[[a][ab]] = a
(since ab = ba).
It holds for all cases; dominant a yields
[] = []
while recessive a yields
=
and both cases of b yield
a = a
This equation is not presented in LoF as a primary arithmetic theorem, nor does it seem to be a consequence of the algebra.
This suggests that it is a new primary arithmetic theorem and might, therefore, be a new, third initial of the algebra as well, opening a whole new world of consequences.
(The theorem interpreted as logic might read " 'a AND ( a OR b )' is equivalent to 'a' ".)
Should the theorem prove indeed to be new, it is suggested that it be called "Volution", its forward step
[[a][ab]] -> a
"devolution", and its reverse
[[a][ab]] <- a
"evolution": This nomenclature is itself suggested by the reflection (LoF primary algebraic consequence C1) and internal iteration (C5) implicit as first steps in the latter, which recall the genetic duplication so critical to biological evolution, allowing mutation of a duplicate gene without losing the function of the protein encoded by the original.
The theorem might be written
If successive spaces sn, sn+1, sn+2 are arranged so that sn, sn+1 are distinguished by one cross, and sn+1, sn+2 are distinguished by two crosses, sn+2 being thus in two divisions, both divisions sharing an expression, one containing an unshared expression, then the value of the resultant expression in sn is the shared expression.
Note that
[[ab][ac]] = a
does not hold for recessive a or dominant b or c.
Finally, the above arose from a continuing skeptical consideration of LoF Theorem 16, "The bridge".
Keywords: binarics, binaries, enarics, enaries, George Spencer Brown, Laws of Form, LoF
[[pr][qr]] = [[p][q]]r
(square-bracket is preferable to paren notation in text, since it doesn't conflict with the literary use of parentheses).
(The theorem interpreted as logic—see Appendix 2 of LoF—might read " '( p OR r ) AND ( q OR r )' is equivalent to '( p AND q ) OR r'.)
The theorem itself is proven in LoF by application of Theorem 16, "The bridge" (see the final remark below), showing it holds in either case or value of r, reducing to
[] = []
when r is dominant ( r = [] ) and
[[p][q]] = [[p][q]]
when r is recessive ( r = ).
Let p be dominant instead, and the theorem reduces to
qr = qr
while letting q be dominant yields
pr = pr
But let p be recessive, and the theorem reduces to
[[r][qr]] = r
while letting q be recessive yields
[[pr][r]] = r
The left-hand expressions of these equations are of course equiformal, since
[[r][qr]] = [[qr][r]]
and the equations therefore represent a single form of equation.
This equation can be rewritten as
[[a][ab]] = a
(since ab = ba).
It holds for all cases; dominant a yields
[] = []
while recessive a yields
=
and both cases of b yield
a = a
This equation is not presented in LoF as a primary arithmetic theorem, nor does it seem to be a consequence of the algebra.
This suggests that it is a new primary arithmetic theorem and might, therefore, be a new, third initial of the algebra as well, opening a whole new world of consequences.
(The theorem interpreted as logic might read " 'a AND ( a OR b )' is equivalent to 'a' ".)
Should the theorem prove indeed to be new, it is suggested that it be called "Volution", its forward step
[[a][ab]] -> a
"devolution", and its reverse
[[a][ab]] <- a
"evolution": This nomenclature is itself suggested by the reflection (LoF primary algebraic consequence C1) and internal iteration (C5) implicit as first steps in the latter, which recall the genetic duplication so critical to biological evolution, allowing mutation of a duplicate gene without losing the function of the protein encoded by the original.
The theorem might be written
If successive spaces sn, sn+1, sn+2 are arranged so that sn, sn+1 are distinguished by one cross, and sn+1, sn+2 are distinguished by two crosses, sn+2 being thus in two divisions, both divisions sharing an expression, one containing an unshared expression, then the value of the resultant expression in sn is the shared expression.
Note that
[[ab][ac]] = a
does not hold for recessive a or dominant b or c.
Finally, the above arose from a continuing skeptical consideration of LoF Theorem 16, "The bridge".
Keywords: binarics, binaries, enarics, enaries, George Spencer Brown, Laws of Form, LoF
Tuesday, September 13, 2011
First Forms:
The Enary Plenary Parenthetics
In 1969, George Spencer Brown published Laws of Form (abbreviated among the cognoscenti as “LoF”), the classic and exhaustive study of the simplest possible analysis, involving two indexes or indices and transition between those indices, providing an elegant and powerful calculus for such analysis; extending it to the corresponding binary arithmetic and algebra; treating questions both fundamental and advanced about such analysis, calculus, arithmetic and algebra; and applying that algebra in Appendix 2 to the binary resolution or analysis of propositional logical arguments and to set analysis.
The book has gone through many editions since, and deservedly so.
Since the publication of LoF, there has been no comparable study of the next least complex or most simple analysis, three-index or ternary analysis, still less four-index or quaternary, five- or quinary, and least of all the generalized N-ary or enary (nor should one discount the validity and eventual elucidation of the transfinary).
The companion volume to this one, Enaries (now, July ’11, in progress), begins the study of enarics or enary analysis, by examining the enary plenaries, the fundamental forms that become available as order of analysis increases, from unary to binary, binary to ternary, ternary to quaternary, and so on, up to the septenary (N = 7), each new plenary form using each and every index available at its order once and only once.
The companion data CD to this volume and that, The Enaries CD (also available as a download) (also in progress), contains various digital forms of both volumes, the separate figures, graphics and tables, extended graphics and tables, and the software used to generate all of the above (generally more or less simple original programs written in the very simple, ancient, beautiful and powerful C programming language).
This volume simply presents the enary plenaries, as parenthetics, from the unary (N = 1) to the terdecenaries (N = 13), simply enumerated and tabulated, for their beauty alone, certainly, but also for examination of their fundamental characteristics and properties and consequent taxonomizings.
Note that each abstract plenary form here represents N! equiforms; for example, the ternary plenaries “(())” and “()()” each represents 3! = 3*2*1 = 6 possible forms, eg "A(B(C))", "A(C(B))", "B(A(C))", "B(C(A))", "C(A(B))" and "C(B(A))" in the first case, and "A(B)(C)", "A(C)(B)", "B(A)(C)", "B(C)(A)", "C(A)(B)" and "C(B)(A)" in the second.
The Enary Plenary Parenthetics
(As usual, this research could use some financial support.)
[20110927 Jim Snyder-Grant of the Yahoo group lawsofform has found several missing enary plenaries in the nonaries, eg "(()()())((()()))", so that means the posted list is incomplete.
[The list will be expanded as soon as possible.]
keywords: enarics, enaries, enary analysis, foundations of analysis, George Spencer Brown, laws of form, LoF, mathematics, philosophy, science
The book has gone through many editions since, and deservedly so.
Since the publication of LoF, there has been no comparable study of the next least complex or most simple analysis, three-index or ternary analysis, still less four-index or quaternary, five- or quinary, and least of all the generalized N-ary or enary (nor should one discount the validity and eventual elucidation of the transfinary).
The companion volume to this one, Enaries (now, July ’11, in progress), begins the study of enarics or enary analysis, by examining the enary plenaries, the fundamental forms that become available as order of analysis increases, from unary to binary, binary to ternary, ternary to quaternary, and so on, up to the septenary (N = 7), each new plenary form using each and every index available at its order once and only once.
The companion data CD to this volume and that, The Enaries CD (also available as a download) (also in progress), contains various digital forms of both volumes, the separate figures, graphics and tables, extended graphics and tables, and the software used to generate all of the above (generally more or less simple original programs written in the very simple, ancient, beautiful and powerful C programming language).
This volume simply presents the enary plenaries, as parenthetics, from the unary (N = 1) to the terdecenaries (N = 13), simply enumerated and tabulated, for their beauty alone, certainly, but also for examination of their fundamental characteristics and properties and consequent taxonomizings.
Note that each abstract plenary form here represents N! equiforms; for example, the ternary plenaries “(())” and “()()” each represents 3! = 3*2*1 = 6 possible forms, eg "A(B(C))", "A(C(B))", "B(A(C))", "B(C(A))", "C(A(B))" and "C(B(A))" in the first case, and "A(B)(C)", "A(C)(B)", "B(A)(C)", "B(C)(A)", "C(A)(B)" and "C(B)(A)" in the second.
The Enary Plenary Parenthetics
(As usual, this research could use some financial support.)
[20110927 Jim Snyder-Grant of the Yahoo group lawsofform has found several missing enary plenaries in the nonaries, eg "(()()())((()()))", so that means the posted list is incomplete.
[The list will be expanded as soon as possible.]
keywords: enarics, enaries, enary analysis, foundations of analysis, George Spencer Brown, laws of form, LoF, mathematics, philosophy, science
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