Naturalism Philosophy Forum Group Participants,

Part II. of II. of F.E.D. Vignette #4, "The Goedelian Dialectic of the Standard Arithmetics", was just posted to the --

www.dialectics.org

-- website.

The scope and content of Part II. is rather replete, but, for the purposes of this post, I want to quote its discussion of just one topic, that shows how accession to the dialectical operations stage of human adult cognitive development -- transcending the last Piagettian "formal operations" stage -- an accession exhibited so abundantly by Plato, Hegel, and Marx, can open, among many other new vistas, new vistas of knowledge representation condensation.

Below I have reproduced Part II., pages II-58 through II-59, as best as the available typography here will allow.

The definitions of the symbols for the axioms-systems of arithmetic used in the extract below are as follows:

N# connotes the axioms-system of the arithmetic of the "Natural Numbers", N;

W# connotes the axioms-system of the arithmetic of the "Whole Numbers", W;

Z# connotes the axioms-system of the arithmetic of the "Integers" [in German,

<<Zahlen>>], Z;

Q# connotes the axioms-system of the arithmetic of the "Rational, or Quotient,

Numbers", Q;

R# connotes the axioms-system of the arithmetic of the "Real Numbers", R;

C# connotes the axioms-system of the arithmetic of the "Complex Numbers", C;

H# connotes the axioms-system of the arithmetic of the Hamilton "Quaternions",

H;

O# connotes the axioms-system of the arithmetic of the Cayley/Graves

"Octonions", O;

K# connotes the axioms-system of the arithmetic of the "William Kingdon Clifford

Numbers", K;

G# connotes the axioms-system of the arithmetic of the Grassmann "Geometric

Numbers", G, and;

X# connotes the axioms-system of the arithmetic of the [unknown] "next"

arithmetic, X, . . .

"Symbolic Economy, Semantic Density / Semantic Productivity, and Mnemonic Power.

The `Dialectical Equation' that constitutes our `meta-model' of the systems of

the ```Standard Arithmetics''' --

#)-|-(s = (N#)^(2^s)

-- functions also as our Encyclopedia Dialectica definition of ```Standard

Arithmetic'''. [Part I. of this essay contains two formulas modeling two models

for the Value-Form Dialectic of the three volumes of Marx's Capital, with

similar Knowledge Representation Condensation Ratios to those of the above

'meta-model']

As such, it is a dialectically ```open-ended''' kind of definition.

No final term, no ultimate `meta-meristem', no closing ```culminant''', is

specified in this `meta-model', by its `Dialectical Equation'.

It remains a `"potentially infinite"' [cf. Aristotle] sequence of series, though

one which is always, at any given moment of Terran human history, actually

[meta-]finite in terms of that part of its infinite potential which has been

actualized by Terran humanity so-far.

This `dialectical definition' of ```Standard Arithmetic''' is therefore not

simply N#, or W#, or even C#, or H#, or O#.

It is, on the contrary, the entire `meta-system-atic', dialectical and cognitive

movement from N# to W#, from W# to Z#, from Z# to Q#, from Q# to R#, from R# to

C#, from C# to H#, from H# to O#, and beyond, that is summarized by, and

"contained in", that `Dialectical Equation' --

N# ---) W# ---) Z# ---) Q# ---) R# ---) C# ---) H# ---) O# ---) K# ---) G#...

s: 1 ---) 2 ---) 3 ---) 4 ---) 5 ---) 6 ---) 7 ---) 8 ---) 9 ---) 10 ...

-- both actually [e.g., to K# and to G#.], and potentially [to arithmetics

beyond those that have been actualized -- codified or axiomatized -- by Terran

humanity to-date], given the "incompletability or inexhaustibility" of

mathematics in general, and of arithmetics in particular, established by Gödel.

However, confronted with the potential infinity of such encyclopedic

dialectical-equational definitions, we must grapple with issues of the ease and

compactness of their `representability' via our 'dialectical arithmetics', and

via their `dialectical algebras'.

Doing so, we find that our situation is, indeed, quite favorable in that regard.

If we strip the `Dialectical Equation meta-model' that forms the core of this

essay down to its bare essentials, stripping off all of the helpful but

inessential taxonomical locator epithets, or `dialectical diacritical marks',

then our most condensed concentration of the meaning of this entire essay

requires just four symbols, or symbolic elements, namely, the elements `_', `N',

`2', and `6', arrayed as follows --

N^(2^6)

-- such that the 4 symbolic-elements above [given font infrastructure supporting

superscripts, and superscripts of superscripts], so arranged, can replace, e.g.,

the entire 64-term, ~641 symbolic-element expression that concludes the core

section of this essay. They can do so in this sense: the entire 64-term series

can be re-constituted and recovered, from the 4 symbol, `semantically

concentrated' version, simply by repeatedly applying 3 simple rules -- i.e.,

just 3 of the 9 core axioms of the NQ space of dialectical arithmetic, as given

herein within section B.i. -- namely, Axioms §7, §8, and §9 --

§7. For all n in N: qn + qn = qn.

§8. For all j, and k, both in N: If j is quantitatively unequal to k, then qj +

qk is qualitatively unequal to qx for any x in N.

§9. For every j and k, both in N: qk x qj = qj + qk+j.

-- and by one or more applications of the `Organonic Algebraic Method' to

```re-solve-for''' any once-known but no-longer-known-remembered terms, when the

meanings of some of them are forgotten subsequent to reading this essay.

If we take the ```replacement rate''' -- the percent-ratio of the count of the

number of terms replaced to that of the symbolic elements so replacing -- as a

crude metric for the degree of `semantic compression', or of

`knowledge-representation-condensation', achieved, then the `semantic density'

improvements that we are achieving by using the stripped down, dialectical,

Dyadic Seldon Function formulations, are impressive, viz. --

· out to R# and its «aporia», N^(2^5): 32/4 = 8 = 800% `semantic condensation

rate';

· out to C# and its «aporia», N^(2^6): 64/4 = 16 = 1,600% `semantic

condensation rate';

· out to H# and its «aporia», N^(2^7): 128/4 = 32 = 3,200% `semantic

condensation rate';

· out to O# and its «aporia», N^(2^8): 256/4 = 64 = 6,400% `semantic

condensation rate';

· out to K# and its «aporia», N^(2^9): 512/4 = 128 = 12,800% `semantic

condensation rate';

· out to G# and its «aporia», N^(2^10): 1,024/5 ~ 204 = 20,400%

`semantic condensation rate';

· out to X# and its «aporia», N^(2^11): 2,048/5 ~ 410 = 41,000%

`semantic condensation rate';

Using the `minimalized' Seldon Function format -- a^(n^s) -- the systematic(s)

core of a discourse: of a whole lecture, or of a whole text -- paper, essay,

book, multi-«buch»/multi-volume treatise, etc. ... -- can be mnemonically

summarized, using as few as four symbolic elements, in an expression which, with

the application of three rules, & of the `organonic method', if needed, can, at

will, be quickly reconstituted into a series/sum/cumulum of tens, or hundreds,

or thousands,... of terms, capturing, in systematically-ordered detail, the gist

of the content of that discourse.

That `minimalized' Seldon Function format can formulate condensed,

`re-implicitized', `connotationally curtailed', or `darkened', ```black

[w]holes''' of information, from which ```white [w]holes''' of outpouring

```[w]holistic'''/mnemonic re-elaboration and reconstitution of that information

are ever ready to be `re-unfolded', to be `re-unfurled', to be `rotely'

`re-burgeoned', by those who know the 3 axiomatic rules [and the `organonic

method'].

The mere assertion of a category, within a specific, interpreted

progression/sum, or `[ac]cumulum', of categories, is not, in itself, the

delineation and articulation, or `explicitization', of the detailed content --

of the progression/sum/cumulum of sub-categories and of sub-sub-categories...

which are implicit in that category when it is asserted as an unarticluated,

undelineated, undivided, univocal whole.

But the assertion of that undivided category does serve as a collective name

for, and as a reminder of -- an intimation of -- the content of that category in

its more fully articulated detail, as experienced/conducted in the past, and as

still `rememberable', to some degree, by the user, presently.

Of course, in the last analysis, the `categorogram' or `category ideogram'

symbols, that constitute these dialectical progression expressions, are

"intensional symbols", not "extensional symbols". Each is a `connotogram', not

an explicit list of symbols in 1-to-1 correspondence with "every last" element

of meaning of the [ideo-]ontological category that it represents.

The meanings of those `categorograms' are not "all there in the symbols", and

such they never can be. What each is, is a `mnemonic trigger', an `associational

catalyst', to remind the user of, and to help [re-]evoke in the user, the rich

totality of `implicit semanticities' that these "intensional" symbols intend.

The richer the web of associations, of previously constructed and `re-member-éd'

knowledge -- of remembered experience in general -- that the user brings to

those symbols, evoked in the user's past, and retained in mind, i.e., in the

user's `meme-ory', ever since, the richer, then, the totality of meanings that

these `semantically densified' and `semantically concentrated' symbols

```hold''' for that user, and the better the odds for that user to evoke that

richness in and for others."

Enjoy Part II.!

Regards,

Miguel