F.E.D. Dialectics: "William of Ockham's Dialectic of Science...", Rendered in this Blog's Native Typography.

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I have reproduced the content of the new F.E.D. Vignette, below, in the typography native to this blog.

For the official rendition, using the full regalia Encyclopedia Dialectica symbols palette, see the Vignettes Page at the www.dialectics.org website --

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http://www.dialectics.org/dialectics/Vignettes_files/Aoristos_Dyosphainthos,v.1.0,F.E.D._Vignette_13,William_of_Ockham%27s_Dialectic_of_Science,posted_24MAY2013.pdf

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Regards,

Miguel

F.E.D. Vignette #13 --

William of Ockham’s Dialectic of Science --

A Dialectical Episode in Early Medieval Philosophy

by Aoristos Dyosphainthos

Author’s Preface. The purpose of F.E.D. Vignette #13 is to present an E.D. ‘Dialectical Model Equation’ for a systematic dialectical method of presentation of the Early Medieval Philosophy of Science of William of Ockham.

This ‘Dialectical Model Equation’ also serves as a “worked”, “cook-book” example, and sample, of the application of the _NQ dialectical algebra to help organize, and to compactly encode, systematic expositions for ‘‘‘synchronic snapshots’’’ of the structure/process of the living self-sustenance of systems -- ‘physio-systems’, ‘ideo-systems’, and hybrids of the two, such as the system of Science -- that are presently in existence; expositions that explicate and assess the present content of such systems, without explicitly expositing their ‘“diachronic chronicles”’ -- neither explicitly reconstructing the prior, diachronic history that constituted that content, nor explicitly ‘pre-constructing’ any presently-predicted future, successor system of such. [Note: the ordering Mythopoeia, Religion, Philosophy, Science, Psychohistory is, per the E.D. first Psychohistorical-Dialectical ‘Meta-Equation’, that of Human Ideology/Knowledge ‘Meta-Evolution’. See http://www.dialectics.org/dialectics/Aoristoss_Blog/Entries/2012/5/19_The_F.E.D._Psychohistorical_Equations.html ].

A Note about the On-Line Availability of Definitions of F.E.D. Key Technical Terms. Definitions of Encyclopedia Dialectica technical terms, including of E.D. ‘neologia’, are available on-line via the following URLs --

http://www.dialectics.org/dialectics/Glossary.html

https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/ClarificationsArchive.htm

-- by clicking on the links associated with each such term, listed, in alphabetic order, on the web-pages linked-to above.

Definitions of the Encyclopedia Dialectica special terms most fundamental to this vignette are linked-to below --

«arché»

https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arche/Arche.htm

«arithmos» and «arithmoi»

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arithmos/Arithmos.htm

«aufheben»

https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Aufheben/Aufheben.htm

Diachronic vs. Synchronic

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Synchronic/Synchronic.htm

_NQ dialectical arithmetic/algebra

http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf

Systematic or Synchronic Dialectics

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/SystematicDialectics/SystematicDialectics.htm

-- and we plan to expand these definitions resources as the Encyclopedia Dialectica Dictionary Project unfolds.

[Note: ‘‘‘Arithmetical Quantifiers’’’ vs. ‘Arithmetical Qualifiers’. In the phrase “3 apples”, we term “3” the “arithmetical [“pure”-]quantifier”, and “apples” the ‘‘‘ontological’’’ -- or kind of thing -- ‘‘‘qualifier’’’. In the phrase “3 pounds of apples”, we term “pounds” the ‘metrical[-unit] qualifier’ -- or ‘‘‘unit of measure qualifier’’’ -- quantified by the 3, which, together, ‘quanto-qualify’ the ‘ontological qualifier’, “apples”. A key use-value of the dialectical arithmetics is to provide algorithmic, ideographical-symbolic systems for the various kinds of ‘arithmetical qualifiers’, both with and without the co-presence of ‘‘‘arithmetical quantifiers’’’.].

I. Introduction to the Subject Systematic Dialectic within Medieval Occidental Philosophy. The present F.E.D. vignette sets forth a dialectical-ideographical model of William of Ockham’s circa 1323 C.E. philosophical theory / ‘ideo-taxonomy’ of the sciences. Ockham lived circa 1290 to 1349 C.E. [Common Era].

This vignette contains this example of dialectical-mathematical models in general, formulated using a specific dialectical-mathematical language -- that of the “purely-qualitative”, “purely connotative” mathematics of the ‘algorithmic-heuristic algebra’ of the _NQ ‘arithmetic of dialectics’, which is an axiomatized algebraic dialectical logic.

For further background on the algorithmics and the axiomatics of its ideographical, or “symbolic” dialectical logic -- or ‘algebraic logic of dialectics’’ -- see the www.dialectics.org website’s Briefs Page as well as section II. of this vignette.

The core of Ockham’s philosophical theory of the sciences is as follows.

Ockham gave to terms of language that refer to ‘non-term’ “physical facts”, the descriptor [the term] “categorematic”.

Thus, the “term”, or word, “rock” refers to that “external-to-mind reality”, and is therefore a “categorematic” term.

Ockham gave to terms of language that refer to, e.g., “categorematic” terms, i.e., to other terms of language itself, the descriptor [the term] “syncategorematic”.

Thus, the “terms”, or words, “all”, “some”, “none”, and “not” are all examples of such “word-referring words”, or “term-referring-terms”.

Ockham also labeled terms that refer to things belonging to the reality outside of [the reality of] language, by the phrase “terms of the first intention”.

He labeled terms that refer to terms of the first intension by the phrase “terms of the second intention”.

In the most direct sense, then, “syncategorematic” terms are “terms of the second intention”.

However, Ockham distinguishes mere “terms” of language, in the sense of individual words, from those higher-level /- higher ‘qualo-fractal scale’ language constructions [constructions that we of F.E.D. call ‘meta-terms’, or ‘meta-words’, each one made up out of a [usually] heterogeneous multiplicity of mere “terms”, of mere individual words] which are named “complete sentences”, or “well-formed propositions”.

Ockham does not “reduce” the latter ‘ideo-ontology’ to the former.

Thus, for Ockham, categorematic terms, in ‘sentence-ial’ combination with syncategorematic terms, in the form of well-formed sentences or propositions, and, thus, functioning together, refer to, and describe, more adequately, typically, the things belonging to the reality outside of human language, than do single, isolated, “mere” terms, or single, isolated words, such as those terms/words which are named “names”, or “nouns”.

The sentence/proposition -- “All natural objects are transitory.” -- is thus first intentional. Note: this sentence “contains” [combines] both categorematic terms, like “natural objects”, and also syncategorematic terms, like “All”.

The sentence/proposition -- “All second-order-formal-logic axiomatizations of “Natural Numbers” arithmetic are either formally inconsistent or formally incomplete.” -- is thus second intentional [at least!]. Note that this sentence “contains” no terms which refer to non-linguistic, “physical” objects, but only terms which refer to other terms, or to the names of ‘‘‘mental objects’’’/‘‘‘idea-objects’’’, to ‘meme-ic’ or ‘memetic’, human-Phenomic, ‘psychohistorical materialities’.

On the basis of these distinctions, Ockham divides the «Genos» of the “Sciences” into two «species», or ‘sub¹-«Gene» --

«species» 1: «Scientia realis», encompassing the sciences of “real things” [i.e., of external-to-mind, physical things].

For example, the science of biology would form a sub¹-«species» of this «species».

This «species» is first intentional.

«species» 2: «Scientia rationalis», encompassing the sciences of ‘noetic-things’ [i.e., of internal-to-mind, ‘Phenomic’ things].

For example, mathematics would form a sub¹-«species» of this «species».

This «species» is second intentional.

For further background on Ockham’s theory, see W. L. Reese, Dictionary of Philosophy and Religion, Eastern and Western Thought, Humanities Press [Atlantic Heights, NJ: 1980], pp. 627 ff.

The two passages from the source cited above that are most relevant to this vignette are extracted below:

“...Ockham makes an important distinction between categorematic and syncategorematic terms [A.D.: a distinction in logic which harks back at least to the Stoics; cf. also the work of Ockham’s predecessor, William of Sherwood, circa 1210 to 1270 C.E.].”

“Terms which refer to reality are categorematic, while terms which refer to categorematic terms are syncategorematic.”

“Most common nouns are categorematic, while words such as “not,” “all,”, and “some” are syncategorematic.”

“...Terms that refer to [A.D.: external-to-language/-mind] things are called terms of first intention.”

“Terms referring to terms of first intention are called terms of second intention.”

“In one sense the syncategorematic terms mentioned above are second intentional; but, as Ockham wishes to use the distinction, categorematic terms and syncategorematic terms, functioning together, refer to [A.D.: external-to-language/-mind] things. “All men are mortal” is thus first intentional.”

“But when we use terms such as “genus,” “species,” and “difference” we are using terms of second intention.”

“Propositions utilizing such terms refer not to the world [A.D.: i.e., to the physical world, external to the human-language/-mind “world”] but to terms of first intention.”

“...On the basis of the distinction just mentioned, Ockham divides the sciences into two types [A.D.: into two «species»].

“Scientia rationalis, or rational science, is second intentional. Logic is a science of this type.”

“Scientia realis is first intentional. Physics is an example of a science of real [A.D.: external-to-language/-mind] things.”

The same source gives, elsewhere [p. 563], a useful excursion into the ‘philosophical etymology’ of the term “syncategorematic”:

“From the Greek syn (“together”) and categorema (“predicate”).”

“The derivation may refer to those terms which go together with the predicates or [to] those terms which hold the predicates together; and thus, [to] those terms which cannot be [intelligibly] used by themselves but only in conjunction with other terms: e.g., “all,” taken alone, has no definite reference to anything, but “all men” has.

“ “All” is syncategorematic and “men” is categorematic.”

“In addition to the quantifiers “all,” “some,” and “none,” included among syncategorematic terms are “not,” “if...then,” “either...or,” and “both...and.”

II. E.D. Standard Interpretations for the Initial Generic Ordinal Qualifiers of the _NQ Arithmetic. Perhaps a bit surprisingly, upon first apprehension, the first four, first-order-logic, Dedekind-Peano Postulates for the “Natural” Numbers focus on their ordinality, not on their cardinality, viz. --

1. 1 is a “Natural Number”.

2. The successor of any “Natural Number” is also a “Natural Number”.

3. No two, distinct “Natural Numbers” have the same successor.

4. 1 is not the successor of any “Natural Number”, i.e., 1 has no ancestor within the “Natural Numbers”.

These postulates thus identify the essence of the “Natural Numbers”, explicitly, in terms of [apparently purely-] quantitative ordinality.

In keeping with this focus on the ordinal, Seldon defines the system of the _NQ dialectical arithmetic -- the first ‘antithesis-system’, or ‘contra-system’, to the “Natural Numbers” as «arché»-system -- in terms of qualitative ordinality.

The_NQ, which he also calls the ‘meta-Natural meta-Numbers’, are, in their simplest, least-interpreted essence, a consecutive sequence of ‘meta-numeral’ ideograms representing the successive qualities, not the quantities, of ordinality -- the quality of ‘first-ness’, followed by the quality of ‘second-ness’, followed by the quality of ‘third-ness’, and so on... -- satisfying the four first-order-logic ‘contra-Peanic’, ‘Qualo-Peanic’ axioms:

1q. The ordinal qualifier for the quality of ‘first-ness’ is an element of the ‘consecuum’ of generic ordinal qualifiers.

2q. The successor of any element of the ‘consecuum’ of generic ordinal qualifiers is also an element of same.

3q. Any two, distinct ordinal qualifiers have qualitatively unequal successors.

4q. The ordinal qualifier for the quality of ‘first-ness’ is «arché»: not the successor of any element of its ‘consecuum’.

The symbols, or ‘meta-numerals’, that stand for the ‘meta-numbers’ of the _NQ ‘archeonic consecuum’ are derived, syntactically, in a way which represents the semantic ‘self-subsumption’, ‘self-subordination’, or ‘self-demotion’ [dialectical, self-«aufheben» self-negation] of the “Natural Numbers”.

That derivation is part of the positive fruition of the dialectical, immanent self-critique of the “Natural Numbers”, which divulges the _NQ as the implicit, most extreme known opposite, “Non-Standard Model” of the “Standard”, Peano “Natural Numbers”.

It involves the turning of the ‘generic ordinal quantifiers’ of the “Natural Numbers” into the ‘generic ordinal qualifiers’ of the _NQ ‘meta-Natural meta-Numbers’.

The conceptual ‘self-subsumption’ of the quantitative ordinality intended by the Dedekind-Peano Postulates surfaces the _NQ as their hitherto hidden, implicit ‘intra-dual’, based upon the generic quality of ordinality, a ‘‘‘genericity’’’ which we represent by the ‘meta-numeralic’ ideogram ‘q’.

That ‘meta-numeral component’ represents ‘qualitative ordinality’, or ‘ordinal quality’, in general: just ‘q’, or, more fully expressed, just q_N.

To fully express, ‘meta-numeral-y’, or ideographically, the ‘consecuum’ of specific ordinal qualities, namely --

_NQ = {‘first-ness’; ‘second-ness’; ‘third-ness’, etc.}

-- we must add a second ‘meta-numeral component’, via ‘‘‘subordinating’’’ specific “Natural Numbers”, as specific ‘ordinal quantifiers’, to the generic ordinal qualifier symbol ‘q’, by ‘subscripting’ those specific “Natural Numbers” to a ‘script-level’ ‘q’, ‘‘‘above’’’ them, yielding --

_NQ = { q₁, q₂, q₃, ... }

[in which each ‘meta-number’ is a minimal, «genos»/«species» «arithmos eidetikos» in its own right],

vs.

N = { 1, 2, 3, ... }.

Note that this opposition of an arithmetical system of purely-quantitative ordinality, based upon the N, versus an arithmetical system of purely-qualitative ordinality, based on the _NQ, is not a radical dualism, imagined as an absolute, irreconcilable diremption between an absolute quantitative and an absolute qualitative.

This opposition is, on the contrary, a dialectical antithesis-sum.

The N quantifiers are still there, as ‘specifiers’ -- still present -- in, or ‘‘‘under’’’, the generic qs of the _NQ qualifiers, though subsumed, subordinated, demoted -- demoted to being their mere subscripts or denominators: The N quantifiers are still “contained” in[side] [‘‘‘beneath’’’] the _NQ qualifiers.

That is, each _NQ qualifier is an «aufheben» determinate negation / conservation / elevation-into-one-step-higher-generality of an individual N quantifier.

The joint «aufheben» elevation of the quantitative ordinals «species» yields the «genos» ‘q’, of the qualitative ordinals.

For this first layer of interpretation of these “purely-qualitative” _NQ ‘meta-numbers’ -- which does not yet make explicit their universal interpretability for the modeling of dialectical progressions -- this is all that they represent: abstract temporality; [abstract chronological] order; generic ‘ordered-ness’; the consecutive succession of ‘qualitative ordinality’; the ‘consecuum’ of order quality or of order qualities.

But even here, at this minimally-interpreted stage of the construction of the _NQ dialectical arithmetic, there is already a kind of generic ‘connotative entailment’ at work.

True, it is but a shadow, and but a ‘pre-vestigial’ harbinger, of the richness of the kind of particularity of categorial followership that drives forward, intuitively, the dialectical, purely-qualitative logic of the more concrete, more specific dialectical-algebraic interpretations thereof.

A case in point is exemplified in the very _NQ algebraic model Ockham’s dialectic of Science constructed herein.

This generic ‘connotative entailment’ can be formulated as follows: ‘second-ness’ follows -- and even follows from --‘first-ness’; ‘third-ness’ follows [from] ‘second-ness’, and so on.

In the next section, the construction, by iterated interpretations layering, of the Seldonian first dialectical arithmetic will advance from this harbinger of ‘connotative entailment’ to the following, still generic, but at last also explicitly dialectical, form of ‘connotative entailment’ ordinality: ‘ first full antithesis follows from the self-interaction of [«arché»-]thesis; first full synthesis follows from the mutual interaction of first full thesis and first full antithesis’, and so on.

III. Dyadic Seldon Function Interpretation of the Initial Generic _NQ Ontological Category Qualifiers.

The generic ‘dialectical model meta-equation’ form for the functions-family of the Seldon Functions is that of a generic cumulum symbol [‘|-|-|’] on the LHS [Left-Hand Side] of the ‘dialectical meta-equation’, equated to an RHS expression representing ‘self-reflexive operation’ of an [«arché», ‘‘‘seed’’’, ‘‘‘cell-form’’’, or ‘ultimate ancestor’ ontological category symbol [represented, generically, by [‘q1’] -- indicating its recurring ‘self-reflexion’ via a ‘meta-exponentiated’, monotonically increasing whole-number-valued ‘‘‘independent variable’’’ [‘h’ ] -- on the RHS of the generic Seldon Function equation, viz. [with ‘generic-ness’ connoted by the “rectangular” motif of the symbols-set]:

|-|-|_h = |[ q1 ]|v^h, for h in { 0, 1, 2, 3, . . . }, v in {2, 3}.

If v = 2, the Generic Seldon Function above is said to belong to the Dyadic Seldon Function sub-family.

If v = 3, the Generic Seldon Function above is said to belong to the Triadic Seldon Function sub-family.

Our remarks herein are concentrated on the Dyadic Seldon Functions, because the main ‘dialectical-mathematical meta-equations’, modeling the dialectical-systematic presentation of Ockham’s Theory of Science, exposited herein, are of the v = 2 variety.

With v = 2, and selecting that special generic Dyadic Seldon Function form that we reserve for a systematic dialectic -- for the dialectical-systematic method of presentation, the form of the ‘model meta-equation’ to be constructed herein becomes, more specifically --

)-|-(s = () qa ()2^s, for s in { 0, 1, 2, 3, . . . }.

-- wherein the symbol s, replacing the more generic symbol h, takes on ‘‘‘systematic’’’, ‘‘‘taxonomic’’’ connotations, representing the well-ordered, successive steps of a systematic exposition, and wherein, in general, the “curvilinear” motif of the entire symbols-set used is to connote the systematic dialectic domain of ‘dialectical modeling’.

The Seldon Functions bring with them a further, second layer of interpretation of the _NQ qualifiers,

{ q₁, q₂, q₃, ... },

by which they are interpreted as qualifiers that stand for generic dialectical ontological categories, e.g., for ‘‘‘thesis’’’ categories, or for full or partial ‘contra-thesis’ categories, or for full or partial ‘uni-thesis’ categories.

If we assign [ ‘[----)’ ] the «arché»-thesis category, qa, to the generic _NQ qualifier ‘meta-number’, q₁, as signed by ‘qa[----) q₁’, and if we can discern that qa, and all of its successor-categories, and their cumula, as generated by its successive, cumulative, ‘Seldon-functional self-operations’, connote «aufheben» operators, that is, dialectical negation operators, then the Dyadic Seldon Function is seen to signify, under the axioms of the system of arithmetic of the _NQ ‘meta-numbers’ [ http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf ], a ‘self-iterated’, cumulative recurrence of dialectical “negations of [the] negations”.

With every [unit] increase in s, the Dyadic Seldon Function ‘formulaic recipe’ calls for the dyadic self-operation of the result of the previous dyadic self-operation, i.e., for a negation of the negation of the result of the previous negation of the negation.

Only for s = 0 -- only for the case in which no self-operation occurs -- is the “result” a singleton [ideo-] ontological category symbol, the symbol for the «arché» [ideo-]ontological category alone, instead of that “result” taking the form of a cumulum of [powers-of-]two [or more] such symbols, i.e., a “non-amalgamative sum” [cf. Musès], or an «a-sumbletoi» sum [cf. Plato], of ‘[ideo-]ontological category’ symbols,

since 2⁰ = 1,

and since qa¹ = qa --

)-|-(0 = () qa ()2⁰ = () qa ()1 =

() qa () = qa.

For example, if we take epoch s = 1, and denote the «arché» ontological category simply by a, for syntactical convenience, then the Dyadic Seldon Function calls for the following, as per the _NQ axioms, since 2¹ = 2 --

)-|-(1 = () a ()2¹ = () a ()2 = () a () x () a () =

a + b.

-- wherein

a [----)_q1connotes the «arché» category or ‘‘‘thesis’’’,

and where

b [----)q2 connotes the first ‘contra-category’ or ‘‘‘antithesis’’’,

with ‘+’ standing for a generalized addition operation, that encompasses the addition of qualitatively distinct terms, and with ‘x’ standing for a generalized multiplication operation, that encompasses the multiplication operation specific to the _NQ qualifiers.

If we take ‘() a () x () a ()’ with () a () connoting the category to be dialectically negated, and with ‘() a ()’ connoting the category doing that dialectically [self-]negating, with ‘() a ()’ as the dialectical, determinate ‘negation-sign’ specific to () a () as the object of the so-indicated dialectical, determinate negation operation, then we have ‘() a () x () a ()’, as a whole, as connoting the first dialectical negation, by the negation-operation () a () of the self-same negation operation, () a (), that is also the operand of the negation-operator () a (), forming what Seldon calls an ideographical

‘subject [() a ()]-verb[() a ()]-object[() a ()] identical’,

and yielding the first dyad:

‘‘‘thesis + antithesis’’’, or

‘«arché» category + first contra-category’, viz. --

() a () x () a () = qa + qaa =

qa + qb = a + b

-- which, in terms of the generic, minimally-interpreted _NQ arithmetic, is a dialectical interpretation of the generic --

|-|-|1    =    |[ q1 ]|2¹   =     |[ q1 ]|2   =


|[ q1 ]| |x| |[ q1 ]|    =


|[ q1 |+| q1+1 ]|    =

q1  |+| q2.

The second iteration of this dialectical negation of the negation, corresponding to the consecutively next value of s namely, s = 2, for the _NQ arithmetic interpreted for [psycho]historical dialectics, yields the following, ontologically-expanded cumulum of ontological categories -- a ‘dyad of dyads’, consisting of 4 consecutive ontological categories:

)-|-(2 =   () a ()2²   =    () a ()4    =    ()    () a ()2 ()2    =

() a + b ()2           =

() a + b ()   x   () a + b () =

a + b + c + d.

The additional 2 ontological category-symbols above are dialectically interpreted, per the E.D. standard, as follows:

c = third ontological category, first full uni-category;

d = fourth ontological category, second contra-category.

We will not here pursue this E.D. standard dialectical interpretation of the ontological categories generated by the generic Dyadic Seldon Function beyond t = 2, because the ‘‘‘solution’’’ -- or ‘semantification’ -- of the category-terms generated by the ‘dialectical-mathematical meta-model meta-equation’ constructed in this vignette [whose terms are generated, initially, as algebraic unknowns, terms of unknown meaning], as presented herein, does not extend beyond that second ‘self-iteration’ for that ‘meta-model’.

The ‘purely-qualitative calculations’ illustrated above describe our expectations for this ‘meta-model’ in terms of generic characterizations of the successive, consecutive dialectical categories.

The next section addresses the heart of this ‘meta-model’ -- the specific meanings of the generic dialectical categories as applied to the special case of the systematic dialectic of the Early-Medieval Philosophy of Science of William of Ockham.

IV. E.D. Solution for the ‘Meta-Model Meta-Equation’ of Ockham’s Dialectic of Science. The content of this vignette is a recapitulation, and an immanent extension, of our 16 June 2009 exposition of this topic, available via the Correspondence Page of the www.dialectics.org website: http://www.dialectics.org/dialectics/Correspondence_files/Letter18-16JUN2009.pdf .

To begin our model of Ockham’s dialectical taxonomy of this universe-of-discourse of human ideas, we use the dialectical logic axiomatized in an earlier letter available via the www.dialectics.org Correspondence Page --

http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf

-- and we enact the ‘‘‘mental action’’’ / ‘‘‘mental operation’’’ signified by the ideogram ‘(----]’. That is, we “interpret” or “assign” q₁, the first of the generic ‘meta-Natural meta-Numbers’ of our _NQ,‘unquantifiable Qualifier’ ‘dialectical ideography’, or ‘dialectical symbolic logic’ [‘dialectical-ideographic logic’], as follows --

q₁(----] q_r, or q₁(----] r

-- such that r(----) “reality”, the universe of “real things”, outside of human language, outside of the human mind.

Our ‘dialectical-ideographic model’ then captures a sequential, systematically-ordered, ‘‘‘ordinal-ized’’’ presentation of Ockham’s theory of the sciences, using the connotations of the series of ideographical “shorthand” symbols that it generates algorithmically.

It does so via r as the recurrently ‘self-operating operator’, ‘self-operation’, ‘self-function’, ‘self-refluxive function’, ‘‘‘self-reflexive function’’’, or recurrently self-applied, connotationally specific, determinate dialectical, «aufheben»-negation operation at the heart of the ‘self-iteration’ formula --

)-|-(s = () r ()2^s

-- wherein s, as the higher exponent of the exponent 2, counts the “steps” or “stages” of that systematic presentation of the ‘idea-ontology’, or ‘ideo-ontology’-- of the ‘ideo-taxonomy’ -- of the philosophical categories, and wherein )-|-(s denotes the ‘cumulum’ -- meaning the connotative and non-amalgamative, ‘‘‘qualitative sum’’’, or ‘‘‘superposition’’’ -- of the categories presented / ‘acCumulated’ as of step s of this “shorthand” ‘‘‘systematic-dialectical’’’ ‘present-ation’.

Let us, then, unfold this model step-by-step, starting with step s = 0, then moving on to step s = 1, and thence onward to step s = 2, and, finally, to step s = 3, a step which immanently extends Ockham’s theory to a category of The Sciences that Ockham, as far as we know, did not envision.

Step 0. The 0th step of this ‘presentation-model’, i.e., for s = 0,

since

2⁰ =

2⁺¹/2⁺¹ =

2⁺¹ x 2^-¹ =

2⁺¹^-¹ = 1,

yields --

)-|-(0 = () r ()2⁰ =

r¹ = r

-- which simply reiterates the starting point -- the «arché» ‘ideo-taxonomic category’ -- of this ‘present-ation’.

¿Does this categorial-singleton of step 0 grow into a ‘categorial-cumulum’ in step 1?

Step 1. For step s = 1 of this ‘presentation-model’, since 2¹ = 2⁺¹ = 2, we have --

)-|-(1 = () r ()2¹ = r² =

r() r ()   = r “of” r   =


r “squared”

-- which connotes the ‘self-reflexive function’, or ‘self-operation’, of the dialectical [«aufheben»] operation denoted by r.

Syntactically, per the axioms of the _NQ ‘meta-numbers’, we have the following values as the content of step 1:

)-|-(1 = () r ()2¹= r² =

qr² = qr + qrr

[----)

q1 |+| q1+1 = q1 |+| q2.

Our challenge, for this step, is to “solve for” the new category, q_rr, that is, to determine the meaning, the connotation, of that new ‘categorogram’, given the already known/assigned meaning, or connotation, of the ‘categorogram’ r = q_r.

So, let us apply the general ‘‘‘canon of interpretation’’’ of such dialectical categories to the specific case of this example.

Per that ‘‘‘canon’’’, the «arché» ‘categorogram’, r, is to be interpreted, as stipulated, as connoting an «arithmos» of «monads» -- an assemblage of units -- each one of which is a “fact”, a physical, empirical “reality”, “outside of” language and of the human mind.

Also per that ‘‘‘canon’’’, a symbol like q_rr connotes the new category/«arithmos» that results from the self-«aufheben» of the q_r category/«arithmos», and, quite often, that results from the self-«aufheben» ‘meta-monadization’ of some of the «monads» of the q_r category/«arithmos».

That is, each «monad» or unit of the q_rr «arithmos» / category / assemblage of units should be a ‘meta-«monad»’, made up out of a multiplicity -- made up out of a ‘sub-«arithmos»’ -- of the «monads» of the q_r «arithmos» / category.

So, the units of the q_rr assemblage should be, in this specific case, ‘meta-realities’, ‘realities of second degree’, each one made up out of a heterogeneous multiplicity of ‘realities of first degree’.

Also, the category q_rr should be another category -- other than and to the physical/empirical “realities” category; a ‘meta-category’ to, and a ‘contra-category’ of, the r category.

Moreover, the q_rr symbol should point to a category that is already known to us, that is part of our synchronic, present conceptualized experience of the totality of The Sciences, and to a category that is ontologically different than, and that is richer in determinations than -- that is “more determinate” than -- that is more “complex” and more ‘thought-concrete’ than, the r category.

On the bases set forth above, we propose that our solution -- our ‘semantification’ -- of the syntactic term q_rr should be t, connoting the «arithmos»/category of terms of human language:

q_rr = q_t = t [----) q₂.

We propose this solution, in part, because the terms of human language, as units, are meta-units of the units of empirical/physical “reality”.

Each term of human language refers to, and represents, a heterogeneous multiplicity of similar physical “facts”.

The term “apple” maps to a large, heterogeneous multiplicity of similar, physical apples.

The term “orange” maps to a large, heterogeneous multiplicity of similar, physical oranges.

That is, “terms” are general relative to physical “facts.

One “term” typically stands for a large, although still finite, set of similar physical situations.

Each “term” is an «aufheben» ‘meta-«monad»’ of a large number of similar “fact”-«monads».

Our solution for q_rr is that q_rr connotes the «arithmos»/category which has terms of human language as its «monads».

In an historical, diachronic sense, r “of” r also connotes the natural-historical result of the ‘self-reflexion’, of the ‘bending-back-upon-self’, of “reality”; of the turning back, upon itself, of physical “reality”, to form physical objects -- new parts of that physical “reality” -- that are capable of conscious reflection upon physical “realities”, including upon themselves.

In that historical, diachronic context, this would mean the immanent emergence, within pre-human[oid] Nature, from out of pre-human[oid] Nature -- as part of the dialectic, or of the self-development, of Nature itself -- of “observing eyes”, and of “observing sensoria” in general; of observing and thinking minds, as also “things” / objects / bodies within Nature -- i.e., the emergence of natural ‘[ev]entities’ which produce languages, hence “terms”, to name / communicate their observations / experiences among themselves, as a crucial part of their “struggle for existence” -- of their struggle for successful expanded human[oid]-social reproduction.

But this consideration does not belong to synchronic, systematic dialectics in itself.

It only represents the diachronic implications -- the pointers to past history, and to future history -- that are implicit in synchronic dialectics, but which do not, as such -- as diachronic -- form an explicit component of the explanatory apparatus of synchronic, systematic dialectics.

Marx’s treatises on the dialectical, immanent critique of the capitalist, ideology-ridden science of “political economy”, are works of synchronic, systematic dialectics.

But, in Marx’s view, systematic dialectics is not absolutely dirempt from diachronic, historical dialectics.

Marx discussed this interconnexion between systematic dialectics and historical dialectics, as it arises in his extension of previous concepts of dialectics -- an interconnexion which is, e.g., inaccessible to the eternally systematic, Parmenidean dialectics of Plato’s immutable «arithmos eidetikos» -- in a crucial methodological passage from the Grundrisse:

“...our [A.D.: systematic-dialectical] method indicates the points where historical investigation must enter in, or where bourgeois economy as a merely historical form of the production process points beyond itself to earlier historical modes of production.”

“In order to develop the laws of bourgeois economy, therefore, it is not necessary to write the real history of the relations of production.”

“But the correct observation and deduction of these laws, as having themselves become in history, always leads to primary equations -- like the empirical numbers e.g. in natural science -- which point towards a past lying behind this system.”

“These indications [Andeutung], together with a correct grasp of the present, then also offer the key to the understanding of the past -- a work in its own right which, it is hoped, we shall be able to undertake as well.”

“This correct view likewise leads at the same time to points at which the suspension [at which the «aufheben» -- A.D.] of the present form of production relations gives signs of its becoming -- foreshadowings of the future.”

“Just as, on one side the pre-bourgeois phases appear as merely historical, i.e. suspended [i.e., ‘«aufheben»-ed’ -- A.D.] pre-suppositions, so do the contemporary conditions of production likewise appear as engaged in suspending themselves [i.e., in ‘«aufheben»-ating themselves’ -- A.D.] and hence in positing the historic presuppositions for a new state of society.”

[Karl Marx, Grundrisse: Foundations of the Critique of Political Economy (Rough Draft), M. Nicolaus, transl., Penguin Books [Middlesex: 1973], pp. 460-461, italics only emphasis by Marx; combinations of italic, underlined, bold, and color emphasis have been added by A.D.].

Accepting this solution, our step s = 1 dialectical-model equation thus becomes [with ‘D’ for the systematic-dialectic context’s ‘curvilinear, ontological delta operator’, signifying the addition of an increment of new ‘ideo-ontology’] --

)-|-(1 = () r ()2¹=   r²   = q_r² =

r + Dr  = r + t

[----)

q₁ |+| q₂

-- and we have the meaning that, as of step 1, s = 1, our model,

)-|-(s = () r ()2^s,

posits the possible existence of two philosophical categories -- (1) the category of physical “realities”, ‘‘‘plus’’’ [signed by ‘+’, for connotatively “interpreted”, or “assigned”, ‘ontological qualifiers’, in the ‘curvilinear symbols motif’ context of systematic dialectics], (2) its ‘first contra-category’, of ‘‘‘terms of language’’’, both together comprising (1) the «physis»-«monads», or ‘fact-units’, of the physical realities’ «arithmos aisthetos», or sensorially-perceived “facts”-assemblage, superposed with, and ‘‘‘named’’’ -- in a many-to-one relationship -- by (2) the ‘trans-Platonian’ «arithmos eidetikos» of terms-units, or of terms-«monads».

¿How does this two-dimensional ‘possibility-space’, or categorial ‘ideo-ontology cumulum’, expand in step 2?

Step 2. For step s = 2 of this presentation-model, since 2² = 2 x 2 = 4, we have --

)-|-(2 = () r ()2² = r⁴ = r²^x² =

() r² ()² = () r + t ()²

-- which connotes the ‘self-reflexive function’, or ‘self-operation’, of the collective dialectical operation -- of the two-category, two-dimensional categories-cumulum operation -- denoted by

() r + t ().

Syntactically, per the axioms of the _NQ ‘meta-numbers’, we have the following values as the content of step 2:

)-|-(1 = () r ()2²=   () r + t ()²   =

() r + t () x () r + t ()  =

() r + t () + D() r + t ()  =

() t   x () r + t () () [using Miguel’s shortcut] =

q_r + q_rr + q_tr + q_tt   =

r + t + q_tr + q_tt

[----)

q₁ |+| q₂ |+| q₂₊₁ |+| q₂₊₂ =

q₁ |+| q₂ |+| q₃ |+| q₄.

Our challenge, for this step, is to “solve for” the two new categories, q_t_r and q_tt, that appear for the first time in this step, that is, to determine the meanings, the connotations, the definitions, of each of these new ‘categorograms’, as ‘connotative entailments’ of the already solved / known / assigned meanings / connotations /definitions of the ‘categorograms’ r = q_r and t = q_rr.

So, let us apply the general ‘‘‘canon of interpretation’’’ of such dialectical categories to the specific case of this example.

Per that ‘‘‘canon’’’, the ‘categorogram’ q_t_r connotes a new ‘uni-category’/hybrid «arithmos» -- the synthesis category -- that results from the «aufheben» of the q_r category/«arithmos», by the q_t category/«arithmos».

This often means a category/«arithmos» each of whose «monads» is a hybrid of the «monads» of the q_r category/«arithmos» with the «monads» of the q_t category/«arithmos».

Also per that ‘‘‘canon’’’, a ‘categorogram’ like q_t_r might also connote ‘‘‘conversion’’’, of «monads» of the q_r category/«arithmos», into «monads» of the q_t category/«arithmos».

Also, per that ‘‘‘canon’’’, the q_t_r symbol should point to a category that is already known to us, that is part of our synchronic, present conceptualized experience of the totality of The Sciences, and to a category that is ontologically different than, and that is richer in determinations than -- that is “more determinate” than -- that is more “complex” and more ‘thought-concrete’ than, all of the categories evoked in the preceding steps.

And, per that ‘‘‘canon’’’, a symbol like q_tt connotes the new category/«arithmos» that results from the self-«aufheben» of the q_t category/«arithmos», and, quite often, that results from the self-«aufheben» ‘meta-monadization’ of some of the «monads» of the q_t category/«arithmos».

That is, each «monad», or unit, of the q_tt «arithmos»/category should be a ‘meta-«monad»’, or ‘meta-unit’, of «monads» or units of the q_t «arithmos» / category.

So, the units of the q_tt assemblage should be, in this specific case, ‘meta-terms’, ‘terms of second degree’, each one made up out of a heterogeneous multiplicity of ‘terms of first degree’.

Also, the category q_tt should be another category -- other than all of the categories previously evoked, in the preceding steps; a ‘contra-category’ of each of those categories, and a ‘meta-category’ / ‘meta-«arithmos»’ to the q_t category / «arithmos».

Moreover, the q_tt symbol should point to a category that is already known to us, that is part of our synchronic, present conceptualized experience of the totality of The Sciences, and to a category that is ontologically different than, and that is richer in determinations than -- that is “more determinate” than -- that is more “complex” and more ‘thought-concrete’ than, all of the categories evoked in the preceding steps.

On the bases set forth above, we propose that our solution -- our ‘semantification’ -- of the syntactic term q_t_r should be f, connoting the «arithmos»/category of Ockham’s propositions of the first intention:

q_t_r = q_f = f [----) q₃.

We propose further that our solution/‘semantification’ of the syntactic term q_tt should be s, connoting the «arithmos» / category of Ockham’s propositions of the second intention:

q_tt = q_s = s [----) q₄.

In proposing these two ‘semantifications’, we are in resonance with Ockham in invoking here, not just a subdivision of the universe of terms of language, i.e., of single words of language, into two sub-universes, of “categorematic” words vs. of “syncategorematic” words, and with “categorematic” words corresponding to q_t_r, and with “syncategorematic” words corresponding to q_tt, so that q_t_r and q_tt would fail to differ, in terms of the ontology of their «monads», from q_t.

We hold that the ‘‘‘conversion’’’, i.e., the apt description, of the «arithmos»/assemblage of physical “facts”, by “terms”, q_t_r, as of the «arithmos»/assemblage of “terms”, by “terms”, q_tt, requires, in each case, something more than mere single words.

Human language is far more than the mere utterance of single word-sounds. Human language is, for starters, an «arithmos»/assemblage of sentences -- of propositions, that is, of well-formed ‘meta-words’, each one made up out of a heterogeneous multiplicity of words.

We thus hold that q_t_r must connote a multiplicity of “terms” -- both “categorematic” and “syncategorematic” alike -- combined, into well-formed sentences, i.e., into propositions, that describe our knowledge of the physical world, q_r.

So:

q_t_r = q_f = f [----) q₃.

[cf. Boole’s category of “Primary Propositions”, Laws of Thought, Chapter IV, Proposition I.].

We thus hold also that q_tt must connote a multiplicity of “terms” -- both “categorematic” and “syncategorematic” alike -- combined, into well-formed sentences, i.e., into propositions, that describe our knowledge of the secondary world of our own creation, of that core component of ‘The Human Phenome’ which is the world of human language itself, q_t.

So:

q_tt = q_s = s [----) q₄.

[cf. Boole’s “Secondary Propositions”, Laws of Thought, Chapter IV, Proposition I.].

Thus, f connotes an «arithmos»/assemblage of “true” proposition-units, or of proposition-«monads», that are about -- of empirically-apt propositions which describe -- the physical world.

And, s connotes an «arithmos»/assemblage of “true” proposition-units, or of proposition-«monads», that are about -- of cognitively-apt propositions which describe -- the world of human language, as an expression of the world of human thought.

But the assemblage of “true” propositions about the “real” world, external to human thought, q_t_r, is what ‘‘‘the sciences of the real’’’ “contain”; is the ‘‘‘content’’’ of those sciences; is what Ockham means by ‘‘‘the sciences of the real’’’, by the category he names «scientia realis», which we will connote by l: f = l.

And the assemblage of “true” propositions about the “terms” world, the world human language/thought, q_tt, is what ‘‘‘the sciences of the rational’’’ “contain” for Ockham; is the ‘‘‘content’’’ of those sciences; is what Ockham means by ‘‘‘the sciences of the rational’’’, by the category he names «scientia rationalis», which we will connote by n: s = n.

Accepting these solutions, our step s = 2 dialectical-model equation thus becomes --

)-|-(1 = () r ()2²= r⁴ = () r + t ()² =

() r + t () x () r + t ()=

() r + t () + D() r + t () =

() t x () r + t () () [using Miguel’s shortcut] =

q_r + q_rr + q_tr + q_tt   =

r + t + l + n

[----)

q₁ |+| q₂ |+| q₃ |+| q₄.

-- and we have the meaning that, as of step 2, s = 2, our model,

)-|-(s = () r ()2^s,

posits the possible existence of four philosophical categories -- (1) the «arché» category of physical “realities”, ‘‘‘plus’’’; (2) its ‘first contra-category’, of ‘‘‘terms of language’’’, ‘‘‘plus’’’; (3) the ‘first uni-category’ of the preceding two categories, whose units or «monads» are, as propositions, hybrids, combinations, or syntheses of “categorematic” and “syncategorematic” terms, and, in a sense, as “synthetic” [cf. Kant] propositions, hybrids of the units of physical “reality’ with the units of terms, thus constituting the category of propositions of the first intention, constituting Ockham’s «scientia realis», ‘‘‘plus’’’; (4) the ‘second contra-category’, the category of propositions of the second intention, constituting Ockham’s «scientia rationalis», ‘meta-monadic’ with respect to the «monads» of the ‘first contra-category’, of ‘‘‘terms of language’’’, given that these propositions, which are the units or «monads» of the «scientia rationalis», constitute, each one, a ‘meta-term’, or ‘term of the second degree’, each one made up out of a heterogeneous multiplicity of ‘terms of the first degree’, which are the units or «monads» of the category/«arithmos»/assemblage of units herein connoted by t.

All four categories together, as an ‘ideo-cumulum’, comprise --

(1) the «physis»-«monads», or ‘fact-units’, of the physical realities’ «arithmos aisthetos», or sensorially-perceived “facts”-assemblage, superposed with, and named -- in a many-to-one relationship -- by;

(2) the ‘trans-Platonian’ «arithmos eidetikos» of terms-units, or of terms-«monads», both together superposed with;

(3) the ‘trans-Platonian’ «arithmos eidetikos» of the propositions-units, or of propositions-«monads», of the Ockhamian «scientia realis», all three together superposed with;

(4) the ‘trans-Platonian’ «arithmos eidetikos» of the propositions-units, or of propositions-«monads», of the Ockhamian «scientia rationalis».

¿But is there a next step, a step 3, for this classificatory, taxonomic, systematic-dialectic presentation of the Sciences?

Our step 2 models where this dialectic completed for Ockham, and for his times; for what was synchronic/extant in his experience, and in the experience of his contemporaries.

Ockham saw, as far as we know, no other sub-categories of Science besides «scientia realis» & «scientia rationalis».

¿But is there another sub-category of Science that was not synchronic/extant for Ockham, but that is synchronic for us?

¿How could this now four-dimensional ‘possibility-space’ of categorial ‘ideo-ontology’ expand in a possible step 3?

Let’s see what arises if we iterate this model one more step, i.e., for s = 3.

Step 3. For step s = 3 of this presentation-model, since 2³ = 2 x 2 x 2 = 8, we have --

)-|-(3 = () r ()2³ = r⁸ = r⁴^x² = () r⁴ ()² =

() r + t + l + n ()²

-- which connotes the ‘self-reflexive function’, or ‘self-operation’, of the collective dialectical operation -- of the four-category, four-dimensional categories-cumulum operation -- denoted by

() r + t + l + n ().

Syntactically, per the axioms of the _NQ ‘meta-numbers’, we have the following values as the content of step 3:

)-|-(3 = () r ()2³= () r + t + l + n ()² =

() r + t + l + n () x () r + t + l + n () =

() r + t + l + n () + D() r + t + l + n () =

() n x () r + t + l + n () ()[using Miguel’s shortcut] =

q_r + q_rr + q_tr + q_tt + q_nr + q_nt + q_n_l + q_nn =

r + t + l + n + q_nr + q_nt + q_n_l + q_nn

[----)

q₁ |+| q₂ |+| q₃ |+| q₄ |+| q₄₊₁ |+| q₄₊₂ |+| q₄₊₃ |+| q₄₊₄ =

q₁ |+| q₂ |+| q₃ |+| q₄ |+| q₅ |+| q₆ |+| q₇ |+| q₈.

Our challenge, for this step, is to “solve for” the four new categories, q_n_r, q_n_t, q_n_l, and q_nn, that is, to determine the meanings, the connotations, the definitions, of each of these new ‘categorograms’, as ‘connotative entailments’ of the already solved / known / assigned meanings or connotations of the ‘categorograms’ r = q_r, t = q_rr, l = q_t_r, and n = q_tt.

Here, at this juncture, we wish to venture a conjecture as to the ‘connotatively-entailed’ meaning-solutions, or definition-solutions, of these four new categories, q_n_r, q_n_t, q_n_l, and q_nn.

In general, we see each of them as connoting the fruition of the application of the «scientia rationalis» -- i.e., of advanced formal, mathematical logic / dialectic -- to each of the already-defined domains, resulting in their ‘‘‘subsumption’’’ by the «scientia rationalis».

We conjecture, more specifically, as follows --

q_n_r = the «scientia rationalis» ‘‘‘conversion’’’ of the «arithmos aisthetos» of physical facts; the rational-scientific immanent critique, explanation [including theory of [past] perceptual error], and reconstruction of human perception of the «realis» realm, extending to the formation of a universal, systematic-dialectical taxonomy of this realm, related to the project of the «Natur» division of Hegel’s Encyclopedia of the Philosophical Sciences;

q_n_t = the «scientia rationalis» ‘‘‘conversion’’’ of the «arithmos eidetikos» of terms of human language; the rational-scientific immanent critique, explanation [including theory of [past] terminological error], and reconstruction of human language, including the formation of a rational universal language, comprising a dialectical phonetic language, or ‘dialectical phonogramy’, as well as of a ‘dialectical pictogramy’, and of a ‘dialectical ideogramy’, related to Leibniz’s vision of a «characteristica universalis»; also related to the project of the «Logik» division of Hegel’s Encyclopedia of the Philosophical Sciences;

q_n_l = the «scientia rationalis» ‘‘‘conversion’’’ of the «arithmos» of the propositions of the «scientia realis»; the rational-scientific immanent critique, explanation [including theory of [past] scientific theories error / ideology], and reconstruction of the «scientia realis», via the formation of ‘theories’ accounting for the differences in the history of different theories/ideologies, or systems of propositions, regarding the same “real” facts / physical phenomena, e.g., of different theories/ideologies of the same facts arising from different historical epochs/human-social formations, related to the project of the «Natur» division of Hegel’s Encyclopedia of the Philosophical Sciences, and to that of Marx’s Grundrisse, A Contribution to the Critique of Political Economy, and Capital: A Critique of Political Economy.

q_nn = the «scientia rationalis» ‘‘‘conversion’’’ of the «arithmos» of the propositions of the «scientia rationalis» themselves; the rational-scientific immanent self-critique, self-explanation [including theory of [past] scientific theories error / ideology], and self-reconstruction of the «scientia rationalis» by the «scientia rationalis» themselves, including the formation of ‘meta-theories’ accounting for the differences in the history of different theories/ideologies, or systems of propositions, regarding the same mathematical and other “noetic” facts/phenomena, e.g., of different theories/ideologies of “the same” ‘‘‘noeta’’’ , or objects of thought, as they arise in different historical epochs/human-social formations, extending to development of a comprehensive mathematics of dialectics itself; related to the project of Hegel’s Science of Logic, and especially to its second volume.

Note that Marx’s A Contribution to the Critique of Political Economy, and Capital: A Critique of Political Economy, provide a specific example, and, mostly implicitly, a generic paradigm, for a core component of q_n_l as conjecturally defined above.

Nothing like Marx’s works was extant in Ockham’s time, but those works are ‘synchronic’ for us, today.

In those works, Marx addresses “...the evolution of the economic formation of society ... as a process of natural history” [Marx, Preface to the First German Edition of Capital], and, thus, addresses the human species and its ‘Phenome’ as a natural object, as a “real” fact/object, a part of the purview of «scientia realis».

The subsidiary objects which Marx addresses in those two treatises -- values, commodities, monies, capitals, and their “law of [surplus-]value” -- are tied to objects of physical “reality” -- packages of tea, bank notes, steam-powered looms, etc. -- but they are not simply physical objects.

They are also noetic objects, ‘meme[t]ic objects’, ‘‘‘mental objects’’’, ‘‘‘ideal objects’’’, ‘cultural objects’; ‘psyche-ic objects’, parts of the human-social “intersubjectivity”; parts of ‘The Human Phenome’ -- human, “social relations of production” [Marx].

Such ‘category-[idea[l]-]objects’, such ideas, such memes, residing within ‘The Human Phenome’, the very «arithmos [eidetikos]» of meme «monads», are “material” not only in the sense that they are germane, that they are salient, for Science, but also in that, being “intersubjective” -- held mentally, and behaviorally, in common among human actors -- they are also objective.

Such is the paradigm that we call ‘psychohistorical materialism’, pioneered by Marx.

Note also that this conjectured solution models the research programme actually pursued by our co-founder, Karl Seldon.

The universe of discourse of our model of the Ockhamian dialectic of Science, as updated, for our time, above --

)-|-(3 = () r ()2³=

r + t + l + n + q_nr + q_nt + q_n_l + q_nn

-- is that of ‘the elements [ facts and words ], and the sub-divisions, of knowledge [science]’.

If we abbreviate this universe of discourse, restricting it to just “The Sciences”, leaving its elements [ facts and words ] implicit therein, instead of making them explicit, as before, then perhaps we can encompass the most essential content of this systematic dialectic, or ‘synchronic ideo-taxonomy’, by means of a Triadic Seldon Function of only three terms --

)-|-(1 = () L ()3¹=

L + N + Y

-- instead of the earlier eight algebraic terms, but one whose third and final algebraic term, Y, we must now define.

For the terms of this new model, we make the following assignments to, or interpretations of, the terms of the old --

1. L (----) l;

2. N (----) n;

3. Y (----) () q_n_r + q_nt + q_n_l + q_nn ().

The ‘cumulum’ -- the superposition -- of the conjectured definitions of the ideographical terms q_n_r, q_n_t, q_n_l, and q_nn, taken together, amount to a definition of the knowledge-field which we name ‘[socio-politico-economico-]psYchohistory, or ‘psYchohistorical dialectics’, for short, and ‘Y’ is our standard symbol for that knowledge-field.

It is conjectured here as a ‘‘‘complex unity’’’, or dialectical synthesis, of N & L, of logico-mathematical-cognitive-psYcheological science & physical science, including human social science as part of the science of natural history in Marx’s sense:

Y = q_N_L.

The fuller development of Y is the intended target and focus of the work of Foundation Encyclopedia Dialectica.

We can illustrate this proposed abbreviation, ‘dialectical-pictogramically’, as --

The connotations of this Y encompass the connotative content of the earlier model’s algebraic terms q_n_r, and q_n_l, and also q_n_t, since L , itself ‘proxying’ l -- and as the operand of N, ‘proxying’ n, in the formation of Y -- implicitly connotes / "contains" the elements r and t.

However, it can be well-argued that the ‘self-reflexive moment’ of the earlier model’s s = 3 step -- namely, its final algebraic term, q_nn -- is missing from, this “abbreviated” model, which thus has a gaping ‘homeomorphic defect’ as a model of that earlier model.

To get back this crucial component of the Seldonian concept of psychohistory, while still retaining some of the second model’s succinctness, we can convert the Triadic Seldon Function model back into a Dyadic Seldon Function model, still using the same «arché».

Then the new dyadic model’s s = 1 step gives us Ockham’s dialectic, and its s = 2 step gives the Seldonian ‘extention’, with

Y = Y₁+ Y₂,

viz. --

V. Solution-Implicit Issue: ¿Did Ockham Intentionally Construct a DIALECTIC of Science?

Ockham was certainly no stranger to concepts of “the dialectic”.

In the first place, medieval scholarship was permeated by notions of, and discourses upon, dialectic, to an extent which has yet to be fully explored and appreciated by modern scholarship.

In the introduction to her book Dialectic and Its Place in the Development of Medieval Logic, Eleonore Stump wrote:

“Since 1975 my work in medieval logic has concentrated on dialectic.”

“I have tried to trace scholastic treatments of dialectic to discussions of it in the work of Aristotle, the Greek commentators on Aristotle, and the Latin rhetorical tradition. But I have been especially interested Boethius [F.E.D.: 480-525 C.E.], whose discussions of dialectic were among the most important influences on scholastic treatments of the subject.”

“Accounts of dialectic based ultimately on Boethius’s views continued to play a fundamental role in philosophy through the fourteenth century.”

“The earliest scholastic logician whose work we know, Garlandus Compotista [F.E.D.: Dialectica, circa 1054 C.E.], devoted a great deal of attention to Boethian dialectic, and I have tried to follow the development of scholastic dialectic from Garlandus through various twelfth-century logicians (including Abelard) and thirteenth century terminists into the fourteenth century... .”

“The scholastic literature on dialectic is a vast treasure for scholarship, which even now is largely unmined ... .”

[Eleonore Stump, Dialectic and Its Place in the Development of Medieval Logic, Cornell University Press [Ithaca: 1989], p. 1].

In the second place, Ockham himself was a major fourteenth century innovative contributor to medieval scholastic dialectic, principally via his circa 1323 treatise Summa logicae, or “Summa[ry] of All Logic”, including to the logic of “obligations”*. Some of his innovative contributions were reviewed by Eleonore Stump in the following terms --

“... there are also contexts (notably that of dialectical disputation, discussed in obligations treatises) in which types of inferences or rules for inferences that would otherwise be perfectly acceptable produce paradoxical results.”

“...for instance, Ockham discussed cases in which the otherwise acceptable rule ‘From the impossible anything follows’ does not hold.”

“In the burgeoning study of consequences and the growing interest in obligations the rather narrow formalization of logic brought about by the terminists was stretched and broadened. Ockham included a discussion of obligations within his treatment of inferences, and he also considered a broad variety of dialectical inferences.”

“In fact, he blurred the boundary between dialectic and demonstration, thereby elevating dialectic above the second-class status accorded it by the terminists.”

-- And --

“...contrary to the views of some contemporary scholars, there is nothing in Ockham’s criteria for valid inferences which corresponds to the notion of material implication in contemporary analytic philosophy.”

“... syllogisms, Topics, obligations, and insolubles are woven together in Ockham’s work into one general account of inferences which is at once richer and broader than that of the thirteenth-century terminists.”

“The terminists tend to force all nonsyllogistic arguments, including dialectical arguments, into syllogistic form. By contrast, Ockham’s account of inferences recognizes a variety of nonsyllogistic inferences, including dialectical and obligational inferences, without trying to mold them into syllogistic form; and it includes syllogisms as one species of consequences among others. The result, which clearly represents an advance, is a development toward a logic which is both formal and able to account for many sorts of inferences.” [Eleonore Stump, ibid., pp. 3; 8-9].

*[We have an ‘‘‘obligation’’’ here to define “obligation” as a term of logic. A respondent in a formal disputation is said to hold an obligation to avoid being forced into formally contradictory [“impossible”] assertions due to respondent logical error, given that the original assertion upheld by the respondent is a “possible” one.].

However, Ockham’s work on dialectics was informed primarily by the dialectics of Aristotle, specifically that of Aristotle’s Topics and Rhetoric, and thus only indirectly, not directly, by the dialectics of Plato.

We therefore cannot infer, with any certainty, that Ockham consciously intended his ‘ideo-taxonomy’ of the Sciences to be anything like a Platonian «arithmos eidetikos», especially given Ockham’s “nominalist” views, rejecting Platonian and other hypostatizations or reifications of “universals”.

More likely, Ockham’s ‘ideo-«genos»’ of the Sciences, and its two ‘ideo-«species»’ of «scientia realis» and « scientia rationalis », were conceived within the prevailing Aristotelian «genus»/«species» ‘ideo-taxonomical’ tradition, which, though distantly related to Plato’s «arithmoi eidetikoi» dialectic, represents a very attenuated form thereof, in the aftermath of Aristotle’s critique and critical rejection of Plato’s «arithmoi eidetikoi».