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__Vignette__

**--**

__#13__**’**

*William of Ockham*

*s*

__Dialectic__

*of*

*Science***--**

*A*__Dialectical__Episode in Early Medieval Philosophy*by*

*Aoristos Dyosphainthos*

__Author__*. The purpose of*

__’__**s Preface****.**

__F__**.**

__E__**.**

__D__**Vignette**

**#13**

**is to present an**

**.**

__E__**.**

__D__*‘*for a

__Dialectical__**Model****Equation**’**of the**

*systematic*__dialectical__method of presentation**of William of Ockham.**

*Early Medieval Philosophy of Science*
This

*‘*also serves as a “worked”, “cook-book” example, and sample, of the application of the__Dialectical__**Model****Equation**’_{N}__Q__**to help organize, and to compactly encode,**__dialectical__algebra**expositions for***systematic**‘‘‘*of the structure/process of the living self-sustenance of systems -- ‘**synchronic****snapshots**’’’*physio-*systems’,*‘ideo-*systems’, and**of the two, such as the system of***hybrids***-- that are***Science***; expositions that explicate and assess the***presently in existence***content of such systems, without explicitly expositing their***present**‘“*-- neither explicitly**diachronic chronicles**”’*reconstructing*the**,***prior***that constituted that content, nor explicitly***diachronic history**‘pre-constructing’*any*presently**-*predicted**,***future**successor system*of such. [__Note__: the ordering**,**__Mythopoeia__**,**__Religion__**,**__Philosophy__**,**__Science__**is, per the**__Psychohistory__**.**__E__**.**__D____first__*Psychohistorical**-*, that of**‘**__Dialectical__**Meta**-**Equation**’*Human Ideology**/*. See http://www.dialectics.org/dialectics/Aoristoss_Blog/Entries/2012/5/19_The_F.E.D._Psychohistorical_Equations.html ].**Knowledge**‘**Meta**-**Evolution**’

__A Note about the On__

__-__

__Line Availability of Definitions of__

__F____.__

__E____.__

__D____.__

**. Definitions of**

__Key Technical Terms__**technical terms, including of**

__Encyclopedia Dialectica__**.**

__E__**. ‘neologia’, are available on-line via the following URLs --**

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

Definitions of the

**special terms most fundamental to this vignette are linked-to below --**

__Encyclopedia Dialectica__
«

**»***archÃ©*
«

**» and «***arithmos***»***arithmoi*
«

**»***aufheben*

__Dia__**vs.**

*chronic*

__Syn__

*chronic*

_{N}

__Q__

__dialectical__

*arithmetic*

*/*

*algebra***or**

*Systematic*

*Synchronic*

__Dialectics__

-- and we plan to expand these definitions resources
as the

__Encyclopedia Dialectica__*Dictionary***unfolds.***Project*
[

__Note__:__‘‘‘Arithmetical Quantifiers’’’ vs. ‘Arithmetical Qualifiers’__. In the phrase “**3**apples”, we term “**3**” the “arithmetical [“pure”-]**ifier”, and “apples” the**__quant__*‘‘‘**ontological**’’’*-- or**of thing -- ‘‘‘**__kind__**ifier’’’. In the phrase “**__qual__**3**pounds of apples”, we term “pounds” the*‘***[-unit]***metrical***ifier’ -- or ‘‘‘unit of measure**__qual__**ifier’’’ --**__qual__**ified by the***quant***3**, which, together,*‘*__quant__*o**-*__qual__*ify**’*the*‘**ontological***ifier’, “apples”. A key use-value of the**__qual____dialectical__**is to provide algorithmic, ideographical-symbolic systems for the various kinds of ‘arithmetical***arithmetics***ifiers’, both with and without the co-presence of ‘‘‘arithmetical**__qual__**ifiers’’’.].**__quant__

__I____.__

**Introduction****. The present**

__to the Subject__*Systematic Dialectic*within*Medieval Occidental Philosophy***.**

__F__**.**

__E__**.**

__D__**vignette sets forth a**

__dialectical__*-*of William of Ockham’s

**ideographical model***circa*

**1323**C.E.

**/**

*philosophical theory**‘*of

**ideo**-**taxonomy**’**. Ockham lived**

*the sciences**circa*

**1290**to

**1349**C.E. [

__C__ommon

__E__ra].

This vignette contains this
example of

__dialectical__*-***in***mathematical models***ral, formulated using a**__gene__**fic**__speci____dialectical__*-***-- that of the “purely-***mathematical language***itative”, “purely connotative” mathematics of the**__qual__*‘**algorithmic**-**heuristic algebra**’*of the_{N}__Q__*‘**arithmetic of*__dialectics__*’*, which is an*axiomatized algebraic*__dialectical__**.***logic*
For further background on
the algorithmics and the axiomatics of its

**, or***ideographical**“**symbolic**”*__dialectical__**-- or***logic**‘**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***is as follows.***the sciences*
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

**eality”, and is therefore a**__r__*“**categorematic**”*term.
Ockham gave to

**erms of language that refer to, e.g.,**__t__*“**categorematic**”*terms, i.e., to other terms of language itself, the descriptor [the term]*“*__syn__*categorematic**”*.
Thus, the “terms”, or words,
“all”, “some”, “none”, and “not” are all examples of such “word-referring
words”, or “term-referring-

**erms”.**__t__
Ockham also labeled terms
that refer to things belonging to the

**eality outside of [the reality of] language, by the phrase**__r__*“terms of the*__f__*irst**intention**”*.
He labeled

**erms that refer to**__t__*terms of the*__f__**by the phrase “***irst intension***erms**__t__*of the*__s__*econd intention**”*.
In the most direct sense,
then,

*“*__syn__*categorematic**”***erms are “**__t__**erms**__t__*of the*__s__*econd intention**”*.
However, Ockham
distinguishes

*mere*“**erms” of language, in the sense of**__t__*individual words*, from those higher-level /- higher ‘qualo-fractal scale’ language constructions [constructions that we of**.**__F__**.**__E__**. call**__D__*‘**meta**-**terms**’*, or*‘**meta**-**words**’*,**[usually]***each one made up out of a**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***erms, in ‘sentence-ial’**__t__**with***combination*__syn__*categorematic***erms, in the form of well-formed**__t__**or***sentences***, and, thus,***propositions***, refer to, and describe,***functioning together***, typically, the things belonging to the***more adequately***eality outside of human language, than do single, isolated,**__r__*“mere”***erms, or single, isolated words, such as those**__t__**erms/words which are named “names”, or “nouns”.**__t__
The

**/***sentence***-- “All natural objects are transitory.” -- is thus***proposition*__f__**. Note: this sentence “contains” [***irst intentional***] both***combines**categorematic***erms, like “natural objects”, and also**__t____syn__*categorematic***erms, like “All”.**__t__
The

**/***sentence***-- “All second-order-formal-logic axiomatizations of “***proposition***N**atural Numbers” arithmetic are either formally inconsistent or formally incomplete.” -- is thus__s__**[at least!]. Note that this sentence “contains” no***econd intentional***erms which refer to**__t____non__*-*linguistic,*“**physical**”***, but only***objects***erms which refer to other**__t__**erms, or to the names of**__t__*‘‘‘**mental**objects**’’’*/*‘‘‘**idea**-**objects**’’’*, to ‘meme-ic’ or ‘memetic’, human-Phenomic,*‘**psychohistorical**materialities**’*.
On the basis of these
distinctions, Ockham divides the «

**» of the**__Genos__*“*__Sciences__*”*into two «**», or***species**‘**sub*^{1}*-*«**» --**__Gene____«__

__species____»__

**:**

__1____«__

__Scientia__

__rea__

__l__

__is____»__, encompassing the

**of “**

*sciences***ea**

__r__**things” [i.e., of**

__l__**ternal-to-mind, physical things].**

__ex__
For example, the science of

This «

**would form a***biology*__sub__^{1}*-*«**» of this «***species***».***species*This «

**» is***species*__f__**.***irst intentional*__«__

__species____»__

**:**

__2____«__

__Scientia__

__ratio__

__n__

__alis____»__, encompassing the

**of ‘**

*sciences***oetic**

__n__*-*things’ [i.e., of

**ternal-to-mind, ‘Phenomic’ things].**

__in__For example,

**would form a**

*mathematics*

__sub__

^{1}*-*«

**» of this «**

*species***».**

*species*This «

**» is**

*species*

__s__**.**

*econd intentional*For further background on Ockham’s theory, see W. L. Reese,

**,**

__Dictionary of Philosophy and Religion__**, Humanities Press [Atlantic Heights, NJ:**

*Eastern and Western Thought***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*

__Gene__

__ric Ordinal__

__Qual__

__ifiers__

__of the__

_{N}

__Q__**. Perhaps a bit surprisingly, upon first apprehension, the first four, first-order-logic, Dedekind-Peano Postulates for the “**

__Arithmetic__**N**atural” Numbers focus on their

**,**

*ordinality***on their**

__not__**, viz. --**

*cardinality***1**.

**1**is a “Natural Number”.

**2**. The

**of any “Natural Number” is also a “Natural Number”.**

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

**.**

*successor***4**.

**1**is

**the**

__not__**of any “Natural Number”, i.e.,**

*successor***1**has

__no__**within the “Natural Numbers”.**

*ancestor*These postulates thus identify the essence of the “

**N**atural Numbers”, explicitly, in terms of [apparently purely-]

__quant__*itative*

**.**

*ordinality*In keeping with this focus on

**, Seldon defines the system of the**

*the ordinal*

_{N}

__Q__

__dialectical__**-- the first**

*arithmetic**‘*

__anti__

*thesis**-*

*system**’*, or

*‘*

__contra__*-*

*system**’*, to the “

**N**atural Numbers” as «

**»**

*archÃ©**-*-- in terms of

**system**

__qual__*itative*.

**ordinality**The

_{ N}

__Q__**, which he also calls the**

*‘*

*meta**-*

**N**

*atural*

*meta**-*

*Numbers*

*’*, are, in their simplest, least-interpreted essence, a consecutive sequence of ‘meta-numeral’ ideograms representing the successive

__qual__*ities*,

**the**

__not__

__quant__*ities*, of

**-- the**

*ordinality*

__qual__*ity*of

*‘*

*first**-*ness’, followed by the

__qual__*ity*of

*‘*

*second**-*ness’, followed by the

__qual__*ity*of

*‘*

*third**-*ness’, and so on... -- satisfying the four first-order-logic

*‘*

__contra__*-*

*Peanic**’*,

*‘*

__Qual__

*o**-Peanic*

*’*axioms:

**1**

**q**. The

*ordinal*

__qual__*ifier*for the

__qual__*ity*of

*‘*

*first**-*ness’ is an element of the

*‘*

*consecuum**’*of generic

*ordinal*

__qual__*ifiers*.

**2**

**q**. The

**of any element of the**

*successor**‘*

*consecuum**’*of generic

*ordinal*

__qual__*ifiers*is also an element of same.

**3**

**q**. Any two, distinct

*ordinal*

__qual__*ifiers*have

__qual__*itatively*

*equal*

__un__**.**

*successors***4**

**q**. The

*ordinal*

__qual__*ifier*for the

__qual__*ity*of

*‘*

*first**-*ness’ is «

**»:**

*archÃ©***the**

__not__**of any element of its**

*successor**‘*

*consecuum**’*.

The symbols, or

*‘*

*meta**-*

*numerals**’*, that stand for the

*‘*

*meta**-*

*numbers**’*of the

_{N}

__Q__

*‘*

__arche__*onic*

*consecuum**’*are derived,

*syntactically*, in a way which represents the

*semantic*

*‘*

*self**-*

*subsumption**’*,

*‘*

*self**-*

*subordination**’*, or

*‘*

*self**-*

*demotion**’*[

**,**

__dialectical__

*self**-*«

**»**

*aufheben*

*self**-*] of the “

**negation****N**atural Numbers”.

That derivation is part of the positive fruition of the

**,**

__dialectical__

*immanent*

__self__*-*

**of the “**

*critique***N**atural Numbers”, which divulges the

_{N}

__Q__**as the implicit, most extreme known**

**,**

*opposite**“*Standard Model” of the “Standard”, Peano “

__Non__-**N**atural Numbers”.

It involves the turning of the

*‘*

*gene**ric*

*ordinal*

__quant__*ifiers*

*’*of the “

**N**atural Numbers” into the

*‘*

*gene**ric*

*ordinal*

__qual__*ifiers*

*’*of the

_{N}

__Q__

*‘*

*meta**-*

**N**

*atural*

*meta**-*

*Numbers*

*’*.

The conceptual

*‘*

*self**-*

*subsumption**’*of the

__quant__**intended by the Dedekind-Peano Postulates surfaces the**

*itative ordinality*

_{N}**as their hitherto hidden, implicit**

__Q__*‘*

*intra**-*

*dual**’*, based upon the

__gene__

__ric__

__qual__*ity of*

**, a**

*ordinality**‘‘‘*

*generic**ity’’’*which we represent by the

*‘*

*meta**-*

*numeral**ic*

*’*ideogram ‘

**’.**

__q__That

*‘*

*meta**-*

*numeral*

*component**’*represents

*‘*

__q__

*ual**itative*

*ordinality**’*, or

*‘*

*ordinal*

__q__

*ual**ity*

*’*,

__in__**: just ‘**

__general__**’, or, more fully expressed, just**

__q__

__q__**.**

_{N}
To fully express,

*‘**meta**-**numeral**-**y**’*, or ideographically, the*‘**consecuum**’*of__speci____fic__*ordinal***, namely --**__qual__ities

_{N}

__Q__

**=**

**{**

*‘*

__first__*-*ness’;

*‘*

__second__*-*ness’;

*‘*

__third__*-*ness’, etc.

**}**

-- we must add a second

*‘**meta**-**numeral**component**’*, via*‘‘‘*__sub__*ordinating**’’’*__speci__**“**__fic__**N**atural Numbers”, as__speci____fic__*‘**ordinal*__quant__*ifiers**’*, to the__gene____ric__*ordinal*__qual__*ifier*symbol ‘**’, by**__q__*‘*__sub__*scripting**’*those__speci__**“**__fic__**N**atural Numbers” to a ‘__script__-level’ ‘**’, ‘‘‘above’’’ them, yielding --**__q__

_{N}

__Q__

**=****{**

__q__**,**

_{1}

__q__**,**

_{2}

__q__**,**

_{3}**... }**

[in which each

*‘*

*meta**-*

*number**’*is a minimal, «

**»**

*genos***/**«

**» «**

*species***» in its own right],**

*arithmos eidetikos*
vs.

**N**

**=**

**{ 1**,

**2**,

**3**,

**... }**.

Note that this

This

The

That is, each

**of an arithmetical system of***opposition**purely**-*__quant__*itative***, based upon the***ordinality***N**,**an arithmetical system of**__versus__*purely**-*__qual__*itative***, based on the***ordinality*_{N}__Q__**, is not a****, imagined as an***radical dualism***,***absolute***between an***irreconcilable diremption**absolute*__quant__*itative*and an*absolute*__qual__*itative*.This

**is, on the contrary, a***opposition*__dialectical__*anti**thesis**-***.***sum*The

**N**__quant__*ifiers*are still there, as*‘**speci**fiers**’*-- still present -- in, or ‘‘‘under’’’, the generic**s of the**__q___{N}__Q____qual__*ifiers*, though**sumed,**__sub__**ordinated,**__sub__**moted -- demoted to being their mere**__de__**scripts or**__sub__**: The***denominators***N**__quant__*ifiers*are still*“**contained**”*in[side] [‘‘‘beneath’’’] the_{N}__Q____qual__*ifiers*.That is, each

_{N}__Q____qual__*ifier*is an «**»***aufheben**determinate***/***negation***/***conservation**elevation**-into-one-step-higher-*of an individual**gene**rality**N**__quant__*ifier*.
The joint «

**»***aufheben***of the***elevation*__quant__*itative*«**ordinal**__s__**» yields the «***species***» ‘***genos***’, of the**__q____qual__*itative***.***ordinal*__s__
For this first layer of
interpretation of these

*“purely-*__qual__*itative**”*_{N}__Q__*‘**meta**-**numbers**’*-- which does not yet make explicit their universal interpretability for the modeling of__dialectical__**-- this is all that they represent:***progressions***; [***abstract temporality**abstract chronological*]**;***order**gene**ric**‘**order**ed**-ness**’*; the consecutive succession of*‘*__qual__*itative**ordinality**’*; the*‘**consecuum**’*of*order*__qual__*ity*or of*order*__qual__*ities*.
But even here, at this
minimally-interpreted stage of the construction of the

True, it is but a shadow, and but a ‘pre-vestigial’ harbinger, of the richness of the kind of particularity of

A case in point is exemplified in the very

_{N}__Q____dialectical__**, there is already a kind of***arithmetic**gene**ric**‘**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

**that drives forward, intuitively, the***categorial followership***,**__dialectical__*purely**-**qual**itative***of the more concrete, more***logic***fic***speci*__dialectical__*-***interpretations thereof.***algebraic*A case in point is exemplified in the very

_{N}__Q__**Ockham’s***algebraic model*__dialectic__*of***constructed herein.***Science*
This

*gene**ric**‘**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__**will advance from this harbinger of***arithmetic**‘**connotative entailment**’*to the following, still__gene__**, but at last also explicitly**__ric__**, form of**__dialectical__*‘**connotative entailment**’***:***ordinality**‘**first**full**anti***follows from the***thesis*__self__*-***of [«***interaction***»***archÃ©**-*]**;***thesis**first***follows from the***full synthesis***of***mutual interaction***and***first full thesis***’, and so on.***first full antithesis*

__III____.__

__Dyad__

__ic Seldon Function__

__Interpretation of the__*Initial*

__Generic__

_{N}

__Q__

__Ontological Category__

__Qual__**.**

__ifiers__**ric**

*gene**‘*

__dialectical__

*model meta**-*

*equation**’*form for the functions-family of the Seldon Functions is that of a

**ric**

*gene***symbol [‘**

__cumulum__**’] on the LHS [**

__|-|-|____L__eft-

__H__and

__S__ide] of the

*‘*

__dialectical__

*meta**-*

*equation**’*, equated to an RHS expression representing

*‘*

*self**-*

*reflexive operation**’*of an [«

**»,**

*archÃ©**‘‘‘*

*seed**’’’*,

*‘‘‘*

*cell**-*, or

**form**’’’*‘*

*ultimate ancestor**’*

**symbol [represented,**

*ontological category***rically, by [‘**

*gene*

__q__**1**’] -- indicating its recurring

*‘*

*self**-*

*reflexion**’*via a

*‘*

*meta**-*exponentiated’,

*monotonically increasing*whole-number-valued ‘‘‘independent variable’’’ [‘

**h**’ ] -- on the RHS of the

**ric**

*gene***, viz. [with**

*Seldon Function equation**‘*

**ric-ness’ connoted by the “rectangular”**

*gene**motif*of the symbols-set]:

__|-|-|__

_{h}**=**

**|**

**[**

__q__**1**

**]**

**|v**

**, for**

^{h}**h**in

**{**

**0**,

**1**,

**2**,

**3**,

**.**

**.**

**.**

**}**,

**v**in

**{**

**2**,

**3**

**}**.

If

If

Our remarks herein are concentrated on the

**v =****2**, the**ric***Gene***above is said to belong to the***Seldon Function***sub-family.**__Dyad__ic Seldon FunctionIf

**v =****3**, the**ric***Gene***above is said to belong to the***Seldon Function***sub-family.**__Triad__ic Seldon FunctionOur remarks herein are concentrated on the

**, because the main**__Dyad__ic Seldon Functions*‘*modeling the**-**__dialectical__**mathematical meta**-**equations**’,__dialectical__*-***systematic****of***presentation**Ockham**’*, exposited herein, are of the**s****Theory of Science****v =****2**variety.With

**v =**

**2**, and selecting that special generic

__Dyad__

*ic*

*Seldon Function**that we reserve for a*

__form__**-- for the**

__systematic____dialectic__

__dialectical__*-*, the

**systematic method of presentation***of the*

__form__*‘*

*model meta**-*

*equation**’*to be constructed herein becomes, more specifically --

__)-|-(__**s**

****

**=**

**(**)

__q__**a**(

**)**

**2**

^{s}

^{ }****, for

**s**in

**{**

**0**,

**1**,

**2**,

**3**,

**.**

**.**

**.**

**}**.

-- wherein the symbol

**s**, replacing the more**ric symbol***gene***h**, takes on ‘‘‘**ystematic’’’, ‘‘‘taxonomic’’’ connotations, representing the well-ordered,**__s__**uccessive**__s__**teps of a**__s__**ystematic exposition, and wherein, in general, the “curvilinear”**__s__*motif*of the entire symbols-set used is to connote the**domain of**__systematic____dialectic__*‘*.__dialectical__**modeling**’
The

by which they are interpreted as

**bring with them a further,***Seldon Functions***layer of interpretation of the***second*_{N}__Q____q__*ual**ifiers*,**{**__q__**,**_{1}__q__**,**_{2}__q__**,**_{3}**... }**,by which they are interpreted as

__q__*ual**ifiers*that stand for__gene__*ric*__dialectical__**, e.g., for***ontological categories**‘‘‘*__thesis__*’’’***, or for full or partial***categories**‘*__contra__*-**thesis**’***, or for full or partial***categories**‘*__uni__*-**thesis**’***.***categories*
If we

**[ ‘***assign***[----)**’ ] the «__a__**»***rchÃ©**-*category,**thesis**__q__**a**, to the*generic*_{N}__Q____q__*ual**ifier**‘*,**meta**-**number**’__q__**, as signed by ‘**_{1}__q__**a**_{ }**[----)**__q__**’, and if we can discern that**_{1}__q__**a**, and all of its successor-**, and their***categories***, as generated by its successive,***cumula***, ‘Seldon-functional self-operations’, connote «***cumulative***»***aufheben***, that is,***operators***, then the**__dialectical____negation__operators**is seen to signify, under the axioms of the system of arithmetic of the**__Dyad__ic Seldon Function_{N}__Q__*‘*[ http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf ], a**meta**-**numbers**’*‘*,**-**__self__**iterated**’**recurrence of***cumulative*__dialectical__*“*[**negation**__s__of**]***the**negation*__s__*”*.
With every [unit] increase in

Only for

since

and since

**s**, the**‘formulaic recipe’ calls for the**__Dyad__ic Seldon Function*dyadic**self**-*of the result of the previous**operation***dyadic**self**-*, i.e., for a**operation****of the result of the previous***negation of the negation***.***negation of the negation*Only for

**s****=****0**-- only for the case in which__no__*self**-*occurs -- is the “result” a singleton [ideo-]**operation***ontological category***, the***symbol***for the «***symbol*__a__**» [ideo-]***rchÃ©***alone, instead of that “result” taking the form of a***ontological category***of [powers-of-]***cumulum***[or more] such symbols, i.e., a “non-amalgamative sum” [cf. MusÃ¨s], or an «***two*__a__*-*» sum [cf. Plato], of ‘[ideo-]__sum__bletoi*ontological category**’***,***symbols*since

**2**^{0}**=****1**,and since

__q__**a**^{1}**=**__q__**a**--

__)-|-(__**0**

**=**

**(**)

__q__**a**(

**)**

**2**

^{0}**=**

**(**)

__q__**a**(

**)**

**1**

**=****(**)

__q__**a**(

**)**

**=**

__q__**a**.

For example, if we take epoch

**s**

**=**

**1**, and denote the «

__a__**»**

*rchÃ©***simply by**

*ontological category***, for syntactical convenience, then the**

__a__**calls for the following, as per the**

__Dyad__ic Seldon Function

_{N}**axioms, since**

__Q__**2**

^{1}**=**

**2**--

__)-|-(__**1**

**=**

**(**)

**(**

__a__**)**

**2**

^{1}**=**

**(**)

**(**

__a__**)**

**2**

**=**

**(**)

**(**

__a__**)**

**x**

**(**)

**(**

__a__**)**

**=**

__a__

**+**

**.**

__b__
-- wherein

and where

with ‘

__a__**[----)**_{ }**connotes the «**_{q1 }__a__**»***rchÃ©***or***category**‘‘‘*,**thesis**’’’and where

__b__**[----)**_{ }__q__**2**connotes the*first**‘*or**contra**-**category**’*‘‘‘*,**antithesis**’’’with ‘

**+**’ standing for a**ralized***gene***operation, that encompasses the***addition***of***addition*__q__*ual**itatively*distinct terms, and with ‘**x**’ standing for a**ralized***gene***operation, that encompasses the***multiplication***operation***multiplication***fic to the***speci*_{N}__Q____q__*ual**ifiers*.
If we take ‘

‘subject [

and yielding the

‘‘‘

**(**)**(**__a__**)****x****(**)**(**__a__**)**’ with**(**)**(**__a__**)**connoting the*category**to be***, and with ‘**__dialectically__negated**(**)**(**__a__**)**’ connoting the*category**doing that***[**__dialectically____self__*-*]**, with ‘***negating***(**)**(**__a__**)**’ as the**,**__dialectical__*determinate**‘*sign’**negation**-**fic to***speci***(**)**(**__a__**)**as the object of the so-indicated**,**__dialectical__**, then we have ‘***determinate negation operation***(**)**(**__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__**+****’’’, or**__antithesis__
‘

__«__**archÃ©**»**category****+**__first__**’, viz. --**__contra-category__**(**)

**(**

__a__**)**

**x**

**(**)

**(**

__a__**)**

**=**

__q__**a**

**+**

__q__**aa**

**=**

__q__**a**

**+**

__q__**b**

**=**

__a__**+**

__b__

-- which, in terms of the

**ric, minimally-interpreted**

*gene*

_{N}

__Q__

**, is a**

*arithmetic*

__dialectical__**of the**

*interpretation***ric --**

*gene*

__|-|-|__**1**

**=**

**|**

**[**

__q__**1**

**]**

**|**

**2**

^{1}**=**

**|**

**[**

__q__**1**

**]**

**|**

**2**

**=**

**|**

**[**

__q__**1**

**]**

**|**

**|**

__x__**|**

**|**

**[**

__q__**1**

**]**

**|**

**=**

**|**

**[**

__q__**1**

**|**

__+__**|**

__q__**1**

**+**

**1**

**]**

**|**

**=**

__q__**1**

**|**

__+__**|**

__q__**2**.

The

**iteration of this**

*second***, corresponding to the consecutively**

__dialectical__negation of the negation**value of**

*next***s**namely,

**s**

**=**

**2**, for the

_{N}

__Q__**interpreted for [**

*arithmetic***]**

*psycho*

*historical***, yields the following, ontologically-expanded**

__dialectics__**of**

*cumulum***-- a**

*ontological categories**‘*, consisting of

**dyad of dyads**’**4**consecutive

**:**

*ontological categories*

__)-|-(__**2**

**=**

**(**)

**(**

__a__**)**

**2**

^{2}**=**

**(**)

**(**

__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**, per the**__dialectically__interpreted**.**__E__**. standard, as follows:**__D__

__c__

**=**

*third***,**

*ontological category*

*first*

__full__

*uni**-*

**;**

*category*

__d__

**=**

*four***,**

*th ontological category*

*second*

*contra**-*

**.**

*category*

We will not here pursue this

**.**__E__**. standard**__D__**of the**__dialectical__interpretation**generated by the generic***ontological categories***beyond**__Dyad__ic Seldon Function**t****=****2**, because the ‘‘‘solution’’’ -- or*‘*-- of the category-terms generated by the**semantification**’*‘*constructed in this vignette [whose terms are generated, initially, as**-**__dialectical__**mathematical meta**-**model meta**-**equation**’**, terms of**__algebraic____un__knowns**meaning], as presented herein, does not extend beyond that**__un__known*second**‘*for that**self**-**iteration**’*‘*.**meta**-**model**’
The

*‘*illustrated above describe our expectations for this**purely**-**qual**itative**calculations**’*‘*in terms of**meta**-**model**’__gene__*ric*characterizations of the successive, consecutive**.**__dialectical__categories
The next section addresses the heart of this

*‘*-- the**meta**-**model**’__speci__*fic*meanings of the*gene**ric*as applied to the__dialectical__categories__speci__*al**case*of*the systematic*__dialectic__of the Early*-*of William of Ockham.**Medieval****Philosophy of Science**

__IV____.__

__E____.__

__D____.__

__Solution for the__

__‘__

__Meta__

__-__

__Model__

__Meta__

__-__

__Equation__

__’__

__of Ockham____’__

**. The content of this vignette is a recapitulation, and an immanent extension, of our**

__s__*Dialectic of Science***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

**of this universe-of-discourse of human ideas, we use the**__dialectical__taxonomy**axiomatized in an earlier letter available via the www.dialectics.org Correspondence Page --**__dialectical__logic
-- and we enact the ‘‘‘mental action’’’ / ‘‘‘mental
operation’’’ signified by the ideogram ‘

**(----]**’. That is, we “interpret” or “assign”__q__**, the**_{1}**of the***first***ric ‘meta-***gene***N**atural meta-Numbers’ of our_{N}**,**__Q__*‘*ifiable__unquant____Q__**ifier’**__ual__*‘*, or**’**__dialectical__ideography*‘*[**’**__dialectical__symbolic logic*‘*], as follows --**-**__dialectical__**ideographic logic**’

__q__

_{1 }**(----]**

__q__**, or**

_{r}

__q__

_{1 }**(----]**

__r__
-- such that

__r___{ }**(----)**“**eality”, the universe of “**__r__**eal things”, outside of human language, outside of the human mind.**__r__
Our

*‘*then captures a sequential, systematically-ordered, ‘‘‘ordinal-ized’’’**-**__dialectical__**ideographic model**’**of Ockham’s***presentation***, using the connotations of the series of ideographical “shorthand” symbols that it***theory of the sciences***rates algorithmically.***gene*
It does so via

**as the**__r__*‘self-operating operator’, ‘self-operation’, ‘self-function’, ‘self-refl*__recurrently__**xive function’, ‘‘‘self-refl**__u__**xive function’’’, or**__e__*self-applied, connotationally specific, determinate*__recurrently__**, «**__dialectical__**»***aufheben**-*at the heart of the**negation operation***‘*formula --**-**__self__**iteration**’

__)-|-(__**s**

****

**=**

**(**)

**(**

__r__**)**

**2**

^{s}****

-- wherein

**s**, as the higher exponent of the exponent

**2**, counts the “

**teps” or “**

__s__**tages” of that**

__s__**of the**

*systematic presentation**‘idea-*ontology’, or

*‘ideo-*ontology’-- of the

*‘ideo-taxonomy’*-- of the philosophical categories, and wherein

__)-|-(__**s**denotes the

*‘cumul*um’ -- meaning the connotative and non-amalgamative,

*‘‘‘*itative sum’’’, or ‘‘‘superposition’’’ -- of the

__qual__**presented / ‘ac**

*categories**Cumul*ated’ as of

**tep**

__s__**s**of this “shorthand” ‘‘‘

__systematic__*-*’’’

__dialectical__*‘*.

**present**-**ation**’
Let us, then, unfold this model

**tep-by-**__s__**tep, starting with**__s__**tep**__s__**s****=****0**, then moving on to**tep**__s__**s****=****1**, and thence onward to**tep**__s__**s****=****2**, and, finally, to**tep**__s__**s****=****3**, a**tep which**__s__**Ockham’s theory to a***immanently extends***of***category***that Ockham, as far as we know, did not envision.***The Sciences*__Step__

**. The**

__0__**0**th

**tep of this**

__s__*‘*, i.e., for

**presentation**-**model**’**s**

**=**

**0**,

since

**2**

^{0}

**=**

**2**

^{+}

^{1}**/2**

^{+}

^{1}

**=**

**2**

^{+}

^{1}**x**

**2**

^{-}

^{1}

**=**

**2**

^{+}

^{1}

^{-}

^{1}

**=**

**1**,

yields --

__)-|-(__**0**

**=**

**(**)

**(**

__r__**)**

**2**

^{0}

^{ }**=**

__r__

^{1}**=**

__r__-- which simply reiterates the starting point -- the «

**»**

*archÃ©**‘*-- of this

**ideo**-**taxonomic****category**’*‘*.

**present**-**ation**’**Does this categorial-singleton of**

*¿***tep**

__s__**0**

**grow into a ‘categorial-**

*cumulum**’*in

**tep**

__s__**1**

*?*__Step__

**. For**

__1__**tep**

__s__**s**

**=**

**1**of this

*‘*, since

**presentation**-**model**’**2**

^{1}

**=**

**2**

^{+}

^{1}

**=**

**2**, we have --

__)-|-(__**1**

**=**

**(**)

**(**

__r__**)**

**2**

^{1}

^{ }**=**

__r__

^{2}

^{ }

**=**

__r__**(**)

**(**

__r__**)**

^{ }**=**

__r__*“*

**of**”

__r__

^{ }**=**

__r__*“*

**squared**”-- which connotes the

*‘*, or

**-**__self__**refl**’__e__xive function*‘*, of the

**-**__self__**operation**’**[«**

__dialectical__**»]**

*aufheben***denoted by**

*operation***.**

__r__
Syntactically, per the axioms of the

_{N}__Q__**‘meta-numbers’, we have the following values as the content of****tep**__s__**1**:

__)-|-(__**1**

**=**

**(**)

**(**

__r__**)**

**2**

^{1}

^{ }**=**

__r__

^{2}

^{ }**=**

__q__

**r**

^{2}

^{ }**=**

__q__

**r****+**

__q__

**r**

**r****[----)**

__q__

**1****|**

__+__**|**

__q__

**1+1****=**

__q__

**1****|**

__+__**|**

__q__**.**

**2**Our challenge, for this

**tep, is to “solve for” the new category,**

__s__

__q__**, that is, to determine the meaning, the connotation, of that new ‘categorogram’, given the already known/assigned meaning, or connotation, of the ‘categorogram’**

_{rr}

__r__

**=**

__q__**.**

_{r}
So, let us apply the

**ral ‘‘‘canon of interpretation’’’ of such**__gene__**to the**__dialectical__categories**fic case of this example.**__speci__
Per that ‘‘‘canon’’’, the «

Also per that ‘‘‘canon’’’, a symbol like

**» ‘categorogram’,***archÃ©***, is to be interpreted, as stipulated, as connoting an «**__r__**» of «***arithmos***» -- an***monads***of***assemblage***-- each one of which is a***units**“*, a**fact**”**, empirical “***physical***eality”, “outside of” language and of the human mind.**__r__Also per that ‘‘‘canon’’’, a symbol like

__q__**connotes the new**_{rr}**/«***category***» that results from the***arithmos*__self__*-*«**» of the***aufheben*__q___{r}**/«***category***», and, quite often, that results from the***arithmos*__self__*-*«**»***aufheben**‘*of some of the «**-**__meta__**monad**ization’**» of the***monads*__q___{r}**/«***category***».***arithmos*
That is, each «

**» or***monad***of the***unit*__q__**«**_{rr}**» /***arithmos***/***category***of***assemblage***should be a***units**‘*«**-**__meta__**»’, made up out of a multiplicity -- made up out of a ‘sub-«***monad***»’ -- of the «***arithmos***» of the***monads*__q__**«**_{r}**» /***arithmos***.***category*
So, the

**of the***units*__q___{rr}**should be, in this***assemblage***fic case, ‘meta-**__speci__**ealities’, ‘**__r__**ealities**__r__*of second degree**’*, each one made up out of a heterogeneous multiplicity of ‘**ealities**__r__*of first degree**’*.
Also, the

Moreover, the

*category*__q__**should be**_{rr}*an*__other__**--***category***than and to the**__other__**/empirical “***physical***ealities”**__r__**; a***category**‘*to, and a**-**__meta__**category**’*‘*of, the**-**__contra__**category**’**category.**__r__Moreover, the

__q__**symbol should point to a**_{rr}**that is already known to us, that is part of our***category***,***synchronic***conceptualized experience of the totality of***present***, 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***The Sciences***category.**__r__
On the bases set forth above, we propose that our
solution -- our ‘semantification’ -- of the syntactic term

__q__**should be**_{rr}**, connoting the «**__t__**»/***arithmos***of***category***erms of human language:**__t____q___{rr}**=**__q___{t}**=**__t__**[----)**__q__**.**_{2}
We propose this solution, in part, because the

Each

The

The

That is, “

One “

Each “

Our solution for

**erms of human language, as**__t__**, are***units*__meta__*-*of the**units****of empirical/***units***“***physical***eality”.**__r__Each

**erm of human language refers to, and represents, a heterogeneous multiplicity of similar**__t__*physical**“*.**facts**”The

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

**erm “**__t__**orange**” maps to a large, heterogeneous multiplicity of similar, physical**oranges**.That is, “

**erms” are**__t__**ral relative to***gene**physical**“*.**facts**One “

**erm” typically stands for a large, although still finite, set of similar**__t__**situations.***physical*Each “

**erm” is an «**__t__**»***aufheben**‘*«**-**__meta__**»***monad**’*of a large number of similar “*fact**”*-«**».***monads*Our solution for

__q__**is that**_{rr}__q__**connotes the «**_{rr}**»/***arithmos***which has***category***erms of human language as its «**__t__**».***monads*
In an

In that

**,***historical***sense,**__dia__chronic__r__*“***of**”**also connotes the natural-historical result of the ‘self-reflexion’, of the ‘bending-back-upon-self’, of “**__r__**eality”; of the turning back, upon itself, of physical “**__r__**eality”, to form**__r__**objects -- new parts of that***physical***“***physical***eality” -- that are capable of conscious reflection upon**__r__**“***physical***ealities”, including upon**__r__**.***themselves*In that

**,***historical***context, this would mean the immanent emergence, within pre-human[oid]**__dia__chronic**, from out of pre-human[oid]***Nature***-- as part of the***Nature***, or of**__dialectic__*the self**-*, of**development****itself -- of “observing eyes”, and of “observing sensoria” in***Nature***ral; of observing and thinking minds, as also “things” / objects / bodies within***gene***-- i.e., the emergence of natural ‘[ev]entities’ which produce languages, hence “***Nature***erms”, 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.**__t__
But this consideration does not belong to

**,***synchronic***in itself.***systematic*__dialectics__
It only represents the

**implications -- the pointers to***diachronic*__past__**, and to***history*__future__**-- that are***history***plicit in**__im__*synchronic***, but which do**__dialectics__**, as such -- as**__not__**-- form an***diachronic***plicit component of the explanatory apparatus of**__ex__**,***synchronic***.***systematic*__dialectics__
Marx’s treatises on the

But, in Marx’s view,

**,**__dialectical__**of the capitalist, ideology-ridden***immanent critique***of “political economy”, are works of***science***,***synchronic***.***systematic*__dialectics__But, in Marx’s view,

**is***systematic*__dialectics__**absolutely dirempt from**__not__**,***diachronic***.***historical*__dialectics__
Marx discussed this

**between***interconnexion***and***systematic*__dialectics__**, as it arises in his extension of previous concepts of***historical*__dialectics__**-- an**__dialectics__**which is, e.g., inaccessible to the***interconnexion*__eternally__**,***systematic**Parmenidean***of Plato’s**__dialectics__**«**__immutable__**» -- in a crucial methodological passage from the***arithmos eidetikos***:**__Grundrisse__**“**...our [A.D.:

*systematic**-*] method indicates the points where

__dialectical__**investigation must enter in, or where bourgeois economy as a merely**

*historical***form of the production process**

*historical*

*points beyond itself to*__earlier__**modes of production.”**

*historical*
“In order to develop the laws of bourgeois economy,
therefore, it is

**necessary to write the**__not__*real*.”**history**of the relations of production
“But the correct observation and deduction of these
laws,

**, always leads to primary equations -- like the empirical numbers e.g. in natural science -- which***as having themselves*__become__in history**lying***point towards a*__past__**this system.”***behind*
“These

**[***indications**Andeutung*], together with a correct grasp of the**, then also offer the key to the understanding of the**__present__**-- a work in its own right which, it is hoped, we shall be able to undertake as well.”**__past__
“This correct view likewise leads at the same time to
points at which the

**[at which the «***suspension***» -- A.D.] of the***aufheben***form of production relations gives signs of its***present***--**__becoming__**.”***foreshadowings of the*__future__
“Just as, on one side the pre-bourgeois phases appear
as

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

**:**__Grundrisse__**(***Foundations of the Critique of Political Economy***), M. Nicolaus, transl., Penguin Books [Middlesex:***Rough Draft***1973**], pp.**460**-**461**,*italics*only emphasis by Marx;**of**__combinations__*italic*,__underlined__,**bold**, and color emphasis have been added by A.D.].Accepting this solution, our

**tep**

__s__**s**

**=**

**1**

__dialectical__*-*

**thus becomes [with ‘**

*model equation***’ for the**

__D__

*systematic**-*context’s ‘curvilinear, ontological

__dialectic__**operator’, signifying the addition of an increment of new ‘ideo-ontology’] --**

__delta__

__)-|-(__**1**

**=**

**(**)

**(**

__r__**)**

**2**

^{1}

^{ }**=**

__r__

^{2}

^{ }**=**

__q__

_{r}

^{2}

^{ }**=**

__r__**+**

__D__

__r__

^{ }**=**

__r__**+**

__t__**[----)**

__q__

_{1}**|**

__+__**|**

__q___{2}
-- and we have the meaning that, as of

posits the

**tep**__s__**1**,**s****=****1**, our model,__)-|-(__**s****=****(**)**(**__r__**)****2****,**^{s}posits the

*possible*existence of**philosophical categories --***two***(1)**the category of physical “**ealities”, ‘‘‘plus’’’ [signed by ‘**__r__**+**’, for connotatively “interpreted”, or “assigned”,*‘*, in the ‘curvilinear symbols**ontological**’__qual__ifiers*motif’*context of**],***systematic*__dialectics__**(2)**its ‘first contra-category’, of ‘‘‘**erms of language’’’,**__t__**together comprising***both***(1)**the «**»***physis**-*«**», or***monads**‘*, of the**fact**-**units**’**cal**__physi__**ealities’ «**__r__**», or sensorially-perceived***arithmos aisthetos**“*, superposed with, and**facts**”-**assemblage***‘‘‘*’’’ -- in a__named__*many**-*to*-*relationship --**one**__by__**(2)**the ‘trans-Platonian’ «**» of***arithmos eidetikos***erms-**__t__**, or of***units***erms-«**__t__**».***monads***How does this**

*¿*

*two**-*dimensional ‘possibility-space’, or categorial ‘ideo-ontology cumulum’, expand in

**tep**

__s__**2**

*?*__Step__

**. For**

__2__**tep**

__s__**s**

**=**

**2**

**of this presentation-model, since**

**2**

^{2}

**=**

**2**

**x**

**2**

**=**

**4**, we have --

__)-|-(__**2**

**=**

**(**)

**(**

__r__**)**

**2**

^{2}

^{ }**=**

__r__

^{4}

^{ }**=**

__r__

^{2}

^{ }

^{x}

^{ }

^{2}

^{ }**=**

**(**)

__r__**(**

^{2}**)**

^{2}

^{ }**=**

**(**)

__r__**+**

**(**

__t__**)**

^{2}

^{ }-- which connotes the

*‘*, or

**-**__self__**refl**’__e__xive function*‘*, of the

**-**__self__**operation**’*collective*

__dialectical__**-- of the**

*operation*

*two**-*category,

*two**-*dimensional

*categories**-*-- denoted by

__cumulum__operation**(**)

__r__**+**

**(**

__t__**)**.

Syntactically, per the axioms of the

_{N}

__Q__**‘meta-numbers’, we have the following values as the content of**

**tep**

__s__**2**:

__)-|-(__**1**

**=**

**(**)

**(**

__r__**)**

**2**

^{2}

^{ }**=**

**(**)

__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}

**[----)**

__q__

_{1}**|**

__+__**|**

__q__

_{2}**|**

__+__**|**

__q__

_{2}

_{+}

_{1}**|**

__+__**|**

__q__

_{2}

_{+}

_{2}

^{ }**=**

__q__

_{1}**|**

__+__**|**

__q__

_{2}**|**

__+__**|**

__q__

_{3}**|**

__+__**|**

__q__**.**

_{4}
Our challenge, for this

**tep, is to “solve for” the**__s__**new categories,***two*__q___{t}**and**_{r}__q__**, that appear for the first time in this**_{tt}**tep, 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’**__s____r__**=**__q__**and**_{r}__t__**=**__q__**.**_{rr}
So, let us apply the

**ral ‘‘‘canon of interpretation’’’ of such**__gene__**to the**__dialectical__categories**fic case of this example.**__speci__
Per that ‘‘‘canon’’’, the ‘categorogram’

This often means a

Also per that ‘‘‘canon’’’, a ‘categorogram’ like

__q___{t}**connotes a new**_{r}*‘*/**-**__uni__**category**’**«***hybrid***» -- the***arithmos***category -- that results from the «***synthesis***»***aufheben***the**__of____q___{r}**/«***category***»,***arithmos***the**__by____q___{t}**/«***category***».***arithmos*This often means a

**/«***category***» each of whose «***arithmos***» is a***monads***of the «***hybrid***»***monads***the**__of____q___{r}**/«***category***»***arithmos***the «**__with__**» of the***monads*__q___{t}**/«***category***».***arithmos*Also per that ‘‘‘canon’’’, a ‘categorogram’ like

__q___{t}**might also connote**_{r}*‘‘‘*,**’’’**__conversion__**«**__of__**» of the***monads*__q___{r}**/«***category***»,***arithmos***«**__into__**» of the***monads*__q___{t}**/«***category***».***arithmos*Also, per that ‘‘‘canon’’’, the

__q__

_{t}**symbol should point to a**

_{r}**that is already known to us, that is part of our**

*category***,**

*synchronic***conceptualized experience of the totality of**

*present***, 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**

*The Sciences***teps.**

__s__
And, per that ‘‘‘canon’’’, a symbol like

__q__**connotes the new**_{tt}**/«***category***» that results from the***arithmos*__self__*-*«**» of the***aufheben*__q___{t}**/«***category***», and, quite often, that results from the***arithmos*__self__*-*«**»***aufheben**‘*of some of the «**-**__meta__**monad**ization’**» of the***monads*__q___{t}**/«***category***».***arithmos*
That is, each «

**», or***monad***, of the***unit*__q__**«**_{tt}**»/***arithmos***should be a***category**‘*«**-**__meta__**»***monad**’*, or*‘*, of «**-**__meta__**unit**’**» or***monads***of the***units*__q__**«**_{t}**» /***arithmos***.***category*
So, the

**of the***units*__q___{tt}**should be, in this***assemblage***fic case, ‘meta-**__speci__**erms’, ‘**__t__**erms**__t__*of second degree**’*, each one made up out of a heterogeneous multiplicity of ‘**erms**__t__*of first degree**’*.Also, the

*category*

__q__**should be**

_{tt}

*an*__other__**--**

*category***than**

__other__**of the categories previously evoked, in the preceding**

__all__**teps; a**

__s__*‘*of each of those categories, and a

**-**__contra__**category**’*‘*«

**-**__meta__**category**’**/**‘**-**__meta__**»’ to the**

*arithmos*

__q__

_{t}**/ «**

*category***».**

*arithmos*Moreover, the

__q__**symbol should point to a**

_{tt}**that is already known to us, that is part of our**

*category***,**

*synchronic***conceptualized experience of the totality of**

*present***, 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**

*The Sciences***teps.**

__s__On the bases set forth above, we propose that our solution -- our ‘semantification’ -- of the syntactic term

__q__

_{t}**should be**

_{r}**, connoting the «**

__f__**»/**

*arithmos***of Ockham’s**

*category*

__propositions__of the

__f__

*irst***:**

*intention*

__q__

_{t}

_{r}

**=**

__q__

_{f}

**=**

__f__**[----)**

__q__**.**

_{3}
We propose further that our solution/‘semantification’
of the syntactic term

__q__**should be**_{tt}**, connoting the «**__s__**» /***arithmos***of Ockham’s***category*__propositions__of the__s__*econd***:***intention*__q___{tt}**=**__q___{s}**=**__s__**[----)**__q__**.**_{4}
In proposing these two ‘semantifications’, we are in
resonance with Ockham in invoking here,

**just a subdivision of the universe of**__not__**erms of language, i.e., of**__t__*single***of language, into two sub-universes, of**__words__*“categorematic”*words vs. of*“syncategorematic”*words, and with*“***categorematic**”**corresponding to***words*__q___{t}**, and with**_{r}*“***syncategorematic**”**corresponding to***words*__q__**, so that**_{tt}__q___{t}**and**_{r}__q__**would fail to differ, in terms of the**_{tt}**of their «**__ontology__**», from***monads*__q__**.**_{t}
We hold that the

*‘‘‘*, i.e., the**’’’**__conversion__*apt***, of the «***description***»/***arithmos***of***assemblage**physical**“*, by**facts**”*“*,**terms**”__q___{t}**, as of the «**_{r}**»/***arithmos***of***assemblage**“*, by**terms**”*“*,**terms**”__q__**, requires, in each case, something more than mere**_{tt}*single***.**__words__
Human language is far more than the mere utterance of

*single***-sounds. Human language is, for starters, an «**__word__**»/***arithmos***of***assemblage***-- of**__sentences__**, that is, of well-formed**__propositions__*‘*, each one made up out of a heterogeneous multiplicity of**meta**-**words**’**.***words*
We thus hold that

So:

[cf. Boole’s category of “Primary Propositions”,

__q___{t}**must connote a multiplicity of “**_{r}**erms” -- both “categorematic” and “syncategorematic” alike -- combined, into**__t__*well**-*, i.e., into**formed sentences****, that describe our knowledge of the**__propositions__**,***physical world*__q__**.**_{r}So:

__q___{t}_{r}**=**__q___{f}**=**__f__**[----)**__q__**.**_{3}[cf. Boole’s category of “Primary Propositions”,

**, Chapter IV, Proposition I.].**__Laws of Thought__
We thus hold also that

So:

[cf. Boole’s “Secondary Propositions”,

__q__**must connote a multiplicity of “**_{tt}**erms” -- both “categorematic” and “syncategorematic” alike -- combined, into**__t__*well**-*, i.e., into**formed**__sentences__**, that describe our knowledge of the**__propositions__**of our own creation, of that core component of ‘The Human Phenome’ which is***secondary world***,***the world of human language itself*__q__**.**_{t}So:

__q___{tt}**=**__q___{s}**=**__s__**[----)**__q__**.**_{4}[cf. Boole’s “Secondary Propositions”,

**, Chapter IV, Proposition I.].**__Laws of Thought__
Thus,

**connotes an «**__f__**»/***arithmos***of “true”***assemblage*__proposition__*-*, or of**units**__proposition__*-*«**», that are***monads***-- of***about**empirically-apt***which**__propositions__**--***describe***.***the physical world*
And,

**connotes an «**__s__**»/***arithmos***of “true”***assemblage*__proposition__*-*, or of**units**__proposition__*-*«**», that are***monads***-- of***about**cognitively-apt***which**__propositions__**--***describe***, as an expression of***the world of human language***.***the world of human thought*
But the

**of “true”***assemblage***about the**__propositions__*“*world, external to human thought,**real**”__q___{t}**, is what**_{r}*‘‘‘***the**__sciences__of the rea__l__*’’’*“contain”; is the ‘‘‘content’’’ of those sciences; is what Ockham means by*‘‘‘***the**__sciences__of the rea__l__*’’’*, by the category he names «*scientia rea*__l__**», which we will connote by***is***:**__l____f__**=****.**__l__
And the assemblage of “true”

**about the**__propositions__*“*world, the world human language/thought,**terms**”__q__**, is what**_{tt}*‘‘‘***the sciences of the ratio**__n__*al**’’’*“contain” for Ockham; is the ‘‘‘content’’’ of those sciences; is what Ockham means by*‘‘‘***the sciences of the ratio**__n__*al**’’’*, by the category he names «*scientia ratio*__n__**», which we will connote by***alis***:**__n____s__**=****.**__n__
Accepting these solutions, our

**tep**__s__**s****=****2**__dialectical__*-***thus becomes --***model equation*

__)-|-(__**1**

**=**

**(**)

**(**

__r__**)**

**2**

^{2}

^{ }**=**

__r__

^{4}

^{ }**=**

**(**)

__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__

**[----)**

__q__

_{1}**|**

__+__**|**

__q__

_{2}**|**

__+__**|**

__q__

_{3}**|**

__+__**|**

__q__**.**

_{4}-- and we have the meaning that, as of

**tep**

__s__**2**,

**s**

**=**

**2**, our model,

__)-|-(__**s**

**=**

**(**)

**(**

__r__**)**

**2**

**,**

^{s}posits the

*possible*existence of

**philosophical categories --**

*four***(1)**the «

**» category of**

*archÃ©***“**

*physical***ealities”, ‘‘‘plus’’’;**

__r__**(2)**its

*‘*, of ‘‘‘

**first****-**__contra__**category**’**erms of language’’’, ‘‘‘plus’’’;**

__t__**(3)**the

*‘*of the preceding two categories, whose

**first****-**__uni__**category**’**or «**

*units***» are, as**

*monads***,**

__propositions__**,**

*hybrids***, or**

*combinations***of “categorematic” and “syncategorematic”**

__syntheses__**erms, and, in a sense, as**

__t__*“*[cf. Kant]

**”**__synthetic__**,**

__propositions__**of the**

*hybrids***of**

*units***“**

*physical***eality’ with the**

__r__**of**

*units***erms, thus constituting the category of**

__t__

__propositions__of the

__f__

*irst***, constituting Ockham’s «**

*intention*

*scientia rea*

__l__**», ‘‘‘plus’’’;**

*is***(4)**the

*‘*, the category of

**second****-**__contra__**category**’

__propositions__of the

__s__

*econd*

**, constituting Ockham’s «**

*intention*

*scientia ratio*

__n__**»,**

*alis**‘*ic’ with respect to the «

**-**__meta__**monad****» of the**

*monads**‘*, of ‘‘‘

**first****-**__contra__**category**’**erms of language’’’, given that these**

__t__**, which are the**

__propositions__**or «**

*units***» of the «**

*monads*

*scientia ratio*

__n__**», constitute, each one, a**

*alis**‘*, or

**-**__meta__**term**’*‘*, each one made up out of a heterogeneous multiplicity of

**term of the second degree**’*‘*

__t__

*erms*

*of the first degree**’*, which are the

**or «**

*units***» of the**

*monads***/«**

*category***»/**

*arithmos***of**

*assemblage***herein connoted by**

*units***.**

__t__
All

**categories together, as an***four**‘*, comprise --**ideo**-**cumulum**’**(1)**the «

**»**

__physi__s*-*«

**», or**

*monads**‘*, of the

**fact**-**units**’**cal**

__physi__**ealities’ «**

__r__**», or sensorially-perceived**

*arithmos aisthetos**“*, superposed with, and

**facts**”-**assemblage****-- in a**

__named__

*many**-*to

*-*relationship --

**one****;**

__by__**(2)**the ‘trans-Platonian’ «

**» of**

*arithmos eidetikos***erms-**

__t__**, or of**

*units***erms-«**

__t__**»,**

*monads***together superposed with;**

*both***(3)**the ‘trans-Platonian’ «

**» of the**

*arithmos eidetikos*

__propositions__*-*, or of

**units**

__propositions__*-*«

**», of the Ockhamian «**

*monads*

*scientia rea*

__l__**», all**

*is***together superposed with;**

*three***(4)**the ‘trans-Platonian’ «

**» of the**

*arithmos eidetikos*

__propositions__*-*, or of

**units**

__propositions__*-*«

**», of the Ockhamian «**

*monads*

*scientia ratio*

__n__**».**

*alis***But is there a**

*¿***step, a**

*next***tep**

__s__**3**, for this

**,**

*classificatory***,**

*taxonomic*

*systematic**-*

__dialectic__**of**

*presentation*

*the Sciences?*
Our

**tep**__s__**2**models where this**completed for Ockham, and for his times; for what was**__dialectic__**/extant in his experience, and in the experience of his***synchronic***.***contemporaries*
Ockham saw, as far as we know, no other sub-categories
of

**besides «***Science**scientia rea*__l__**»***is***&**«*scientia ratio*__n__**».***alis***But is there another sub-category of**

*¿***that was**

*Science*

__not__**/extant for Ockham, but that**

*synchronic*

__is__**for us**

*synchronic*

*?***How could this now**

*¿*

*four**-*dimensional ‘possibility-space’ of categorial ‘ideo-ontology’ expand in a possible

**tep**

__s__**3**

*?*
Let’s see what arises if we iterate this model one
more

**tep, i.e., for**__s__**s****=****3**.__Step__

**. For**

__3__**tep**

__s__**s**

**=**

**3**

**of this**

*presentation**-*, since

**model****2**

^{3}

**=**

**2**

**x**

**2**x

**2**

**=**

**8**, we have --

__)-|-(__**3**

**=**

**(**)

**(**

__r__**)**

**2**

^{3}

^{ }**=**

__r__

^{8}

^{ }**=**

__r__

^{4 }

^{x}

^{ }

^{2}

^{ }**=**

**(**)

__r__**(**

^{4}**)**

^{2}

^{ }**=**

##
**(**) __r__ **+ **** **__t__
**+ **** **__l__ **+**** **__n__ (**)**^{2}^{ }

__r__

__t__

__l__

__n__

^{2}

^{ }

-- which connotes the

*‘*, or

**-**__self__**refl**’__e__xive function*‘*, of the

**-**__self__**operation**’*collective*

__dialectical__**-- of the**

*operation*

*four**-*category,

*four**-*dimensional

*categories**-*-- denoted by

__cumulum__operation**(**)

__r__**+**

__t__**+**

__l__**+**

**(**

__n__**)**.

Syntactically, per the axioms of the

_{N}**‘meta-numbers’, we have the following values as the content of**__Q__**tep**__s__**3**:

__)-|-(__**3**

**=**

**(**)

**(**

__r__**)**

**2**

^{3}

^{ }**=**

**(**)

__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}**[----)**

__q__

_{1}**|**

__+__**|**

__q__

_{2}**|**

__+__**|**

__q__

_{3}**|**

__+__**|**

__q__

_{4}**|**

__+__**|**

^{ }

__q__

_{4}

_{+}

_{1}**|**

__+__**|**

__q__

_{4}

_{+}

_{2}**|**

__+__**|**

__q__

_{4}

_{+}

_{3}**|**

__+__**|**

__q__

_{4}

_{+}

_{4}**=**

__q__

_{1}**|**

__+__**|**

__q__

_{2}**|**

__+__**|**

__q__

_{3}**|**

__+__**|**

__q__

_{4}**|**

__+__**|**

^{ }

__q__

_{5}**|**

__+__**|**

__q__

_{6}**|**

__+__**|**

__q__

_{7}**|**

__+__**|**

__q__**.**

_{8}
Our challenge, for this

**tep, is to “solve for” the**__s__**new categories,***four*__q___{n}**,**_{r}__q___{n}**,**_{t}__q___{n}**, and**_{l}__q__**, 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’**_{nn}__r__**=****,**__q___{r}__t__**=**__q__**,**_{rr}__l__**=**__q___{t}**, and**_{r}__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

**new categories,**

*four*

__q__

_{n}**,**

_{r}

__q__

_{n}**,**

_{t}

__q__

_{n}**, and**

_{l}

__q__**.**

_{nn}In general, we see each of them as connoting the fruition of the

**of the «**

__application__

*scientia ratio*

__n__**» -- i.e., of advanced**

*alis***,**

*formal***/**

*mathematical logic***-- to each of the already-defined domains, resulting in their**

__dialectic__*‘‘‘*by the «

**subsumption**’’’

*scientia ratio*

__n__**».**

*alis*
We conjecture, more specifically, as follows --

__q___{n}_{r}the «**=***scientia ratio*__n__»*alis**‘‘‘*of the «**conversion**’’’» of*arithmos aisthetos*; the*physical facts**rational**-*immanent**scientific**,__critique__[including__explanation__[past]*theory of**perceptual*], and*error*of human perception of the «__reconstruction____r__» realm, extending to the formation of a*ealis*,*universal**systematic**-*__dialectical__of this realm, related to the project of the «*taxonomy*» division of Hegel’s*Natur*;__Encyclopedia of the Philosophical Sciences__

__q___{n}_{t}the «**=***scientia ratio*__n__»*alis**‘‘‘*of the «**conversion**’’’» of*arithmos eidetikos*erms of human language; the__t__*rational**-*immanent**scientific**,__critique__[including__explanation__[past]*theory of**terminological*], and*error*of human language, including the formation of a__reconstruction__*rational**universal*, comprising a*language*__dialectical__, or*phonetic language**‘*, as well as of a’__dialectical__phono__gram__y*‘*, and of a’__dialectical__picto__gram__y*‘*, related to Leibniz’s vision of a «’__dialectical__ideo__gram__y»; also related to the project of the «*characteristica universalis*» division of Hegel’s*Logik*;__Encyclopedia of the Philosophical Sciences__

__q___{n}_{l}the «**=***scientia ratio*__n__»*alis**‘‘‘*of the «**conversion**’’’» of the*arithmos*of the «__proposition__s*scientia rea*__l__»; the*is**rational**-*immanent**scientific**,__critique__[including__explanation__[past]*theory of**scientific theories*], and*error / ideology*of the «__reconstruction__*scientia rea*__l__», via the formation of*is**‘*accounting for the**theories**’in the history of*differences**different*, or*theories/ideologies*, regarding*systems of*__propositions__*the same**“***real**”/*facts*, e.g., of*physical phenomena**different*of the same*theories/ideologies*arising from different historical epochs/human-social formations, related to the project of the «*facts*» division of Hegel’s*Natur*, and to that of Marx’s__Encyclopedia of the Philosophical Sciences__,__Grundrisse__, and__A Contribution to the Critique of Political Economy__:__Capital__.*A Critique of Political Economy*

__q___{nn}the «**=***scientia ratio*__n__»*alis**‘‘‘*of the «**conversion**’’’» of the*arithmos*of the «__propositions__*scientia ratio*__n__» themselves; the*alis**rational**-***scientific***immanent*__self__*-*,__critique____self__*-*[including__explanation__[past]*theory of**scientific theories*], and*error / ideology*__self__*-*__reconstruction__the «*of**scientia ratio*__n__»*alis*the «*by**scientia ratio*__n__» themselves, including the formation of*alis**‘*accounting for the-__meta__**theories**’in the history of*differences**different*, or*theories/ideologies*, regarding*systems of*__propositions__*the same*and other*mathematical**“***noetic**”/*facts*, e.g., of*phenomena**different*of “the same”*theories/ideologies**‘‘‘*, or objects of thought, as they arise in different historical epochs/human-social formations, extending to development of a comprehensive**noeta**’’’itself; related to the project of Hegel’s*mathematics of*__dialectics__, and especially to its second volume.__Science of Logic__

Note that Marx’s

**, and**

__A Contribution to the Critique of Political Economy__**:**

__Capital__**, provide a**

*A Critique of Political Economy***fic example, and, mostly implicitly, a**

*speci***ric paradigm, for a core component of**

*gene*

__q__

_{n}**as conjecturally defined above.**

_{l}Nothing like Marx’s works was extant in Ockham’s time, but those works

__are__*‘*for us, today.

**synchronic**’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

**], and, thus, addresses the human species and its ‘Phenome’ as a**

__Capital__**, as a “**

*natural object*

__r__**”**

*eal***/**

*fact***, a part of the purview of «**

*object*

*scientia*

__r__**».**

*ealis*The subsidiary objects which Marx addresses in those two treatises -- values,

**,**

*commodities***,**

*monies***, and their “law of [surplus-]value” -- are tied to**

*capitals***of**

*objects***“**

*physical*

__r__*eality*” --

**,**

*packages of tea***,**

*bank notes*

*steam**-*, etc. -- but they are

**powered looms****simply**

__not__**.**

*physical objects*They are also

**, ‘meme[t]ic objects’, ‘‘‘mental objects’’’, ‘‘‘ideal objects’’’, ‘cultural objects’;**

*noetic objects**‘*ic objects’, parts of the human-social

**psyche**-*“*; parts of ‘The Human Phenome’ -- human, “social relations of production” [Marx].

**intersubjectivity**”
Such ‘category-[idea[l]-]objects’, such ideas, such
memes, residing within ‘The Human Phenome’, the very «

Such is the paradigm that we call

**[***arithmos***]» of meme «***eidetikos***», are***monads**“*not only in the sense that they are**material**”**, that they are***germane***, for***salient***, but also in that, being***Science**“*-- held mentally, and behaviorally,**inter**”__subjective__**among human actors -- they are also***in common***.**__objective__Such is the paradigm that we call

*‘*, pioneered by Marx.**’**__psycho__historical materialism
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

**, as updated, for our time, above --**__dialectic__of Science

__)-|-(__**3**

**=**

**(**)

**(**

__r__**)**

**2**

^{3}

^{ }**=**

__r__**+**

__t__**+**

__l__**+**

__n__**+**

__q___{nr}**+**

__q___{nt}**+**

__q___{n}

_{l}**+**

__q___{nn}-- is that of ‘

**[**

*the*__elements__**and**

*f*__acts__**],**

*words*

*and the sub**-*,

**divisions****[**

*of knowledge***]’.**

*science*If we abbreviate this universe of discourse, restricting it to just

*“*, leaving its

**The Sciences**”**[**

__elements__**and**

*f*__acts__**]**

*words***plicit therein, instead of making them**

__im__**plicit, as before, then perhaps we can encompass the most essential content of this**

__ex__**, or**

*systematic*__dialectic__*‘*, by means of a

**synchronic****ideo**-**taxonomy**’**ic Seldon Function of only three terms --**

*Triad*

__)-|-(__**1**

**=**

**(**)

**(**

__L__**)**

**3**

^{1}

^{ }**=**

__L__**+**

__N__**+**

__Y__
-- instead of the earlier eight algebraic terms, but one
whose third and final algebraic term,

**, we must now define.**__Y__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

*‘*-- the superposition -- of the conjectured definitions of the ideographical terms

**cumulum**’

__q__

_{n}**,**

_{r}

__q__

_{n}**,**

_{t}

__q__

_{n}**, and**

_{l}

__q__**, taken together, amount to a definition of the**

_{nn}

*knowledge**-*which we name ‘[

**field**

*socio**-*

**politico**-**economico**-**]**, or

*ps*__Y__*chohistory**‘*

**ps**

__Y__*, for short, and ‘*

**chohistorical****’**__dialectics__**’ is our standard symbol for that**

__Y__

*knowledge**-*.

**field**It is conjectured here as a

*‘‘‘*, or

**complex unity**’’’**, of**

__dialectical__synthesis

__N__

**&**

**, of**

__L__

*logico**-*

**mathematical**-cognitive-__ps__

__Y__

__che__ological**science****&**

*physical***, including**

*science***as part of**

*human social science***in Marx’s sense:**

*the science of natural history*

__Y__

**=**

__q__

_{N}**.**

_{L}
The fuller development of

**is the intended target and focus of the work of**__Y__**Foundation**__Encyclopedia__**.**__Dialectica__
The connotations of this

However, it can be well-argued that the

To get back this crucial component of the Seldonian concept of

Then the new

viz. --

**encompass the connotative content of the earlier model’s algebraic terms**__Y____q___{n}**, and**_{r}__q___{n}**, and also**_{l}__q___{n}**, since**_{t}**, itself ‘proxying’**__L__**-- and as the operand of**__l__**, ‘proxying’**__N__**, in the formation of**__n__**-- implicitly connotes / "contains" the elements**__Y____r__**and****.**__t__However, it can be well-argued that the

*‘self-refl*of the earlier model’s**xive moment’**__e__**s****=****3****tep -- namely, its final algebraic term,**__s____q__**-- is missing from, this “abbreviated” model, which thus has a gaping ‘homeomorphic defect’ as a model of that earlier model.**_{nn}To get back this crucial component of the Seldonian concept of

**, while still retaining some of the second model’s succinctness, we can convert the***psychohistory***model back into a**__Triad__ic Seldon Function**model, still using the same «**__Dyad__ic Seldon Function**».***archÃ©*Then the new

**model’s**__dyad__ic**s****=****1****tep gives us Ockham’s**__s__**, and its**__dialectic__**s****=****2****tep gives the Seldonian ‘extention’, with**__s____Y__**=**__Y___{1 }**+**__Y__**,**_{2}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,

**, to an extent which has yet to be fully explored and appreciated by modern scholarship.**__dialectic__In the introduction to her book

**, Eleonore Stump wrote:**

__Dialectic and Its Place in the Development of Medieval Logic__
“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,

**, Cornell University Press [Ithaca:**__Dialectic and Its Place in the Development of Medieval Logic__**1989**], p.**1**].
In the second place, Ockham himself was a major
fourteenth century

*innovative*contributor to medieval scholastic**, principally via his**__dialectic__*circa***1323**treatise**, or**__Summa logicae__*“*[__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,

**., pp.***ibid***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

**to avoid being forced into formally contradictory [**__obligation__*“*possible”] assertions__im__**, given that the original assertion upheld by the respondent is a “possible” one.].***due to respondent logical error*However, Ockham’s work on

**was informed primarily by the**

__dialectics__**of Aristotle, specifically that of Aristotle’s**

__dialectics__**and**

__Topics__**, and thus only indirectly, not directly, by the**

__Rhetoric__**of Plato.**

__dialectics__
We therefore cannot infer, with any certainty, that
Ockham consciously intended his ‘ideo-taxonomy’ of

**to be anything like a Platonian «***the Sciences***», especially given Ockham’s “nominalist” views, rejecting Platonian and other hypostatizations or reifications of***arithmos eidetikos**“*.**universals**”
More likely, Ockham’s ‘ideo-«

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

__Encyclopedia Dialectica__**special terms deployed in the discourse above --**
«

**»***arithmos aisthetos*
«

**»***arithmos eidetikos*

*categorial*

*category**‘*

**cumulum**’

__dialectical__

*categorial progression presentations*

*homeomorphic defect*
[

**]***The**Human Phenome*

*immanent*

*immanent critique*
«

**»***monad*

*ontological category*

*ontology*

*psychohistory*

*psycho*

*historical*

__dialectics__

__qual__

*o**-*

*fractal*

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

__qual__

*o**-*

*Peanic*

*Seldon Functions**‘*or

**self**-**meta**-**monad**-**ization**’*‘*or

**self**-**meta**-**individual**-**ization**’*‘*

**self**-**meta**-**holon**-**ization**’
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