‘The Grassmann Family Phenome’. -- Part 12: Seldon’s Worldview Series.

 

‘The Grassmann Family Phenome’.

 

-- Part 12: Seldon’s Worldview Series.

 

 

 

 

 

Dear Reader,

 

 

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release, by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.

 

The twelfth release in this new such series is posted below [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

 

In this 12th installment, Seldon reviews the psychohistorical phenomenon of an early-modern variant of dialectic, one featuring synthesis-less multi-directional dualisms, but one which explicitly addressed a dialectic within mathematics, and that even encompassed an arithmetic/algebra that anticipated a key axiom plus a fundamental theorem of the Seldonian ‘first arithmetic/algebra for dialectical logic’ – the one its core ‘contra-Boolean’ theorem, the other an axiom that it shares with the later [Jevonian-]Boolean arithmetic/algebra for formal logic.

 

 

 

 

 

 

 

 

 

 

 

Seldon –

 

 

“The ‘Human-Phenome-in-general’ versus ‘Family Phenomes’.

 

In the Encyclopedia Dialectica parlance, ‘the human phenome’ names the counterpart of “the human genome” in the ‘‘‘complex unity’’’ or ‘‘‘dialectical synthesis’’’ that constitutes humanity, that characterizes the category of the human species.  The ‘human phenome’ names the totality of human ‘‘‘memes’’’ and of human “phenotypes”, the historically-cumulating cultural endowment of non-chromosomally-encoded, “acquired” – “learned” – characteristics that co-define the human species, as elaborated in the text-image below.”

 

“By a ‘family phenome’, we name those psychohistorical phenomena whereby certain memetic content, not [yet] extant in the human phenome of the human society surrounding a given human family – e.g., a prevenient memes-complex -- has arisen in that family, and has been passed on in that family, from parent(s) to child(children), from one generation to the next, across at least two generations, i.e., from at least from one parent to one child.”

 

“Our focus here is the psychohistorical, ‘psycho-artefact’ evidence of several interconnected proto-dialectical-mathematical memes reproduced across two generations of the Grassmann family, transmitted by the father, Justus Grassmann, to his children, brothers Hermann Grassmann and Robert Grassmann.”

 

 

“The Grassmanns’, Proto-Mathematical-Dialectical ‘Family Phenome’.”

 

 

“Hermann Grassmann’s discoveries.

Hermann Grassmann is famous, at least in mathematical/historical circles, for his development of the Grassmannian “hypercomplex” numbers, in the ‘ arithmetic for “n-dimensional geometry” ’ that he discovered, and the presentation of applications for which he pioneered.  The Grassmannian arithmetic/algebra is recently finding applications in “particle physics”, because of the comprehensively nilpotent nature of the Grassmann hypernumbers.”

 

 

“Robert Grassmann’ discoveries.

Robert Grassmann is less well-known than Hermann.  However, for us, his importance may be the greater of the two, because he was more involved in writings about the explicitly [proto-]dialectical aspects of the theory of mathematics, and of the specific new forms of mathematics – especially of mathematics for logics -- that he and his brother co-developed.”

 

“Robert Grassmann independently discovered the axiomatic core of a 0 & 1 logic-arithmetic and logic-algebra, a ‘Jevons-ish’ later variant of the algebraic logic now known as “Boolean algebra”, as a model of formal logic, and which, together with the Leibniz-discovered binary arithmetic per se, forms the conceptual and engineering foundation of today’s (0 & 1)-logic-and-arithmetic based ‘Boolean Computers’ [as opposed to, e.g., “analog computers”, and to (0 & 1 & [1|0])-based “quantum computers”, as proto-dialectical computers, etc.].”

 

“A similar (0 & 1)-based ‘Boolean logic-arithmetic’ variant was also discovered much earlier than Boole’s and Grassmann’s independent discoveries of such, by Leibniz, as part of his quest to develop his envisioned «Characteristic Universalis», or ‘universal mathematics’ [as a ‘character-language’, i.e., a ‘letter-language’ or algebra] of thought.”

 

“Moreover, Robert Grassmann also discovered an alternative arithmetic-and-algebra for logic, in contrast to his ‘Boolean-Jevonian arithmetic for logic’, the latter of which we have formulated as within a category of «aufheben» ‘supplementary opposition’ axioms-systems to the category of the Boolean axioms-systems variants.”

 

“Robert Grassmann’s contrasting ‘algebraic logic’ included a ‘Jevonian-Boolean-arithmetic’-echoing axiom of the Encyclopedia Dialectica _NQ ‘first arithmetic for dialectics’ [axiom 7], plus the fundamental ‘contra-Boolean’ theorem of the _NQ arithmetic, but as an axiom.  This is the only anticipation of our Encyclopedia Dialectica ‘mathematics for dialectics’ that is known to us as of this writing.”

 

 

 

“Justus Grassmann’s discoveries.

The following excerpts from the writings of Justus Grassmann already express subtle germs of both Hermann Grassmann’s ‘arithmetic for abstract, n-dimensional geometry’ and the view of “dialectic” which both Hermann and Robert shared –

“…[Hermann] Grassmann stated concerning geometrical multiplication that the idea came from his father and in a footnote referred the reader to two of his father’s books, “Raumlehre Theil II, p. 194” and “Trigonometrie, p. 10.”

“Justus Günther Grassmann made the following statements in a footnote in the first of these books:

     The rectangle itself is the true geometrical product, and the construction of it, as it appears in §53, is really geometrical multiplication.  If the concept of multiplication is taken in its purest and most general sense, then one comes to view a construction as something constructed from elements already constructed and in fact constructed in the same way.  Thus multiplication is only a construction of a higher power.  In geometry the point is the original “producing” element”; from it through construction the line emerges; if we take as the basis of a new construction the finite line constructed from the point, and if we treat it in the same manner as we formerly treated the point, then the rectangle emerges.  Just as the line comes from the point, so the rectangle comes from the line.

 

The situation is the same in arithmetic.  In this case the unit is the original “producing” element.  The unit must simply be viewed as given.  From this, through counting (arithmetic construction), number appears [K.S.: As for the ancients, a single «monad» does not an «arithmos» make.].  If this number is taken as the basis of a new counting (setting it in place of the unit), then arises the arithmetic connection to multiplication, which is nothing else but a number of a higher order, a number of which the unit is also a number.  Thus it could perhaps be said that the rectangle is a (finite) line in which, in place of the “producing” points, a finite line has been substituted.  Thus the following laws may be suggested: A rectangle is the geometrical product of its base and height and this product behaves in the same way as the arithmetic product.    

 

In the second book (the Trigonometrie) the elder Grassmann wrote in a footnote:

     If the concept of product is taken in its most pure and general sense, then it is viewed as the result of a synthesis in which an element (produced from an earlier synthesis) is set in the place of the original element and treated in the same way.  The product must arise likewise from what resulted from the first synthesis, just as this arose from the original elements.  In arithmetic the unit [K.S.: «monad»] is the element, counting is the synthesis, and the result is number [K.S.: «arithmos»].  If this number, as the result of the first synthesis, is set in the place of the unit, and treated in the same way (i.e., counted), then the arithmetic product appears, and this may be viewed as a number of a higher order than a number of which unity is already a number. 

In geometry the point is the element, the synthesis is the motion of the point in some direction, and the result, the path of the point, is the line.  If this line, produced by the first synthesis, is set in the place of the point and treated in the same way (i.e., moved in some other direction), then a surface is produced from the path of the line.  This is a true geometrical product of two linear factors and appears in the first place as a rectangle, insofar as  the first direction shares nothing with the second.  If the surface is set in place of the point, a geometrical solid is produced as the product of three factors.  This is as far as one can go in geometry since space [K.S.: empirical physical space] has only three dimensions; no such limitation appears in arithmetic. ...”

[Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Dover, 1994, pp. 58-59].”

 

“These outstanding passages, exemplars, among other logical principles’, of the principle of reasoning by analogy [‘analogic’], do suffer from some – relatively minor – imprecisions and incoherencies, the most consequential of which is the implicit and unnecessary restriction of conceptual, abstract, mathematical space to the apparent restrictions of our experiential physical space.”

 

 

“Hermann Grassmann’s n-dimensional ‘Arithmetic for Abstract Geometry’ as an Actualization of Justus Grassmann’s Insights into ‘Generalized Multiplication/Product-tion’, & as a precursor to Key Features of _NQ.

 

The son surpassed the father in that regard.   Hermann Grassmann’s ‘arithmetic for geometry’ is ‘n-dimensional’.  In a simplified* account of it, we can describe it as expanding from 0-dimensional to n-dimensional, for any finite n > 0, by means of the operations of ‘geometrical multiplication’ within it, 0-dimensional points ‘produc(t)-ing’ 1-dimensional line-segments, 1-dimensional line-segments ‘produc(t)-ing’ 2-dimensional, finite ‘plane-segments’, 2-dimensional ‘plane-segments’ produc(t)-ing 3-dimensional, finite ‘solid-segments’, 3-dimensional ‘solid-segments’ ‘‘‘producing’’’ higher-dimensional, n > 3, finite ‘hyper-solid segments’, and so on.”

 

 

“Comparisons: Hermann Grassmann’s ‘Geometric Arithmetic’, ‘Boolean Arithmetic’, and the _NQ ‘Arithmetic for Dialectics’.

 

But note the characteristics which this ‘geometrical products-expandable/ “multiplications-expandable”, ‘‘‘dimensionally [self-]expanding’’’ space share with the spaces of the _NQ ‘first arithmetics for dialectic’ category-of-variants.”

 

“First of all, note the [self-]reflexivity and 'ideo-«autokinesis»' character of this ‘geometrical arithmetic’, which we had noted early on as a characteristic of Marxian [and of Hegelian] dialectic, and which is exhibited explicitly in the _NQ dialectical arithmetics.”

 

“This observation must be carefully qualified, if it is to retain its truth-value, because of the [quadratic] “nilpotent” character of Hermann Grassmann’s ‘arithmetic for geometry’, a ‘quadratic nilpotency’ [i.e., a presence of "proper square-roots of zero"; square-roots of 0 which are not themselves also 0] which is absent -- or, at least, not explicit -- in Justus Grassmann’s descriptions of ‘geometrical product-tion’, quoted above.”

 

“In a simplified* account of Hermann Grassmann’s realization of his father’s insights into ‘multiplication-in-general’, and into “geometric multiplication” in particular, two 0-D, distinct points “multiply” each other by filling-in the finite line-segment spanning between them, thus “producing” that 1-D line-segment as the “product” of the two points.  But a single point, so “multiplying” itself, produces no gain, and so this ‘self-product’ is a ‘nil product’; is equated to zero, is “nil” –

If P1 ¹ P2, then P1 x P2 = L.

But, e.g., P1 x P1 = P1² = 0 [“nilpotency”].”

 

“Self-similarly, two distinct but parallel [‘||’] line segments “multiply” each other by filling-in the finite ‘plane-segment’ 2-D Surface, spanning between them, thus “producing” that ‘plane-segment’ as the “product” of the two line-segments.  But a single line-segment, so “multiplying” itself, “produces” nothing more, nothing new, and so this ‘self-product’ is a ‘nil product’ –

If L1 ¹ L2, and L1 || L2, then L1 x L2 = S.

But, e.g., L1 x L1 = L1² = 0 [“nilpotency”].”

 

“Likewise, two distinct, but ‘‘‘parallel’’’ 2-D surfaces “multiply” each other by filling-in the finite ‘solid-segment’ 3-D Block, spanning between them, thus “producing” that ‘block-segment’ as the “product” of the two ‘plane-segments’.  But a single ‘plane-segment’, so “self-multiplying”, “produces” nil, and so this ‘self-product’ is a ‘nil product’ –

If S1 ¹ S2, and S1 ‘||’ S2, then S1 x S2 = B.

But, e.g., S1 x S1 = S1² = 0 [degree 2 “nilpotency”]

– and so on, into the production of n-dimensional abstract-geometrical idea-objects of ideal dimensionality n > 3, for any such finite value of that n.”

 

“On the contrary, in the _NQ ‘arithmetic for modeling dialectics’, if x or q_x is in the space _NQ, then –

x² = x [+] Delta-x, or q_x² = q_x [+] q_xx

–     which states our ‘fundamental law of dialectical thought’, or ‘fundamental [in]equation of dialectics’.  It asserts the ‘expanded reproduction of ideas’, by idea self-action/self-reflexion’, opposing but also «aufheben»-containing, or ‘conserving/negating’, as 

x² ~=  x, 

Boole’s “fundamental law of [formal-logical] thought”, or “law 

of duality” –

x² = x

given that x is in the set {0, 1}, or, more precisely, in the set {0/1, 1/1, 0/0, 1/0}.”

 

“This Boolean “fundamental law of thought” asserts the immediate reduction of [degree 2] ‘logical nonlinearity’ to [degree 1; x¹ = x] ‘logical linearity’, the ‘simple reproduction of ideas’, per which a human mind’s reflection of an idea upon itself, e.g., as an immanent critique, or self-critique, of that idea, yields nothing new.”

 

“Note also that both the space of Hermann Grassmann’s ‘arithmetic for abstract geometry’, and the space of, the abstract geometrical model of, the standard _NQ ‘arithmetic for dialectics’, share a feature which is exceedingly rare in all other mathematics today: these two spaces are not of any fixed dimensionality.  Their dimensionality is dynamical – is ‘self’-expanding – in a dimensional dynamism and expansion which is driven by the operations, the generalized multiplications, the interactions, which can go on within them, of the operations or operators or ‘hypernumbers’/‘meta-numbers’ that are defined within those spaces. 

 

Note that it is not simply that these spaces [self-]expand purely-quantitatively, i.e., that their n-dimensional volume increases, for some finite, fixed dimensionality, D = n.  When such a space [self-]expands, from, e.g., 2-dimensional to 3-dimensional, its 2-dimensional ‘“volume”’ [“area”], a squared value, measured in, say, square[d] centimeters, cm.², becomes a 3-dimensional “volume”, a cubed value, measured, say, in cubic centimeters, cm.³.  The ‘unit square units’ that make up a 2-D ‘‘‘volume’’’ are dimensionally, qualitatively different from the ‘unit cube units’ that make up a 3-D “volume”.  Thus the [self-]expansion of these spaces constitutes primarily a qualitative, not a quantitative, kind of change.

 

In other words, these mentally-constructible, mentally-perceivable abstract spaces exhibit the ‘ideo-phenomenon’ of ‘operator prevalence’, and, therefore, also exhibit ‘arithmetical and spatial ‘non-closure’, or ‘openness’, because their internal operational interactions drive the escalation of their dimensionalities.”

Morris Kline, in his monumental 3-volume history of mathematics, spanning in coverage from remote antiquity to the late 20^th century, described Hermann Grassmann’s ‘arithmetic for abstract geometry’ as follows –

While Hamilton was developing his quaternions, another mathematician, Hermann Günther Grassmann (1809-77), who showed no talent for mathematics as a youth and who had no university education in mathematics but later became a teacher of mathematics in the gymnasium (high school) at Stettin, Germany, as well as an authority on Sanskrit, was developing an even more audacious generalization of complex numbers.  Grassmann had his ideas before Hamilton but did not publish until 1844, one year after Hamilton announced his discovery of quaternions.  In that year he [Grassmann] published his Die lineale Ausdenungslehre (The Calculus of Extension). …

Though Grassmann’s exposition was almost inextricably bound up with geometrical ideas – he was in fact concerned with n-dimensional geometry – we shall extract the algebraic notions… .  His basic notion, which he called an extensive quantity (extensive Grösse), is one type of hypernumber with n components.  To study his ideas we shall discuss the case n = 3.

Consider two hypernumbers

A = a₁e₁ + a₂e₂ + a₃e₃  and  B = b₁e₁ + b₂e₂ + b₃e₃,

Where the a_i and b_i are real numbers [K.S.: elements of the set R] and where e₁, e₂, and e₃ are primary or qualitative units represented geometrically by direct[ed] line segments of unit length drawn from [K.S.: extending from] a common origin so as to determine a right-handed orthogonal system of axes.  The a_ie_i are multiples of the primary units [K.S.: or “qualitative units”] and are represented geometrically by lengths a_i along the respective axes, while A is represented by a directed line segment in space whose projections on the axes are the lengths a_i.  The same is true for the b_i and B.  Grassmann called these directed line segments or line-vectors Strecke.

The addition and subtraction of these hypernumbers are defined by

A ± B  =  (a₁ ± b₁)e₁ + (a₂ ± b₂)e₂ + (a₃ ± b₃)e₃.

Grassmann introduced two kinds of multiplications, the inner product and the outer product. … For the outer product

(9) [e_ie_j]  =  -[e_je_i] [K.S.: a behavior called “anti-commutativity”; assumes j ¹ i],

[e_ie_i]  =  0 [K.S.: a behavior called [degree 2] “nilpotency”].

These brackets are called units of the second order and are not reduced by Grassmann (whereas Hamilton does) to units of the first order, that is, to the e_i. …

With the aid of the outer product rule (9) the outer product P of the hypernumbers A and B can be expressed as follows:  

P  =  [AB]  = 

(a₂b₃ - a₃b₂)[e₂e₃] +

(a₃b₁ - a₁b₃)[e₃e₁] +

(a₁b₂ - a₂b₁)[e₁e₂].  

This product is a hypernumber of the second order and is expressed in terms of independent units [K.S.: i.e., in terms of new, higher “qualitative units” which are units of higher intrinsic dimensionality than are the one-dimensional, directed unit-length line-segment “qualitative units” of “first order”, e₁, e₂, and e₃, from which those higher units are created, by Grassmannian multiplication] of the second order.

[Morris Kline, Mathematical Thought from Ancient to Modern Times, Volume 2, Oxford U. Press, 1972, pp. 782-783; Greek algebraic letter substitution by corresponding bolded English letters, and bold italic and underscored emphases, added by K.S.].

 

“In the case of our ‘abstract-geometrical’ representation of the _NQ arithmetic, its ‘‘‘analytical geometry’’’, we can start with an arbitrary but 'unpartitionable', unit-length line-segment, aligned in any arbitrary direction, as the 1-dimensional geometry representing the _NQ-arithmetical ‘meta-number’ q₁ alone.

 

“In the _NQ space, unlike Grassmann-space, self-multiplication is not “nilpotent”, but, on the contrary, is ‘hyper-potent’, or ‘meta-potent’, as we shall soon see.”

 

“If, next, that q₁ value operates upon itself, self-reflexively, i.e., ‘‘‘self-multiplies’’’, or undergoes ‘self-involution’, then the _NQ space expands dimensionally, from having been 1-dimensional, to being 2-dimensional, as the product or result of this self-multiplication, in the sense that this self-multiplication «aufheben»-conserves the q₁ unit line-segment, but also accrues a new unit-length line-segment, geometrically representing q₂, or q₁₊₁, perpendicular to/independent of the unit-length line-segment geometrically representing q₁.  In _NQ-arithmetical formulation, this process of the _NQ space’s expansion in its dimensionality is expressible as follows, per the _NQ axioms [especially its axioms 7, 8 and 9] –

 

q₁ [x] q₁  =  q₁²  =  

q₁ [+] q₁ [+] q₁₊₁ 

=  q₁ [+] q₂.”**

 

 “If we interpret this abstract _NQ geometry and arithmetic so as to mathematically model the more concrete example of ‘the Dialectic of Nature’, so that we assign the ‘cosmo-ontological’ category of ‘pre-atomic’ “particle” units to q₁, then q₂ can be assigned by solution to the results of “Big Bang nucleosynthesis”, as the ‘cosmo-ontological’ category of, e.g., Hydrogen and Helium atom units, which are qualitatively, ontologically different units in relation to pre-atomic “particle” units, even though typical atom units are each made up out of a heterogeneous multiplicity of ‘pre-atomic’ “particle” units – e.g., electrons, protons, and neutrons.”

 

“The relation of mutual perpendicularity or orthogonality between the q₁ unit line-segment and the q₂ unit line-segment represents that qualitative, ‘cosmo-ontological’ difference geometrically.”

 

“If, next, the line-segment representing the ‘meta-number’ q₂ ‘‘‘multiplies’’’/«aufheben»-operates with the line-segment representing the ‘meta-number’ q₁, then the abstract space _NQ expands dimensionally, from having been 2-dimensional, to being 3-dimensional, as the product or result of this mutual-multiplication, in the sense that this mutual-multiplication «aufheben»-conserves the q₁ and the q₂ unit line-segments, but also cumulatively accrues a new unit-length line-segment, geometrically representing q₃, or q₂₊₁, also perpendicular to/independent of both of those two, already mutually-perpendicular unit-length line-segments geometrically representing q₁ and q₂.  In _NQ-arithmetical formulation, this process of the _NQ space’s further [self-]expansion of its dimensionality is expressible as follows, per the _NQ axioms –

 

q₂ [x] q₁  =  q₂ [+] q₁ [+] q₂₊₁ 

=  q₁ [+] q₂ [+] q₃.”

 

“If we interpret this abstract _NQ geometry and arithmetic so as to model the more concrete example of ‘the dialectic of Nature’, so that we assign the ‘cosmo-ontological’ category of ‘pre-atomic’ “particle” units to q₁, and q₂ to the ‘cosmo-ontological’ category of atom units, then we obtain the «aufheben»-conservation of the q₁ and q₂ ‘‘‘axes’’’, together with the accrual of a third ‘‘‘axis’’’, assigned to q₃, also perpendicular to both of the q₁ and q₂ ‘‘‘axes’’’, which solves as the ‘cosmo-ontological’ category of those units which combine, into a ‘‘‘complex unity’’’, atom units [e.g., Helium atoms] and/with “particle” units [e.g., stellar-core ionized Hydrogen atoms, or “naked” proton “particle” units], e.g., those units known as ‘‘‘first generation’’’ star units, which are qualitatively, ontologically different units in relation to both pre-atomic “particle” units and atom units, despite the fact that each, e.g., star unit is made up out of both “particle” units and atom units.”

 

“If, next, that q₂ value operates upon itself, self-reflexively, i.e., ‘‘‘self-multiplies’’’, or undergoes ‘self-involution’, then the space _NQ space expands dimensionally, from having been 3-dimensional, to being 4-dimensional, as the product or result of this self-multiplication, in the sense that this self-multiplication «aufheben»-conserves the q₂ unit line-segment, but also accrues a new unit-length line-segment, geometrically representing q₄, or q₂₊₂, perpendicular to/‘‘‘linearly-independent’’’ of all three of the unit-length line-segments geometrically representing the _NQ ‘meta-numbers’ q₁, q₂, and q₃.  In _NQ-arithmetical formulation, this process of the _NQ space’s expansion in its dimensionality is expressible as follows, per the _NQ axioms –

 

q₂ [x] q₂  =  q₂²  =  

q₂ [+] q₂ [+] q₂₊₂ 

=  q₂ [+] q₄.”

 

 “If we interpret this abstract _NQ geometry and arithmetic so as to model the ‘the dialectic of Nature’, then q₄ assigns, per our solution, to the ‘cosmo-ontological’ category of molecule units, e.g., in the 'intra-galactic, inter-stellar' "molecular clouds" from which new stellar/planetary systems are born, which are qualitatively, ontologically different units in relation to pre-atomic “particle” units, assigned to q₁, atom units, assigned to q₂, and, e.g., ‘‘‘first-generation’’’ star units, etc., assigned to q₃, even though each typical molecule unit is made up out of a heterogeneous multiplicity of “atom” units.”

 

“To this stage, the ontologies-content of the ‘historical dialectic of Nature’ can be summarily represented as the following ‘non-amalgamative sum’ [Plato: «asumbletoi» sum] and ‘qualitative superposition’ of ‘cosmo-ontological’ categories  –

 

“particles” <+> atoms <+> first stars <+> molecules.

 

– which also represents an “applied” version of the 4-dimensional stage of the self-expansion/‘self-involution’ of the _NQ abstract space/geometry.”

 

“In our dialectical interpretation, ontological category 1, named “particles”, is the ‘first thesis category’ of this dialectic, category 2, named atoms the ‘first antithesis category’ of this dialectic, category 3, named first stars the ‘first synthesis category’ of this dialectic, and category 4, named molecules the ‘second antithesis category’, of this historical dialectic of Nature dialectical-mathematical model.”

 

 

“The Grassmannian-Schleiermacherian “Dialectic”.

 

The kind of “dialectic” that forms the later core of the ‘Grassmann family phenome’ is one which we have characterized as a ‘synthesis-less dialectic’, a “dialectic” that never escapes or resolves its prominent multiplicities of ‘dualisms’, categorial oppositions, or “contrasts”, and which, thus, is perhaps not a ‘‘‘dialectic’’’, in our terms, at all.”

 

“If there is any semblance of dialectical synthesis in this Grassmannian “dialectic”, it is that of the original unity of a given category, whose internal, dualistic “contrasts” or oppositions are brought into 'explicitude' via the Grassmannian “dialectical” method of analysis of that category.”

 

“The best known example of Grassmannian “dialectical” method is that of Robert and Hermann’s analysis of the category of the totality of “mathematics” itself, which is explicated in Hermann Grassmann’s famous 1844 book, and exposition of “A New Branch of Mathematics”, whose name, in English translation, is “Linear Extension Theory” [«Ausdenungslehre»].  In that book, in its introduction, Hermann wrote --

The principal division of the sciences is into the real and the formal.  The real represent the existent in thought as existing independently of thought, and their truth consists in the correspondence of the thought with the existent.  The formal on the other hand have as their object what has been produced by thought alone, and their truth consists in the correspondence between the thought processes themselves. …

 

The formal sciences treat either the general laws of thought or the particular as established by means of thought, the former being dialectic…, the latter, pure mathematics. 

 

The contrast between the general and the particular thus produces the division of the formal sciences into dialectics and mathematics.  The first is a philosophical science, since it seeks the unity in all thought, while mathematics has the opposite orientation in that it regards each individual thought as a particular. …

 

Each particular existent brought to be by thought…can come about in one of two ways, either through a simple act of generation or through a twofold act of placement and conjunction.  That arising in the first way is the continuous form, or magnitude in the narrow sense, while that arising in the second way is the discrete or conjunctive form. …

 

Each particular existent becomes such through the concept of the different, whereby it is coordinated with other particular existents, and through this with the equal. Whereby it is subordinated to the same universal with other [K.S: mathematical, particular] existents.  That arising from the equal we may call the algebraic form, that from the different as the combinatorial form. …

 

From the interaction of these two oppositions, the former of which is related to the type of generation, the latter to the elements of generation, arise the four species of [K.S.: mathematical] form and the corresponding branches of the [K.S.: mathematical] theory of forms; thus the discrete form separates into number and combination [K.S.: combinatorics].  Number is the algebraic discrete form, that is, it is the unification of those established as equal.  Combination is the combinatorial discrete form; that is, it is the unification of those established as different.  The [K.S.: mathematical] sciences of the discrete are number theory and combination theory (relation theory). …

 

In precisely the same way, the continuous form or magnitude separates into the algebraic continuous form, or intensive magnitude and the combinatorial continuous form or extensive magnitude.  The intensive magnitude is thus that arising through the generation of equals, the extensive magnitude or extension that arising through generation of the different.  As variable magnitudes the former constitute the foundation of function theory, that is differential and integral calculus, the latter the foundation of extension theory [K.S.: i.e., of the “new branch of mathematics” introduced in this very text by Hermann Grassmann, as a replacement for the empirically-constrained – the “real existent” constrained – traditional mathematical field of, e.g. Euclidean, geometry].

 

[From Hermann Grassmann, A New Branch of Mathematics, translated by Lloyd C. Kannenberg, Open Court, 1995, pp. 23-27].”

 

“A better idea of what Hermann is getting at in the discourse extracted, in his [obsolete] attempt at a “dialectical” classification, into four contrasting/-opposing “species”, of the totality and unity of the entire field of mathematics, can be gleaned from a kind of table which is a typical device of the Grassmannian-Schleiermacherian “dialectic” –

	Discrete Forms	Continuous Forms
Algebraic Forms	Number Theory	Function Theory [differential and integral calculus]
Combinatorial Forms	Combinatorics	Linear Extension Theory [Hermann Grassmann’s “New Branch of Mathematics”]

-- The ‘content-structure’ instanced above is called, in this tradition of “dialectic”, a “positive double relative opposition” and a “chiasma”.”

 

“Note that, per this depiction, the categorial unity or unit that is the [«Genos»] category of MATHEMATICS divides, per our reckoning, into six pairs of dualisms of opposing “species”.”

 

“Two pairs rest in the vertical direction for the “discrete” and “continuous” columns, two in the horizontal direction of the “algebraic” and “combinatorial” rows, and two more in the diagonal directions, upper left to lower right, and lower left to upper right.”

 

“Of this “dialectic” content-structure, Robert Grassmann wrote: “A concept and its contrasting term are generated from a unity by an opposition.  The mind posits oppositions within unities. … A unity can be transformed into two contrasting terms when the two sides of two oppositions respectively are present in every one of the two oppositions, but connected in the first, and opposed in the second, in short, connected crosswise.  This amounts to saying that the two contrasting terms in the unity are generated by a chiasma of two oppositions, by connecting two oppositions crosswise.”

[Hans-Joachim Petsche, Hermann Graßmann: Biography, Birkhäuser, 2009, p. 154].

 

Friedrich Schleiermacher stated his view of the relationship of mathematics and dialectics, as summarized, as follows –

“…mathematics comes onto the scene once thinking is treated technically and in the light of confusions that have arisen in it, so as “to order thinking in a well-defined manner.” 

 

Moreover, “this ordering is the fruit of mathematics, for only at this point can dialectical procedure begin, and without mathematics it is difficult to awaken consciousness. 

 

Plato’s claim simply goes to the natural succession of knowledge in every individual person.  In every instance of knowing, there is only so much true knowing and so much knowing permeated in accordance with its idea as dialectic and mathematics are present in it – that is to say, dialectic to the degree that it appertains to speculative form and mathematics to the degree that it appertains to empirical form.

 

The two do not permit of being separated, if we do not want to lose knowing itself.  The character of knowing is grounded only in their joining and ever stronger interpenetration.  This last statement is the general canon of all sciences, if people’s will is to value and advance them”.  Thus does Schleiermacher close his 1822 lectures on dialectic.”

[Friedrich Schleiermacher, Dialectic, translated by Terrence Tice, Scholars Press, 1966, pp. 73-74n.].”  

 

 

“Robert Grassmann’s Version of the _NQ ‘Arithmetic for Dialectical Logic’.

We have two sources to cite on that one of Robert Grassmann’s algebraic logics that anticipated the _NQ ‘arithmetic/algebra for dialectical logic’.”

 

“First, from the paper by Volker Peckhaus in the anthology on Hermann Grassmann’s work, edited by Hans-Joachim Petsche, et al. –

 

The different results of connecting pins [K.S.: “«Stifte»”; values] with themselves give the criteria for distinguishing between special parts of the theory… .

 

The “theory of concepts or logic” (“Gegriffslehre oder Logik”) is the first part, “the most simple and, at the same time, the most inward part”, as Grassmann calls it… .  Inner joining e + e = e, and inner weaving 

ee = e are valid [K.S.: Thus, Robert Grassmann’s variant of Boolean-Jevonian ‘formal-logic arithmetic’].

 

In the “theory of binding or theory of combinations” (“Bindelehre oder Combinationslehre”) as the second part of the theory of [K.S.: thought-]forms, inner joining e + e = e and outer weaving ee ¹ e are valid… . [K.S.: Thus, Robert Grassmann’s variant of the _NQ ‘contra-Boolean dialectical-logic arithmetic’].

 

[from “Robert and Hermann Grassmann’s influence on the history of formal logic”, in Hermann Graßmann From Past to Future: Graßmann’s Work in Context, edited by Petsche, Lewis, Liesen and Rush, Birkhauser: 2011, p. 224].”

 

Robert Grassmann’s description, above, of his _NQ-like “Bindelehre” arithmetic/algebra as belonging to the “theory of combinations” was insightful, from our point of view, because we interpret the _NQ ‘arithmetic/algebra for dialectical logic’ as modeling ‘categorial combinatorics’, proxying ‘monadic combinatorics’, or ‘unitic combinatorics’.

“Second, from the anthology edited by Gert Schubring –

The work of the two brothers was closely linked; but it was Robert who explicitly applied Hermann’s Ausdenungslehre to logic.  A group of five little books collectively called Die Formenlehre oder Mathematik appeared in 1872.  In this remarkable work, written ‘in pure German’ [K.S: note the German bourgeois nationalist psychohistorical «mentalité» in evidence thereby] with all foreign words avoided, he went well beyond Hermann in generality.  To start, Formenlehre laid out the laws of ‘strong scientific thought’ of ‘Groesen’(sic, following his avoidance of ‘ss’) denoting any ‘object of thought’; each could be composed as a sum of basic ‘pegs’ (‘Stifte’) e – that is, the expression…with the coefficients set to unity.  He admitted, Hermann-style, two means of ‘connection’ between pegs, ‘inner’ and ‘outer’, symbolized by ‘+’ and ‘´’ [K.S.: respectively].  But then he defined four special kinds of Formenlehre, of which Hermann’s Ausdenungslehre was only an example of the last.

 

The members of the quartet were distinguished by the basic laws which their pegs obeyed, under suitable interpretations of them and their means of connection --

‘Begriffslehre’: [K.S.: formal] logic: e + e = e, e ´ e = e [K.S.: Booleanish, with the Jevonian modifications] …

‘Bindelehre’, theory of combinations: e + e = e, e ´ e ¹ e [_NQ-like, E.D. dialectical logic] …

‘Zahlenlehre’: arithmetic: e + e ¹ e, e ´ e = e [ K.S.: e.g., 1 x 1 = 1]…

‘Ausenlehre’ (sic): "exterior" objects: e + e ¹ e, e ´ e ¹ e …

 

The [K.S.: formal] logic was presented in the first book of the succeeding quartet, as ‘the simplest and also most central’ kind of Formenlehre, in 43 pages and under the title Die Begriffslehre oder Logik.  The three parts covered, in turn, the development of concepts, judgements and deductions.  The theory itself looks like Boole’s in much of its contents; but the latest reference in the historical preface is 1825, and he seems not to have known either of Boole or of Boole’s first commentator (in 1863), Stanley Jevons.

 

… to have achieved so much in apparent isolation, and within the conception of Formenlehre, which had no rivals for generality at that time, is a fine achievement.  It is well overdue for a detailed study.

 

…the quantity and range of Robert’s production is amazing, surpassing even Hermann’s.  For just one example, the revived version of the 1872 booklets appeared as the book Die Logik und die anderen logischen Wissenschaften (1890) of around 220 pages; it was reprinted as Die Logik ten years later.  In this version logic was the ‘lower analytic’ of a quartet of ‘logical sciences’ of which Bindelehre was its lower synthetic companion.  But this volume was the second half of the second Book (Denklehre) of the second part of the first section of his 10-section 5000-page Das Gebaeude des Wissens [The Edifice of Knowledge] (1882-1899), which otherwise handled a companion quartet of mathematical sciences (including the Ausdenungslehre as lower synthetic) and elsewhere treated ethics, physics, chemistry, animal and plant physiology, religion, theology, war sciences and technology.”

 

[from “Where Does Grassmann Fit in the History of Logic”, in Hermann Günther Graßmann (1809-1877): Visionary Mathematician, Scientist and Neohumanist Scholar, Gert Schubring, editor, Kluwer Academic Publishers, 1996, pp. 212-214; emphases added by K.S.].”

 

“Note that the “quartet” of “special kinds of Formenlehre”, listed above, form their own Grassmannian-Schleiermacherian “dialectic”, i.e., another “positive double reverse opposition” –

	“inner connection”	“outer connection”
‘Additive Equality’	e + e = e, e ´ e = e Begriffslehre [» Boole]	e + e = e, e ´ e ¹ e Bindelehre [» NQ ]
‘Additive Inequality’	e + e ¹ e, e ´ e = e Zahlenlehre [» Z]	e + e ¹ e, e ´ e ¹ e Ausenlehre [» Hermann’s]

 

– or “chiasma”, encompassing six opposing [‘~’] or “contrasting” pairs –

 

2 pairs of vertical/downward oppositions, 

namely [e + e = e] ~ [e + e ¹ e] 

for the “inner connection” column, and, once again, the  

[e + e = e] ~ [e + e ¹ e] opposition 

for the “outer connection” column, and;

 

2 pairs of horizontal/rightward oppositions, 

namely [e ´ e = e] ~ [e ´ e ¹ e]

for the “Additive Equality” row, and, once again, 

the [e ´ e = e] ~ [e ´ e ¹ e] opposition for the “Additive Inequality” row, and;

 

 

2 pairs of diagonal/rightward oppositions, namely

the [e + e = e, e ´ e = e] ~ [e + e ¹ e, e ´ e ¹ e] opposition for the Upper Left/Lower Right downward diagonal, and;

the [e + e ¹ e, e ´ e = e]  ~ [e + e = e, e ´ e ¹ e] opposition for the Lower Left/Upper Right upward diagonal.”

 

 

 

“We would not have chosen the Grassmanns – German nationalists, victims of a ‘Christianoid’ ideology, dualistic “dialecticians”, and Constitutional Monarchists – to be our progenitors in the discovery of 'the mathematics of dialectics' and of 'the dialectic of mathematics'.

However, we also know well the study of Psychohistory, which teaches us that breakthroughs in human “universal labor” [Marx] can, and quite often do, first irrupt in the most unexpected places!”

 

 

“*[This ‘‘‘simplified’’’ version of Hermann Grassmann’s ‘arithmetic for abstract geometry’, averts mention, e.g., of the “oriented” and therefore “signed” – ‘+’ or ‘-’ – nature of the geometric idea-objects of various dimensionality that interpret the Grassmannian “hypernumbers” – the “qualitative units” – of Hermann’s ‘geometrical arithmetic’, as well as the reductive tendency to derive all of the geometrical objects/“qualitative arithmetical units” of higher intrinsic dimensionality from the non-oriented geometrical points, g_i -- i.e., the geometric objects/“qualitative arithmetical units” of intrinsic dimensionality D = 0 -- e.g., to define:

 

 

g₁ x g₂ as a 1-D line-segment,

 

g₁ x g₂ x g₃ as a 2-D triangle, and

 

g₁ x g₂ x g₃ x g₄ as a 3-D tetrahedron.],

so that, as Michael J. Crowe wrote [ibid., p. 72], "The product of N such [K.S.: idea-]elements was considered to be an [K.S.: idea-]entity of the Nth order [K.S.: and of dimensionality D = N-1]".”

 

“**[For this example, we are actually applying a different, commutative product axiom variant of the _NQ arithmetic than the one specified in the core axioms’ ‘text-image’, above [for a product named 'the triple-conservation meta-genealogical «aufheben» evolute product of categories'.]. –

 

(§9’) [for all j & k in N; for all q_j & q_k in _NQ]:

           

[ q_j [x] q_k = q_j [+] q_k [+] q_k+j ].  

 

Bibliography:  

On the Grassmann Family's Intellectual Legacy --

Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Dover, 1994, pp. 54-96.

Hermann Grassmann, A New Branch of Mathematics, translated by Lloyd C. Kannenberg, Open Court, 1995.

Hermann Grassmann, Extension Theory, translated by Lloyd C. Kannenberg, American Mathematical Society, 2000.

Hermann Grassmann, Rig-Veda (in German), Kessinger Publishing.  

Morris Kline, Mathematical Thought from Ancient to Modern Times, Volume 2, Oxford U. Press, 1972, pp. 782-785.

Giuseppe Peano, Geometric Calculus: According to the Ausdenungslehre of H. Grassmann, translated by Lloyd C. Kannenberg, Birkhäuser, 2000.

Hans-Joachim Petsche, Hermann Graßmann: Biography, Birkhäuser, 2009.

Hans-Joachim Petsche et al., editors, Hermann Graßmann From Past to Future: Graßmann’s Work in Context, Birkhauser, 2011.

Hans-Joachim Petsche et al., editors, Hermann Graßmann: Roots and Traces, Birkhäuser, 2009.

Friedrich Schleiermacher, Dialectic, translated by Terrence Tice, Scholars Press, 1966.

Gert Schubring, editor, Hermann Günther Graßmann (1809-1877): Visionary Mathematician, Scientist and Neohumanist Scholar, Kluwer Academic Publishers, 1996.

 

 

 

 

 

 

 

 

 

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Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

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