‘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 qx is in the space NQ,
then –
x2 = x [+] Delta-x, or qx2 = qx
[+] qxx
– 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 20th 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 ai and bi 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 aiei are multiples of the primary units [K.S.:
or “qualitative units”] and are represented geometrically by lengths ai along the respective axes, while A is
represented by a directed line segment in space whose projections on the axes
are the lengths ai. The same is true for the bi 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) [eiej]
=
-[ejei]
[K.S.: a behavior called “anti-commutativity”; assumes j ¹ i],
[eiei] = 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 ei.
…
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,
gi -- 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 qj & qk in NQ]:
[ qj [x]
qk = qj [+] qk [+] qk+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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updates regarding
F.E.D. content, please
see -- www.dialectics.info .
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‘poster-ized’ visualizations of many of our hypotheses -- specimens of ‘dialectical art’
-- see:
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¡ENJOY!
Regards,
Miguel
Detonacciones,
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];
Elected Member, F.E.D. General Council;
Participant, F.E.D. Special Council for Public Liaison;
Officer,
F.E.D. Office of Public Liaison.
