Ok, I have had a another, and longer look at your work. [Unfortuately, I know very little about art!]
Anyway, I think your work is excellent. I like your use of colour and composition, and the variety of your subjects.
Results 21 to 40 of 547
Ok, I have had a another, and longer look at your work. [Unfortuately, I know very little about art!]
Anyway, I think your work is excellent. I like your use of colour and composition, and the variety of your subjects.
Thanks -- I soak up compliments like a paper towel.... Also please feel free to comment on any of the *substance* of the framework-type compositions that deal with abstracted concepts.
RevLeft Sciences Forum Participants,
FYI:
The FIRST of the promised "Postlude Letters", entitled --
Platonian Dialectic: Statical Versus Dynamical
-- an excerpt from the new book on the Foundation Encyclopedia Dialectica [F.E.D.] [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] "Dialectical Meta-Numbers", which are used to construct the 17-Symbol dialectical-mathematical model of cosmological natural history, the F.E.D. "Dialectic of Nature" Equation -- was recently posted at:
http://www.dialectics.org/dialectics/Welcome.html
http://www.dialectics.org/dialectics...spondence.html
http://www.dialectics.org/dialectics...,22FEB2011.pdf
Regards,
Miguel
Last edited by Miguel Detonnaciones; 8th August 2011 at 15:38.
RevLeft Science Forum Participants,
FYI:
The SECOND of the promised "Postlude Letters", entitled --
The Stellar Paradigm and the Somatic Paradigm of Dialectic
-- an excerpt from the new book on the F.E.D. "Dialectical Meta-Numbers", and on their "contra-Boolean algebra", used to formulate the new 17-symbol "Theory of Everything Equation", a Cosmos-History "Meta-Model" formulating a multi-epochal, Marxian "Dialectic of Nature as a whole", and a dialectical algebra also used to express their versions of "The Psycho-Historical Equations", formulating multi-epochal "dialectical meta-models" of the "self-meta-evolutions" [or "self-revolutions"] of [the] Human [part of] Nature, also generating predictions of future epochs in both cases -- is now available for free download via --
http://www.dialectics.org/dialectics/Welcome.html
http://www.dialectics.org/dialectics...spondence.html
http://www.dialectics.org/dialectics...,27APR2011.pdf
F.E.D.'s new "Mathematics of Psycho-Historical Dialectics" was also used to derive the "Equitism" / "Generalized Equity" / "Political-Economic Democracy" model of the new social relation of production forming the societal self-reproductive foundation of the successor system to the present, terminal, suicidal, capital-relation-based socio-politico-economic system, which, without such replacement, will, per F.E.D., soon result in a new, and, this time, likely FINAL Global DARK AGE --
http://www.equitism.org/Equitism/Equitism-entry.htm
http://www.equitism.org/Equitism/Theory/Theory.htm
http://www.equitism.org/Equitism/The...cDemocracy.htm
Regards,
Miguel
Last edited by Miguel Detonnaciones; 4th August 2011 at 22:44.
Shouldn't this be in the Theory forum? If it's going to remain in Science and Environment can you rename it to something that isn't as misleading as "Theory of Everything", since it appears to be about politics, society, history, etc. not physics.
Sciences and Environment Forum Participants,
This post belongs under this Forum, IMO, because, for example, the "17-symbol Dialectical Theory of Everything" Equation-Model of the Dialectic of Nature, which this post reports upon, is a Cosmological Model, a Model of "Natural History" as the Total History of "Nature", "Cosmos", or "Universe", taken as the maximal Dialectical Totality, starting from the original cosmological population of pre-nuclear "particles", and summarizing that History in terms of the highest level "ontological categories" that describe the "kinds of being" so far extent, as well as those "kinds of being", predicted, by that Dialectical Equation-Model, to be next to "irrupt" in the Universe.
These ontological categories quite naturally extend to, and include, human-social categories.
Natural History, or the total history of Nature as "The Totality", includes the evolutions and "meta-evolutions" of the human[oid] species of Nature, and, therefore, includes human history, human society, human economic history, the history of human ideologies, etc.
That is, the work on which I am reporting echoes Marx in rejecting ideologies such as the Christian / Kantian / Lukacsian ideology of an absolute diremption, or radical dualism, of <<Natur>> vs. <<Geist>>.
[Clarification: I referred above to the "Dialectical Theory of Everything Model".
The work upon which I am reporting uses the term 'Meta-Model' instead.
I believe that the reason for this is that, by "Model", that work means, e.g., an ideogramic symbolic metaphor -- such as a nonlinear total-differential equation, or such as a system of such equations -- that expresses the "law of motion" historically-specific to a given epoch of natural history, and that describes mainly the quantitative "evolution" within a fixed set of qualities, or ontologies -- e.g., as represented by a fixed set of metrics, state-variables and control parameters -- that characterizes and defines that epoch.
Such a "model" is valid only up to the "resonance" and/or "depletion" "singularity" that bounds its given epoch on its later time-side, and that opens that epoch's successor epoch, with a "meta-evolution" that "irrupts" new, hitherto unprecedented qualities -- new, unprecedented ontology; new, unprecedented kinds of being.
Such "singularities" represent "self-revolutions" of and in Nature, such as the "irruption" of molecules in a universe previously organized only up to the level of atoms, or the "irruption" of prokaryotic living cells in a universe previously organized only up to molecules.
Such "singularities" typically involve "finite time infinite values", e.g., an infinite magnitude in the value of a key equational metric -- state-variable / control parameter -- arising suddenly at a finite value of the time parameter, t, and brought about by a zero value in a key denominator for that t value, i.e., by a zero division.
This makes "purely quantitative" models" break down -- become "infinitely wrong" quantitatively -- at and after that critical t value.
The work on which I am reporting has shown how to correct the "infinite model errors" produced by such "singularities", and to obtain an accurate evaluation of such equation-models at such critical t values, by "re-qualifying" those equations, using "meta-numerals" -- "metrical ["dimensional"] qualifiers", and/or "ontological qualifiers" -- that arise in that work's dialectical meta-model of the dialectical, categorial progression of the dialectical arithmetics themselves.
Thus, "models", in that work's view, echoing Marx's view, are bounded, limited in their validity, on both temporal sides, by ontological "singularity", or "natural self-revolution", boundaries.
"Models", e.g., partial-differential "evolution equations", describe mainly the purely quantitative "evolution" -- e.g., the self-expanding self-reproduction -- of a given set of ontological categories/"kinds of being" populations, which is qualitatively, ontologically fixed between such "self-revolutions", and fixed in the terms of reference of the "model specifications" of such "models".
A "Meta-Model", in this view, is a symbolic, e.g., ideogramic, metaphor, such as a "dialectical equation", which is "historically generic" and "quanto-qualitative", or "quanto-ontological", in that it describes a succession of "meta-evolutions" as well as of "evolutions", i.e., a progression of epochs, and of their historically-specific dynamical laws of "evolution", as well as describing the "self-revolutions in Nature", the "resonant" and "depletion" "singularities" that form the "meta-evolutionary" transitions from one such epoch to its successor epoch, all of this in a single symbolic, "quanto-qualitative", or even "purely-qualitative" / "purely-ontological" "dialectical-mathematical" expression.
Of such latter kind is their "17-Symbol Dialectical Theory of Everything Meta-Model".]
Regards,
Miguel
F.E.D. definitions for special terms applied above --
dialectical equations, dialectical equation-models
no definition is as yet available in the Clarifications Archive
[The] Dialectic of Nature
no definition is as yet available in the Clarifications Archive, but see pp. B-20 to B-22 in --
http://www.dialectics.org/dialectics...%20v.2_OCR.pdf
historically-generic
no definition is as yet available in the Clarifications Archive
historically-specific
no definition is as yet available in the Clarifications Archive
meta-evolution, meta-evolutionary
no definition is as yet available in the Clarifications Archive
meta-model
no definition is as yet available in the Clarifications Archive
ontological category
http://www.point-of-departure.org/Po.../Onto/Onto.htm
<<physis>>
no definition is as yet available in the Clarifications Archive
prokaryotic living cells
no definition is as yet available in the Clarifications Archive, but see pp. c-1 through c-2 in --
http://www.dialectics.org/dialectics...%5D.w3_OCR.pdf
pre-nuclear "particles"
no definition is as yet available in the Clarifications Archive
quanto-ontological
no definition is as yet available in the Clarifications Archive
quanto-qualitative, qualo-quantitative
no definition is as yet available in the Clarifications Archive
radical dualism
no definition is as yet available in the Clarifications Archive
self-revolutions in Nature
no definition is as yet available in the Clarifications Archive
[dialectical] totality
no definition is as yet available in the Clarifications Archive
zero-division singularities of nonlinear differential equations, ontological singularity, resonant singularities, depletion singularities
no definition is as yet available in the Clarifications Archive, but see pp. A-9 through A-17 in --
http://www.dialectics.org/dialectics...%20A-1_OCR.pdf
Last edited by Miguel Detonnaciones; 29th June 2011 at 18:09.
RevLeft Science Forum Participants,
FYI:
The THIRD of the promised "Postlude Letters", entitled "The Story of the Self-Creation of the Cosmos, Told via Self-Reflexive Sentences" -- is now available for free download via --
http://www.dialectics.org/dialectics/Welcome.html
http://www.dialectics.org/dialectics...spondence.html
http://www.dialectics.org/dialectics...,19MAY2011.pdf
F.E.D.'s new "Mathematics of Psycho-Historical Dialectics" was also used to derive the "Equitism" / "Generalized Equity" / "Political-Economic Democracy" model of the new social relation of production forming the societal self-reproductive foundation of the successor system to the present, terminal, suicidal, capital-relation-based socio-politico-economic system, which, without such replacement, will, per F.E.D., soon result in a new, and, this time, likely FINAL Global DARK AGE --
http://www.equitism.org/Equitism/Equitism-entry.htm
http://www.equitism.org/Equitism/Theory/Theory.htm
http://www.equitism.org/Equitism/The...cDemocracy.htm
http://www.equitism.org/Equitism/Ame...mentXXVIII.pdf
Regards,
Miguel
Last edited by Miguel Detonnaciones; 16th June 2011 at 08:20.
Sciences Forum Participants,
I want to provide you with a simplified account of the F.E.D. [Foundation Encyclopedia Dialectica] "First Dialectical Arithmetic", and its "Algebra" [which differs in special ways from standard algebras which are purely-quantitative, or which are otherwise not explicitly dialectical].
If you want the full-regalia, complex account of these matters, you'll have to go to the F.E.D. Primers.
Suppose we take q to denote a generic symbol, an "order qualifier", for qualitative "ordinality", or "ordered-ness" in general; for the "genus" of "consecutive ordered-ness".
Suppose we take q/1 to denote the specific ordinal quality "species" of "first-ness", and to denote also the "purely-qualitative division" of "purely-quantitative" Natural Number 1 into the "purely qualitative" generic "order qualifier" q [called "qualitative" division because numerator and denominator differ in quality].
Since q and 1 are mutually qualitatively different, or heterogeneous, q/1 is a division, or fraction, which does not reduce.
Suppose also that we take q/2 to denote another non-reductive division/fraction, and the specific ordinal quality species of "second-ness".
Suppose as well that we take q/3 to denote yet a third "ordinal qualifier", denoting yet another -- the next -- specific ordinal quality species, the quality of "third-ness".
Then the generic q/1, q/2, and q/3 can be applied as symbols for generic ontological qualifiers, standing for qualitatively distinct ontological categories, assignable to more specific successive terms in a "dialectical model", according to the following "dialectical interpretation" of these three purely-qualitative "meta-numerals", formed by the aufheben negation / conservation / elevation of the purely quantitative numerals 1, 2, and 3, respectively:
q/1 -- assign to the "first, or arche', thesis" of the first triad of a specific dialectical progression / succession / "consecuum";
q/2 -- assign to the "first anti-thesis", or "first contra-thesis", of the first triad of a specific dialectical progression / succession / "consecuum";
q/3 -- assign to the "first [full] synthesis", or "first [full] uni-thesis", of the first triad of a specific dialectical progression / succession / "consecuum".
Thereafter, q/4, q/5, q/6, q/7, etc., can be applied to denote, e.g., the second "contra-thesis", the "first partial uni-thesis", the "second partial uni-thesis", and the "second full uni-thesis", respectively, etc.
Given the rules for "ontological qualifier" "addition" and "multiplication", the formula that F.E.D. calls "The Dyadic Seldon Function", generates a succession of "epochs" or "stages", s = 0, 1, 2, 3, 4, ..., in which a cumulative heterogeneous / non-amalgamative / non-reducing sum of all of these "ontological category" "qualifiers" increasingly and consecutively extends itself as the value of s rises:
[q/1]^(2^s) = q/1 + . . . + q/(2^s) .
KEY RULE: Repeated interpreted symbols in the denominators of the q numerators represent "revolutionary" irruptions of new, higher ontology / ontological categories, relative to that ontology which is represented by the category-symbol that is repeated in that denominator, in its single / non-repeated occurrence(s).
To take the "social relations of production" progression presented in Marx's Kapital as an example, suppose that we assign q/1 to q/C, or just C, standing for the human-social-relations-of-production-ontology category of "Commodities", as the beginning, or "arche' ", thesis".
Then the following dialectical model captures the "value-form" categorial progression of Marx's method of presentation as follows --
stage 0, s = 0, [C]^(2^0) = C^1 = C;
stage 1, s = 1, [C]^(2^1) = C^2 = q/C + q/CC = q/C + q/M = C + M,
such that M or q/M connotes the new social-relations-of-production-ontology category of "Monies", not yet extant in stage 0, but first-irrupted in stage 1;
stage 2, s = 2, [C]^(2^2) = C^4 = q/C + q/M + q/MC + q/MM = C + M + q/MC + K,
such that --
q/MC connotes "Money-Commodities" / conversion of Commodities into Monies / "real subsumption" of the Commodity barter-relation by the Money-relation / production of Commodities for exchange for Money, rather than for barter / Money-mediated Circulation of Commodities, not yet extant in stages 0 and 1, but first irrupted in stage 2;
K or q/K connotes the social-relations-of-production-ontology-category of "Kapitals", not yet extant in stages 0 and 1, but first irrupted in stage 2, and signifying the "formal subsumption" of all previous social-relations-of-production by the Kapital-relation [and therefore connoting the "antediluvian" species of capital -- usurers' capital; mercantile capital; latifundial, slavery-based agricultural production capital; merchant-orchestrated household-crafts manufacturing capital, etc., but not yet the "real subsumption by Capital" species of the Capital-relation -- industrial capital];
stage 3, s = 3, [C]^(2^3) = C^8 = C + M + q/MC + K + q/KC + q/KM + q/KMC + q/KK = C + M + q/MC + K + q/KC + q/KM + q/KMC + S,
such that --
q/KC = Commodity-Kapital / conversion of Commodities into Kapitals / "real subsumption" of the Commodity-relation by the Kapital-relation, not yet extant in stages 0 , 1, and 2, but first irrupted in stage 3;
q/KM = Money-Kapital / conversion of Monies into Kapitals / "real subsumption" of the Money-relation by the Kapital-relation, not yet extant in stages 0 , 1, and 2, but first irrupted in stage 3;
q/KMC = Conversion, by the Kapital-relation, of the total C-M-C'-... Money-mediated-Circulation of Commodities process into the process of the Circulation of Kapitals, not yet extant in stages 0 , 1, and 2, but first irrupted in stage 3, and signifying the completion of the "real subsumption" of all previously-irrupted social relations of production by the [fixed-capital-intensive, industrial-]Kapital-relation;
q/KK = q/S = S = the self-conversion / self-expropriation / self-fetter-bursting of the Kapital-relation; the revolutionary irruption of the Association-relation of the Associated Producers / the irruption of Socialism, not yet extant in stages 0 , 1, and 2, but first irrupted in stage 3, and signifying the "formal subsumption" of all previously-irrupted social relations of production by the "Associated-Producers-relation", but not yet the "real subsumption" of those predecessor social relations of production by the "Associated-Producers-relation".
For those intrepid enough to want to check for themselves the above calculations, using the "Dyadic Seldon Function", the main rules ["axioms"] of the "Rules System" of the F.E.D. "First Dialectical Arithmetic", that govern these "purely qualitative", yet, nevertheless, "algorithmic", calculations, can be found at --
http://www.dialectics.org/dialectics/Welcome.html
http://www.dialectics.org/dialectics...spondence.html
http://www.dialectics.org/dialectics...-06JUN2009.pdf
-- or at --
http://www.adventures-in-dialectics....tics-entry.htm
http://www.adventures-in-dialectics....ters-entry.htm
http://www.adventures-in-dialectics....ightAxioms.pdf
Regards,
Miguel
F.E.D. definitions of special terms applied above --
<<arche'>>
http://www.point-of-departure.org/Po...rche/Arche.htm
[producers'] Association-relation
no definition is as yet available in the Clarifications Archive, but see --
http://www.equitism.org/Equitism/The...cDemocracy.htm
<<aufheben>>
http://www.point-of-departure.org/Po...n/Aufheben.htm
[archeonic] consecuum
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...%5D.w3_OCR.pdf
Capital-[social-]relation[-of-production] as human socio-ontological category
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...%5D.w3_OCR.pdf
[barterable] Commodity-[social-]relation[-of-production] as human socio-ontological category
no definition is as yet available in the Clarifications Archive, but see p. B-29 in --
http://www.dialectics.org/dialectics...%20v.2_OCR.pdf
contra-thesis
no definition is as yet available in the Clarifications Archive, but see pp. 2 through 4 in --
http://www.dialectics.org/dialectics...act1-1_OCR.pdf
dialectical arithmetic
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...act1-1_OCR.pdf
dialectical [meta-]model
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...act1-1_OCR.pdf
http://www.dialectics.org/dialectics...iefing_OCR.pdf
meta-numerals
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
Money-[social-]relation[-of-production] as human socio-ontological category
no definition is as yet available in the Clarifications Archive
ontological qualifier
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
ordinality
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
purely-qualitative
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
purely-quantitative
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
[purely-]qualitative addition / sum
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
[purely-]qualitative division
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.org/dialectics...AY2008_OCR.pdf
[Dyadic, Triadic] Seldon Functions
http://www.point-of-departure.org/Po...nFunctions.htm
[human-]social-relations-of-production [neo-]ontology
no definition is as yet available in the Clarifications Archive, but see pp. B-24 through B-33 in --
http://www.dialectics.org/dialectics...%20v.2_OCR.pdf
subsumption, formal, real
no definition is as yet available in the Clarifications Archive
<<arche'>>-thesis
no definition is as yet available in the Clarifications Archive, but see pp. 2 through 4 in --
http://www.dialectics.org/dialectics...act1-1_OCR.pdf
uni-thesis, partial, full
no definition is as yet available in the Clarifications Archive, but see pp. 2 through 4 in --
http://www.dialectics.org/dialectics...act1-1_OCR.pdf
Last edited by Miguel Detonnaciones; 29th June 2011 at 18:55.
Sciences Forum Participants,
Thought it might be useful to reproduce here the core "axioms" -- the main rules -- of the "rules-system", or "axioms-system" ["axiomatic system"] of the "First Dialectical Arithmetic".
In this post, I am basing my account on the following F.E.D. writings --
http://www.dialectics.org/dialectics...spondence.html
http://www.dialectics.org/dialectics...-06JUN2009.pdf
http://www.dialectics.org/dialectics...deography.html
http://www.dialectics.org/dialectics...iefing_OCR.pdf
[pages I-144 through I-150 especially].
http://www.dialectics.org/dialectics/Primer.html
http://www.dialectics.org/dialectics...%20A-1_OCR.pdf
[pages A-1 through A-35].
These eleven dialectical-arithmetical rules, or axioms, are rendered below in a language closer to everyday English -- less in the language of symbolic logic than in the sources cited above, or in my immediately-previous post to this thread, and with some commentary from yours truly.
Throughout, I am using spectrum-order color-coding -- red, orange, yellow, green, blue, indigo, violet -- to call attention to relative, dialectical, qualitative ordinality.
The "Peano Postulates" are axioms for the Standard Natural Numbers, the "set" or "space" [FONT=Arial Black]N[/FONT] = {1, 2, 3, . . . }.
An arithmetic that follows these postulates, yet is qualitatively different from the Standard Natural Numbers, is called a "Non-Standard Model" of the Peano Postulates.
Such "Non-Standard Models" have been relatively little explored in academic mathematics.
Such a "Non-Standard Model" is the F.E.D. "First Dialectical Arithmetic", whose "meta-number" space they denote by [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] = { q/1, q/2, q/3, . . . }.
A key concept for the "Peano Postulates" is that of the successor function, which, for the [FONT=Arial Black]N[/FONT], can be expressed, for n in [FONT=Arial Black]N[/FONT], as s(n) = n + 1.
The successor function for [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] incorporates the function s in its own version of the successor function, denoted by s:
s[q/n] = q/(s(n)) = q/(n+1).
1. q/1 is an element of [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT]. [the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] Non-Standard version of Peano 1]
2. For any n in [FONT=Arial Black]N[/FONT], if q/n is in [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT], then s(q/n) = q/(n+1) is in[FONT=Arial Black] N[/FONT][FONT=Arial]Q[/FONT] as well. [i.e., the successor of any element of [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] is also an element of [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT]].
[[FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] version of Peano 2].
3. For any j and k, both in [FONT=Arial Black]N[/FONT]: If q/j is not equal to q/k, then the successor of q/j is not equal to the successor of q/k. [i.e., no two, distinct elements of [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] have the same successor].
[[FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] version of Peano 3].
4. For all x in N: There is no q/x in [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] such that the successor of q/x is q/1. [i.e., q/1 has successors in [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT], but no predecessor(s); q/1 is the <<arche'>> of the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] "meta-numbers"]. [[FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] version of Peano 4].
5. For every n in [FONT=Arial Black]N[/FONT], q/n is in [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT]. [this states the <<aufheben>> tie between the [FONT=Arial Black]N[/FONT] and the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT]. The rest of the rules below state "Non-Standard" aspects of the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] relative to the "Standard" [FONT=Arial Black]N[/FONT]].
6. For any j and k, both in [FONT=Arial Black]N[/FONT]: If j is quantitatively unequal to k, then q/j is qualitatively unequal to q/k.
[This axiom expands the "trichotomy principle" of the "Standard" arithmetics -- the principle that for any arithmetical objects a and b, just one of the following three relations obtains:
a < b, or a = b, or a > b
-- to a "tetrachotomy principle", that adds a fourth possibility, that of qualitative inequality, to the basic possible relations between any pair of arithmetical objects].
7. For all n in N: q/n + q/n = q/n.
[the kind of "addition of likes" specified by this rule is called "idempotent addition". It is true not only for the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] dialectical logic, but also for the modern Boolean algebra of formal logic, in which not only does 0 + 0 = 0, but also 1 + 1 = 1. In the context of dialectically interpreted [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] expressions, it means that repeated "summed" occurrences of th same ontological category symbol is redundant, i.e., per our previous post, C + C = C, not "2C". This is the rule that makes the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] arithmetic "non-quantitative", or "purely qualitative". The "count" of the presence(s) of any ontological category symbol can never exceed "1". This rule is crucial to the calculations summarized in that previous post].
8. For all j, and k, both in [FONT=Arial Black]N[/FONT]:
If j is quantitatively unequal to k, then
q/j + q/k is qualitatively unequal to q/x
for any x in [FONT=Arial Black]N[/FONT].
[This is the rule that makes dialectically interpreted [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] expressions "ontologically anti-reductionist". It means that a "heterogeneous sum" of ontological categories, like C + M in the previous post, does not reduce or collapse into any single ontological category at the same level of generalization as its summands. This rule too is crucial to the calculations summarized in that previous post].
9. For every j and k, both in [FONT=Arial Black]N[/FONT]: [q/k] x [q/j] = [q/j] + [q/(k+j)].
[This rule is most crucial of all to the "purely-qualitative, algorithmic calculations" presented in the previous post. It defines what "multiplication" means among the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] "dialectical meta-numbers". It is called, by F.E.D., "the double-aufheben evolute product rule".]
10. For all j, and k, both in [FONT=Arial Black]N[/FONT]:
q/j + q/k = q/k + q/j
[This rule -- the commutative "law" of addition -- is shared, by the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] dialectical arithmetic, with most of the most familiar "Standard" arithmetics. The indifference of the value of an [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] "qualitative sum" to the order of its terms does not mean, however, that there is not a preferred order for sums of [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] "meta-numbers" / "dialectors". That preferred order -- the "progressive order" -- is their "ordinal order":
q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8 + . . .]
11. For all i, j, and k, all in [FONT=Arial Black]N[/FONT]:
[ q/i + q/j ] + q/k = q/i + [q/k + q/j ]
[This rule -- the associative "law" of addition -- is shared, by the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] dialectical arithmetic, with most of the most familiar "Standard" arithmetics.]
Extended Commentary.
This product rule is called "double <<aufheben>>", I think, because the j denominator/subscript of the "multiplicand" or "operand", q/j, is "<<aufheben>>-conserved" twice, in both the first, "Boolean", term of the product, and, once again, in the second, "ontology-increment" term of the product, where it is also, simultaneously "<<aufheben>>-negated/determinately-changed" and <<aufheben>>-elevated", by the addition, to that j denominator/subscript, of the k denominator/subscript from the "multiplier" or "operator", q/k.
This product rule is called "evolute" because it encodes, in generic syntax, an empirical observation of great generality: when a new, higher level of organization irrupts within a given universe[-of-discourse], not all still-extant units [if any; if not completely "extincted"] of the previously-irrupted levels of organization are converted into units of the new level [as they would be in "convolute product-tion"]. Some of the previous levels' units remain unconverted to the new, higher level units.
E.g., not all Commodity units disappear / are converted into Money units, once the concentrated ferment of Commodity-barter reaches the quantitative threshold where qualitative/ontological change irrupts; where the new, previously-unprecedented units of Money-Commodit(y)(ies) irrupt, forming the next <<arithmos>>-of-<<monads>> [assemblage-of-units] of social-relations-of-production human-social ontology. Units at the Commodity level of organization remain present, "even" after the "Money-ontology revolution" irrupts in any locus of human history.
E.g., not all atoms disappear / are converted into molecules, once the concentrated ferment of atoms-formation reaches the quantitative threshold where qualitative/ontological change irrupts; where the new, previously-unprecedented units called "molecules" irrupt, forming the next <<arithmos>>-of-<<monads>> [assemblage-of-units of cosmological, dialectic-of-nature ontology]. Units at the atom level remain present, "even" after the "molecules-ontology revolution" irrupts in any locus of cosmological natural history; of the "meta-evolution" of the cosmos.
In the Boolean algebra for formal logic, x times x = x.
In fact, Boole called this rule "the fundamental law of [undialectical] thought".
The "Boolean term" is still there in this rule 9., for dialectical logic.
The "multiplicand", q/j, returns in the "aufheben" product.
That is a part of the "aufheben" operation, a "conservation" moment.
However, this rule for dialectical logic not only includes but also goes beyond that for formal logic in adding a second term -- a term that simultaneously "conserves, determinately changes/negates, and elevates" the "multiplicand", or "operand", q/j.
That second term, qualitatively distinct from q/j, per Rule 6., and capable of representing the <<aufheben>> revolutionary irruption of new ontology, of a new ontological category, is q/(k+j), which still "contains"/"conserves" j in its denominator, but, also, adds k to it, thereby also changing / determinately "negating" and "elevating" that denominator vis-a-vis that of q/j.
If a q/j -- say q/1 -- is "interpreted" for, or "assigned" to, a specific ontological category, for example, to C, denoting the Commodities <<arithmos>>-of-<<monads>>[ = assemblage of Commodity units], then --
[q/1] x [q/1] = [q/1] + [q/(1+1)] = q/1 + q/2
-- stands for the "interpreted" dialectical-ontological process --
C x C = [q/C] x [q/C] = [q/C] + [q/CC] = [q/C] + [q/M] = C + M.
The "interpreted" expression can be read as --
"the self-reflexion / self-confrontation / self-operation of a sufficiently numerous and spatially-concentrated "population" of Commodities continues to expandedly-reproduce itself, but also <<aufheben>>-irrupts a new ontological category/<<arithmos>>/"population": that of "Monies"; an assemblage of a new kind of units; whose units are Money units".
Each unit of Money is, in the minds of the human agents of the Money-social-relation-of-production, a meta-Commodity unit.
Each unit of Money, as <<meme>>, is made up out of a heterogeneous multiplicity of the Commodities that Money will trade-for, a mental "Money-price-list" for the various purchasable Commodities, a la the lists in "value-form" sections C. and D. in Chapter I. of Volume I. of Marx's Capital.
Another example: Assigning q/4 to the ontological category "atoms", denoted a, in an [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] model of the dialectic of nature, then we have --
[q/4] x [q/4] = [q/4] + [q/(4+4)] = [q/4] + [q/8]
-- standing for the "interpreted" dialectical-ontological process --
a x a = [q/a] x [q/a] = [q/a] + [q/aa] = [q/a] + [q/m] = a + m.
The "interpreted" expression can be read as --
-- "the self-reflexion / self-confrontation / self-operation of a sufficiently numerous and sufficiently spatially-concentrated, or "self-densified", "population" of atoms continues to expandedly-reproduce itself, but also <<aufheben>>-irrupts a new ontological category/<<arithmos>>-of-<<monads>>/"population": that of "molecules"; the assemblage of a new kind of unit, of a new kind of assemblage, whose units are molecules".
Each typical molecule unit is a meta-atom.
Each typical molecule unit is made up out of a heterogeneous multiplicity of its predecessor, atom, units.
Regards,
Miguel
F.E.D. definitions for special terms applied in the narrative above --
<<arithmos>>
http://www.point-of-departure.org/Po...s/Arithmos.htm
<<aufheben>>
http://www.point-of-departure.org/Po...n/Aufheben.htm
convolute
http://www.point-of-departure.org/Po.../Convolute.htm
evolute
http://www.point-of-departure.org/Po...te/Evolute.htm
[the] dialectic of nature
no definition is as yet available in the Clarifications Archive, but see pp. B-20 through B-22 in --
http://www.dialectics.org/dialectics...%20v.2_OCR.pdf
dialectical meta-numbers
no definition is as yet available in the Clarifications Archive
dialectical logics
http://point-of-departure.org/Point-...icalLogics.htm
formal logics
http://point-of-departure.org/Point-...ics/Logics.htm
<<meme>>
no definition is as yet available in the Clarifications Archive
<<monad>> [unit]
http://point-of-departure.org/Point-...onad/Monad.htm
ontological reductionism, and [ontological] anti-reductionism
no definition is as yet available in the Clarifications Archive
ontological category
http://www.point-of-departure.org/Po.../Onto/Onto.htm
ontology
http://point-of-departure.org/Point-...y/Ontology.htm
[neo-]ontology-increment term
no definition is as yet available in the Clarifications Archive
[dialectical, qualitative] ordinality
no definition is as yet available in the Clarifications Archive
purely-qualitative
no definition is as yet available in the Clarifications Archive
units [<<monads>>]
http://www.point-of-departure.org/Po...onad/Monad.htm
Last edited by Miguel Detonnaciones; 13th August 2011 at 18:18.
Sciences Forum Participants,
Here is a shortcut "rule of thumb" for mapping back from "interpreted", specifically "assigned" dialectical "meta-numbers" as "ontological-categorial qualifiers", to "uninterpreted" dialectical "meta-numbers" [or, really, contra Hilbert, to "minimally-interpreted" "meta-numbers", interpreted only as "ordinality qualifiers"], expressed using the main example of the immediately previous post --
If C or q/C connotes the Marxian socio-ontological category, or <<arithmos>> [assemblage-of-units], named "Commodities", "assigned" to the minimally-interpreted "ordinal qualifier" q/1, then the C in q/C has the ordinal value 1, and, given '( )' in ( )X = (X), as the ordinal-value extraction-function, then (C) = (q/C) = q/(C) = q/1;
If q/CC = q/M = M connotes the Marxian socio-ontological category of "Monies", then (q/CC) = q/(CC) = q/(1+1) = q/2 = (M) = (q/M) = q/(M) = q/2;
If q/MM = q/K = K connotes the Marxian socio-ontological category of "Kapitals", then (q/MM) = q/(MM) = q/(2+2) = q/4 = (K) = (q/K) = q/(K) = q/4;
If q/KK = q/S = S connotes the Marxian, predicted future social-relations-of-production, socio-ontological category of "ASsociations-of-Producers", then (q/KK) = q/(KK) = q/(4+4) = q/8 = (S) = (q/S) = q/(S) = q/8.
Therefore also --
For q/MC, connoting the Marxian socio-ontological sub-category of the subsumption of "Commodities" by "Monies", or of "Money-mediated Commodity-Circulations", (q/MC) = q/(MC) = q/(2+1) = q/3;
For q/KC, connoting the Marxian socio-ontological sub-category of "Commodity-Kapital", (q/KC) = q/(KC) = q/(4+1) = q/5;
For q/KM, connoting the Marxian socio-ontological sub-category of "Money-Kapital", (q/KM) = q/(KM) = q/(4+2) = q/6;
For q/KMC, connoting the Marxian socio-ontological sub-category of the "Kapitals Circulations-Processes as a Whole", (q/KMC) = q/(KMC) = q/(4+2+1) = q/7.
Summary rule-of-thumb: Juxtaposition of denominator-located "connotograms", such as the C, M, K and S of this example, once they have been resolved into their ordinal-number pure-quantitative residues, such as 1, 2, 4, and 8, respectively, for the example used above, calls for, not the multiplication of these residue values, as per the syntax of typical modern algebras [e.g., aa = a x a = a squared], but calls for, instead, their addition, as per the syntax of typical ancient arithmetics [e.g., in the case of "Roman Numerals", XX = "X + X" = 20].
For example --
(CC) = (C) + (C) = 1 + 1;
(MC) = (M) + (C) = 2 + 1;
(MM) = (M) + (M) = 2 + 2;
(KC) = (K) + (C) = 4 + 1;
(KM) = (K) + (M) = 4 + 2;
(KMC) = (K) + (M) + (C) = 4 + 2 + 1;
(KK) = (K) + (K) = 4 + 4.
Regards,
Miguel
F.E.D. definitions of special terms applied in the narrative above --
associations of producers
no definition for this term is available as yet in the Archive, but see Section 5 in --
http://www.equitism.org/Equitism/Ame...mentXXVIII.pdf
<<arithmos>>
http://www.point-of-departure.org/Po...s/Arithmos.htm
<<aufheben>>
http://www.point-of-departure.org/Po...n/Aufheben.htm
connotograms
no definition for this term is available as yet in the Archive
[dialectical] meta-numbers
no definition for this term is available as yet in the Archive, but see --
http://www.dialectics.org/dialectics...iefing_OCR.pdf
minimally-interpreted [symbols] [contra Hilbert]
no definition for this term is available as yet in the Archive
ontological-categorial qualifiers
no definition for this term is available as yet in the Archive
ontological category
http://www.point-of-departure.org/Po.../Onto/Onto.htm
ordinality
no definition for this term is available as yet in the Archive
ordinal number
no definition for this term is available as yet in the Archive
ordinality qualifiers
no definition for this term is available as yet in the Archive
qualifiers
no definition for this term is available as yet in the Archive
socio-ontological category
no definition for this term is available as yet in the Archive
subsumption
no definition for this term is available as yet in the Archive
uninterpreted [symbols] [cf. Hilbert]
no definition for this term is available as yet in the Archive
Last edited by Miguel Detonnaciones; 30th June 2011 at 15:43.
RevLeft Sciences Forum Participants,
[In this post, I am basing my account on the following F.E.D. writings --
http://www.dialectics.org/dialectics/Primer.html
http://www.dialectics.org/dialectics...%20A-1_OCR.pdf
[pages A-1 through A-35, especially page A-32] ].
In an earlier post to this thread, I identified the F.E.D. "First Dialectical Arithmetic" as a "Non-Standard Model" of "Natural" Numbers Arithmetic.
The purpose of this post is to sketch the remarkable story of how, while such "Non-Standard Models" have been relatively little explored in academic mathematics, their ineluctable inherence -- their immanence -- in the "Standard Model", of "first order", of the "Natural" Numbers Arithmetic was established long before any at all of such "Non-Standard Models" were known, by three of the deepest theorems ever to emerge, so far, in the modern inflorescence of mathematical formal logic, launched by Boole, Peano, Frege, Russell, Goedel, and others.
These three theorems are --
1. The Goedel Completeness Theorem
2. The Goedel First Incompleteness Theorem
3. The Lowenheim-Skolem Theorem
Key Background.
The meaning of "first order" as in "first order mathematical logic". "First order" mathematical logic is symbolic formal logic [or "ideographic" formal logic; formal logic expressed in "ideograms", as distinct from "phonograms" or "pictograms"] that "quantifies over" -- that makes assertions about "some" or "all" of -- only "logical individuals", i.e., about individual elements of the set of all elements, the universe of discourse, of a given logical [axiomatic] mathematical system.
In the context of the system of the "[FONT=Arial Black]N[/FONT]atural" Numbers, [FONT=Arial Black]N[/FONT] = {1, 2, 3, . . .}, a first order "axiomatization" makes assertions about only individual "[FONT=Arial Black]N[/FONT]atural" numbers, not about the qualities of shared by and defining groups [non-empty, non-singleton "sub-sets"] of such numbers.
A "second order" axiomatization for the "[FONT=Arial Black]N[/FONT]atural" Numbers would "quantify over", or make assertions about, the qualities shared by groups of "Natural" numbers, e.g., via ideographic formulas that would translate into English sentences like "all odd numbers are...", or "all even numbers are...".
And so on, to "third order" logic, and beyond.
A. The Goedel Completeness and First Incompleteness Theorems.
[FONT=Helvetica][FONT=Arial]The inescapability of "Non-Standard Models" of the "first order" axioms of [FONT=Arial Black][FONT=Arial]"[/FONT]N[/FONT]atural" arithmetic, given only the assertion of their "Standard Model", was "predicted" [rigorously implied] by the joint applicability of the Gödel Completeness and Incompleteness theorems to the "first order" axiomatic system of the Peano[/FONT][/FONT][FONT=Helvetica][FONT=Arial]"Standard" "[FONT=Arial Black]N[/FONT]atural numbers" arithmetic:
[/FONT][/FONT]
[FONT=Helvetica][FONT=Arial]"Most discussions of Gödel's proof [of his '''First Incompleteness Theorem''' -- M.D.] ... focus on its quasi-paradoxical nature.[/FONT]
It is illuminating, however, to ignore the proof and ponder the implications of the theorems themselves.
It is particularly enlightening to consider together both the completeness and incompleteness theorems and to clarify the terminology, since the names of the two theorems might wrongly be taken to imply their incompatibility.
The confusion arises from the two different senses in which the term "complete" is used within logic.
In the semantic sense, "complete" means "capable of proving whatever is valid", whereas in the syntactic sense, it means "capable of proving or refuting [i.e., of "deciding" -- M.D.] each sentence of the theory".
Gödel's completeness theorem states that every (countable) [and ω-consistent -- M.D.] first-order theory, whatever its non-logical axioms may be, is complete in the former sense: Its theorems coincide with the statements true in all models of its axioms.
The incompleteness theorems, on the other hand, show that if formal number theory is consistent, it fails to be complete in the second sense.
The incompleteness theorems hold also for higher-order formalizations of number theory [wherein the Godel completeness theorem no longer holds at all, neither semantically nor syntactically — M.D.].
If only first-order formalizations are considered, then the completeness theorem applies as well, and together they yield not a contradiction, but an interesting conclusion.
Any sentence of arithmetic that is undecidable must be true in some models of Peano's axioms (lest it be formally refutable [as it would be were it true in no models of the Peano axioms -- M.D.]) and false in [some] others (lest it be formally provable [as it would be were it true in all models of the Peano axioms -- M.D.]).
In particular, there must be models of first-order Peano arithmetic whose elements do not "behave" the same as the natural numbers.
Such non-standard models were unforeseen and unintended but they cannot be ignored, for their existence implies that no first-order axiomatization of number theory can be adequate to the task of deriving as theorems exactly those statements that are true of the ["Standard" -- M.D.] natural numbers."
[John W. Dawson, Jr., Logical Dilemmas: The Life and Work of Kurt Godel[/FONT][FONT=Arial], A. K. Peters [Wellesley, MA: 1997], pages 67 to 68, emphases added by M.D.].[/FONT]
[FONT=Arial][FONT=Helvetica]
[/FONT][/FONT]
[FONT=Arial][FONT=Helvetica]B. The Lowenheim-Skolem Theorem.
[/FONT][/FONT]
[FONT=Arial][FONT=Helvetica]The [/FONT][FONT=Helvetica]Löwenheim-Skolem theorem[/FONT][FONT=Helvetica] also, by itself, implies the inseparability of "Standard" and "Non-Standard" Models of the "first order" axioms of arithmetics:[/FONT]
[/FONT][FONT=Arial]"[FONT=Helvetica]The research begun in 1915 by Leopold Löwenheim (1878-c. 1940), and simplified and completed by Thoralf Skolem (1887-1963) in a series of papers from 1920 to 1933, [/FONT][FONT=Helvetica]disclosed new flaws in the structure of mathematics[/FONT][FONT=Helvetica].
The substance of what is now known as the [/FONT][FONT=Helvetica]Löwenheim-Skolem theory[/FONT][FONT=Helvetica] is this.
Suppose one sets up axioms, logical and mathematical, for a branch of mathematics [/FONT][FONT=Helvetica]or for set theory as a foundation for all of mathematics[/FONT][FONT=Helvetica].
The most pertinent example is the set of axioms for the whole numbers.
One intends that these axioms should [/FONT][FONT=Helvetica]completely[/FONT][FONT=Helvetica] describe the [/FONT][FONT=Helvetica]positive whole[/FONT][FONT=Helvetica] numbers[/FONT][FONT=Helvetica] [i.e., the "[/FONT][FONT=Arial Black]N[/FONT][FONT=Helvetica]atural" numbers, [/FONT][FONT=Arial Black]N[/FONT] -- M.D.[FONT=Helvetica]] and [/FONT][FONT=Helvetica]only[/FONT][FONT=Helvetica] the whole numbers. But, [/FONT][FONT=Helvetica]surprisingly, one discovers that one can find interpretations — models — that are drastically different and yet satisfy the axioms[/FONT][FONT=Helvetica].
Thus, whereas the set of whole numbers is countable, or, in Cantor's notation, there are only [/FONT][FONT=Helvetica]aleph-subscript-0 [spoken as either aleph-sub[script]-zero, [/FONT][FONT=Helvetica]aleph-null[/FONT][FONT=Helvetica], or [/FONT][FONT=Helvetica]aleph-nought[/FONT][FONT=Helvetica] -- M.D.[/FONT][FONT=Helvetica]] of them [i.e., there are only the [/FONT][FONT=Helvetica]minimal[/FONT][FONT=Helvetica] infinite number of them — only an «[/FONT][FONT=Helvetica]arché[/FONT][FONT=Helvetica]» infinity of them — according to Cantor's theory of an endlessly-escalating progression of "actual" infinities, starting with the infinity that he denoted by the ideogram[FONT=Arial] for aleph-sub-zero[/FONT][/FONT][FONT=Helvetica], thence progressing to[FONT=Arial] aleph-sub-one[/FONT][/FONT][FONT=Helvetica], then to[FONT=Arial] aleph-sub-two[/FONT][/FONT][FONT=Helvetica], etc. -- M.D.[/FONT][FONT=Helvetica]], there are [/FONT][FONT=Helvetica]interpretations[/FONT][FONT=Helvetica] that contain as many elements as the real numbers [ = aleph-sub-one [/FONT][FONT=Helvetica]elements, per the "Cantor Continuum Hypothesis" -- M.D.[/FONT][FONT=Helvetica]], and even sets larger in the transfinite sense.
The converse phenomenon also occurs.
That is, suppose one adopts a system of axioms for [/FONT][FONT=Helvetica]a theory of sets[/FONT] and one intends that these axioms should permit and indeed characterize non-denumerable [FONT=Helvetica]collections of sets[/FONT][FONT=Helvetica].
One can, nevertheless, find a countable (denumerable) collection of sets that satisfies the system of axioms and other transfinite interpretations quite apart from the one intended.
In fact, every consistent set of ['''first-order''' -- M.D.[/FONT][FONT=Helvetica]] axioms has a countable [/FONT][FONT=Helvetica]model[/FONT][FONT=Helvetica] [from a '''finitist/constructivist''' point-of-view, a model [/FONT][FONT=Helvetica]potentially[/FONT][FONT=Helvetica], but never "actually", involving[FONT=Arial] aleph-sub-zero[/FONT][/FONT][FONT=Helvetica] "logical individuals" in its '''universe''' [of discourse], but no more than that -- M.D.[/FONT][FONT=Helvetica]]
.... In other words, [/FONT][FONT=Helvetica]axiom systems[/FONT][FONT=Helvetica] that are designed to characterize a unique class of mathematical objects do not do so[/FONT][FONT=Helvetica].
Whereas [/FONT][FONT=Helvetica]Gödel's incompleteness theorem[/FONT][FONT=Helvetica] tells us that a set of axioms is not adequate to prove all the theorems belonging to the branch of mathematics that the axioms are intended to cover, [/FONT][FONT=Helvetica]the Löwenheim-Skolem theorem[/FONT][FONT=Helvetica] tells us that a set of axioms permits many more essentially different [/FONT][FONT=Helvetica]['[/FONT][FONT=Helvetica]qualitatively different', 'ideo-ontologically different', [/FONT][FONT=Helvetica]unequal in a[/FONT][FONT=Helvetica] non-quantitative[/FONT][FONT=Helvetica] sense -- M.D.[/FONT][FONT=Helvetica]] [/FONT][FONT=Helvetica]interpretations[/FONT][FONT=Helvetica] than the one intended[/FONT][FONT=Helvetica].
The axioms do not limit the [/FONT][FONT=Helvetica]interpretations[/FONT][FONT=Helvetica] or [/FONT][FONT=Helvetica]models[/FONT][FONT=Helvetica] [uniquely to the model intended -- M.D.]
Hence [/FONT][FONT=Helvetica]mathematical reality[/FONT][FONT=Helvetica] cannot be unambiguously incorporated in axiomatic systems[/FONT][FONT=Helvetica].*
*Older texts did "prove" that the basic systems were [/FONT][FONT=Helvetica]categorical[/FONT][FONT=Helvetica]; that is, all the [/FONT][FONT=Helvetica]interpretations[/FONT][FONT=Helvetica] of any basic axiom system are isomorphic -- they are essentially the same but differ in terminology.
But the "proofs" were loose in that logical principles were used that are not allowed in Hilbert's metamathematics and the axiomatic bases were not as carefully formulated then as now. [/FONT][FONT=Helvetica]
No set of axioms is categorical, despite "proofs" by Hilbert and others[/FONT][FONT=Helvetica]....
One reason that [/FONT][FONT=Helvetica]unintended[/FONT][FONT=Helvetica] interpretations[/FONT][FONT=Helvetica] are possible is that each axiomatic system contains [/FONT][/FONT][FONT=Arial][FONT=Helvetica]undefined[/FONT][FONT=Helvetica] terms[/FONT][/FONT][FONT=Arial][FONT=Helvetica].
Formerly, it was thought that the axioms "defined" these terms [/FONT][FONT=Helvetica]implicitly[/FONT][FONT=Helvetica].
But the axioms do not suffice.
Hence the concept of [/FONT][FONT=Helvetica]undefined[/FONT][FONT=Helvetica] terms[/FONT][FONT=Helvetica] must be altered in some as yet unforeseeable way. [/FONT][FONT=Helvetica]
The Löwenheim-Skolem theorem[/FONT][FONT=Helvetica] is as startling as [/FONT][FONT=Helvetica]Gödel's incompleteness theorem[/FONT][FONT=Helvetica].
It is another blow to the axiomatic method which from 1900 even to recent times seemed to be the only sound approach, and is still the one employed by logicists, formalists, and set-theorists."
[Morris Kline, [/FONT][FONT=Helvetica]Mathematics: The Loss of Certainty,[/FONT][FONT=Helvetica] Oxford University Press [NY: 1980], pages 271 to 272, emphases added by M.D.[/FONT]].
[/FONT]
Thus, the first-order Peano axioms for the "[FONT=Arial Black]N[/FONT]atural" numbers can span a range of "models" which, at one extreme -- the "Standard" extreme -- describes the arithmetic of the "pure, unqualified quantifiers" of --
N = {1, 2, 3, ...} ["dialectical THESIS" system of arithmetic]
-- and, at another -- opposite -- extreme, describes the arithmetic of the "pure, unquantifiable qualifiers" of --
N[FONT=Arial]Q[/FONT] = {q/1, q/2, q/3, ...} ["dialectical ANTI-THESIS" system of arithmetic].
Terminological clarification, for present and future reference:
In the sentence "I picked three oranges.", F.E.D. terms the word "three" an "[ontological] quantifier", and the word "oranges" an "ontological category name" and an "ontological qualifier", or "kind-of-being qualifier".
In the sentence "I picked three pounds of oranges.", F.E.D. terms the word "three" a "[metrical] quantifier", and the word "pounds" a "metrical unit(s) name", and a "metrical qualifier", and the word "oranges" an "ontological category name" and an "ontological qualifier".
Regards,
Miguel
F.E.D. definitions of special terms utilized in the narrative above --
<<arche'>>
http://point-of-departure.org/Point-...rche/Arche.htm
dialectical antithesis
no definition for this term is as yet available in Clarifications Archive
dialectical thesis
no definition for this term is as yet available in Clarifications Archive
difference, ideo-ontological
no definition for this term is as yet available in Clarifications Archive
difference, qualitative
no definition for this term is as yet available in Clarifications Archive
difference, quantitative
no definition for this term is as yet available in Clarifications Archive
ideogram
no definition for this term is as yet available in Clarifications Archive
immanence
http://point-of-departure.org/Point-...t/Immanent.htm
logic, first order; second order; third order . . .
no definition for this term is as yet available in Clarifications Archive
metrical qualifier
no definition for this term is as yet available in Clarifications Archive
non-standard model [of the "Natural" Numbers]
no definition for this term is as yet available in Clarifications Archive
ontological qualifier
no definition for this term is as yet available in Clarifications Archive
phonogram
no definition for this term is as yet available in Clarifications Archive
pictogram
no definition for this term is as yet available in Clarifications Archive
quantifier
no definition for this term is as yet available in Clarifications Archive
standard model [of the "Natural" Numbers]
no definition for this term is as yet available in Clarifications Archive
unqualified quantifiers
no definition for this term is as yet available in Clarifications Archive
unquantifiable qualifier
no definition for this term is as yet available in Clarifications Archive
Last edited by Miguel Detonnaciones; 10th August 2011 at 15:46.
RevLeft Science Forum Participants,
FYI:
The FOURTH of the promised "Postlude Letters", entitled --
"The Dialectic of the[Finitary] Set of All Sets -- The Inescapability of Mathematical Ideo-<<Auto-Kinesis>> Instantiated"
-- an excerpt from the initial Chapter of the new book on the F.E.D. "Dialectical Meta-Numbers", and on their "contra-Boolean algebra" for Dialectical Logic, used to formulate their new 17-symbol "Theory of Everything Equation" Cosmos-History "Meta-Model", and used to express their versions of "The Psycho-Historical-Dialectical Equations" -- is now available for free download via --
http://www.dialectics.org/dialectics/Welcome.html
http://www.dialectics.org/dialectics...spondence.html
http://www.dialectics.org/dialectics...,11JUN2011.pdf
F.E.D.'s new "mathematics of Psycho-Historical Dialectics" was also used to derive the "Equitism" / "Political-Economic Democracy" model of the expected new social relation of production, predicted by F.E.D. to form the social self-reproductive foundation of the successor system to the present, plutocratic, terminal, suicidal, [state-]Capital-based, [state-]Capital-centered socio-politico-economic system --
http://www.equitism.org/Equitism/Equitism-entry.htm
http://www.equitism.org/Equitism/Theory/Theory.htm
http://www.equitism.org/Equitism/The...cDemocracy.htm
http://www.equitism.org/Equitism/Ame...overLetter.pdf
http://www.equitism.org/Equitism/Ame...mentXXVIII.pdf
Regards,
Miguel
Last edited by Miguel Detonnaciones; 16th June 2011 at 08:29.
Miguel,
I'm appreciative of your passion for dialectics and the work you did on your post. I consider an efficacious materialist dialectic to be the key to a human future.
I'm amazed no one has mentioned Bertell Ollman in the conversation. He is the foremost dialectician in the US and was the victim while teaching at Univ of Maryland of the nation's most notorious denial of acaemic freedom. He is now 75 years old and teaching at NYU.
Ollman conclusively establishes that Marx's organic perception of life, society, and mind and the materialist dialectic were taken from Hegel's philosophy of internal relations and its abstraction process. This philosophy sees the world as an internally related whole and is in agreement (an astounding prescience) with the new sciences of the nature and organization of life and the cosmos.
Ollman has written a number of works. I recommend Dance of the Dialectic (2003).
Rosa notwithstanding, Hegel is essential to Marxism. Here is what the young Marx had to say on this matter in an 1837 letter to his father:
"There are moments in one's life which are like frontier posts marking the completion of a period but at the same time clearly indicating a new direction....
"I had read fragments of Hegel's philosophy, the grotesque craggy melody of which did not appeal to me. Once more I wanted to dive into the sea, but with the definite intention of establishing that the nature of the mind is just as necessary, concrete and firmly based as the nature of the body....
"For some days my vexation made me quite incapable of thinking; I ran about madly in the garden by the dirty Spree, which 'washes souls and dilutes the tea' [Heine]. I even joined my landlord in a hunting excursion, rushed off to Berlin and wanted to embrace every street-corner loafer....
"While I was ill I got to know Hegel from beginning to end, together with most of his disciples....I came across a Doctor's Club [radical Hegelians]....In controversy here, many conflicting views were expressed and I became ever more firmly bound to the modern world philosophy from which I had thought to escape."
So yes, Marx was a Hegelian, and he turned Hegel right side up to create a materialist dialectic. And I firmly believe bringing this materialist dialectic to life and practice is the key to a human future.
I'll be visiting the website you recommended, Miguel. However, I have problems with a mathematical dialectic. Hell, I don't have a clue, and I don't see how it could be popularly usable, unless it is brought down to Earth in common language.
In any case, I'm looking forward to exploring the dialectic with you and others. Thanks for all the work you have done on this post.
RevLeft Sciences Forum Participants,
Dear Mr. Natural,
You are most welcome, and I thank you as well!
I agree with you that bringing "an efficacious materialist dialectic . . . to life and practice is the key to a human future".
I also agree with you that the work of Bertell Ollman has contributed mightily to the resurrection of work on Marxian dialectic, after the fiasco of the Stalinist state-capitalist ideology of "DiaMat", and I thank you again for bringing the consideration of Prof. Ollman's work into this thread.
His recent book that you cite -- The Dance of the Dialectic: Steps in Marx's Method -- is, indeed, that one of his works that is most focussed on Marxian dialectic, to my knowledge, but even one of his earliest books, Alienation: Marx's Conception of Man in Capitalist Society, is, as I recall, also already very good on dialectics as internal relations theory.
Ollman recently co-edited a book on Marxian dialectic with Tony Smith, another recent worker on Marxian dialectic, who, I think, has influenced F.E.D., in regard to the modeling of systematic-dialectical methods of presentation of theorized totalities -- such as of Marx's theory of Capital[ism] -- by means of "category-sums" -- "categorial progressions", or "superpositions" ["qualitative, non-amalgamative sums"] of "ontological categories". That book is Dialectics for the New Century.
A good source on Marxian systematic dialectics -- which Tony Smith sees as Marx's entire method of presentation in Das Kapital -- is Tony Smith's book The Logic of Marx's Capital: Replies to Hegelian Criticisms.
Bertell Ollman has a website, dialectics net, where some of his writings are available for free off-printing and/or downloading --
southerndomains com dialectics.
I see Ollman's account of "internal relations" as articulating a kind of "holographic" -- or, better, "holonomic" -- model, in which not only is every "part", or "sub-totality", of a dialectical totality, "contained by" that totality, but in which every such part also, in turn, "contains" its whole -- that entire totality -- but in a unique, particularized, "sided", "partial", or partially incomplete and deformed/distorted manner.
I see some of that "holonomic" characteristic coming into the F.E.D. models of dialectic as well.
It comes into them in the way that the oldest, "origin-al", or <<arche'>>, ontological category of any given universe[-of-discourse] categorial progression model, is "ever-present" in all ontological categories that irrupt later in that progression, and is so "ingredient" at ever-deeper levels for each such succeeding "ontological category".
This happens by means of the concrete materiality of the aufheben process, of "self-meta-monad-ization", or of "self-meta-unit-ization", by which a part of the <<monads>>, or "units" -- the "individuals" -- of each predecessor major historical <<arithmos>>, or "number" ["population"], of "units", as modeled by a given ontological category, internalize themselves to form, relative to themselves, "meta-monads", or "meta-units", which become the mere <<monads>> or "units" of their successor <<arithmos>>, which their very "self-meta-monad-ization" creates -- e.g., atoms are the sub-units of molecules, molecules the sub-units of prokaryotic living cells, prokaryotic living cells the sub-units of "endo-symbiotic" eukaryotic living cells, eukaryotic living cells the sub-units of "multicellular" plants and animals -- the so-called "meta-phyta" and "meta-zoa" -- etc., etc.
The progress of Nature thus grows -- self-produces, in its "quanto-qualitatively" self-expanding, ontologically-expanding "expanded self-reproduction" -- a [qualo-quanto-]fractal "content-structure": a "content-structure" characterized by "[meta-]finite scaled self-similarity"; a "trans-Leibnizian, Meta-Monadology".
It is as Errol E. Harris -- a Hegelian dialectician from whom [as from Hegel himself] Marxians can learn, despite his idealism -- writes, in his book Formal, Transcendental, and Dialectical Thinking: Logic & Reality: "what humanity sees as nature is its own self in becoming...".
I also agree with you that Marx was, in his earliest awakening as himself, and throughout the rest of his life, a -- critical -- Hegelian, indeed, an immanently critical Hegelian, who performed a dialectical, immanent critique of the Hegelian dialectic.
Finally, I agree with you about the problems of communicating the F.E.D. "mathematics of dialectics" to the broader producers-class.
I suspect that the F.E.D. membership is too steeped in the natural and social sciences, and in mathematics -- including in modern mathematical ["symbolic"] logic -- to be readily capable of communicating their discoveries to a non-specialist audience.
They do need to be steeped in these topics in order to fulfill the project that constitutes my main interest in their contributions -- their project of a Marxian, dialectical, immanent critique of the capitalist ideology in all of modern science and mathematics.
It is like when the programmer of a software application system tries to write the manual for the users of that application system.
However, I have at least a foot in both worlds -- the world of "mathematico-science", and the world of revolutionary political activism.
I hope that I can therefore learn to make some contribution to communicating the F.E.D. discoveries to our class at large, and to rendering the F.E.D. "First Dialectical Arithmetic", in particular, a powerful heuristic tool for the many, one that has all sorts of useful applications in "everyday human praxis", as well as at the frontiers of the natural and social sciences, and of mathematics, just as does "phonetic writing", and the "Natural Numbers", which were both once also the exclusive property of a tiny so-called "elite".
However, the F.E.D. opus is far too rich, and far too deep -- like the opus of Marx himself, which, I think, inspired it -- integrating themes from ancient philosophy, ancient science, and ancient mathematics, as well as from modern, for any one human background to be capable of presenting its riches in full.
I will do my best to critically present that material.
But the communication of that material will not be an adequate one unless there are eventually many others also so doing.
Regards,
Miguel
F.E.D. definitions for special terms utilized in the narrative above --
<<arche'>>
http://point-of-departure.org/Point-...rche/Arche.htm
<<arithmos>>
http://point-of-departure.org/Point-...s/Arithmos.htm
<<aufheben>>
http://point-of-departure.org/Point-...n/Aufheben.htm
category-sums
no definition for this term is as yet available in the Clarifications Archive
categorial progression
no definition for this term is as yet available in the Clarifications Archive
content-structure
no definition for this term is as yet available in the Clarifications Archive
fractal
no definition for this term is as yet available in the Clarifications Archive
heuristic tool
no definition for this term is as yet available in the Clarifications Archive
holographic
no definition for this term is as yet available in the Clarifications Archive
holonomic
no definition for this term is as yet available in the Clarifications Archive
immanent critique
no definition for this term is as yet available in the Clarifications Archive
living cells, eukaryotic
no definition for this term is as yet available in the Clarifications Archive
living cells, prokaryotic
no definition for this term is as yet available in the Clarifications Archive
mathematico-science
no definition for this term is as yet available in the Clarifications Archive
mathematics of dialectics
no definition for this term is as yet available in the Clarifications Archive
meta-phyta
no definition for this term is as yet available in the Clarifications Archive
meta-zoa
no definition for this term is as yet available in the Clarifications Archive
<<monad>>
http://point-of-departure.org/Point-...onad/Monad.htm
non-amalgamative sums
no definition for this term is as yet available in the Clarifications Archive
ontological category
http://point-of-departure.org/Point-.../Onto/Onto.htm
qualitative sum
no definition for this term is as yet available in the Clarifications Archive
quanto-qualitative
no definition for this term is as yet available in the Clarifications Archive
self-meta-<<monad>>-ization
http://point-of-departure.org/Point-.../Meta/Meta.htm
http://point-of-departure.org/Point-...nadization.htm
state-capitalism
no definition for this term is as yet available in the Clarifications Archive
systematic dialectics
http://point-of-departure.org/Point-...Dialectics.htm
Last edited by Miguel Detonnaciones; 15th July 2011 at 08:49.
Damn you, Miguel, you're much too smart and you are going to make me work much too hard. But let's do it.
I'm going to have to digest your detailed response. I'm a simple sort, and my specialty is bringing the complexity of life down to its organizational essentials. In this, my "bible" is Fritjof Capra's Web of Life (1996). This book is the hub from which I journeyed in many directions into the new sciences of the processes, patterns, and organization of life and the universe. I then apply the organization of life and the cosmos to human social systems. Well, are we not life?
Oh, joy! Someone else is familiar with Bertell Ollman. And I didn't know of his website. I am incompetent on computers. I have yet to figure out how to transfer quotes from one post to another, for instance.
Late last year I "found" Ollman and sent him missives imploring him to look at Capra's Web of Life as the scientific embodiment of the materialist dialectic. Ollman then e-mailed back that he was changing his graduate seminar this year to combine Capra and Marx, and that he would get back to me.
Well, six month have gone by without any further communication. Damn! Damn, damn, damn, damn, damn!
But in the meantime I found revleft and got a computer.
I'll get back to you (and others, I hope) in a couple of days. My brain has popped.
No. They kicked her out.
[QUOTE=Miguel Detonnaciones;2136342]
I agree with your theory semantically but nevertheless it tends to reproduce the secondary and, therefore, the precariat view of, as you say, "an efficacious materialist dialectic [...]"
Bertell Ollman might have re-created the syntax of the state and thus inadvertently "resurrected" the Stalinist ideologico-economico-politico project; certainly Stalin contributed to Ollman's dialectical analysis.
RevLeft Sciences Forum Participants,
Dear ar734,
I am not aware of any explicitly dialectical content -- or of any scientific content of any kind -- in Stalin's writings on his "DiaMat", or on any other topic.
I do not detect any substantial traces of Stalinism in Bertell Ollman's writings on dialectics.
Regards,
Miguel
You beat me to a response, Miguel, but you answered ar734's gross mischaracterization of Ollman much better than I could.
ar734 and others on this site: I challenge you to actually read Ollman, as I don't believe you can understand Marx or the materialist dialectic without understanding that Marx developed his organic perception of "nature, human society, and thought" from Hegel's philosophy of internal relations, which sees life and society as an internally related whole.
This philosophy of internal relations gives life to the materialist dialectic and is in general agreement with the new sciences of life's (thus society's) processes, patterns, and organization.
Forum: Don't discount Miguel Detonnaciones. I only know him from this exchange, but I am experiencing him as an unusually deep and sensitive thinker.
Come on, comrades! Get real! Ollman is important! His Dance of the Dialectic (2003) is comprehensive.
Mr. Natural sends his red-green very best to all. Now do your homework.