Full Title: Part II. Introduction. The Dialectic of Nature.
Dear Reader,
The present blog-entry is the second blog-entry in this series on the dialectical ‘meta-modeling’ of the natural history of the total cosmos, as presently known to science [except that the ‘meta-model’ narrated herein does not yet explicitly address “Dark Energy” and “Dark Matter”].
Part I. of this series is available via the following link --
http://feddialectics-miguel.blogspot.com/2014/04/part-i-prolegomena-dialectic-of-nature.html
Regards,
Miguel
Part II.: Narration -- Dialectical-Mathematical Meta-Model Story of our Cosmos --
Introduction.
Preliminary Considerations on Dialectical-Cosmological Meta-Modeling” --
Recounting the Story of the Ontological Self-Revolutions
of Pre-Human Cosmological Nature
Dialectical-Mathematically, as well as Narratively.
Please Note: Throughout this part, we employ two key ancient scientifico-philosophical terms.
Their resurrection in the present also serves well to catalyze an urgently needed future transition, to a ‘trans-modern’, dialectical science --
1. « Monad », which means, e.g., “qualitative Unit”, “qualitative logical Individual”, or “qualitative Element” -- a single one of the many like-kind constituents represented, collectively, by a “category”, and;
2. « Arithmos » [plural: «arithmoi»], which means a “category”, e.g. an “ensemble”, a “multitude”, a ‘“population”’, or an “assemblage” of «monads» -- i.e., of “qualitative units”, of “qualitative logical individuals”, or of “qualitative elements” -- all of which «monads» share a common quality, or a common predicate, i.e., a «categorema» in common; a like “kind”.
Please Note Also: Many of the special, technical words employed below are also hypertext links to definitions of those words [often for the first occurrence of each such word only].
The Dialectical ‘Theory of Everything’ ‘Meta-Equation’ --
The Historical-Dialectical ‘Meta-Monadology’ of Pre-Human Natural Formation(s), and Beyond.
The dialectical ‘meta-equation meta-model’ of the total cosmos which we will be exploring, and iterating for escalating values of its “discrete time” parameter, or ‘epoch parameter’, tALL, and whose story we will be narrating throughout this series, is the following --
ALL>-|-<tALL = < ALLr >2tALL
-- wherein ALLr denotes the primordial ‘cosmo-ontological category’ of pre-nuclear “particles”,
i.e., the category of “elementary” and composite bosons and fermions, belongs to a general family of such historical-dialectical ‘meta-models’, whose various features can be defined as follows --
The F.E.D. solution to this ‘Dialectical Theory of Everything meta-equation’ exhibits an «aufheben» progression which features the ‘Qualo-Peanic ’, ‘meta-fractal’, ‘meta-«monad»-ic’ ‘archéonic consecuum’ process / structure which characterizes dialectics in general, and the dialectics of the F.E.D. ‘Dialectical Theory of Everything meta-equation’ in particular.
This ‘meta-equation’, so interpreted, constitutes a ‘dialectical meta-model’, and one which also tells the story of our cosmos as natural-historical totality, to-date, and also one with the wherewithal to generate predictions of a next future ‘cosmo-ontological revolution’. This categorial-progression story “follows from” its initial category, ALLr, by ‘connotational entailment’, once the principles of
dialectical interpretation of the product of the initial, self-multiplication / ‘self-reflexion’ of category ALLr, and, later, of the products of that product with itself, are set forth.
This ‘dialectical meta-model’ is, in part, one of a ‘physical dialectic’, or ‘«physis» dialectic’ — indeed, models ‘The Dialectic of the Universal «Physis», from which, at length, human consciousness is formed.
This ‘dialectical meta-model’ describes the ‘dialectic of the successive and progressive neo-ontological formation(s)’ of cosmological Nature, in terms of the «monads», and of the «arithmoi», of the «phys»-ical cosmos.
However, this ‘dialectical meta-model’ is also one of ‘«psyche»-ic dialectic’, of ‘cognitive dialectic’, or of ‘ideo-dialectic’, because its rising levels of ‘“complexity/consciousness”’ [cf. Chardin], implicit in the ‘«phys»io-ontological categories’ that it describes, eventually arrive at -- and, predictively, surpass -- present-day, Terran human[oid] self-awareness.
The ‘meta-dynamics’ of the ‘meta-evolution’ of the ‘meta-dynamical meta-systems’ of such natural formation(s) constitute(s) an «autokinesis», and an « auto-onto-dynamasis » at the level of Nature’s ontology -- a creation of new kinds of being, from out of the interactions, and self-interactions, of the preceding kinds.
The systems-progression, or ‘diachronic meta-system’, of these successive “natural formations”, is grasped as a ‘self-«aufheben» self-progression’ of ‘Qualo-Peanic’, ‘meta-fractal’, ‘meta-«monad»-ic’, ‘archeonic consecuum’ processes / structures, when we grasp each of its successive pre-human[oid] «arithmoi» of natural formation «monads» [e.g., the global assemblages / «arithmoi» of living atoms units, or of molecules units, or of living pre-eukaryotic cells units, etc., as of some epoch in cosmos-history when any one or more of them are extant] as a ‘pre-consciousness, unified, collective natural subject[-ivity], verb, and object’, or as a ‘pre-consciousness, collective natural agent[-ivity]’.
This natural systems self-progression is therefore one that qualifies as a[n] ‘historical-dialectical process’ per F.E.D.’s definition.
The reader is referred to Supplement B (Part III, pages B-20 through B-22) of the F.E.D. Introductory Letter for the classical NQ_ ‘formulaic’ rendition of the ‘dialectical
meta-model’ re-rendered narratively in this series --
link: http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf.
Parsimony. The cosmological story that the narrative, in the ensuing blog-entries, recounts, is “unembellished” -- it invokes no more of the details of this cosmological natural history than is given explicitly in the F.E.D. standard solutions for the categorial terms that it narrates.
‘Helicity’. The forthcoming narrative of this series of blog entries instantiates a “helical narrative”, and is also close to being a “model-generated” narrative, emphasizing the recurring, self-similar aspects of the story encoded in the meta-equation, and of the ‘temporal qualo-fractal’ that the meta-equation generates, repeating the form of the narrative account as much as possible for each epoch / whorl of the helical course of the self-development of cosmological nature.
Nonetheless, the cumulative, unprecedented, non-cyclical aspects of this “greatest story”, and the overall progressive gain in ontological complexity / richness / “determinateness” from epoch to epoch, also demand their due in the course of the apt description of the connotations of the meta-equation, whose equations differ qualitatively, ontologically, in each of its successive epoch / whorl, thus refuting any ideological claim of an ‘ontologically-statical’, or merely “flat-cyclical”, merely circular natural-historical motion.
This helical ‘qualo-fractal’ ‘content-structure’ should not be mistaken as one which can be fitted to any helical graph-trajectory, confined to a single three-dimensional mathematical space with purely-quantitative axes, whether of the R, or of the Q, or of the Z, or of the W, or even of the N variety.
This kind of helix transcends such confinement.
Each whorl of such a standard-number-spaces-transcendent helix, though ‘qualo-fractally’, generically similar to each of its predecessor whorls [if any], and to each of its successor whorls [if any], is also qualitatively, ontologically different from each of them, as are they from it.
No doubt metrics can often be defined, that quantify generic common features of a whole succession of such whorls, and which, for each such generic feature, map back into a helical trajectory in, e.g., an R3, purely-quantitative mathematical space.
But each such mapping, by itself, will fail to capture the ontologically-dynamical, quantity-transcending ‘qualo-fractal’ helix in its totality.
‘Heuristicity’. HYPOTHESIS: The algebra of an arithmetical language that is limited to the expression of “unquantifiable” ordinal “qualifiers”, interpreted as representing ontological categories, cannot be other than an “algorithmic heuristic” algebra, and that is what we have in the NQ_ algebra as a tool of cognition.
The algorithmic layer of this “algorithmic heuristic”, the layer of the “minimally-interpreted” -- “ordinal qualifier”-interpreted -- generic NQ_ arithmetic, exhibits only a doubly-relentless generic qualitative ordinality, denoted by q, species told by a subsumed N numeral, n, in qn --
{ q1, q2, q3, . . . } =
{the quality of first-ness, the quality of second-ness, the quality of third-ness, . . . }
-- relentless, both, first, at its subscript level, and, second, at its superscript level. That relentless ‘subscriptal’ ordinality is presented horizontally, in the rightward direction, below, and that relentless ‘superscriptal’ ordinality is presented vertically, in the downward direction, below --
q11 = q1;
q12 = q1 + q2;
q13 = q1 + q2 + q3;
q14 = q1 + q2 + q3 + q4; . . ., etc.
The logic, the “followership”, so far, is strictly ordinal.
However, when the generic ordinal qualifier for “the quality of first-ness”, q1, is “interpreted’ or “assigned” -- identified with -- the specific «arché», or “ultimate ancestor”, ontological category of a specific categorial-progression “meta-genealogy” -- in this case, with the primordial universal ontological category” of human social formation, the foraging “pre-nuclear particles”, connoted herein by ALLr -- then the symbol q1, and its “followers”, may take on new meaning, new
“intension”, new connotations.
And, thereby, a new level of “followership” -- of their special ‘‘‘logic’’’ -- emerges, beyond that of the mere “qualitative ordinality” of the generic, algorithmic arithmetic; a special ‘‘‘logic’’’ which is a heuristic, intuitive, connotative logic -- a logic of “connotative entailment”.
For an NQ_ ‘meta-model’ to “work”, the meanings of the category-representing terms of
its categorial progression must follow from one another, and from their own subscripted, interpreted epithets, specifically, connotatively, semantically, not just generically, algorithmically, syntactically.
The solution of a t-specific, interpreted N Q_ equation’s “poly-qualinomial”, or ontological
categorial progression/cumulum, is a determination of a meaning, of an intension, of a connotation, for each «arché»-subsequent term in that heterogeneous sum of category-symbol-terms, a meaning that intuitively follows from the given meaning of the «arché» / first term, and that also intuitively follows from the meaning of every already so “solved-for”, predecessor term of the term now being “solved-for”, all the way back to that «arché», or originating, meaning-given term, in accord with the “canons of interpretation” codified in the procedure-narrative of the F.E.D. solution-method, the “organonic algebraic method” for solving Seldon Function meta-equations.
That is, in this case, if generic q1 is identified with specific ALLr, with the “physio-ontology” of
the of “elementary” and composite bosons and fermions, then, for a user of the
NQ_ cognitive tool who is also versed and immersed in knowledge about -- in the
reconstructed phenomena / phenomenology of -- pre-human, cosmological natural formation(s), a meaning, a solution, for the next specific term, for the term that corresponds to the generic q2, for the term deltar = qrr, must suggest itself, if the model is to “work”.
This means that, when such a user “self-inquires” in the form of “self-asking” the question --
“¿What known, past pre-human natural formation corresponds to the algorithm-generated description / definition: “The term qrr designates a “particle of particles”, an «arithmoi» of “meta1-particles” natural formation units, such that each such unit is made up out of a heterogeneous multiplicity of “particle” units as its sub-units”?
-- that a user-known pre-human cosmological natural-historic formation -- in this case, “atoms” --
qrr = a
-- must come to mind as the answer to that question; as the solution for that category-describing term.
For an NQ_ model to “work”, such apt, symbol-connotation-evoked “comings-to-mind”
must continue, from epoch t = 1, all the way out to epoch t = max., i.e., the maximal ordinal epoch needed to reconstruct all of the incremental ontology “begat” by the «arché» in question in cosmological natural history so far -- in this case, in the natural history of natural formation to-date, i.e., to this meta-model’s cosmological epoch t = 7.
This criterion of model success applies most unequivocally to the solution / “semantification” of the “self-hybrid” or “auto-hybrid” terms -- the terms of the form deltax = qxx.
Thomas K. Simpson describes, as follows, the process by which James Clerk Maxwell derived the dynamical equations of the electromagnetic field, using the Lagrange Equations.
Maxwell did so by honing down the full possible ensemble of terms of the latter to those that were actual for electromagnetic field dynamics --
Of course, this solution-method, as a heuristic method, and as a “semantic” method, will, even more so than the methods of mathematical logic, of formal-logical “followership”, involve differences of opinion about solutions.
It would be naive to expect otherwise.
And «vive les differences»!
Civil dialogue about such differences evokes new insights, and new and fruitful hypotheses.
The NQ_ “algorithmic heuristic” method can conduce to greater clarity in such
dialogues.
Dear Reader,
The present blog-entry is the second blog-entry in this series on the dialectical ‘meta-modeling’ of the natural history of the total cosmos, as presently known to science [except that the ‘meta-model’ narrated herein does not yet explicitly address “Dark Energy” and “Dark Matter”].
Part I. of this series is available via the following link --
http://feddialectics-miguel.blogspot.com/2014/04/part-i-prolegomena-dialectic-of-nature.html
Regards,
Miguel
Part II.: Narration -- Dialectical-Mathematical Meta-Model Story of our Cosmos --
Introduction.
Preliminary Considerations on Dialectical-Cosmological Meta-Modeling” --
Recounting the Story of the Ontological Self-Revolutions
of Pre-Human Cosmological Nature
Dialectical-Mathematically, as well as Narratively.
Please Note: Throughout this part, we employ two key ancient scientifico-philosophical terms.
Their resurrection in the present also serves well to catalyze an urgently needed future transition, to a ‘trans-modern’, dialectical science --
1. « Monad », which means, e.g., “qualitative Unit”, “qualitative logical Individual”, or “qualitative Element” -- a single one of the many like-kind constituents represented, collectively, by a “category”, and;
2. « Arithmos » [plural: «arithmoi»], which means a “category”, e.g. an “ensemble”, a “multitude”, a ‘“population”’, or an “assemblage” of «monads» -- i.e., of “qualitative units”, of “qualitative logical individuals”, or of “qualitative elements” -- all of which «monads» share a common quality, or a common predicate, i.e., a «categorema» in common; a like “kind”.
Please Note Also: Many of the special, technical words employed below are also hypertext links to definitions of those words [often for the first occurrence of each such word only].
The Dialectical ‘Theory of Everything’ ‘Meta-Equation’ --
The Historical-Dialectical ‘Meta-Monadology’ of Pre-Human Natural Formation(s), and Beyond.
The dialectical ‘meta-equation meta-model’ of the total cosmos which we will be exploring, and iterating for escalating values of its “discrete time” parameter, or ‘epoch parameter’, tALL, and whose story we will be narrating throughout this series, is the following --
ALL>-|-<tALL = < ALLr >2tALL
-- wherein ALLr denotes the primordial ‘cosmo-ontological category’ of pre-nuclear “particles”,
i.e., the category of “elementary” and composite bosons and fermions, belongs to a general family of such historical-dialectical ‘meta-models’, whose various features can be defined as follows --
The F.E.D. solution to this ‘Dialectical Theory of Everything meta-equation’ exhibits an «aufheben» progression which features the ‘Qualo-Peanic ’, ‘meta-fractal’, ‘meta-«monad»-ic’ ‘archéonic consecuum’ process / structure which characterizes dialectics in general, and the dialectics of the F.E.D. ‘Dialectical Theory of Everything meta-equation’ in particular.
This ‘meta-equation’, so interpreted, constitutes a ‘dialectical meta-model’, and one which also tells the story of our cosmos as natural-historical totality, to-date, and also one with the wherewithal to generate predictions of a next future ‘cosmo-ontological revolution’. This categorial-progression story “follows from” its initial category, ALLr, by ‘connotational entailment’, once the principles of
dialectical interpretation of the product of the initial, self-multiplication / ‘self-reflexion’ of category ALLr, and, later, of the products of that product with itself, are set forth.
This ‘dialectical meta-model’ is, in part, one of a ‘physical dialectic’, or ‘«physis» dialectic’ — indeed, models ‘The Dialectic of the Universal «Physis», from which, at length, human consciousness is formed.
This ‘dialectical meta-model’ describes the ‘dialectic of the successive and progressive neo-ontological formation(s)’ of cosmological Nature, in terms of the «monads», and of the «arithmoi», of the «phys»-ical cosmos.
However, this ‘dialectical meta-model’ is also one of ‘«psyche»-ic dialectic’, of ‘cognitive dialectic’, or of ‘ideo-dialectic’, because its rising levels of ‘“complexity/consciousness”’ [cf. Chardin], implicit in the ‘«phys»io-ontological categories’ that it describes, eventually arrive at -- and, predictively, surpass -- present-day, Terran human[oid] self-awareness.
The ‘meta-dynamics’ of the ‘meta-evolution’ of the ‘meta-dynamical meta-systems’ of such natural formation(s) constitute(s) an «autokinesis», and an « auto-onto-dynamasis » at the level of Nature’s ontology -- a creation of new kinds of being, from out of the interactions, and self-interactions, of the preceding kinds.
The systems-progression, or ‘diachronic meta-system’, of these successive “natural formations”, is grasped as a ‘self-«aufheben» self-progression’ of ‘Qualo-Peanic’, ‘meta-fractal’, ‘meta-«monad»-ic’, ‘archeonic consecuum’ processes / structures, when we grasp each of its successive pre-human[oid] «arithmoi» of natural formation «monads» [e.g., the global assemblages / «arithmoi» of living atoms units, or of molecules units, or of living pre-eukaryotic cells units, etc., as of some epoch in cosmos-history when any one or more of them are extant] as a ‘pre-consciousness, unified, collective natural subject[-ivity], verb, and object’, or as a ‘pre-consciousness, collective natural agent[-ivity]’.
This natural systems self-progression is therefore one that qualifies as a[n] ‘historical-dialectical process’ per F.E.D.’s definition.
The reader is referred to Supplement B (Part III, pages B-20 through B-22) of the F.E.D. Introductory Letter for the classical NQ_ ‘formulaic’ rendition of the ‘dialectical
meta-model’ re-rendered narratively in this series --
link: http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf.
Parsimony. The cosmological story that the narrative, in the ensuing blog-entries, recounts, is “unembellished” -- it invokes no more of the details of this cosmological natural history than is given explicitly in the F.E.D. standard solutions for the categorial terms that it narrates.
‘Helicity’. The forthcoming narrative of this series of blog entries instantiates a “helical narrative”, and is also close to being a “model-generated” narrative, emphasizing the recurring, self-similar aspects of the story encoded in the meta-equation, and of the ‘temporal qualo-fractal’ that the meta-equation generates, repeating the form of the narrative account as much as possible for each epoch / whorl of the helical course of the self-development of cosmological nature.
Nonetheless, the cumulative, unprecedented, non-cyclical aspects of this “greatest story”, and the overall progressive gain in ontological complexity / richness / “determinateness” from epoch to epoch, also demand their due in the course of the apt description of the connotations of the meta-equation, whose equations differ qualitatively, ontologically, in each of its successive epoch / whorl, thus refuting any ideological claim of an ‘ontologically-statical’, or merely “flat-cyclical”, merely circular natural-historical motion.
This helical ‘qualo-fractal’ ‘content-structure’ should not be mistaken as one which can be fitted to any helical graph-trajectory, confined to a single three-dimensional mathematical space with purely-quantitative axes, whether of the R, or of the Q, or of the Z, or of the W, or even of the N variety.
This kind of helix transcends such confinement.
Each whorl of such a standard-number-spaces-transcendent helix, though ‘qualo-fractally’, generically similar to each of its predecessor whorls [if any], and to each of its successor whorls [if any], is also qualitatively, ontologically different from each of them, as are they from it.
No doubt metrics can often be defined, that quantify generic common features of a whole succession of such whorls, and which, for each such generic feature, map back into a helical trajectory in, e.g., an R3, purely-quantitative mathematical space.
But each such mapping, by itself, will fail to capture the ontologically-dynamical, quantity-transcending ‘qualo-fractal’ helix in its totality.
‘Heuristicity’. HYPOTHESIS: The algebra of an arithmetical language that is limited to the expression of “unquantifiable” ordinal “qualifiers”, interpreted as representing ontological categories, cannot be other than an “algorithmic heuristic” algebra, and that is what we have in the NQ_ algebra as a tool of cognition.
The algorithmic layer of this “algorithmic heuristic”, the layer of the “minimally-interpreted” -- “ordinal qualifier”-interpreted -- generic NQ_ arithmetic, exhibits only a doubly-relentless generic qualitative ordinality, denoted by q, species told by a subsumed N numeral, n, in qn --
{ q1, q2, q3, . . . } =
{the quality of first-ness, the quality of second-ness, the quality of third-ness, . . . }
-- relentless, both, first, at its subscript level, and, second, at its superscript level. That relentless ‘subscriptal’ ordinality is presented horizontally, in the rightward direction, below, and that relentless ‘superscriptal’ ordinality is presented vertically, in the downward direction, below --
q11 = q1;
q12 = q1 + q2;
q13 = q1 + q2 + q3;
q14 = q1 + q2 + q3 + q4; . . ., etc.
The logic, the “followership”, so far, is strictly ordinal.
However, when the generic ordinal qualifier for “the quality of first-ness”, q1, is “interpreted’ or “assigned” -- identified with -- the specific «arché», or “ultimate ancestor”, ontological category of a specific categorial-progression “meta-genealogy” -- in this case, with the primordial universal ontological category” of human social formation, the foraging “pre-nuclear particles”, connoted herein by ALLr -- then the symbol q1, and its “followers”, may take on new meaning, new
“intension”, new connotations.
And, thereby, a new level of “followership” -- of their special ‘‘‘logic’’’ -- emerges, beyond that of the mere “qualitative ordinality” of the generic, algorithmic arithmetic; a special ‘‘‘logic’’’ which is a heuristic, intuitive, connotative logic -- a logic of “connotative entailment”.
For an NQ_ ‘meta-model’ to “work”, the meanings of the category-representing terms of
its categorial progression must follow from one another, and from their own subscripted, interpreted epithets, specifically, connotatively, semantically, not just generically, algorithmically, syntactically.
The solution of a t-specific, interpreted N Q_ equation’s “poly-qualinomial”, or ontological
categorial progression/cumulum, is a determination of a meaning, of an intension, of a connotation, for each «arché»-subsequent term in that heterogeneous sum of category-symbol-terms, a meaning that intuitively follows from the given meaning of the «arché» / first term, and that also intuitively follows from the meaning of every already so “solved-for”, predecessor term of the term now being “solved-for”, all the way back to that «arché», or originating, meaning-given term, in accord with the “canons of interpretation” codified in the procedure-narrative of the F.E.D. solution-method, the “organonic algebraic method” for solving Seldon Function meta-equations.
That is, in this case, if generic q1 is identified with specific ALLr, with the “physio-ontology” of
the of “elementary” and composite bosons and fermions, then, for a user of the
NQ_ cognitive tool who is also versed and immersed in knowledge about -- in the
reconstructed phenomena / phenomenology of -- pre-human, cosmological natural formation(s), a meaning, a solution, for the next specific term, for the term that corresponds to the generic q2, for the term deltar = qrr, must suggest itself, if the model is to “work”.
This means that, when such a user “self-inquires” in the form of “self-asking” the question --
“¿What known, past pre-human natural formation corresponds to the algorithm-generated description / definition: “The term qrr designates a “particle of particles”, an «arithmoi» of “meta1-particles” natural formation units, such that each such unit is made up out of a heterogeneous multiplicity of “particle” units as its sub-units”?
-- that a user-known pre-human cosmological natural-historic formation -- in this case, “atoms” --
qrr = a
-- must come to mind as the answer to that question; as the solution for that category-describing term.
For an NQ_ model to “work”, such apt, symbol-connotation-evoked “comings-to-mind”
must continue, from epoch t = 1, all the way out to epoch t = max., i.e., the maximal ordinal epoch needed to reconstruct all of the incremental ontology “begat” by the «arché» in question in cosmological natural history so far -- in this case, in the natural history of natural formation to-date, i.e., to this meta-model’s cosmological epoch t = 7.
This criterion of model success applies most unequivocally to the solution / “semantification” of the “self-hybrid” or “auto-hybrid” terms -- the terms of the form deltax = qxx.
For
the “merely hybrid”, or “allo-hybrid” terms, of form qyx...,
it has been found that some of them may be rightly “solved” to be “impossibles”,
i.e., to be “inoperative terms”, so named by analogy with
the unused terms often
encountered in specific
applications of the generic
Lagrange Equations.
Thomas K. Simpson describes, as follows, the process by which James Clerk Maxwell derived the dynamical equations of the electromagnetic field, using the Lagrange Equations.
Maxwell did so by honing down the full possible ensemble of terms of the latter to those that were actual for electromagnetic field dynamics --
“...Maxwell approaches the construction of his own electromagnetic theory
with a clear initial vision of the shape it must take. He does not begin
with a collection of basic empirical results and seek a merely complete and
convenient set of equations which will save the appearances.”
“Maxwell knows at the outset that his theory
must take the form of the equations of motion of a moving material system;
these, as we have seen, are Lagrange’s equations of motion, which in Maxwell’s
view simply explicate mathematically our a priori concept of matter in motion.”
“A priori, Maxwell’s equations are merely a special case of Lagrange’s equations.”
“Therefore, Maxwell’s program for a
“dynamical” approach to electromagnetism must be this: beginning with
Lagrange’s equations of motion, identify the generalized
coordinates and velocities which characterize an electromagnetic system, and
then determine by experiment which of the possible
coefficients are actually operative in this particular science,
and what relationships exist among the coefficients and the coordinates.”
“Lagrange’s equations, thus related to
electromagnetism and sifted of
inoperative terms,
will be the basic equations of electromagnetism.”
“At the same time, they will characterize in
broad strokes a particular form of connected system.”
[Thomas K. Simpson, Maxwell’s
Mathematical Rhetoric: Rethinking the Treatise on Electricity and Magnetism, Green Lion Press [Santa
Fe: 2010], pages 272-273, emphasis added].
Of course, this solution-method, as a heuristic method, and as a “semantic” method, will, even more so than the methods of mathematical logic, of formal-logical “followership”, involve differences of opinion about solutions.
It would be naive to expect otherwise.
And «vive les differences»!
Civil dialogue about such differences evokes new insights, and new and fruitful hypotheses.
The NQ_ “algorithmic heuristic” method can conduce to greater clarity in such
dialogues.
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