Part 12: Seldon’s Message Series --
‘Actual Metafinity’ versus Fanciful Infinity:
‘Metafinite’, ‘‘‘Meta-Dynamical’’’ Change and Mathematical “Singularity”.
It is my pleasure, and my honor, as an officer of the Foundation Encyclopedia Dialectica [F.E.D.] Office of Public Liaison, to share with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.
The twelfth such release in this new series is entered below [Some E.D. standard edits have been applied, in the version presented below, to the direct transcript of our co-founder’s discourse].
This instalment addresses the Seldonian view of the relation between the bandying about of fanciful and even mystical “infinities” in descendence-phase “pure” mathematics and even in mathematical physics, vis-a-vis the Seldonian concept of ‘metafinity’.
For more information regarding, and for [further] instantiations of, these Marxian and Seldonian insights, please see --
For ‘poster-ized’ visualizations of many of these Marxian and Seldonian insights -- specimens of ‘dialectical art’ -- see:
Member, Foundation Encyclopedia Dialectica [F.E.D.],
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.
“...leaving aside any extended commentary upon the contra-factual, ‘contra-actual’ axiomatics of Cantorian, “mathematical-Platonist” mysticism, and its deductively rigorous phantasies, for now, this is to register a few remarks regarding the mathematical “infinitism” that invades even the ostensive realism of mathematical physics.”
“The more-realistic, nonlinear partial [integro-]differential equations that abound in modern mathematical physics are prone to predict -- precisely because of their nonlinearity -- bouts of divisions by zero in real finite-time, divisions by zero which can be interpreted as representing infinite values for key physical metrics and variables: so-called “singularities”.”
“Such conjunctures in the modeled time/history of a physical phenomenon or system are rightly regarded as mathematically meaningless by many of the more realistic physicists, and as marking physical conjunctures for which the equations, and the “laws” that they formulate, “break down”.”
“There are, of course, some “mush-minded” ‘mystoids’ among “physicists”, who want us to believe that, e.g., the gravitational collapse singularity of the Einstein General Relativity system of “simultaneous” nonlinear partial differential equations means that there is some kind of ‘“actual, physical infinity”’ at the core of every “black hole”, as well as at the “Big Bang” natural-historical «arché»/core of our cosmos as a whole itself.”
“However, in a metaphoric manner, there is something meaningful about such a zero-division “singularity” in an otherwise physically accurate, highly-predictively-valid equation or equations-system. Instantaneous ‘‘‘infinite quantitative’’’ change, from the very instant in which a denominator-resident dynamical factor-variable takes on the value 0 in such an equation or equations-system, and/or in its/their solution-function(s) [in the few cases in which such functions are known in “closed form” for nonlinear equations], might signify, and might be the only way to signify, a finite but qualitative change, e.g., an ontological change, within a mathematical language which is capable only of “purely” quantitative expression(s).”
“This kind of change is what we call ‘metafinite change’, whereby the also ‘meta-dynamical’ character of an also dynamical system becomes manifest.”
“And, indeed, we find that, quite typically, zero-division singularities in the equations of physics and of engineering occur at critical points in the cumulative course of at first apparently only quantitative change, at which qualitative change, in the form of the self-expansion of the ontology of the phenomenon or system that the equations model, bursts forth. And this irruption of additional ontology typically also coincides with the ‘meta-dynamical’ transition from a predecessor dynamical system to its qualitatively, ontologically different -- ‘‘‘evolutely’’’ expanded -- successor system, that may also «aufheben»- ‘‘‘contain’’’ and incorporate its predecessor system.”
“Division-by-zero singularities typically mark, not [impossible] actual, physical infinities, but transitions from a predecessor system to its successor system: revolutions in natural history.”
“Beginning with the seventh system of dialectical mathematics in the “slow”, “training-wheels” version of our solution to our ‘meta-equation’ for the systems-progression of the axioms-systems of dialectical mathematics, the “purely”-quantitative equations of mathematical physics become ‘[re-]qualified’, by fully-arithmetical, fully-ideographical, fully-algorithmic ‘mathematical qualifiers’. These comprise ‘metrical qualifiers’, that arithmeticize, e.g., the more primitive, “syncopated”, “dimensional analysis” expressions like ‘sec.’, ‘gm.’, & ‘cm.’, as well as ‘ontological qualifier’ mathematical ideograms. ... We say ‘re-qualified’, because these ideographical ‘mathematical qualifiers’ bear a psychohistorical resemblance to, and resonance with, the ‘Monad’ arithmetical qualifier ‘‘‘numeral’’’ of the psychohistorically pivotal ancient proto-algebraic work of Diophantus of Alexandria, entitled The Arithmetica, which launched modern, algebraic mathematics, while still evincing salient ‘psychoartefacts’ of the ancient Mediterranean mathematical «mentalité».’
“Per a key axiom of that seventh dialectical-mathematical axioms-system, and beyond, the multiplication, by ‘empty zero’, 0, of a ‘‘‘[re-]qualified’’’, ‘quanto-qualitative’ mathematical expression, whether that expression be ‘metrically-qualified’ only, ‘ontologically-qualified’ only, or metrically and ontologically ‘co-qualified’, yields a new kind of arithmetical value and ‘meta-numeral’, which we call ‘full zero’.”
“Occurrences of that, ‘full zero’, value, signify, in that axioms-system, and beyond, that a new successor system has emerged, starting from the moment of a sustained such singularity, such that the mathematical language of the old, finite equation(s), that may have “miraculously” well-described and predicted the finite development of the predecessor dynamical system, up to that very moment of division-by-zero singularity, is no longer adequate to describe the ‘new finite-y’ of the successor dynamical system, that has «aufheben»-irrupted from out of the very heart of the predecessor dynamical system, as an expression of the very nature of that predecessor system itself; as an ultimate, immanent, ‘‘‘evolute’’’ self-transcendence of that predecessor system, via the irruption of a new ‘qualo-fractal’ scale of -- still finite, but ontologically-expanded -- ‘cosmo-ontology’. ...”