"The mind is a compact, multiply connected thought mass with internal connections of the most intimate kind. It grows continuously as new thought masses enter it, and this is the means by which it continues to develop."

Bernhard Riemann On Psychology and Metaphysics ca. 1860

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Tuesday, July 03, 2007

The physical and metaphysical application of residues

When Cantor attempted to approach the question of the cardinality of the continuum, there was a fundamental flaw in the very positing of the problem to be solved. Leibniz warned of this when he designated the continuum as a labyrinth. The line of attack was in essence the same blunder that is the faulty axiomatic basis for information theory. That is, a Manichean yes/no, on/off, zero/one reductionism which is radically incommensurate with the actual energetic living and thinking processes which characterize the composition of reality. What Gauss was tackling in his Disquisitions, is an elaboration of the principal that Leibniz adduced that monads never truly are without existence, no matter how much they may be apparently diminished in physical scope. Which is to say, that the posited mechanical principal of entropic heat death is in fact subsumed by a higher reality which is anomalous and contradictory and will always show up as residual in any attempted linear or modular representation. That is indeed why Riemann's study of non Euclidean curvature and Gauss' higher arithmetic are indissolubly connected, and that Cantor's principle of transfinite density functionals is a continuation in the same direction.

Any misguided attempt to create a sort of topological "covering" from a fixed order to an ever potentially evolving substantial universal process is foredoomed to failure and quite frankly madness. As Cusa elaborated for his own time, the microcosm of the human mind or what Leibniz referred to as the thinking monad necessarily recapitulates the entirety of the development of the macrocosmic inert, biophysical, and noetic strata of the universe, no matter how imperfectly and cannot fail to do otherwise except through an obdurate negligent choice. Thus bringing in the element of free will, which can never be accounted for in a system that attempts to reduce or restrict choice to the radically absurd assumption of a simple on/off stochasticism. When the mathematical artifact is taken as the reality then indeed we are confronted with some quite literally insane hypotheses, such as reversing time and hyper-dimensional "m" or string theory.

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