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operations resulting in the surety of proofs. The unresolved tension about what |
believed from intuitive experience and what | was allowed to believe from the logic
of theorem and proof, perhaps not unlike my belief in the transcendent experience
over logical theological argument as Reality, continued throughout my life. For
example, many decades later at HES, | saw the world class dynamical systems
theorist and differential geometer-topologist, Dennis Sullivan, use a projector to
display a computer-generated, intricate and beautiful, mathematical object, the well
known, computer screen saver, Mandelbrot set. It represents the control parameter
plane of the well studied complex analytic map, z > z* + c. Sullivan, pointing to a
small, discrete complicated little part of it that looked like a little version of the whole
of it, from a distance looking like a point, said, “An important Ph.D. dissertation is
waiting to be done on the question: is this (pointing to the little object) really there?”
In the audience of about a hundred professional mathematicians and one amateur, |
was the only one that laughed.
Historians of mathematics point to the successful generalization of Euclidian
geometry via its abstract axioms, postulates and logical operations to a new, not
naturally intuitable, almost nonvisualizable, non-Euclidean geometry (with the new
geometric axiom, parallel lines do meet at infinity), as evidence against the Kantian
idea of the intuitively accessible, a priori status of geometry. This served as an
example of where mathematics naturally resides, and argues in favor of the thought
control imposed by the modern set theoretic and logical rituals of mathematical
theorem and proof. Thom, in a_hereditary-evolutionary biological argument
developed in Semiophysics, said “Objections raised to the Kantian apriority of
Euclidean geometry after the discovery of non-Euclidean geometries, and the
theories of twentieth century physics (restricted and general relativity, quantum
mechanics) appear to me to be irrelevant...they deal with ...the infinitely small and
infinitely large...which lies outside the usual cognitive activity of ancient man.”
In my discussions with him, Thom found equivalence relations between
mental and real world objects and their behaviors. He described what he called an
abstract physicalist truth that describes a psychic universe, which, in turn, simulates
outside things and processes. Much like the transcendent experiential God | have
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