Abstract
In the above chapters, I have set forth a view of immediate awareness and knowing how as highly complex, self-organizing and adaptive categories of human knowing. The bases for these categories were found in ontological as well as epistemological analyses and arguments, carrying forth a tradition established by James and Russell. In particular, I have characterized knowing the unique as a multifaceted set consisting of a hierarchy of primitive epistemic relations of immediate awareness, and suggested that we must look to a broader theory of signs, within an even broader theory of indexicality, to fully understand it. In this last section, I want to consider more abstractly the computability, that is decidability (in principle), of Boundary Set S.
“.. .This non-computational process lies in whatever it is that allows us to become directly aware of something.” Roger Penrose1
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References
Roger Penrose, 1994 [emphasis mine], p. 53.
Dinkar Gupta, “Computer Gesture Recognition: Using the Constellation Method,” in Caltech Undergraduate Research Journal, Vol. 1, April 2001.
As noted above, classical computability theory demands that we define problems or function instances over N, the natural numbers, Z, the integers, or Q, the rationals, or mechanisms encodable into N, a countable domain. As Blum points out [1989], extending beyond N, problems arise when we are tempted, as is prevalent in much current literature, to consider only rational Q skeletons of problem sets, thus confining our focus to points on a rational grid, Q 2. There are “gaps” or “holes” in a rational graph of such problems. For example, a real graph of x3 +y3 =1 in the positive quadrant will look like a quadrant of a circle, which might be one of our complex epistemic sets. However, the corresponding set of rational points on the graph will be empty [See Blum, 1989; also Lay, 1990].
Certainly, as I have made apparent throughout, knowledge that is a set defined over rule-governed, that is R.E., sets. Such sets are the only ones classical epistemology has dealt with, and are the only ones most philosophers currently wish to deal with.
I explore the extensions of classical recursive function theory and its relation to epistemological theory construction and a broader concept of intelligence in my 1995 (in progress). To understand the nature of Boundary Set S, I have utilized the mathematical characterizations of the Mandelbrot and Julia sets, within the broader context of Boolean network theory. The dynamics of our set S become clear by referring to the properties of the Mandelbrot and Julia sets, which also assist in highlighting problems with extant computability theory to address decidability questions on those sets as well as our set S. There are a number of questions which a complete theory must address, which we are unable to address here. These include: (1) What is the nature of rule-governedness and rule-boundedness on that epistemic set 5? More to the point, how are we to use the mathematical characterization of rule-boundedness to make sense of epistemic rule-boundedness? This question entails the following questions: (2) If our epistemic boundary set S is like the filled Julia set in significant mathematical and epistemological respects, how can we use the Julia set to mathematically characterize and understand our own knowing? (3) How do we encode primitive elements of the presentation set, immediate awareness (knowing the unique), and levels of this primitive epistemic set, including touching imagining and moving, to get the dynamics we need to understand our own knowing? (4) How do we mathematically define manner of a performance which is central to the dynamics of our set S? and (5) Since classical computability (decidability) theory is limited to machines (algorithms) over discrete, countable domains, and our boundary set S is definable over continuous, uncountable domains, how do we extend classical recursive function theory to make sense of questions regarding the decidability of our set S? Though I cannot address these questions here, we can at least consider the properties of the Mandelbrot and Julia sets and see that they have a usefulness in a mathematical model of natural knowing systems, specifically a mathematical characterization of our boundary set S.
The set of all real numbers is denoted by R. Tools for arithmetical work on the reals are the operations of addition, multiplication and relations between reals such as equality (=), ‘greater than’ (>), and ‘less than’ (<). Components of vectors will be the real numbers; each such vector will be a member of the set of all vectors, like (a,b) where a and b are arbitrary real numbers. The set of all vectors is denoted by R 2.
For the sake of argument, I assume the epistemic universe is like the filled Julia set of a polynomial map on a Riemann sphere where S=C ? {8} of the form g(z) = z2 + c. The boundary Julia set is the set of points that don’t go off to infinity under iterations of g. [See Blum, 1989].
Bonevac, Daniel, “Ethical Impressionism: A Response to Braybrooke,” in Social Theory and Practice, Volume 17, no. 2, Summer, 1991, pp. 157–173.
Ibid.
Ibid.
Benoit B. Mandelbrot, The Fractal Geometry of Nature, New York: W. H. Freeman and Company, 1977.
Roger Penrose, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics, New York, Oxford: Oxford University Press, 1989.
A.M. Turing, “On Computable Numbers, With An Application to the Entscheidungsproblem,” in Proceedings of the London Mathematical Society, Volume 42, 1937, pp. 230–265.
Penrose, Ibid., pp. 121–122. 17 See Penrose, Ibid., pp. 74–79.
See Simon Singh, Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem, New York: Walker & Co., 1997
Kurt Gödel, “Über Formal Unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, ” in Monatshefte für Mathematik und Physik, vol. 38, 1931, pp. 173–198.
Lenore Blum and S. Smale (1990). The Gödel Incompleteness Theorem and Decidability Over a Ring, Technical Report, Berkeley, California: International Computer Science Institute.
G. J. Chaitin, “Information Theoretical Limitations of Formal Systems,” Journal of the Association of Computing Machinery, Volume 21, 1974, pp. 403–424
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Estep, M. (2003). Computability of Boundary Set S . In: A Theory of Immediate Awareness. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0183-9_7
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