Computational Complexity and the Origin of Universals :: Mathematics Mathematical Philosophy Papers

Computational Complexity and the Origin of UniversalsABSTRACT This authorship establishes close relationships between fundamental problems in the philosophic and mathematical theories of listen. It reviews the mathematical concepts of intelligence, including pattern recognition algorithms, neural networks and rule systems. Mathematical difficulties manifest as combinatorial complexity of algorithms are related to the roles of a priori association and adaptative learning, the same issues that have shaped the two-thousand year old debate on the origins of the universal concepts of fountainhead. Combining philosophical and mathematical analyses enables tracing current mathematical difficulties to the contradiction between Aristotelian logic and Aristotelian theory of mind (Forms). Aristotelian logic is shown to be the culprit for the current mathematical difficulties. I will also discuss connections to Gdels theorems. The conclusion is that fuzzy logic is a fundamental require ment for cartel adaptivity and apriority. Relating the mathematical and philosophical helps clarifying both and helps analyzing future research directions of the mathematics of intelligence. I. Introduction mathematics and PhilosophyThe two-thousand year old debate on the origins of universal concepts of mind was about the roles of adaptivity or learning from ascertain vs. the a priori knowledge (the intrinsic or God-given). It is closely related to the epistemological problem of the origins of knowledge. The problem of combining adaptivity and a-priority is fundamental to computational intelligence as well as to judgment human intelligence. There is an interrelationship among concepts of mind in mathematics, psychology, and philosophy, which is much encompassing(prenominal) than currently thought among scientists and philosophers of today. From the contemporary point of view, the questions about mind represent by ancient philosophers are astonishingly scientific. A central qu estion to the work of Plato, Aristotle, Avicenna, Maimonides, Aquinas, Occam, and Kant was the question of the origins of universal concepts. Are we born with a priori knowledge of concepts or do we acquire this knowledge adaptively by learning from experience? This question was central to the work of ancient philosophers, medieval theologists, and it was equally classical to theories of Freud, Jung, and Skinner. The different answers they gave to this question are very similar to the answers given by McCulloch, Minsky, Chomsky and Grossberg.When 2300 years ago Plato faced a need to explain our capacity to conceptualize, he concluded that concepts are of a priori origin. The philosophy based on the transcendental, a priori reality of concepts was named realism. During the following 2000 years the concept of a-priority was tremendously strengthened by the development of monotheistic religion in Europe, to the extent that it interfered with observational studies.