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

Computational Complexity and the Origin of Universals ABSTRACT: This paper establishes close relationships between fundamental problems in the philosophical and mathematical theories of mind. 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 knowledge and adaptive learning, the same issues that have shaped the two-thousand year old debate on the origins of the universal concepts of mind. 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 GÃ ¶del’s theorems. The conclusion is that fuzzy logic is a fundamental requirement for combining adaptivity and apriority. Relating the mathemati cal and philosophical helps clarifying both and helps analyzing future research directions of the mathematics of intelligence. I. Introduction: Mathematics and Philosophy The two-thousand year old debate on the origins of universal concepts of mind was about the roles of adaptivity or learning from experience vs. the a priori knowledge (the inborn 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 understanding human intelligence. There is an interrelationship among concepts of mind in mathematics, psychology, and philosophy, which is much closer than currently thought among scientists and philosophers of today. From the contemporary point of view, the questions about mind posed by ancient philosophers are astonishingly scientific. A central question 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 experienc e? This question was central to the work of ancient philosophers, medieval theologists, and it was equally important 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 ability 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 empirical studies.