A Constructivist Approach to the Problem of Induction

Abstract

The unsolved problem of induction is closely linked to unreasonable effectiveness of mathematics in the natural sciences (Wigner 1960) and to the question why the universe is algorithmicly compressible (Davies, 1990). The problem of induction is approached here by means of a constructivist version of the evolutionary epistemology (CEE) considering both, the perceived regularities we condense to the laws of nature and the mathematical structures we condense to axioms, as invariants of inborn cognitive and mental operators. A phylogenetic relationship between the mental operators generating the perceived and the mathematical regularities respectively may explain the high suitability of mathematical tools to extrapolate observed data. The extension of perceptional operators by means of experimental operators (i.e. by means of measurement devices) would lead to the completion of the classical world picture if both the cognitive and the physical operators are commutable in the sense of operator algebra (quantitative extensions). Otherwise the physical operators will have invariants which no longer can be described in classical terms, and, therefore, would require the formation of non-classical theories (qualitative extension), exceeding the classical world picture. The mathematical analogon would be the algorithmic extension of elementary mathematical thinking exceeding the axiomatic basis previously established acording to Godel's incompleteness theorem. As a consequence there will be neither a definitive set of axioms in mathematics, nor will be there a definitive theory of everything in physics.

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