“… against Wigner’s ‘unreasonable effectiveness’ statement (based on success in the physical sciences) one must ask why maths is often so unreasonably ineffective in the human and social sciences of behaviour, psychology, economics, and the study of life and consciousness. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.” ( ibid, p. “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. That expression is intended to have the same meaning elsewhere in this paper.Įugene Wigner writes about “The unreasonable effectiveness of mathematics in the natural sciences”: It is clear that, in this quote, the expression “the world” is intended to mean “everything in the observable universe,” in accordance with normal usage. … Why is the world found to be so unerringly mathematical?” (, bold face added). Mathematics works: as a result we have been tempted to equate understanding of the world with its mathematical encapsulization. “ For some mysterious reason mathematics has proved itself a reliable guide to the world in which we live and of which we are a part. There is a deep and beautiful mystery in this fact: that these superbly accurate theories are also extraordinarily fruitful simply as mathematics” (, bold face added).
“It is remarkable that all the SUPERB theories of Nature have proved to be extraordinarily fertile as sources of mathematical ideas. The fundamental nature of mathematics has been a considerable puzzle to mathematicians and others for many years. There are many potential benefits and applications of the mathematics-as-IC perspective. Also discussed is how the close relation between IC and concepts of probability relates to the established view that some parts of mathematics are intrinsically probabilistic, and how that latter view may be reconciled with the all-or-nothing, “exact,” forms of calculation or inference that are familiar in mathematics and logic. Also discussed is the intimate relation between IC and concepts of probability, with arguments that there are advantages in approaching AI, cognitive science, and concepts of probability via ICMUP. Examples are also given to show how several aspects of the mathematics-related disciplines of logic and computing may be understood as ICMUP. Then, it describes examples showing how variants of ICMUP may be seen in widely used structures and operations in mathematics. This paper shows with an example how mathematics, without any special provision, may achieve compression of information. This perspective on the foundations of mathematics has grown out of a long-term programme of research developing the SP Theory of Intelligence and its realization in the SP Computer Model, a system in which a generalised version of ICMUP-the powerful concept of SP-multiple-alignment-plays a central role.
This new perspective on the foundations of mathematics reflects the facts that mathematics is almost exclusively the product of human brains, and has been developed, as an aid to human thinking, mathematics is likely to be consonant with much evidence for the importance of IC in human learning, perception, and cognition. That is itself a novel approach to IC, couched in terms of nonmathematical primitives, as is necessary in any investigation of the foundations of mathematics.
This paper describes a novel perspective on the foundations of mathematics: how mathematics may be seen to be largely about “information compression (IC) via the matching and unification of patterns” (ICMUP).
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