Abstract
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). 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 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. 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 paper shows with an example how mathematics, without any special provision, may achieve compression of information. Then, it describes examples showing how variants of ICMUP may be seen in widely used structures and operations in mathematics. Examples are also given to show how several aspects of the mathematics‐related disciplines of logic and computing may be understood as ICMUP. 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. 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. There are many potential benefits and applications of the mathematics‐as‐IC perspective.
Highlights
“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
In view of the versatility of the SP System, and the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, and since those AI-related strengths of the SP System are due largely to the versatility of the SP-multiple-alignment construct, there are reasons to believe that the concept of SP-multiple-alignment may prove to be as significant for an understanding of human intelligence as is DNA for biological sciences: it may come to be seen as the “double helix” of intelligence
Achievements with Algorithmic Cognition include the following: “. . . we have offered what we think is an essential and what appears a necessary connection between the concept of cognition and algorithmic information theory
Summary
As noted in the Introduction, much of the thinking in this paper derives from the SP System, meaning the SP eory of Intelligence and its realization in the SP Computer Model. (iii) Versatility in the representation of diverse kinds of knowledge includes the following: the syntax of natural languages; class-inclusion hierarchies (with or without cross-classification); part-whole hierarchies; discrimination networks and trees; if- rules; entity-relationship structures; relational tuples; and concepts in mathematics, logic, and computing, such as “function,” “variable,” “value,” “set,” and “type definition”. In view of the versatility of the SP System, and the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, and since those AI-related strengths of the SP System are due largely to the versatility of the SP-multiple-alignment construct, there are reasons to believe that the concept of SP-multiple-alignment may prove to be as significant for an understanding of human intelligence as is DNA for biological sciences: it may come to be seen as the “double helix” of intelligence. Potential Benefits and Applications of the SP System. e SP System has potential in several areas of application including the following: helping to solve nine problems with
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