Abstract

The aim of this paper is to present a conception of the triple nature of mathematics. It is argued that the nature of mathematics is best served by distinguishing deep ideas (of concepts or propositions), their surface representations (signs which can be perceived by senses) and their formal models (in axiomatic theories). For instance, the concept „number π” has several different models in set theory (those based on Dedekind cuts and on Cantor's equivalence classes of Cauchy sequences) and yet all working mathematicians in the world have the same object π in mind. They have a common deep idea of π. Generally, the deep idea of a concept X is a well-formed mental construction of X which controls reasoning. It manifests itself in a characteristic, definite feeling of purpose, in firm certainty of the meaning of X in various contexts, and in robustness of understanding of X in cases of typical cognitive conflicts. Epistemic deep ideas are intersubjective and have been formed in phylogeny whereas individual deep ideas (or deep intuitions) are formed in ontogeny. In certain situations a deep idea may be described in terms of intuition, of meaning or sense, or of understanding, but none of these terms can provide a satisfactory description fitting all cases