The triple nature of mathematics: deep ideas, surface forms and formal models

Zbigniew Semadeni

Didactica Mathematicae, Tome 46 (2024), pp. 115-171.

Résumé

The purpose of the paper is to develop the idea, outlined by Thom and later by Skemp, that one may speak, in the context of mathematics, of deep structures and surface structures, akin to those in psycholinguistics. The main claim of the present paper is that one should distinguish three basic features of mathematics, namely: 1) deep ideas, which are sound and control reasoning; 2) surface forms (signs of any kind), which are indispensable tools for working on mathematics, applying it, and communicating it; 3) formal models (in axiomatic theories). If A is a name or symbol of a concept, proposition or procedure, then the correspondence between the deep idea of A, surface forms of A and formal models of A is often straightforward, but there are many examples of significant discrepancies. It is shown that in case of a cognitive conflict, deep ideas prevail over formal models (provided that one can avoid formal contradiction or formal vicious circle); the strongest example to support this statement is a cognitive ‘loop’ involving two formal definitions of an ordered pair in an axiomatic set theory. Concepts are categorized as plain or compound. A plain concept (e.g., ‘number 1, ‘point’, ‘straight line’, ‘parabola’, ‘Euclidean space’, ‘sequence’, ‘Lebesgue integral’) is a result of an increasing sequence of generalizations such that intuitive meanings of objects agree with each other. Three basic types of compound concepts are identified: (α) Conceptual doublet, e.g., the sum 5+8 (which is either an arithmetical expression or the value 13 of it); for similar reasons, the fraction ; an angle as either a closed subset of a plane or this subset with a distinguished vertex and/or the sides; cosine as a function defined either on angles or on real numbers. (β) Aggregate — a concept which has a common formal definition comprising objects having distinctly different intuitive meanings, joined together by an important analogy to form a single concept, e.g., a function from a set into a set comprising (a) numeric functions of real variable, (b) sequences, (c) geometric transformations, (d) mappings of finite sets such as permutations, etc.; the integral with respect to an abstract measure μ, comprising integral on [a, b], say, and the intuitively quite different concept of ; abstract group; uniform space. (γ) Conglomerate — a bunch of concepts which includes various intuitively close instances, with the same name (or some variations of the same name), but without any common, formal definition fitting all important special cases, e.g., the concept of an angle, the concept of a polyhedron. Relations between ‘deep idea’ and ‘concept’, between ‘deep idea’ and ‘intuition’ and the epistemological status of ‘deep idea’ are also discussed.

Détail

DOI : 10.14708/dm.v46i0.7380

Mots-clés : deep ideas, surface forms, formal models, meaning in mathematics, intuition, generalization, understanding, ordered pair, plane angles in geometry, infinity