Transpositions

Abstract

Philosophical discussions on 'the nature of mathematical entities' are only relevant if the adopted points of viewinfluence theway in which mathematicians are actually reasoning (Heyting). But what do practicing mathematicians themselves think of the question what mathematics is about? They do not subscribe to the (ironical) view that 'mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true' (Russell). Neither do they hold that mathematical entities 'participate' in a 'sphere' called 'logical reality' (Beth). Mathematicians simply do not pursue the question so deeply; their 'objects' are numbers, points, functions, groups, etc. However, there is a kind of relativity involved: a mathematical theory is not per se about, say, points or 'geometrical objects', for it may happen that a certain problem, allegedly about such things, can be better solved by imagining that it is about other things such as numbers or 'arithmetical' objects, and conversely. It will be argued that such 'transpositions' can have an important heuristic value. Switching from one 'domain' to another, more perspicuous 'field of activity', may facilitate the mathematical problem solving process, accordingly as either the mathematician's 'intuitive' skills, or the computer's 'digital' powers can be better exploited in it. This will be demonstrated by solutions to the problem of finding models for finite (affine and projective) geometries. The argument supports the view that the 'nature of the mathematical objects' may indeed be relevant to the way in which mathematicians are actually reasoning, though in a mundane interpretation which is totally different from its philosophical meaning.