Tacit Models and Mathematical Reasoning

In his famous book "Knowing and Being" Michael Polanyi writes: ... "I shall speak of contributions made to scientific thought by acts of personal judgement which cannot be replaced by the operations of explicit reasoning. I shall try to show that such tacit operations play a decisive part not only in the discovery, but in the very holding of scientific knowledge" [Polanyi, 1969]. The concept of "tacit knowledge" should be considered as a fundamental one for scientific reasoning. In our opinion, cognitive psychology has not dedicated enough attention to it. One may assume that the Piagetian influence and the information processing approach have played a certain role in hindering the progress of systematic research in that direction. In the present paper we intend to focus on one of the main aspects of tacit cognition, namely tacit models, but before continuing a remark is necessary. In Polanyi's conception, the process of sensegiving (conferring a unitary meaning on a certain conglomerate of data) is based on an act of integration which is basically tacit. "No explicit procedure can produce this integration," writes Polanyi [1969, p. 191]. In our opinion, these tacit operations are not, as a matter of fact, inaccessible to an explicit analysis. They are accountable in principle by resorting to adequate means. This is an hypothesis of fundamental practical importance: If the tacit process of integration leads to incorrect solutions, the remedial activity has the option of identifying and analysing these, initially hidden, mechanisms and submitting them to the individual's control "Tacit" does not mean, in our view, mysterious, irrational, genuinely unaccountable. It means only a way of increasing the productivity of the intellectual process. This is, in fact, the basic task of a mental model acting as a substitute for a complex, abstract, difficultly accessible original. A mental model makes an essential contribution in the integration process, in conferring a unitary and directly accessible sense on an ensemble of data. The process is considerably simplified and any conflict is avoided if the model imposes, tacitly, its constraints on the reasoning process. Mathematical concepts and operations are basically abstract, formal constructs. Their meaning and their coherence are not guaranteed by empirical evidence but rather by axiomatic constraints.