Tacit knowledge in mathematical theory. (English) Zbl 0847.00008

Breger, Herbert

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 79-90 (1992).

Many mathematicians feed aversions to logicism, but they have difficulties to express these versions in the form of arguments. Historical texts in mathematics appear to be familiar to the modern reader in dealing with topics which can be found in modern textbooks, but on the other hand they appear to be very remote because of being imbedded into background knowledge not known by modern readers. With the help of computers the theorems of mathematical theories can be derived, but a computer is not able to distinguish between relevant and trivial theorems.
These examples illustrate the rôle of tacit knowledge or “knowing-how” as being constitutive for progress in mathematics beside explicit knowledge or “knowing-that”. The author discusses five types of tacit knowledge, a knowledge “that cannot be made explicit in rules and definitions and accordingly is non-programmable” (p. 81): (1) the insight and understanding of a theory, (2) the know-how for axiomatization, (3) the know-how for the solution of problems, (4) the know-how to find the right definitions, constructions or generalizations, (5) the tacit knowledge of the trivial. Examples are taken from the history of the axiomatization of topology, Descartes’ analytical geometry, the history of number theory (Diophantus, Viète), Fermat’s procedure for the solution of extreme-value problems, Felix Klein’s Erlangen Programme, and the theory of cate.
For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30

Philosophy of mathematics

Keywords:

tacit knowledge; know-how; topology; categories; analytical geometry