The Kinematics and Dynamics of Concept Formation

Analyses of concept formation can be found in disciplines that seem superficially unrelated. The two oldest traditions are in philosophy and mathematics, the one reaching back to Plato and Aristotle and having a continuous history in the theory of knowledge, and the other going back to Eudoxus, Euclid and their several mathematical contemporaries and successors. The logical status of the analyses of concept formation given by philosophers ranging from Aristotle through Hume to Kant and on to Russell has a complex and ambiguous history.It is common in contemporary discussions, for example, to say that Hume was badly confused about the distinction between the logic and psychology of concept formation, but it is also characteristic of the people who say these things that they do not offer a very precise or literal definition of the logic of concepts, and the word “logic” is used by them in a way that is itself tantalizingly vague. In the case of the logical analyses of concepts in a mathematical context, particularly as questions have come to be put in terms of precisely characterized notions of definability, a quite finished logical theory of concept formation has developed. Given any theory and given a concept it is possible to ask in a quite definite and precise way whether or not this concept is definable in the theory or, given the concepts of a theory, it is possible to ask if one of the concepts is definable in terms of the othefs. It is true that problems of definability have often not been discussed as problems of concept formation, and yet it is obvious that there is a close logical relation between the two subjects. If a concept is not definable in terms of a set of other concepts, then in one sense that concept cannot be formed from them. For example, by application of Padoa’s classical method for establishing the independence of concepts it is easy to show that for most standard axiomatizations of classical particle mechanics, the concept of mass cannot be defìned in terms of the concepts of particle and position, Mach’s famous proposal to the contrary. The ordinary-language philosophers who talk about the logic of concepts