Aristotle and Bressan on a number of things

Abstract

Aldo Bressan (1972) used his intensional modal logic to introduce several concepts and distinctions that he saw as useful for work on the formal axiomatization of physics. One of these distinguishes an absolute property from an extensional property. In this paper I use these concepts to clarify an Aristotelian distinction on the word 'number'. I also use a closely related distinction between an extensional property and a quasi-absolute property for some problems in the natural sciences re? lated to number theory. Both applications would be positive instances of Bressan's thesis of "the double use of common nouns". Natural language does not mark the distinction between the extensional and (quasi-) absolute senses of such common nouns as 'number', 'mass point' or 'horse', but formal languages should be able to express the difference. Standard modal logic allows one to express the distinction, but only with fairly complex formulae. Bressan's semantics allows it to become a simple part of empirically based translation decisions. Part II of this paper introduces the Aristotelian distinction on number which I claim is a case of the "double use of a common noun". Part III is an informal presentation of enough of Bressan's semantics to explain the higher order concept absolute. Readers less interested in formal logic can skim this part to pick up some terminology and go on to the applications in Part IV. Bressan's logic is much easier to use than it is to study. Part III is informal in ways that will irritate logicians, but formally correct versions are available and this more readable presentation may encourage their study. Part V introduces the related concept quasi-absolute and considers one problem where the natural sciences intrude on arithmetic. Part VI considers some consequences of these ideas for the philosophy of mathematics. This paper does not recommend an Aristotelian philosophy of mathematics. It recommends a clarified version of a distinction he introduced and an intensional logic for its formal expression.