Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

Abstract

A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: 'omnipresent' and 'multipresent' theories, and 'ubiquitous' notions that form dependent parts, or moments, of theories. The category of 'facets' is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. Mathematics Subject Classification (2000). Primary 00A30, 01A55, 01A60, 01A85, 03-03, 03-99; Secondary 00A35, 00A69, 00A71, 00A79, 03B10, 03C55, 03G05, 30-03, 31-03.Keywords. History and philosophy of pure and applied mathematics, history and philosophy of symbolic logics, parts and moments, generality of theories, theory change, monism and pluralism, sets and multisets, abstract algebras, metamathematics, model theory.

Differential and integral calculus for logical operations. A matrix-vector approach

“…The complex relations between mathematics and logics, and the way many concepts change of status in different epochs, have been reviewed in [20]. This review shows the critical role of invention for producing innovative advances and sustained challenges in mathematics and logic.…”

How Influential Was Mechanics in the Development of Neoclassical Economics? A Small Example of a Large Question

“…Walras claimed that his theory of equilibrium was general, and it was so admired by many successors. The generality was achieved by accumulation: after forming the fairly small theoretical core of maximizing scarcities, he applied versions of it to this topic in economics and then to that one and then … This version of generality differed in degree from that found in many general mathematical theories, which contain a substantial core and are developed in a more implicational way; if (some parts of) the core are assumed, then further parts of the theory are deduced, and from them still further parts follow, with the core frequently in action (Grattan-Guinness 2010). Six important examples are the three traditions in mechanics and in the calculus rehearsed in sections 2 and 4; to note two of them, the core of mathematical analysis is the theory of limits, which is frequently deployed in the further exegesis; similarly, the core of analytical mechanics includes the principle of least action, which is often used in its development.…”