Logica Universalis

, Volume 5, Issue 1, pp 21-73

First online:

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

  • I. Grattan-GuinnessAffiliated withMiddlesex University Business School, The BurroughsCentre for Philosophy of Natural and Social Science, London School of Economics Email author 

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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


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