Logica Universalis

, Volume 5, Issue 1, pp 21–73

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


DOI: 10.1007/s11787-010-0023-0

Cite this article as:
Grattan-Guinness, I. Log. Univers. (2011) 5: 21. doi:10.1007/s11787-010-0023-0


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 00A3001A5501A6001A8503-0303-99Secondary 00A3500A6900A7100A7903B1003C5503G0530-0331-03


History and philosophy of pure and applied mathematicshistory and philosophy of symbolic logicsparts and momentsgenerality of theoriestheory changemonism and pluralismsets and multisetsabstract algebrasmetamathematicsmodel theory

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Middlesex University Business SchoolThe BurroughsHendon, LondonUK
  2. 2.Centre for Philosophy of Natural and Social ScienceLondon School of EconomicsLondonUK