Logica Universalis 5 (1):21-73 (2011)

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
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
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DOI 10.1007/s11787-010-0023-0
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References found in this work BETA

Logical Pluralism.Jc Beall & Greg Restall - 2005 - Oxford University Press.
Philosophy of Logic.W. V. O. Quine - 1970 - Harvard University Press.
Logical Pluralism.Jc Beall & Greg Restall - 2000 - Australasian Journal of Philosophy 78 (4):475 – 493.
Pluralism in Logic.Hartry Field - 2009 - Review of Symbolic Logic 2 (2):342-359.
Beyond the Limits of Thought.Graham Priest - 1995 - Cambridge University Press.

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Citations of this work BETA

The Power of the Hexagon.Jean-Yves Béziau - 2012 - Logica Universalis 6 (1-2):1-43.
A New–Old Characterisation of Logical Knowledge.Ivor Grattan-Guinness - 2012 - History and Philosophy of Logic 33 (3):245 - 290.

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