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The need for closure

Published online by Cambridge University Press:  06 June 2019

Christopher D. Hollings
Christopher D. Hollings*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG The Queen’s College, Oxford OX1 4AW e-mail: [email protected]
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When defining a group, do we need to include closure? This is a detail that is often touched upon when the notion of a group is introduced to undergraduates. Should closure be listed as an axiom in its own right, or should it be regarded as an inherent property of the binary operation? There is no clear answer to this question, although there are firm opinions on both sides. Indeed, a very brief survey of group theory textbooks found in [1, pp. 458-459] suggests that there is a rough 50 : 50 split between authors who include closure explicitly and those who do not. In this Article, we go back to the beginning of the twentieth century to provide some historical perspective on this problem.

Type
Articles
Information
The Mathematical Gazette , Volume 103 , Issue 557 , July 2019 , pp. 248 - 256
Copyright
© Mathematical Association 2019 

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References

Christopher, D. Hollings, ‘Nobody could possibly misunderstand what a group is’: a study in early twentieth-century group axiomatics, Arch. Hist. Exact Sci. 71(5) (2017) pp. 409-481.Google Scholar
Jordan, C. R. and Jordan, D. A., Groups, Edward Arnold (1994).Google Scholar
David, Hilbert, Grundlagen der Geometrie, B. G. Teubner, Leipzig (1899).Google Scholar
Huntington, E. V., Simplified definition of a group, Bull. Amer. Math. Soc. 8(7) (1901-1902) pp. 296-300.10.1090/S0002-9904-1902-00898-7CrossRefGoogle Scholar
Heinrich, Weber, Lehrbuch der Algebra, 2 vols., Vieweg und Sohn, Braunschweig (1895, 1896); 2nd ed., inc. 3rd vol., (1898, 1899, 1908).Google Scholar
Eliakim, Hastings Moore, A definition of abstract groups. Trans. Amer. Math. Soc. 3(4) (1902) pp. 485-492; Erratum: ibid. 5(4) (1904) p. 549.Google Scholar
Huntington, E. V., Two definitions of an abelian group by sets of independent postulates, Trans. Amer. Math. Soc. 4 (1903) pp. 27-30.10.1090/S0002-9947-1903-1500621-2CrossRefGoogle Scholar
Huntington, E. V., Sets of independent postulates for the algebra of logic, Trans. Amer. Math. Soc. 5(3) (1904) pp. 288-309.10.1090/S0002-9947-1904-1500675-4CrossRefGoogle Scholar
Bell, E. T., The development of mathematics, McGraw-Hill (1940).Google Scholar
Edward, V. Huntington, A set of postulates for real algebra, comprising postulates for a one-dimensional continuum and for the theory of groups, Trans. Amer. Math. Soc. 6(1) (1905) pp. 17-41.Google Scholar
Edward, V. Huntington, Note on the definition of abstract groups and fields by sets of independent postulates, Trans. Amer. Math. Soc. 6(2) (1905) pp. 181-197; Errata: ibid. 7(4) (1906) p. 591.Google Scholar
Dickson, L. E., De Séguier’s theory of abstract groups, Bull. Amer. Math. Soc. 11 (1904) pp. 159-162.10.1090/S0002-9904-1904-01201-XCrossRefGoogle Scholar
Raffaella, Franci, On the axiomatization of group theory by American mathematicians: 1902-1905, in Demidov, S. S., Folkerts, M., Rowe, D. E. and Scriba, C. J (eds.), Amphora: Festschrift für Hans Wussing zu seinem 65, Geburtstag, Birkhäuser (1992) pp. 261-277.Google Scholar
Peter, M. Neumann, What groups were: a study of the development of the axiomatics of group theory, Bull. Austral. Math. Soc. 60 (1999) pp. 285-301.Google Scholar
Scanlan, M., Who were the American postulate theorists? J. Symbolic Logic 56(3) (1991) pp. 981-1002.CrossRefGoogle Scholar
Michael, Scanlan, American postulate theorists and Alfred Tarski, Hist. Phil. Logic 24(4) (2003) pp. 307-325.Google Scholar
Dirk, Schlimm, On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff and others, Synthese 183 (2011) pp. 47-68.Google Scholar
Leo, Corry, Modern algebra and the rise of mathematical structures, Birkhäuser (1996) (2nd revised edn., 2004).Google Scholar
Janet, Heine Barnett, Boolean algebra as an abstract structure: Edward V. Huntington and axiomatization, Convergence (July 2013).Google Scholar
Janet, Heine Barnett, An American postulate theorist: Edward V. Huntington, in Zack, Maria and Landry, Elaine (eds.), Research in History and Philosophy of Mathematics, Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques, Springer (2016) pp. 221-235.Google Scholar