A tale of mathematical myth-making: E T Bell and the ‘arithmetization of algebra’

† This article is an expanded version of Section 4.2 of Hollings (2014b), and, as such, contains the outcome of research carried out at the Mathematical Institute of the University of Oxford with the support of research project grant F/08 772/F from the Leverhulme Trust.

Abstract

Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

† This article is an expanded version of Section 4.2 of Hollings (2014b), and, as such, contains the outcome of research carried out at the Mathematical Institute of the University of Oxford with the support of research project grant F/08 772/F from the Leverhulme Trust.

¹ The first two decades of Bell’s life are shrouded in some mystery—a mystery that Bell appears to have cultivated deliberately: see Reid (2001). For a full biography of Bell, see Reid (1993).

² I do not propose to give a comprehensive account of postulational analysis. Fuller surveys of this subject may be found in Corry (2000, §2), Corry (1996, 2nd edition, §3.5), Scanlan (1991) and Schlimm (2011). On the possible nineteenth-century origins of postulational analysis, see Grattan-Guinness (2000) and Parshall (2011).

³ On the theses of Ward and Clifford, see, respectively, Sections 4.3 and 4.5 of Hollings (2014b); on Poole’s dissertation, see the scattered comments in Sections 4.2, 6.5, 6.6 and 8.4. On Ward’s thesis, see also Goodstein and Babbitt (2013, 692). Of the dissertations cited here, it is Ward’s that is the closest in spirit to Bell’s ideas on algebraic arithmetic: it seeks an axiomatization of basic arithmetic, with a particular focus on factorization properties. Worth’s dissertation enumerates various different types of ‘subvarieties’ (in Bell’s terminology) of an abstract field.