Whitehead's early philosophy of mathematics and the development of formalism (Q2910353)

The author (Bulgarian Academy of Sciences, Sofia) seeks to assess the meaning and significance of \textit{A. N. Whitehead}'s [A treatise on universal algebra. Vol. I. Cambridge: The University Press (1898; JFM 29.0066.03)] and to rescue it from scholarly oblivion. In order to do so, the author sketches the history of formalism according to \textit{M. Detlefsen} [``Formalism'', in: The Oxford handbook of philosophy of mathematics and logic. Oxford: Oxford University Press. 236--317 (2005; Zbl 1081.03001)] and focuses especially on Whitehead's \textit{Treatise} as the missing link in Detlefsen's account between \textit{G. Peacock}'s [A treatise on algebra (1830)] and \textit{D. Hilbert}'s [Grundlagen der Geometrie. Leipzig: B. G. Teubner (1899; JFM 30.0424.01)].NEWLINENEWLINESymbolic algebra is introduced by Peacock (1830), who acknowledges that it is a ``science of suggestion'' grounded in ``arbitrary assumptions'' and endowed with laws that are actually ``arbitrary conventions''. Peacock makes furthermore plain its two general requirements: consistency -- hence the ``Principle of Permanence of Equivalent Forms'' (\S132) -- and usefulness. Boole's \textit{Laws of thought} (1854) admits non-interpretable subsidiary algebraic forms such as imaginary numbers but remains dissatisfied with the consistency gap. Hilbert's program, in the making since in his work on the foundation of geometry in the 1890s, is seen by Detlefsen as the direct heir of Peacock.NEWLINENEWLINEThe author argues that Whitehead's \textit{Algebra} clearly anticipates Hilbert in conceiving the mathematical work not as a purely intellectual operation on mental contents but as a physical manipulation of symbols. In order to deal with imaginary numbers, Whitehead enforces the differentiation of ``pure'' and ``applied mathematics''. The former has no ``existential import'' (Whitehead's expression) whereas the latter, by definition, does: concepts of applied mathematics require truth. As the author himself remarks in conclusion, the actual demonstration of his thesis is still to be provided.