Methodological Reflections and Logical Explorations

Abstract Chapter 5 contains methodological reflections and the presentation of some recent mathematical theories relevant to the investigation of mathematical infinity. It discusses the relevance of two conflicting intuitions concerning the size of infinite collections, PART-WHOLE, defended by Bolzano, and ONE-ONE, used by Cantor in his definition of cardinal numbers. The chapter shows that novel mathematical theories of infinity and the continuum (non-standard analysis and the theory of numerosities) now extend our hermeneutical resources and that ideas that could not be made sense of using classical analysis or Cantorian set theory can nonetheless be given a solid footing using these alternative mathematical developments. Through non-standard analysis, it is possible to make sense of Grosseteste’s summations of infinite series as yielding an infinite number, and by means of the applications of the theories of numerosities to measure theory, it is possible to implement Grosseteste’s desiderata concerning the covariation of the numerosity of points in line segments and their lengths.