The aim of the present article is to make the notion of an ontology of fields mathematically rigorous. The conclusion will be that couching an ontology in terms of mathematical bundles and cross‐sections both captures many important intuitions of conventional ontologies, including the universal‐particular paradigm, the connection of universals and their ‘instantiations’, and the notion of ‘possibility’, and makes possible the framing of ontologies without ‘substrata’, bare particulars, and primitive particularizers.

The aim of the present article is to make the notion of an ontology of fields mathematically rigorous. The conclusion will be that couching an ontology in terms of mathematical bundles and cross-sections (i.e. fields) both (1) captures many important intuitions of conventional ontologies, including the universal-particular paradigm, the connection of universals and their ‘instantiations’, and the notion of ‘possibility’, and (2) makes possible the framing of ontologies without ‘substrata’, bare particulars, and primitive particularizers (a goal that trope ontologies, for (...) example, have sought to attain). (shrink)

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...) with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...) those that are more readily available for mathematico-scientific use. (shrink)