(*) in progress
Liber secundus §§18-26: Magnets, Law III, and the Evidence for Universal Gravity*
[Forthcoming in The Evolution of Newton's Principia in 1685. Ed. George E. Smith.]
Force Mereology and Corollary 6 to the Laws of Motion*
[Forthcoming in Newtonian Relativity. Eds. Robert DiSalle and George E. Smith.]
Division of Mass
[Removed for blind review process]
Newton's Argument for Universal Gravitation: A Methodological Interpretation
Against Descartes’ admission of algebraic curves into mathematical practice, Newton’s Principia urges a return to the Euclidean paradigm where geometry is a part of ‘general mechanics’ and geometric objects are the results of quasi-causal motions (e.g. the rectilinear motion of a point generates a line segment). This position borrows from Isaac Barrow’s idea that real definitions of geometric curves may postulate the mechanical causes (tracing mechanisms) of their generation. Though the Principia is deliberately divided into mathematical (Books I and II) and physical (Book III) portions, Newton clearly views this quasi-causal conception of mathematics as central to his method in deriving universal gravity in Book III. But how are the mechanical quasi-causes generating mathematical curves related to the physical causes generating real motions? I offer a reading connecting the two halves of the Principia – mathematical and physical – by illustrating how the success of Newton’s method of successive approximations in Book III may depend upon some quasi-causal features of his conception of mathematics. In particular, I try to connect the plurality of possible tracing mechanisms to Newton’s implicit taxonomy of measures of centripetal force.
Asymmetry of Intuition and Kant's Theory of Physical Space*
Kant resolves the first two ('mathematical') antinomies asymmetrically. Is the world finite or infinite? Our inquiry proceeds ad indefinitum. Is matter composed of simples? Our inquiry proceeds ad infinitum. Yet the indefinite / infinite distinction is an 'empty subtlety' in geometry. This troubles straightforward interpretations of transcendental idealism which simply identify geometric space and physical space. This has led to the impression that the asymmetry has little to do with the nature of transcendental idealism or with the indirect proof. This impression is false. I argue that the asymmetry must arise from the distinctive hylomorphic structure of experience that characterizes Kant's idealism. It is not the form of intuition alone, nor the matter of intuition alone, that explains the asymmetry; it is the matter of some form.
Mereology and the 'Role of Intuition in Geometry' Debate
Kant held that mathematical theorems are grounded in sensibility and essentially involve intuition. Interpretations of why he held this view disagree on the basic meaning of intuition. In the context of geometric reasoning, phenomenological readings argue that intuitions are primarily immediate representations that verify Euclidean axioms, while logical readings argue that intuitions are primarily singular representations that underwrite the introduction and iteration of constructed individuals. Kant's Notes on Kästner reframed the debate and pushed it in a new direction. Here, Kant distinguishes metaphysical space and geometric space: the former is actually infinite and subjectively given, the latter is potentially infinite and constructed. Phenomenological readings saw the Notes as vindication of their view that intuitions present spatial facts immediately to the mind, as in perception; and logical readings attempted to accommodate considerations congenial to the phenomenological viewpoint. In this paper, I contend the responses of both readings to Notes on Kästner are instructive overreaches. The role of intuition in elementary settings like geometry and perception cannot be used straightforwardly to model the role of intuition in experience.
The Metaphysics of Material Substance in the Context of Copernicanism*