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Published online by Cambridge University Press: 18 September 2009
It is absurd that a magnitude should be constituted from non-magnitudes.
Aristotle On Generation and Corruption 1.2Mathematical continuity, at least in the versions of Dedekind, Cantor, and their successors, is clearly not instantiated in experience. This raises the question of the relation of mathematical continuity to experience.
Stephan Korner “Continuity” The Encyclopedia of Philosophy (1967)We have maintained that point-positions and instants are not parts of space and time, respectively. Rather, we have taken the neo-Aristotelean view that such entities are dependent on places and times of higher dimensionality. Thus, we said that a point-position can be a limit of a line, or the place of a corner of a material object, or a place where two spheres touch, and so forth, but a point-position cannot exist apart from a place of higher than zero-dimensionality. Thus, our view has been an antifoundationalist one when it comes to space and time, one aspect of this antifoundationalism being that space and time are not composed of unextended parts.
However, many philosophers, taking their lead from certain mathematicians and, we believe, from the logicist tradition, hold that extended spaces and temporal intervals have a nondenumerable number of zero-dimensional parts. M. J. White has aptly described the contrast between the two views in question:
The tendency of contemporary mathematics, of course, has been to… [treat] continuous magnitudes as constituted of indivisible elements (e.g., sets of points) that are in a certain intuitive sense ‘discrete’. […]
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