- published: 09 May 2016
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Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.
In physics, for example, the space-time continuum model explains space and time as part of the same continuum rather than as separate entities. A spectrum in physics (e.g. of light) is often termed either a 'continuous spectrum' (energy at all wavelengths) or 'discrete spectrum' (energy at only certain wavelengths).
In contrast, quantum mechanics includes quanta, which are distinguished from continuous amounts.
A good introduction to the philosophical issues involved is John Lane Bell's essay in the Stanford Encyclopedia of Philosophy. A significant divide is provided by the law of excluded middle. It determines the divide between intuitionistic continua such as Brouwer's and Lawvere's, and classical ones such as Stevin's and Robinson's. Bell isolates two distinct historical conceptions of infinitesimal, one by Leibniz and one by Nieuwentijdt, and argues that Leibniz's conception was implemented in Robinson's hyperreal continuum, whereas Nieuwentijdt's, in Lawvere's smooth infinitesimal analysis, characterized by the presence of nilsquare infinitesimals: "It may be said that Leibniz recognized the need for the first, but not the second type of infinitesimal and Nieuwentijdt, vice-versa. It is of interest to note that Leibnizian infinitesimals (differentials) are realized in nonstandard analysis, and nilsquare infinitesimals in smooth infinitesimal analysis".