Continuum (theory)

Continuum theories or models explains variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variation using qualitatively different states.

In physics, for example, the space-time continuum model describes space and time as part of the same continuum rather than as separate entities. A spectrum in physics, such as the electromagnetic spectrum, is often termed as either continuous (with energy at all wavelengths) or discrete (energy at only certain wavelengths).

In contrast, quantum mechanics uses quanta, certain defined amounts (i.e. categorical amounts) 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".

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