Lecture
Continuum (from the Latin continuum — continuous) — continuous, unbroken; a continuous totality; continuity amid a diversity of elements; an object possessing certain properties of continuity. Continuity expresses the integral character of an object, the homogeneity and interconnection of its parts (elements) and states. The term «continuum» is often replaced by the noun «continuity», «duration», which coincides with it in meaning. The principle of the continuum is opposed to the principle of atomism.
The term «continuum» is used in relation to the teachings of certain representatives of ancient Greek philosophy. Aleksei Fedorovich Losev, in his work on ancient aesthetics, describes the different approaches to the understanding of the continuous (Greek συνέχεια, synécheia) of the following philosophers: the elementarily-becoming continuum of the Eleatics, the absolute continuum of the Milesians, the general-material continuum of Heraclitus, the organically-living continuum of Empedocles, the intellectual-material continuum of Diogenes of Apollonia.
Continuality (continuitivity) — continuity, the absence of gaps, of quantization, of division into fragments , the transition of one thing into another . The philosopher M. K. Mamardashvili found that the inner space of the binary opposition called the «subject-object relation» is continual. The opposite of continuality — is discreteness.
Continuity is created by abstract thought and can be a property of an ideal object. In real objects, the transition from one element to another occurs suddenly, abruptly, through a «leap».
In philosophy there arises the problem of whether reality is continuous (a single reality) or discontinuous (a multiplicity of realities). A problem which, already in antiquity, gave rise to several aporias, which throughout history have generated a multitude of thought-forms; various forms of monisms and pluralisms, as well as the possibility or impossibility of change or motion.
In physics, for example, the model of the space-time continuum describes space and time as parts of a single continuum, rather than as separate objects. Spectra in physics, such as the electromagnetic spectrum, are often described as either continuous (with energy at all wavelengths) or discrete (with energy only at certain wavelengths).
By contrast, quantum mechanics uses quanta, definite fixed quantities (that is, categorial quantities), which are distinct from continuous quantities.
A good introduction to the philosophical questions involved is provided by John Lane Bell's essay in the Stanford Encyclopedia of Philosophy. An essential division is provided by the law of excluded middle. He defines the gap between intuitionistic continua, such as the continua of Brouwer and Lawvere, and classical continua, such as the continua of Stevin and Robinson. Bell distinguishes two different historical conceptions of the infinitesimal, one belonging to Leibniz and the other to Nieuwentijt, and argues that Leibniz's conception was realized in Robinson's hyperreal continuum, whereas Nieuwentijt's was realized in Lawvere's smooth infinitesimal analysis, characterized by the presence of nilsquare infinitesimals: «One might say that Leibniz recognized the need for the first, but not the second, type of infinitesimal, and Nieuwentijt the opposite. It should be noted that Leibnizian infinitesimals (differentials) are realized in nonstandard analysis, while nilsquare infinitesimals are realized in smooth infinitesimal analysis».
In the social sciences in general, including psychology and psychiatry, data on differences between people, like any data, can be collected and measured using different levels of measurement. These levels include the dichotomous (a person either has personality traits or does not) and the non-dichotomous approach. Whereas the non-dichotomous approach allows one to understand that everyone falls somewhere along a given dimension of personality, dichotomous (nominal-categorical and ordinal) approaches seek only to confirm that a particular person either has or does not suffer from a particular mental disorder.
Expert witnesses, in particular, are trained to help courts translate data into a legal dichotomy (for example, «guilty» or «not guilty»), which is applied to law, sociology and ethics.
In linguistics, the range of dialects spoken across a geographic area, which differ slightly between neighboring areas, is known as a dialect continuum. A language continuum is a similar description for the merging of neighboring languages without a clearly defined boundary. Examples of dialect or language continua include the varieties of Italian or German; and the Romance languages, the Arabic languages, or the Bantu languages.
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