Continuum (: continua or continuums) theories or models explain 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

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 A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

in physics, such as the

electromagnetic spectrum The electromagnetic spectrum is the full range of electromagnetic radiation, organized by frequency or wavelength. The spectrum is divided into separate bands, with different names for the electromagnetic waves within each band. From low to high ...

, is often termed as either continuous (with energy at all wavelengths) or discrete (energy at only certain wavelengths). In contrast,

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

uses quanta, certain defined amounts (i.e. categorical amounts) which are distinguished from continuous amounts.

In mathematics and philosophy

A good introduction to the philosophical issues involved is John Lane Bell's essay in the ''

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...

''. A significant divide is provided by the

law of excluded middle In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...

. It determines the divide between

intuitionistic In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...

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 In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

, one by

Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...

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 The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

, and nilsquare infinitesimals in smooth infinitesimal analysis".

In social sciences, psychology and psychiatry

social science Social science (often rendered in the plural as the social sciences) is one of the branches of science, devoted to the study of societies and the relationships among members within those societies. The term was formerly used to refer to the ...

s in general, psychology and psychiatry included, data about differences between individuals, like any data, can be collected and measured using different

levels of measurement Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to dependent and independent variables, variables. Psychologist Stanley Smith Stevens developed the best-known class ...

. Those levels include dichotomous (a person either has a personality trait or not) and non-dichotomous approaches. While the non-dichotomous approach allows for understanding that everyone lies somewhere on a particular personality dimension, the dichotomous (nominal categorical and ordinal) approaches only seek to confirm that a particular person either has or does not have a particular mental disorder. Expert witnesses particularly are trained to help courts in translating the data into the legal (e.g. 'guilty' vs. 'not guilty')

dichotomy A dichotomy () is a partition of a set, partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothi ...

, which apply to

law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a science and as the ar ...

sociology Sociology is the scientific study of human society that focuses on society, human social behavior, patterns of Interpersonal ties, social relationships, social interaction, and aspects of culture associated with everyday life. The term sociol ...

and

ethics Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...

In linguistics

linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...

, the range of

dialect A dialect is a Variety (linguistics), variety of language spoken by a particular group of people. This may include dominant and standard language, standardized varieties as well as Vernacular language, vernacular, unwritten, or non-standardize ...

s spoken over a geographical area that differ slightly between neighboring areas is known as a

dialect continuum A dialect continuum or dialect chain is a series of Variety (linguistics), language varieties spoken across some geographical area such that neighboring varieties are Mutual intelligibility, mutually intelligible, but the differences accumulat ...

. A language continuum is a similar description for the merging of neighboring languages without a clear defined boundary. Examples of dialect or language continuums include the varieties of Italian or German; and the

Romance languages The Romance languages, also known as the Latin or Neo-Latin languages, are the languages that are Language family, directly descended from Vulgar Latin. They are the only extant subgroup of the Italic languages, Italic branch of the Indo-E ...

, Arabic languages, or

Bantu languages The Bantu languages (English: , Proto-Bantu language, Proto-Bantu: *bantʊ̀), or Ntu languages are a language family of about 600 languages of Central Africa, Central, Southern Africa, Southern, East Africa, Eastern and Southeast Africa, South ...

References

{{reflist

External links

Continuity and infinitesimals
John Bell, Stanford Encyclopedia of Philosophy Concepts in metaphysics Concepts in physics Concepts in the philosophy of science Mathematical concepts