Continuum theories or models explain variation as involving gradual

quantitative Quantitative may refer to: * Quantitative research, scientific investigation of quantitative properties * Quantitative analysis (disambiguation) * Quantitative verse, a metrical system in poetry * Statistics, also known as quantitative analysis ...

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 (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...

in physics, such as the

electromagnetic spectrum The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies. The electromagnetic spectrum covers electromagnetic waves with frequencies ranging fro ...

, 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 a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

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 John Lane Bell (born March 25, 1945) is an Anglo-Canadian philosopher, mathematician and logician. He is Professor Emeritus of Philosophy at the University of Western Ontario in Canada. His research includes such topics as set theory, model theor ...

'
essay
in th
''Stanford Encyclopedia of Philosophy''
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 so-called three laws of thought, along with the law of noncontradi ...

. 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 Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...

's and Lawvere's, and classical ones such as Stevin's and

Robinson Robinson may refer to: People and names * Robinson (name) Fictional characters * Robinson Crusoe, the main character, and title of a novel by Daniel Defoe, published in 1719 Geography * Robinson projection, a map projection used since the 1960 ...

's. Bell isolates two distinct historical conceptions of infinitesimal, one by

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...

and one by

Nieuwentijdt Bernard Nieuwentijt, Nieuwentijdt, or Nieuwentyt (10 August 1654, West-Graftdijk, North Holland – 30 May 1718, Purmerend) was a Dutch philosopher, mathematician, physician, magistrate, mayor (of Purmerend), and theologian. Career As a ph ...

, and argues that Leibniz's conception was implemented in Robinson's

hyperreal Hyperreal may refer to: * Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis * Hyperreal.org, a rave culture website based in San Francisco, US * Hyperreality, a term used in semiotics and po ...

continuum, whereas Nieuwentijdt's, in Lawvere's

smooth infinitesimal analysis Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of bein ...

, 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

In social sciences (in general), psychology and psychiatry

social science Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of so ...

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. 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, which apply to

law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...

sociology Sociology is a social science that focuses on society, human social behavior, patterns of social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of empirical investigation an ...

and

ethics Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concer ...

Linguistics

linguistics Linguistics is the science, scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure ...

, the range of

dialect The term dialect (from Latin , , from the Ancient Greek word , 'discourse', from , 'through' and , 'I speak') can refer to either of two distinctly different types of linguistic phenomena: One usage refers to a variety of a language that is a ...

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 language varieties spoken across some geographical area such that neighboring varieties are mutually intelligible, but the differences accumulate over distance so that widely separated varie ...

. 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, sometimes referred to as Latin languages or Neo-Latin languages, are the various modern languages that evolved from Vulgar Latin. They are the only extant subgroup of the Italic languages in the Indo-European language ...

Arabic languages The varieties (or dialects or vernacular languages) of Arabic, a Semitic language within the Afroasiatic family originating in the Arabian Peninsula, are the linguistic systems that Arabic speakers speak natively. There are considerable vari ...

, or Bantu languages.

References

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External links

Continuity and infinitesimals
John Bell, Stanford Encyclopedia of Philosophy Concepts Concepts in metaphysics Concepts in physics Concepts in the philosophy of science History of mathematics History of philosophy History of science Philosophical concepts Philosophical theories Philosophy of mathematics Philosophy of science Theories