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Continuum (pl.: 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 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.
In mathematics and philosophy
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".
In social sciences, psychology and psychiatry
In social sciences 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, sociology and ethics.
In linguistics
In linguistics, the range of dialects spoken over a geographical area that 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 clear defined boundary. Examples of dialect or language continuums include the varieties of Italian or German; and the Romance languages, Arabic languages, or Bantu languages.
References
- Stevens, S. S. (1946). "On the Theory of Scales of Measurement". Science. 103 (2684): 677–680. Bibcode:1946Sci...103..677S. doi:10.1126/science.103.2684.677. PMID 17750512.
- Bell, John L. (2005-07-27). "Continuity and Infinitesimals". Stanford Encyclopedia of Philosophy. Retrieved 2023-10-03.
External links
- Continuity and infinitesimals, John Bell, Stanford Encyclopedia of Philosophy
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Continuum measurement news newspapers books scholar JSTOR July 2022 Learn how and when to remove this message It has been suggested that this article be split into multiple articles Discuss October 2023 Continuum pl 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 physicsIn 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 In mathematics and philosophyA 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 In social sciences psychology and psychiatryIn social sciences 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 sociology and ethics In linguisticsIn linguistics the range of dialects spoken over a geographical area that 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 clear defined boundary Examples of dialect or language continuums include the varieties of Italian or German and the Romance languages Arabic languages or Bantu languages ReferencesStevens S S 1946 On the Theory of Scales of Measurement Science 103 2684 677 680 Bibcode 1946Sci 103 677S doi 10 1126 science 103 2684 677 PMID 17750512 Bell John L 2005 07 27 Continuity and Infinitesimals Stanford Encyclopedia of Philosophy Retrieved 2023 10 03 External linksContinuity and infinitesimals John Bell Stanford Encyclopedia of Philosophy
