Continuum (theory)
Encyclopedia
Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.
explains space and time as part of the same continuum rather than as separate entities. A spectrum in physics (e.g. of light) is often termed either a 'continuous spectrum' (energy at all wavelengths) or 'discrete spectrum' (energy at only certain wavelengths).
In contrast, quantum mechanics
includes quanta
, which are distinguished from continuous amounts.
'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
".
, theories of mental phenomena can propose discrete differences between individuals (e.g. everyone has certain personality traits and not others) or a continuum (e.g. everyone lies somewhere on a particular personality dimension). This can also apply to fields such as law
or sociology
or ethics
in explaining or judging variation in human behavior.
In clinical psychology
or psychiatry
, categorical models seek to distinguish and define particular mental disorders or illnesses, whilst continuum or dimensional models propose that some people are more extreme than others on particular dimensions.
In physics
In physics, for example, the space-time continuum modelSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
explains space and time as part of the same continuum rather than as separate entities. A spectrum in physics (e.g. of light) is often termed either a 'continuous spectrum' (energy at all wavelengths) or 'discrete spectrum' (energy at only certain wavelengths).
In contrast, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
includes quanta
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...
, which are distinguished from continuous amounts.
In mathematics and philosophy
A good introduction to the philosophical issues involved is John Lane BellJohn Lane Bell
John Bell is Professor of Philosophy at the University of Western Ontario in Canada. He is an outstanding figure in mathematical logic and philosophy...
's essay in the Stanford Encyclopedia of Philosophy. A significant divide is provided by the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
. It determines the divide between intuitionistic
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
continua such as Brouwer
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
's and Lawvere's, and classical ones such as Stevin
Simon Stevin
Simon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...
's and Robinson
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
's.
Bell isolates two distinct historical conceptions of infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
, one by Leibniz and one by Nieuwentijdt, and argues that Leibniz's conception was implemented in Robinson's hyperreal
Hyperreal number
The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
continuum, whereas Nieuwentijdt's, in Lawvere's smooth infinitesimal analysis
Smooth infinitesimal analysis
Smooth infinitesimal analysis is a mathematically rigorous 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 being expressed in terms of discrete...
, 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
Smooth infinitesimal analysis
Smooth infinitesimal analysis is a mathematically rigorous 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 being expressed in terms of discrete...
".
In psychology
In psychologyPsychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
, theories of mental phenomena can propose discrete differences between individuals (e.g. everyone has certain personality traits and not others) or a continuum (e.g. everyone lies somewhere on a particular personality dimension). This can also apply to fields such as law
Law
Law is a system of rules and guidelines which are enforced through social institutions to govern behavior, wherever possible. It shapes politics, economics and society in numerous ways and serves as a social mediator of relations between people. Contract law regulates everything from buying a bus...
or sociology
Sociology
Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...
or ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
in explaining or judging variation in human behavior.
In clinical psychology
Clinical psychology
Clinical psychology is an integration of science, theory and clinical knowledge for the purpose of understanding, preventing, and relieving psychologically-based distress or dysfunction and to promote subjective well-being and personal development...
or psychiatry
Psychiatry
Psychiatry is the medical specialty devoted to the study and treatment of mental disorders. These mental disorders include various affective, behavioural, cognitive and perceptual abnormalities...
, categorical models seek to distinguish and define particular mental disorders or illnesses, whilst continuum or dimensional models propose that some people are more extreme than others on particular dimensions.
External links
- John Bell, Stanford Encyclopedia of Philosophy, Continuity and infinitesimals