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120-507: In the social sciences , distance can refer to a qualitative measurement of separation, such as social distance or psychological distance . The distance between physical locations can be defined in different ways in different contexts. The distance between two points in physical space is the length of a straight line between them, which is the shortest possible path. This is the usual meaning of distance in classical physics , including Newtonian mechanics . Straight-line distance

240-494: A degree in the subject. The classic brief definition of economics, set out by Lionel Robbins in 1932, is "the science which studies human behavior as a relationship between ends and scarce means which have alternative uses". Without scarcity and alternative uses, there is no economic problem . Briefer yet is "the study of how people seek to satisfy needs and wants" and "the study of the financial aspects of human behavior". Economics has two broad branches: microeconomics , where

360-519: A graph , the distance between two vertices is measured by the length of the shortest edge path between them. For example, if the graph represents a social network , then the idea of six degrees of separation can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the Erdős number and the Bacon number —the number of collaborative relationships away

480-506: A ruler , or indirectly with a radar (for long distances) or interferometry (for very short distances). The cosmic distance ladder is a set of ways of measuring extremely long distances. The straight-line distance between two points on the surface of the Earth is not very useful for most purposes, since we cannot tunnel straight through the Earth's mantle . Instead, one typically measures

600-611: A science of society based on historical materialism , becoming recognized as a founding figure of sociology posthumously as the term gained broader meaning. Around the start of the 20th century, the first wave of German sociologists, including Max Weber and Georg Simmel , developed sociological antipositivism. The field may be broadly recognized as an amalgam of three modes of social thought in particular: Durkheimian positivism and structural functionalism ; Marxist historical materialism and conflict theory ; and Weberian antipositivism and verstehen analysis. American sociology broadly arose on

720-416: A statistical manifold . The most elementary is the squared Euclidean distance , which is minimized by the least squares method; this is the most basic Bregman divergence . The most important in information theory is the relative entropy ( Kullback–Leibler divergence ), which allows one to analogously study maximum likelihood estimation geometrically; this is an example of both an f -divergence and

840-457: A "system of rules", as an "interpretive concept" to achieve justice, as an "authority" to mediate people's interests, and even as "the command of a sovereign, backed by the threat of a sanction". However one likes to think of law, it is a completely central social institution. Legal policy incorporates the practical manifestation of thinking from almost every social science and the humanities. Laws are politics, because politicians create them. Law

960-547: A Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are flat manifolds in the corresponding geometry, allowing an analog of the Pythagorean theorem (which holds for squared Euclidean distance) to be used for linear inverse problems in inference by optimization theory . Other important statistical distances include the Mahalanobis distance and

1080-402: A Euclidean space E of dimension n , the choice of a point, called an origin and an orthonormal basis of E → {\displaystyle {\overrightarrow {E}}} defines an isomorphism of Euclidean spaces from E to R n . {\displaystyle \mathbb {R} ^{n}.} As every Euclidean space of dimension n is isomorphic to it,

1200-636: A Euclidean space and E → {\displaystyle {\overrightarrow {E}}} its associated vector space. A flat , Euclidean subspace or affine subspace of E is a subset F of E such that F → = { P Q → | P ∈ F , Q ∈ F } ( {\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}} as

1320-678: A Euclidean space can also be said about R n . {\displaystyle \mathbb {R} ^{n}.} Therefore, many authors, especially at elementary level, call R n {\displaystyle \mathbb {R} ^{n}} the standard Euclidean space of dimension n , or simply the Euclidean space of dimension n . A reason for introducing such an abstract definition of Euclidean spaces, and for working with E n {\displaystyle \mathbb {E} ^{n}} instead of R n {\displaystyle \mathbb {R} ^{n}}

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1440-444: A Euclidean space is the dimension of its associated vector space. The elements of E are called points , and are commonly denoted by capital letters. The elements of E → {\displaystyle {\overrightarrow {E}}} are called Euclidean vectors or free vectors . They are also called translations , although, properly speaking, a translation is the geometric transformation resulting from

1560-419: A Euclidean space that has itself as the associated vector space. A typical case of Euclidean vector space is R n {\displaystyle \mathbb {R} ^{n}} viewed as a vector space equipped with the dot product as an inner product . The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given

1680-406: A Euclidean subspace of direction V → {\displaystyle {\overrightarrow {V}}} . (The associated vector space of this subspace is V → {\displaystyle {\overrightarrow {V}}} .) A Euclidean vector space E → {\displaystyle {\overrightarrow {E}}} (that is, a Euclidean space that

1800-471: A ball thrown straight up, or the Earth when it completes one orbit . This is formalized mathematically as the arc length of the curve. The distance travelled may also be signed : a "forward" distance is positive and a "backward" distance is negative. Circular distance is the distance traveled by a point on the circumference of a wheel , which can be useful to consider when designing vehicles or mechanical gears (see also odometry ). The circumference of

1920-454: A degree of autonomy as practitioners from various disciplines share similar goals and methods. The history of the social sciences began in the Age of Enlightenment after 1651, which saw a revolution within natural philosophy , changing the basic framework by which individuals understood what was scientific. Social sciences came forth from the moral philosophy of the time and were influenced by

2040-422: A degree of difference or separation between similar objects. This page gives a few examples. In statistics and information geometry , statistical distances measure the degree of difference between two probability distributions . There are many kinds of statistical distances, typically formalized as divergences ; these allow a set of probability distributions to be understood as a geometrical object called

2160-421: A diversity of research methods, collecting both quantitative and qualitative data, draw upon empirical techniques, and engage critical theory. Common modern methods include case studies, historical research , interviewing, participant observation , social network analysis , survey research, statistical analysis, and model building, among other approaches. Since the late 1970s, many sociologists have tried to make

2280-662: A foundation to practical social research. Durkheim set up the first European department of sociology at the University of Bordeaux in 1895, publishing his Rules of the Sociological Method . In 1896, he established the journal L'Année sociologique . Durkheim's seminal monograph, Suicide (1897), a case study of suicide rates among Catholic and Protestant populations, distinguished sociological analysis from psychology or philosophy. Karl Marx rejected Comte's positivism but nevertheless aimed to establish

2400-452: A heavy science basis, or heavy social science basis to their degree. If they applied to read the BA . for example, but specialized in heavily science-based modules, then they will still generally be awarded the BA . Sociology is the systematic study of society, individuals' relationship to their societies, the consequences of difference, and other aspects of human social action . The meaning of

2520-921: A humanities-based subject is contested. In the United States the National Endowment for the Humanities includes history in its definition of humanities (as it does for applied linguistics). However, the National Research Council classifies history as a social science. The historical method comprises the techniques and guidelines by which historians use primary sources and other evidence to research and then to write history . The Social Science History Association , formed in 1976, brings together scholars from numerous disciplines interested in social history . The social science of law, jurisprudence, in common parlance, means

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2640-610: A life of its own after World War II, influencing literary criticism and the Birmingham School establishment of cultural studies . Sociology evolved as an academic response to the challenges of modernity , such as industrialization , urbanization , secularization , and a perceived process of enveloping rationalization . The field generally concerns the social rules and processes that bind and separate people not only as individuals, but as members of associations , groups , communities and institutions , and includes

2760-479: A mutual contempt for one another, the former looking down on the latter as unscientific, the latter regarding the former as philistine . […] The difference comes down to the fact that social science really wants to be predictive, meaning that man is predictable, while the humanities say that he is not. The social science disciplines are branches of knowledge taught and researched at the college or university level. Social science disciplines are defined and recognized by

2880-517: A particular point in time), and a great deal of it—partly owing to the influence of Noam Chomsky —aims at formulating theories of the cognitive processing of language. However, language does not exist in a vacuum, or only in the brain, and approaches like contact linguistics, creole studies, discourse analysis , social interactional linguistics, and sociolinguistics explore language in its social context. Sociolinguistics often makes use of traditional quantitative analysis and statistics in investigating

3000-499: A people in detail, using biogenetic, archaeological, and linguistic data alongside direct observation of contemporary customs. In the 1990s and 2000s, calls for clarification of what constitutes a culture , of how an observer knows where his or her own culture ends and another begins, and other crucial topics in writing anthropology were heard. It is possible to view all human cultures as part of one large, evolving global culture. These dynamic relationships, between what can be observed on

3120-468: A person is from prolific mathematician Paul Erdős and actor Kevin Bacon , respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations. In psychology , human geography , and the social sciences , distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience. For example, psychological distance

3240-447: A rule that (unlike a rule of ethics ) is capable of enforcement through institutions. However, many laws are based on norms accepted by a community and thus have an ethical foundation. The study of law crosses the boundaries between the social sciences and humanities, depending on one's view of research into its objectives and effects. Law is not always enforceable, especially in the international relations context. It has been defined as

3360-623: A separate trajectory, with little Marxist influence, an emphasis on rigorous experimental methodology, and a closer association with pragmatism and social psychology . In the 1920s, the Chicago school developed symbolic interactionism . Meanwhile, in the 1930s, the Frankfurt School pioneered the idea of critical theory , an interdisciplinary form of Marxist sociology drawing upon thinkers as diverse as Sigmund Freud and Friedrich Nietzsche . Critical theory would take on something of

3480-439: A shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets ) of the plane should be considered equivalent ( congruent ) if one can be transformed into

3600-459: A strong (or entire) scientific concentration, whereas a BA underlines a majority of social science credits. This is not always necessarily the case however, and in many UK institutions students studying the BPsy , BSc , and BA follow the same curriculum as outlined by The British Psychological Society and have the same options of specialism open to them regardless of whether they choose a balance,

3720-498: A theoretical structure. The development of social science subfields became very quantitative in methodology. The interdisciplinary and cross-disciplinary nature of scientific inquiry into human behaviour, social and environmental factors affecting it, made many of the natural sciences interested in some aspects of social science methodology. Examples of boundary blurring include emerging disciplines like social research of medicine , sociobiology , neuropsychology , bioeconomics and

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3840-569: A vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of + , it suffices to look at the nature of its left argument. The fact that the action is free and transitive means that, for every pair of points ( P , Q ) , there is exactly one displacement vector v such that P + v = Q . This vector v is denoted Q − P or P Q → ) . {\displaystyle {\overrightarrow {PQ}}{\vphantom {\frac {)}{}}}.} As previously explained, some of

3960-445: Is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality". In sociology , social distance describes the separation between individuals or social groups in society along dimensions such as social class , race / ethnicity , gender or sexuality . Most of the notions of distance between two points or objects described above are examples of

4080-400: Is a linear subspace of E → , {\displaystyle {\overrightarrow {E}},} then P + V → = { P + v | v ∈ V → } {\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}} is

4200-423: Is a relatively autonomous term, encompassing the work of practitioners from various disciplines that share in its aims and methods. Social scientists employ a range of methods in order to analyse a vast breadth of social phenomena; from census survey data derived from millions of individuals, to the in-depth analysis of a single agent's social experiences; from monitoring what is happening on contemporary streets, to

4320-500: Is a social science that seeks to analyze and describe the production, distribution, and consumption of wealth. The word "economics" is from the Ancient Greek οἶκος ( oikos , "family, household, estate") and νόμος ( nomos , "custom, law"), and hence means "household management" or "management of the state". An economist is a person using economic concepts and data in the course of employment, or someone who has earned

4440-402: Is a very broad science that is rarely tackled as a whole, major block. Although some subfields encompass a natural science base and a social science application, others can be clearly distinguished as having little to do with the social sciences or having a lot to do with the social sciences. For example, biological psychology is considered a natural science with a social scientific application (as

4560-417: Is also called the direction of F . If P is a point of F then F = { P + v | v ∈ F → } . {\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.} Conversely, if P is a point of E and V → {\displaystyle {\overrightarrow {V}}}

4680-411: Is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are". The distance travelled by an object is the length of a specific path travelled between two points, such as the distance walked while navigating a maze . This can even be a closed distance along a closed curve which starts and ends at the same point, such as

4800-543: Is an academic and applied field involving the study of behaviour and mental processes. Psychology also refers to the application of such knowledge to various spheres of human activity, including problems of individuals' daily lives and the treatment of mental illness . The word psychology comes from the Ancient Greek ψυχή ( psyche , "soul" or "mind") and the suffix logy ("study"). Psychology differs from anthropology, economics, political science, and sociology in seeking to capture explanatory generalizations about

4920-642: Is an arbitrary point (not necessary on the line). In a Euclidean vector space, the zero vector is usually chosen for O ; this allows simplifying the preceding formula into { ( 1 − λ ) P + λ Q | λ ∈ R } . {\displaystyle {\bigl \{}(1-\lambda )P+\lambda Q\mathrel {\big |} \lambda \in \mathbb {R} {\bigr \}}.} A standard convention allows using this formula in every Euclidean space, see Affine space § Affine combinations and barycenter . The line segment , or simply segment , joining

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5040-547: Is an area that is offered at most undergraduate institutions. The word anthropos ( ἄνθρωπος ) in Ancient Greek means "human being" or "person". Eric Wolf described sociocultural anthropology as "the most scientific of the humanities, and the most humanistic of the sciences". The goal of anthropology is to provide a holistic account of humans and human nature. This means that, though anthropologists generally specialize in only one sub-field, they always keep in mind

5160-450: Is clinical medicine), social and occupational psychology are, generally speaking, purely social sciences, whereas neuropsychology is a natural science that lacks application out of the scientific tradition entirely. In British universities, emphasis on what tenet of psychology a student has studied and/or concentrated is communicated through the degree conferred: BPsy indicates a balance between natural and social sciences, BSc indicates

5280-421: Is equal to E → {\displaystyle {\overrightarrow {E}}} ) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector. In a Euclidean space, a line is a Euclidean subspace of dimension one. Since a vector space of dimension one

5400-429: Is formalized mathematically as the Euclidean distance in two- and three-dimensional space . In Euclidean geometry , the distance between two points A and B is often denoted | A B | {\displaystyle |AB|} . In coordinate geometry , Euclidean distance is computed using the Pythagorean theorem . The distance between points ( x 1 , y 1 ) and ( x 2 , y 2 ) in

5520-515: Is philosophy, because moral and ethical persuasions shape their ideas. Law tells many of history's stories, because statutes, case law and codifications build up over time. And law is economics, because any rule about contract , tort , property law , labour law , company law and many more can have long-lasting effects on the distribution of wealth. The noun law derives from the Old English lagu , meaning something laid down or fixed and

5640-434: Is spanned by any nonzero vector, a line is a set of the form { P + λ P Q → | λ ∈ R } , ( {\displaystyle {\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}} where P and Q are two distinct points of

5760-406: Is that it is often preferable to work in a coordinate-free and origin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world. A Euclidean vector space is a finite-dimensional inner product space over the real numbers . A Euclidean space is an affine space over

5880-467: Is the continuous, systematic narrative and research into past human events as interpreted through historiographical paradigms or theories. When used as the name of a field of study , history refers to the study and interpretation of the record of humans , societies, institutions, and any topic that has changed over time. Traditionally, the study of history has been considered a part of the humanities . In modern academia , whether or not history remains

6000-431: Is the holistic "science of man", a science of the totality of human existence. The discipline deals with the integration of different aspects of the social sciences, humanities , and human biology . In the twentieth century, academic disciplines have often been institutionally divided into three broad domains. Firstly, the natural sciences seek to derive general laws through reproducible and verifiable experiments. Secondly,

6120-641: The Age of Revolutions , such as the Industrial Revolution and the French Revolution . The social sciences developed from the sciences ( experimental and applied ), or the systematic knowledge-bases or prescriptive practices, relating to the social improvement of a group of interacting entities . The beginnings of the social sciences in the 18th century are reflected in the grand encyclopedia of Diderot , with articles from Jean-Jacques Rousseau and other pioneers. The growth of

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6240-552: The Latin educare , or to facilitate the realization of an individual's potential and talents. It is an application of pedagogy , a body of theoretical and applied research relating to teaching and learning and draws on many disciplines such as psychology, philosophy , computer science , linguistics, neuroscience , sociology and anthropology. Geography as a discipline can be split broadly into two main sub fields: human geography and physical geography . The former focuses largely on

6360-536: The Platonic solids ) that exist in Euclidean spaces of any dimension. Despite the wide use of Descartes' approach, which was called analytic geometry , the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to

6480-496: The United States and Europe . Another route undertaken was initiated by Émile Durkheim , studying "social facts", and Vilfredo Pareto , opening metatheoretical ideas and individual theories. A third means developed, arising from the methodological dichotomy present, in which social phenomena were identified with and understood; this was championed by figures such as Max Weber . The fourth route taken, based in economics,

6600-411: The academic journals in which research is published, and the learned social science societies and academic departments or faculties to which their practitioners belong. Social science fields of study usually have several sub-disciplines or branches, and the distinguishing lines between these are often both arbitrary and ambiguous. The following are widely-considered to be social sciences: Anthropology

6720-418: The action of a Euclidean vector on the Euclidean space. The action of a translation v on a point P provides a point that is denoted P + v . This action satisfies P + ( v + w ) = ( P + v ) + w . {\displaystyle P+(v+w)=(P+v)+w.} Note: The second + in the left-hand side is a vector addition; each other + denotes an action of

6840-496: The built environment and how space is created, viewed and managed by humans as well as the influence humans have on the space they occupy. This may involve cultural geography , transportation , health , military operations , and cities . The latter examines the natural environment and how the climate, vegetation and life, soil, oceans , water and landforms are produced and interact (is also commonly regarded as an Earth Science ). Physical geography examines phenomena related to

6960-418: The dot product is a Euclidean space of dimension n . Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and R n {\displaystyle \mathbb {R} ^{n}} viewed as a Euclidean space. It follows that everything that can be said about

7080-510: The energy distance . In computer science , an edit distance or string metric between two strings measures how different they are. For example, the words "dog" and "dot", which differ by just one letter, are closer than "dog" and "cat", which have no letters in common. This idea is used in spell checkers and in coding theory , and is mathematically formalized in a number of different ways, including Levenshtein distance , Hamming distance , Lee distance , and Jaro–Winkler distance . In

7200-419: The history and sociology of science . Increasingly, quantitative research and qualitative methods are being integrated in the study of human action and its implications and consequences. In the first half of the 20th century, statistics became a free-standing discipline of applied mathematics. Statistical methods were used confidently. In the contemporary period, Karl Popper and Talcott Parsons influenced

7320-533: The law , education , health , economy and trade , and art . Around the start of the 21st century, the expanding domain of economics in the social sciences has been described as economic imperialism . A distinction is usually drawn between the social sciences and the humanities . Classicist Allan Bloom writes in The Closing of the American Mind (1987): Social science and humanities have

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7440-412: The measurement of earth . As a result of the two subfields using different approaches a third field has emerged, which is environmental geography . Environmental geography combines physical and human geography and looks at the interactions between the environment and humans. Other branches of geography include social geography , regional geography , and geomatics . Geographers attempt to understand

7560-408: The mental function and overt behaviour of individuals, while the other disciplines focus on creating descriptive generalizations about the functioning of social groups or situation-specific human behaviour. In practice, however, there is quite a lot of cross-fertilization that takes place among the various fields. Psychology differs from biology and neuroscience in that it is primarily concerned with

7680-555: The natural sciences as tools for understanding societies, and so define science in its stricter modern sense . Interpretivist or speculative social scientists, by contrast, may use social critique or symbolic interpretation rather than constructing empirically falsifiable theories, and thus treat science in its broader sense. In modern academic practice, researchers are often eclectic , using multiple methodologies (for instance, by combining both quantitative and qualitative research ). The term social research has also acquired

7800-457: The real n -space R n {\displaystyle \mathbb {R} ^{n}} equipped with the standard dot product . Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from

7920-406: The reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces to distinguish them from Euclidean vector spaces. If E is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted E → . {\displaystyle {\overrightarrow {E}}.} The dimension of

8040-629: The theory of relativity , because of phenomena such as length contraction and the relativity of simultaneity , distances between objects depend on a choice of inertial frame of reference . On galactic and larger scales, the measurement of distance is also affected by the expansion of the universe . In practice, a number of distance measures are used in cosmology to quantify such distances. Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics: Many abstract notions of distance used in mathematics, science and engineering represent

8160-844: The 19th and early 20th centuries. Ferdinand Saussure is considered the father of modern linguistics. Political science is an academic and research discipline that deals with the theory and practice of politics and the description and analysis of political systems and political behaviour. Fields and subfields of political science include political economy , political theory and philosophy , civics and comparative politics , theory of direct democracy , apolitical governance, participatory direct democracy, national systems, cross-national political analysis, political development, international relations, foreign policy , international law , politics, public administration, administrative behaviour, public law, judicial behaviour, and public policy . Political science also studies power in international relations and

8280-858: The Earth in terms of physical and spatial relationships. The first geographers focused on the science of mapmaking and finding ways to precisely project the surface of the earth. In this sense, geography bridges some gaps between the natural sciences and social sciences. Historical geography is often taught in a college in a unified Department of Geography. Modern geography is an all-encompassing discipline, closely related to Geographic Information Science , that seeks to understand humanity and its natural environment. The fields of urban planning , regional science , and planetology are closely related to geography. Practitioners of geography use many technologies and methods to collect data such as Geographic Information Systems , remote sensing , aerial photography , statistics , and global positioning systems . History

8400-432: The Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is sometimes called the standard Euclidean space of dimension n . Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an affine space . They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections. Let E be

8520-739: The Euclidean space as a part of the line. It follows that there is exactly one line that passes through (contains) two distinct points. This implies that two distinct lines intersect in at most one point. A more symmetric representation of the line passing through P and Q is { O + ( 1 − λ ) O P → + λ O Q → | λ ∈ R } , ( {\displaystyle {\Bigl \{}O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}} where O

8640-490: The Italian one, sociology slowly affirms itself and experiences the difficulty of affirming a strategic knowledge beyond philosophy and theology. Around the start of the 20th century, Enlightenment philosophy was challenged in various quarters. After the use of classical theories since the end of the scientific revolution, various fields substituted mathematics studies for experimental studies and examining equations to build

8760-402: The abstract sound system of a particular language); however, work in areas like evolutionary linguistics (the study of the origins and evolution of language) and psycholinguistics (the study of psychological factors in human language) cut across these divisions. The overwhelming majority of modern research in linguistics takes a predominantly synchronic perspective (focusing on language at

8880-460: The adjective legal comes from the Latin word lex . Linguistics investigates the cognitive and social aspects of human language. The field is divided into areas that focus on aspects of the linguistic signal, such as syntax (the study of the rules that govern the structure of sentences), semantics (the study of meaning), morphology (the study of the structure of words), phonetics (the study of speech sounds) and phonology (the study of

9000-441: The associated vector space of F is a linear subspace (vector subspace) of E → . {\displaystyle {\overrightarrow {E}}.} A Euclidean subspace F is a Euclidean space with F → {\displaystyle {\overrightarrow {F}}} as the associated vector space. This linear subspace F → {\displaystyle {\overrightarrow {F}}}

9120-405: The basic properties of Euclidean spaces result from the structure of affine space. They are described in § Affine structure and its subsections. The properties resulting from the inner product are explained in § Metric structure and its subsections. For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as

9240-853: The biological, linguistic, historic and cultural aspects of any problem. Since anthropology arose as a science in Western societies that were complex and industrial, a major trend within anthropology has been a methodological drive to study peoples in societies with more simple social organization, sometimes called "primitive" in anthropological literature, but without any connotation of "inferior". Today, anthropologists use terms such as "less complex" societies or refer to specific modes of subsistence or production , such as "pastoralist" or "forager" or "horticulturalist" to refer to humans living in non-industrial, non-Western cultures, such people or folk ( ethnos ) remaining of great interest within anthropology. The quest for holism leads most anthropologists to study

9360-418: The classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions ) on the plane. One is translation , which means

9480-580: The descriptive understanding of the social realm. He proposed that social ills could be remedied through sociological positivism, an epistemological approach outlined in The Course in Positive Philosophy [1830–1842] and A General View of Positivism (1844). Though Comte is generally regarded as the "Father of Sociology", the discipline was formally established by another French thinker, Émile Durkheim (1858–1917), who developed positivism as

9600-560: The difference between two locations (the relative position ) is sometimes called the directed distance . For example, the directed distance from the New York City Main Library flag pole to the Statue of Liberty flag pole has: Social science Social science is one of the branches of science , devoted to the study of societies and the relationships among members within those societies. The term

9720-498: The discipline is as a cluster of sub-fields that examine different dimensions of society. For example, social stratification studies inequality and class structure; demography studies changes in population size or type; criminology examines criminal behaviour and deviance; and political sociology studies the interaction between society and state. Since its inception, sociological epistemologies, methods, and frames of enquiry, have significantly expanded and diverged. Sociologists use

9840-426: The discipline useful for purposes beyond the academy. The results of sociological research aid educators, lawmakers, administrators, developers, and others interested in resolving social problems and formulating public policy, through subdisciplinary areas such as evaluation research , methodological assessment, and public sociology . In the early 1970s, women sociologists began to question sociological paradigms and

9960-753: The distance between them is: d = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 . {\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}.} This idea generalizes to higher-dimensional Euclidean spaces . There are many ways of measuring straight-line distances. For example, it can be done directly using

10080-674: The distance in a "mathematical" space is a number , not something expressed in inches or metres. The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts — the space of translations which is equipped with an inner product . The action of translations makes the space an affine space , and this allows defining lines, planes, subspaces, dimension, and parallelism . The inner product allows defining distance and angles. The set R n {\displaystyle \mathbb {R} ^{n}} of n -tuples of real numbers equipped with

10200-488: The early part of the 20th century, economics has focused largely on measurable quantities, employing both theoretical models and empirical analysis. Quantitative models, however, can be traced as far back as the physiocratic school . Economic reasoning has been increasingly applied in recent decades to other social situations such as politics , law, psychology , history , religion , marriage and family life, and other social interactions. The expanding domain of economics in

10320-540: The examination of the organization and development of human social life. The sociological field of interest ranges from the analysis of short contacts between anonymous individuals on the street to the study of global social processes . In the terms of sociologists Peter L. Berger and Thomas Luckmann , social scientists seek an understanding of the Social Construction of Reality . Most sociologists work in one or more subfields . One useful way to describe

10440-419: The frequency of features, while some disciplines, like contact linguistics, focus on qualitative analysis. While certain areas of linguistics can thus be understood as clearly falling within the social sciences, other areas, like acoustic phonetics and neurolinguistics , draw on the natural sciences. Linguistics draws only secondarily on the humanities, which played a rather greater role in linguistic inquiry in

10560-442: The furtherance of the social sciences. Researchers continue to search for a unified consensus on what methodology might have the power and refinement to connect a proposed "grand theory" with the various midrange theories that, with considerable success, continue to provide usable frameworks for massive, growing data banks; for more, see consilience . The social sciences will for the foreseeable future be composed of different zones in

10680-510: The general laws derived in physics or chemistry, or they may explain individual cases through more general principles, as in many fields of psychology. Anthropology (like some fields of history) does not easily fit into one of these categories, and different branches of anthropology draw on one or more of these domains. Within the United States, anthropology is divided into four sub-fields: archaeology, physical or biological anthropology , anthropological linguistics , and cultural anthropology . It

10800-531: The ground, as opposed to what can be observed by compiling many local observations remain fundamental in any kind of anthropology, whether cultural, biological, linguistic or archaeological. Communication studies deals with processes of human communication , commonly defined as the sharing of symbols to create meaning . The discipline encompasses a range of topics, from face-to-face conversation to mass media outlets such as television broadcasting. Communication studies also examine how messages are interpreted through

10920-473: The humanities generally study local traditions, through their history, literature, music, and arts, with an emphasis on understanding particular individuals, events, or eras. Finally, the social sciences have generally attempted to develop scientific methods to understand social phenomena in a generalizable way, though usually with methods distinct from those of the natural sciences. The anthropological social sciences often develop nuanced descriptions rather than

11040-523: The humanities. As a social science, the discipline often overlaps with sociology, psychology, anthropology, biology, political science, economics, and public policy, among others. From a humanities perspective, communication is concerned with rhetoric and persuasion (traditional graduate programs in communication studies trace their history to the rhetoricians of Ancient Greece ). The field applies to outside disciplines as well, including engineering, architecture, mathematics, and information science. Economics

11160-760: The interaction of mental processes and behaviour, and of the overall processes of a system, and not simply the biological or neural processes themselves, though the subfield of neuropsychology combines the study of the actual neural processes with the study of the mental effects they have subjectively produced. Many people associate psychology with clinical psychology , which focuses on assessment and treatment of problems in living and psychopathology. In reality, psychology has myriad specialties including social psychology , developmental psychology , cognitive psychology , educational psychology , industrial-organizational psychology , mathematical psychology , neuropsychology, and quantitative analysis of behaviour. Psychology

11280-505: The introduction at the end of the 19th century of non-Euclidean geometries , the old postulates were re-formalized to define Euclidean spaces through axiomatic theory . Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by

11400-433: The investigation of ancient historical documents. The methods originally rooted in classical sociology and statistical mathematics have formed the basis for research in other disciplines, such as political science, media studies, and marketing and market research . Euclidean space Euclidean space is the fundamental space of geometry , intended to represent physical space . Originally, in Euclid's Elements , it

11520-562: The invisibility of women in sociological studies, analysis, and courses. In 1969, feminist sociologists challenged the discipline's androcentrism at the American Sociological Association's annual conference. This led to the founding of the organization Sociologists for Women in Society, and, eventually, a new sociology journal, Gender & Society . Today, the sociology of gender is considered to be one of

11640-401: The mathematical idea of a metric . A metric or distance function is a function d which takes pairs of points or objects to real numbers and satisfies the following rules: As an exception, many of the divergences used in statistics are not metrics. There are multiple ways of measuring the physical distance between objects that consist of more than one point : The word distance

11760-453: The most prominent sub-fields in the discipline. New sociological sub-fields continue to appear — such as community studies , computational sociology , environmental sociology , network analysis, actor-network theory , gender studies, and a growing list, many of which are cross-disciplinary in nature. Additional applied or interdisciplinary fields related to the social sciences or are applied social sciences include: The origin of

11880-580: The other by some sequence of translations, rotations and reflections (see below ). In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames , measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions :

12000-446: The physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements , with the great innovation of proving all properties of the space as theorems , by starting from a few fundamental properties, called postulates , which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove ( parallel postulate ). After

12120-462: The physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates , or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry . In 1637, René Descartes introduced Cartesian coordinates , and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra

12240-539: The plane is given by: d = ( Δ x ) 2 + ( Δ y ) 2 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . {\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.} Similarly, given points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) in three-dimensional space,

12360-537: The points P and Q is the subset of points such that 0 ≤ 𝜆 ≤ 1 in the preceding formulas. It is denoted PQ or QP ; that is P Q = Q P = { P + λ P Q → | 0 ≤ λ ≤ 1 } . ( {\displaystyle PQ=QP={\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} 0\leq \lambda \leq 1{\Bigr \}}.{\vphantom {\frac {(}{}}}} Two subspaces S and T of

12480-474: The political, cultural, economic, and social dimensions of their contexts. Communication is institutionalized under many different names at different universities, including communication , communication studies , speech communication , rhetorical studies , communication science , media studies , communication arts , mass communication , media ecology , and communication and media science . Communication studies integrate aspects of both social sciences and

12600-502: The properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic . Therefore, it is usually possible to work with a specific Euclidean space, denoted E n {\displaystyle \mathbf {E} ^{n}} or E n {\displaystyle \mathbb {E} ^{n}} , which can be represented using Cartesian coordinates as

12720-417: The research of, and sometimes distinct in approach toward, the field. The term "social science" may refer either to the specific sciences of society established by thinkers such as Comte, Durkheim, Marx, and Weber, or more generally to all disciplines outside of "noble science" and arts . By the late 19th century, the academic social sciences were constituted of five fields: jurisprudence and amendment of

12840-415: The shortest path along the surface of the Earth , as the crow flies . This is approximated mathematically by the great-circle distance on a sphere. More generally, the shortest path between two points along a curved surface is known as a geodesic . The arc length of geodesics gives a way of measuring distance from the perspective of an ant or other flightless creature living on that surface. In

12960-419: The social sciences has been described as economic imperialism. Education encompasses teaching and learning specific skills, and also something less tangible but more profound: the imparting of knowledge, positive judgement and well-developed wisdom. Education has as one of its fundamental aspects the imparting of culture from generation to generation (see socialization ). To educate means 'to draw out', from

13080-466: The social sciences is also reflected in other specialized encyclopedias. The term "social science" was coined in French by Mirabeau in 1767, before becoming a distinct conceptual field in the nineteenth century. Social science was influenced by positivism, focusing on knowledge based on actual positive sense experience and avoiding the negative; metaphysical speculation was avoided. Auguste Comte used

13200-426: The social sciences, uses methods and techniques that relate to the kinds of inquiries sought: primary sources such as historical documents, interviews, and official records, as well as secondary sources such as scholarly articles , are used in building and testing theories. Empirical methods include survey research, statistical analysis or econometrics , case studies , experiments, and model building. Psychology

13320-525: The survey can be traced back at least as early as the Domesday Book in 1086, while some scholars pinpoint the origin of demography to 1663 with the publication of John Graunt 's Natural and Political Observations upon the Bills of Mortality . Social research began most intentionally, however, with the positivist philosophy of science in the 19th century. In contemporary usage, "social research"

13440-399: The term science sociale to describe the field, taken from the ideas of Charles Fourier ; Comte also referred to the field as social physics . Following this period, five paths of development sprang forth in the social sciences, influenced by Comte in other fields. One route that was taken was the rise of social research. Large statistical surveys were undertaken in various parts of

13560-505: The theory of great powers and superpowers . Political science is methodologically diverse, although recent years have witnessed an upsurge in the use of the scientific method , that is, the proliferation of formal-deductive model building and quantitative hypothesis testing. Approaches to the discipline include rational choice , classical political philosophy, interpretivism , structuralism , and behaviouralism , realism , pluralism, and institutionalism . Political science, as one of

13680-423: The unit of analysis is the individual agent, such as a household or firm, and macroeconomics , where the unit of analysis is an economy as a whole. Another division of the subject distinguishes positive economics, which seeks to predict and explain economic phenomena, from normative economics , which orders choices and actions by some criterion; such orderings necessarily involve subjective value judgments. Since

13800-417: The wheel is 2π × radius ; if the radius is 1, each revolution of the wheel causes a vehicle to travel 2π radians. The displacement in classical physics measures the change in position of an object during an interval of time. While distance is a scalar quantity, or a magnitude , displacement is a vector quantity with both magnitude and direction . In general, the vector measuring

13920-478: The word comes from the suffix -logy , which means "study of", derived from Ancient Greek, and the stem soci- , which is from the Latin word socius , meaning "companion", or society in general. Auguste Comte (1798–1857) coined the term sociology to describe a way to apply natural science principles and techniques to the social world in 1838. Comte endeavoured to unify history, psychology and economics through

14040-419: Was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances. Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n , using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of

14160-557: Was developed and furthered economic knowledge as a hard science . The last path was the correlation of knowledge and social values ; the antipositivism and verstehen sociology of Max Weber firmly demanded this distinction. In this route, theory (description) and prescription were non-overlapping formal discussions of a subject. The foundation of social sciences in the West implies conditioned relationships between progressive and traditional spheres of knowledge. In some contexts, such as

14280-453: Was formerly used to refer to the field of sociology , the original "science of society", established in the 18th century. In addition to sociology, it now encompasses a wide array of academic disciplines , including anthropology , archaeology , economics , geography , linguistics , management , communication studies , psychology , culturology and political science . Positivist social scientists use methods resembling those used in

14400-567: Was the three-dimensional space of Euclidean geometry , but in modern mathematics there are Euclidean spaces of any positive integer dimension n , which are called Euclidean n -spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes . The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling

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