The Condorcet Paris-Aubervilliers Campus , known as Condorcet Campus , is an inter-university campus of the universities of Paris , located between Porte de la Chapelle in Paris and La Plaine Saint-Denis in Aubervilliers and inaugurated in 2019.
118-609: Eleven research universities and institutions are taking part. The campus is dedicated to the humanities and social sciences . The project was born in 2008 from the alliance of the Panthéon-Sorbonne University , the University of Paris 8 , Sorbonne Paris North University , EHESS and PSL University with EPHE and École des Chartes . They are joined by CNRS , INED , the foundation Maison des sciences de l'homme and Sorbonne Nouvelle University . It
236-437: A foreign language usually include study of important works of the literature in that language, as well as the language itself. In everyday language, law refers to a rule that is enforced by a governing institution, as opposed to a moral or ethical rule that is not subject to formal enforcement. The study of law can be seen as either a social science or a humanities discipline, depending on one's perspective. Some see it as
354-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
472-494: A degree in the humanities. By 2010, this figure had risen to 2.6%. This represents a doubling of the number of Americans with degrees in the humanities over a 60-year period. The increase in the number of Americans with humanities degrees is in part due to the overall rise in college enrollment in the United States. In 1940, 4.6% of Americans had a four-year degree, but by 2016, this figure had risen to 33.4%. This means that
590-552: A field called the digital humanities . Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as
708-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
826-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
944-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
1062-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
1180-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
1298-463: 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. For many decades, there has been a growing public perception that a humanities education inadequately prepares graduates for employment. The common belief
SECTION 10
#17327730211131416-433: A particular path. In the liberal arts tradition, musicology is also used to broaden skills of non-musicians by teaching skills, including concentration and listening. Theatre (or theater) (Greek "theatron", θέατρον ) is the branch of the performing arts concerned with acting out stories in front of an audience using combinations of speech, gesture, music, dance, sound and spectacle — indeed any one or more elements of
1534-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
1652-454: A red robe, rather than the modulations of that colour brought about by light, shade and reflection). A characteristic of this style is that the local colour is often defined by an outline (a contemporary equivalent is the cartoon). This is evident in, for example, the art of India, Tibet and Japan. Religious Islamic art forbids iconography, and expresses religious ideas through geometry instead. The physical and rational certainties depicted by
1770-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
1888-421: A significant historical element —as distinguished from the mainly empirical approaches of science . The humanities include the studies of philosophy , religion , history , language arts ( literature , writing , oratory , rhetoric , poetry , etc.), performing arts ( theater , music , dance , etc.), and visual arts ( painting , sculpture , photography , filmmaking , etc.). Some definitions of
2006-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
2124-437: A social science because of its objective and measurable nature, while others view it as a humanities discipline because of its focus on values and interpretation. Law is not always enforceable, especially in the international relations context. Law has been defined in various ways, such as "a system of rules", "an interpretive concept" for achieving justice, "an authority" to mediate between people's interests, or "the command of
2242-602: A source of reliable benchmarks to guide analysis of the state of the humanities in the United States. The 1980 United States Rockefeller Commission on the Humanities described the humanities in its report "The Humanities in American Life": Through the humanities we reflect on the fundamental question: What does it mean to be human? The humanities offer clues but never a complete answer. They reveal how people have tried to make moral, spiritual, and intellectual sense of
2360-575: A sovereign" backed by the threat of punishment. However one likes to think of law, it is a completely central social institution. Legal policy is shaped by the practical application of ideas from many social science and humanities disciplines, including philosophy, history, political science, economics, anthropology, and sociology. Law is politics , because politicians create them. Law 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. Law
2478-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
SECTION 20
#17327730211132596-481: A surface by applying pressure from a tool, or moving a tool across a surface. Common tools are graphite pencils , pen and ink , inked brushes , wax color pencils , crayons , charcoals , pastels , and markers . Digital tools that simulate the effects of these are also used. The main techniques used in drawing are: line drawing, hatching , crosshatching, random hatching, scribbling, stippling , and blending. A computer aided designer who excels in technical drawing
2714-593: A universally accepted definition, but which has variably included all written work; writing that possesses literary merit; and language that emphasizes its own literary features, as opposed to ordinary language . Etymologically the term derives from the Latin word literatura /litteratura which means "writing formed with letters", although some definitions include spoken or sung texts . Literature can be classified as fiction or non-fiction ; poetry or prose . It can be further distinguished according to major forms such as
2832-541: A variety of settings, from small dance studios to large theaters. Definitions of what constitutes dance are dependent on social , cultural , aesthetic , artistic , and moral constraints and range from functional movement (such as Folk dance ) to codified, virtuoso techniques such as ballet . The great traditions in art have a foundation in the art of one of the ancient civilizations, such as Ancient Japan , Greece and Rome , China , India , Greater Nepal , Mesopotamia and Mesoamerica . Ancient Greek art saw
2950-531: A veneration of the human physical form and the development of equivalent skills to show musculature, poise, beauty and anatomically correct proportions. Ancient Roman art depicted gods as idealized humans, shown with characteristic distinguishing features (e.g., Zeus ' thunderbolt). The emphasis on spiritual and religious themes in Byzantine and Gothic art of the Middle Ages reflected the dominance of
3068-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
3186-555: A world where irrationality, despair, loneliness, and death are as conspicuous as birth, friendship, hope, and reason. The Commission on the Humanities and Social Sciences 2013 report "The Heart of the Matter" supports the notion of a broad " liberal arts education ", which includes study in disciplines from the natural sciences to the arts as well as the humanities. Many colleges provide such an education; some require it. The University of Chicago and Columbia University were among
3304-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
3422-640: Is also a specialized form of fine art in which the artists perform their work live to an audience. This is called Performance art . Most performance art also involves some form of plastic art, perhaps in the creation of props . Dance was often referred to as a plastic art during the Modern dance era. Musicology as an academic discipline can take a number of different paths, including historical musicology , music literature, ethnomusicology and music theory . Undergraduate music majors generally take courses in all of these areas, while graduate students focus on
3540-527: Is also economics, because any rule about contract , tort , property law , labour law , company law and many more can have long-lasting effects on how productivity is organised and the distribution of wealth. The noun law derives from the Old English word lagu , meaning something laid down or fixed, and the adjective legal comes from the Latin word LEX . Literature is a term that does not have
3658-529: Is also used to describe methods of non-verbal communication (see body language ) between humans or animals ( bee dance , mating dance), and motion in inanimate objects ( the leaves danced in the wind ). Choreography is the process of creating dances, and the people who create choreography are known as choreographers. Choreographers use movement, music , and other elements to create expressive and artistic dances. They may work alone or with other artists to create new works, and their work can be presented in
Condorcet Campus - Misplaced Pages Continue
3776-404: Is commonly regarded as a social science . Based on current knowledge, it seems that all known cultures, both in the past and present, have some form of belief system or religious practice. While there may be isolated individuals or groups who do not practice any form of religion, it is not known if there has ever been a society that was entirely devoid of religious belief. The definition of religion
3894-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
4012-450: Is contested. While the scientific study of language is known as linguistics and is generally considered a social science , a natural science or a cognitive science , the study of languages is also central to the humanities. A good deal of twentieth- and twenty-first-century philosophy has been devoted to the analysis of language and to the question of whether, as Wittgenstein claimed, many of our philosophical confusions derive from
4130-475: Is cut or shaved down to the desired form, like stone and wood. In the West, the history of the humanities can be traced to ancient Greece, as the basis of a broad education for citizens. During Roman times, the concept of the seven liberal arts evolved, involving grammar , rhetoric and logic (the trivium ), along with arithmetic , geometry , astronomy and music (the quadrivium ). These subjects formed
4248-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
4366-403: Is distinguished from other ways of addressing these issues by its critical, generally systematic approach and its reliance on reasoned argument, rather than experiments ( experimental philosophy being an exception). Philosophy used to be a very comprehensive term, including what have subsequently become separate disciplines, such as physics . (As Immanuel Kant noted, "Ancient Greek philosophy
4484-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
4602-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
4720-623: Is located in Aubervilliers . By 2022, 12,000 researchers and doctoral students will be working there in 116 research units and as many teaching rooms, occupying a dozen buildings (including those of the INED) on more than six hectares, which will also include a library of one million books, known as the “ Humathèque Condorcet ”, designed by architect Elizabeth de Portzamparc . An 8,000 m2 building for PSL 's École Pratique des Hautes Études will be completed by 2025, followed by new premises for
4838-716: Is managed by the Condorcet Foundation, chaired from 2009 to 2016 by Jean-Claude Waquet, formerly President of the École pratique des hautes études . Paris Nanterre University joined the Condorcet project in 2018. The cost is €40 million in 2010, rising to €110 million in 2014. The campus is one of ten selected in July 2008 by the French Ministry of Higher Education as part of the Campus plan. The operation
Condorcet Campus - Misplaced Pages Continue
4956-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
5074-511: Is no formalized register of colors. However, the Pantone system is widely used in the printing and design industry to standardize color reproduction. Modern artists have extended the practice of painting considerably to include, for example, collage . This began with cubism and is not painting in strict sense. Some modern painters incorporate different materials such as sand , cement , straw or wood for their texture. Examples of these are
5192-546: Is no longer statistically significant. Humanities graduates can boost their incomes if they obtain advanced or professional degrees. Humanities majors are sought after in many areas of business, specifically for their critical thinking and problem-solving skills. While often considered "soft skills", humanities majors gain skills such as "include persuasive written and oral communication, creative problem-solving, teamwork, decision-making, self-management, and critical analysis". The Humanities Indicators , unveiled in 2009 by
5310-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
5428-456: Is not universal, and different cultures may have different ideas about what constitutes religion. Religion may be characterized with a community since humans are social animals . Rituals are used to bound the community together. Social animals require rules. Ethics is a requirement of society, but not a requirement of religion. Shinto, Daoism, and other folk or natural religions do not have ethical codes. While some religions do include
5546-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
5664-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
5782-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
5900-415: Is referred to as a draftsman or draughtsman . Literally, painting is the practice of applying pigment suspended in a carrier (or medium ) and a binding agent (a glue ) to a surface (support) such as paper , canvas or a wall. However, when used in an artistic sense, it means the use of this activity in combination with drawing , composition and other aesthetic considerations in order to manifest
6018-515: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
SECTION 50
#17327730211136136-555: Is supported by the local authorities concerned (Aubervilliers, Plaine Commune, City of Paris, Seine-Saint-Denis General Council, Île-de-France region). Construction of the Aubervilliers site begins in 2018, and that of Porte de la Chapelle in 2023. In 2020, the member institutions are: Since June 22, 2022, the President of the Condorcet Campus public body has been Pierre-Paul Zalio. The main Condorcet Campus site
6254-440: Is systematically collected information about the past . 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 , history can occasionally be classified as a social science , though this definition
6372-476: Is that graduates from such programs face underemployment and that incomes are too low for a humanities education to be worth the investment. Humanities graduates find employment in a wide variety of management and professional occupations. In Britain, for example, over 11,000 humanities majors found employment in the following occupations: Many humanities graduates may find themselves with no specific career goals upon graduation, which can lead to lower incomes in
6490-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
6608-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
6726-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
6844-487: The École des Hautes Études en Sciences Sociales and the fondation Maison des Sciences de l'Homme by 2029-2030. This human sciences research cluster will have taken almost two decades and 500 million euros to complete, due to the reluctance of certain institutions to move away from the center of Paris, and the various ups and downs of architectural competitions and site management. Planned to start in 2021, but delayed to 2023 by an unsuccessful initial call for proposals,
6962-608: The American Academy of Arts and Sciences , are the first comprehensive compilation of data about the humanities in the United States, providing scholars, policymakers and the public with detailed information on humanities education from primary to higher education, the humanities workforce, humanities funding and research, and public humanities activities. Modeled after the National Science Board's Science and Engineering Indicators, Humanities Indicators are
7080-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
7198-1058: The Baháʼí Faith . Religions must adapt and change through the generations because they must remain relevant to the adherents. When traditional religions fail to address new concerns, then new religions will emerge. The performing arts differ from the visual arts in that the former uses the artist's own body, face, and presence as a medium, and the latter uses materials such as clay, metal, or paint, which can be molded or transformed to create some art object . Performing arts include acrobatics , busking , comedy , dance , film , magic , music , opera , juggling , marching arts , such as brass bands , and theatre . Artists who participate in these arts in front of an audience are called performers, including actors, comedians , dancers , musicians , and singers . Performing arts are also supported by workers in related fields, such as songwriting and stagecraft . Performers often adapt their appearance , such as with costumes and stage makeup , etc. There
SECTION 60
#17327730211137316-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
7434-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
7552-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
7670-559: The Renaissance , the term "humanities" referred to the study of classical literature and language, as opposed to the study of religion, or " divinity ". The study of the humanities was a key part of the secular curriculum in universities at the time. Today, the humanities are more frequently defined as any fields of study outside of natural sciences , social sciences , formal sciences (like mathematics ), and applied sciences (or professional training ). They use methods that are primarily critical , speculative, or interpretative and have
7788-566: The Western academic tradition, refers to the studies of the cultures of classical antiquity , namely Ancient Greek and Latin and the Ancient Greek and Roman cultures. Classical studies is considered one of the cornerstones of the humanities; however, its popularity declined during the 20th century. Nevertheless, the influence of classical ideas on many humanities disciplines, such as philosophy and literature, remains strong. History
7906-475: The comparative method and comparative research . Other methods used in the humanities include hermeneutics , source criticism , esthetic interpretation , and speculative reason . The word humanities comes from the Renaissance Latin phrase studia humanitatis , which translates to study of humanity . This phrase was used to refer to the study of classical literature and language, which
8024-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
8142-425: The novel , short story or drama ; and works are often categorised according to historical periods, or according to their adherence to certain aesthetic features or expectations ( genre ). Philosophy—etymologically, the "love of wisdom"—is generally the study of problems concerning matters such as existence, knowledge, justification, truth, justice, right and wrong, beauty, validity, mind, and language. Philosophy
8260-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
8378-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
8496-436: The 19th-century Enlightenment were shattered not only by new discoveries of relativity by Einstein and of unseen psychology by Freud , but also by unprecedented technological development. Increasing global interaction during this time saw an equivalent influence of other cultures into Western art. Drawing is a means of making a picture , using a wide variety of tools and techniques. It generally involves making marks on
8614-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
8732-577: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
8850-437: The West, but elsewhere white may be. Some painters, theoreticians, writers and scientists, including Goethe , Kandinsky , Isaac Newton , have written their own colour theories . Moreover, the use of language is only a generalization for a colour equivalent. The word " red ", for example, can cover a wide range of variations on the pure red of the spectrum. Unlike music, where notes such as C or C# are universally accepted, there
8968-529: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
9086-423: The bulk of medieval education, with the emphasis being on the humanities as skills or "ways of doing". A major shift occurred with the Renaissance humanism of the fifteenth century, when the humanities began to be regarded as subjects to study rather than practice, with a corresponding shift away from traditional fields into areas such as literature and history ( studia humaniora ). In the 20th century, this view
9204-405: The church. However, in the Renaissance , a renewed focus on the physical world was reflected in art forms that depicted the human body and landscape in a more naturalistic and three-dimensional way. Eastern art has generally worked in a style akin to Western medieval art , namely a concentration on surface patterning and local colour (meaning the plain colour of an object, such as basic red for
9322-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
9440-417: The concept of deities, others do not. Therefore, the supernatural does not necessarily require the existence of deities. Rather, it can be broadly defined as any phenomena that cannot be explained by science or reason. Magical thinking creates explanations not available for empirical verification. Stories or myths are narratives being both didactic and entertaining. They are necessary for understanding
9558-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
9676-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
9794-658: The development of the Porte de la Chapelle site in Paris, on the site of the former Dubois railway station, is due to be handed over at the start of the 2025 academic year. Its 20,000 m2 of classrooms, library and gardens will accommodate 3,500 undergraduate students at the University of Paris 1 , and was designed by architects Françoise Mauffret and Jean Guervilly. Humanities Humanities are academic disciplines that study aspects of human society and culture , including certain fundamental questions asked by humans . During
9912-487: The early stages of their career. On the other hand, graduates from more career-oriented programs often find jobs more quickly. However, the long-term career prospects of humanities graduates may be similar to those of other graduates, as research shows that by five years after graduation, they generally find a career path that appeals to them. There is empirical evidence that graduates from humanities programs earn less than graduates from other university programs. However,
10030-405: The empirical evidence also shows that humanities graduates still earn notably higher incomes than workers with no postsecondary education, and have job satisfaction levels comparable to their peers from other fields. Humanities graduates also earn more as their careers progress; ten years after graduation, the income difference between humanities graduates and graduates from other university programs
10148-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
10266-463: The expressive and conceptual intention of the practitioner. Painting has been used throughout history to express spiritual and religious ideas, from mythological scenes on pottery to the frescoes of the Sistine Chapel , to body art. Colour is highly subjective, but has observable psychological effects, although these can differ from one culture to the next. Black is associated with mourning in
10384-474: The first schools to require an extensive core curriculum in philosophy, literature, and the arts for all students. Other colleges with nationally recognized, mandatory programs in the liberal arts are Fordham University , St. John's College , Saint Anselm College and Providence College . Prominent proponents of liberal arts in the United States have included Mortimer J. Adler and E. D. Hirsch, Jr. In 1950, 1.2% of Americans aged 22 had earned
10502-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
10620-573: The human predicament. Some other possible characteristics of religion are pollutions and purification, the sacred and the profane, sacred texts, religious institutions and organizations, and sacrifice and prayer. Some of the major problems that religions confront, and attempts to answer are chaos, suffering, evil, and death. The non-founder religions are Hinduism , Shinto , and native or folk religions . Founder religions are Judaism , Christianity , Islam , Confucianism , Daoism , Mormonism , Jainism , Zoroastrianism , Buddhism , Sikhism , and
10738-520: The humanities accounted for 15% of the bachelor's degrees awarded by colleges and universities in the United States. As of 2022, they accounted for less than 9%. Researchers in the humanities have developed numerous large- and small-scale digital corporations, such as digitized collections of historical texts, along with the digital tools and methods to analyze them. Their aim is both to uncover new knowledge about corpora and to visualize research data in new and revealing ways. Much of this activity occurs in
10856-553: The humanities and closer to the formal sciences , becoming much more analytic. Analytic philosophy is marked by emphasis on the use of logic and formal methods of reasoning, conceptual analysis, and the use of symbolic and/or mathematical logic , as contrasted with the Continental style of philosophy . This method of inquiry is largely indebted to the work of philosophers such as Gottlob Frege , Bertrand Russell , G.E. Moore and Ludwig Wittgenstein . Religious Studies
10974-456: The humanities are called humanities scholars or sometimes humanists . The term humanist also describes the philosophical position of humanism , which antihumanist scholars in the humanities reject. Renaissance scholars and artists are also known as humanists . Some secondary schools offer humanities classes usually consisting of literature , history , foreign language , and art . Human disciplines like history and language mainly use
11092-485: The humanities encompass law and religion due to their shared characteristics, such as the study of language and culture. However, these definitions are not universally accepted, as law and religion are often considered professional subjects rather than humanities subjects. Professional subjects, like some social sciences, are sometimes classified as being part of both the liberal arts and professional development education, whereas humanities subjects are generally confined to
11210-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
11328-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
11446-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
11564-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
11682-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
11800-433: The other performing arts. In addition to the standard narrative dialogue style, theatre takes such forms as opera , ballet , mime , kabuki , classical Indian dance , Chinese opera , mummers' plays , and pantomime . Dance (from Old French dancier , perhaps from Frankish ) generally refers to human movement either used as a form of expression or presented in a social , spiritual or performance setting. Dance
11918-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
12036-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
12154-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
12272-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
12390-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
12508-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
12626-457: The total number of Americans with college degrees has increased significantly, resulting in a greater number of people with degrees in the humanities as well. The proportion of degrees awarded in the humanities has declined in recent decades, even as the overall number of people with humanities degrees has increased. In 1954, 36% of Harvard undergraduates majored in the humanities, but in 2012, only 20% took that course of study. As recently as 1993,
12744-422: The traditional liberal arts education . Although sociology , anthropology , archaeology , linguistics and psychology share some similarities with the humanities, these are often considered social sciences . Similarly, disciplines such as finance , business administration , political science , economics , and global studies have closer ties to the social sciences rather than the humanities. Scholars in
12862-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
12980-401: The vocabulary we use; literary theory has explored the rhetorical, associative, and ordering features of language; and historical linguists have studied the development of languages across time. Literature, covering a variety of uses of language including prose forms (such as the novel ), poetry and drama , also lies at the heart of the modern humanities curriculum. College-level programs in
13098-569: The works of Jean Dubuffet or Anselm Kiefer . Modern and contemporary art has moved away from the historic value of craft in favour of concept ( conceptual art ); this has led some e.g. Joseph Kosuth to say that painting, as a serious art form, is dead, although this has not deterred the majority of artists from continuing to practise it either as whole or part of their work. Sculpture involves creating three-dimensional forms out of various materials. These typically include malleable substances like clay and metal but may also extend to material that
13216-473: Was divided into three sciences: physics, ethics, and logic.") Today, the main fields of philosophy are logic , ethics , metaphysics , and epistemology . Still, it continues to overlap with other disciplines. The field of semantics , for example, brings philosophy into contact with linguistics . Since the early twentieth century, philosophy in English-speaking universities has moved away from
13334-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
13452-583: Was in turn challenged by the postmodernist movement, which sought to redefine the humanities in more egalitarian terms suitable for a democratic society since the Greek and Roman societies in which the humanities originated were elitist and aristocratic. 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
13570-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
13688-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
13806-531: Was seen as an important aspect of a refined education in the Renaissance . In its usage in the early 15th century, the studia humanitatis was a course of studies that consisted of grammar, poetry, rhetoric, history, and moral philosophy, primarily derived from the study of Latin and Greek classics. The word humanitas also gave rise to the Renaissance Italian neologism umanisti , whence "humanist", " Renaissance humanism ". Classics , in
13924-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
#112887