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  • Mathematics - 6 | Britannica.com
    are used extensively in linear equation statement that a first degree polynomial that is the sum of a set of terms each of which is the product of a constant and the first power of a variable is equal to a constant Specifically a linear equation in n variables is of the form a 0 a linear programming mathematical modeling technique useful for guiding quantitative decisions in business planning industrial engineering and to a lesser extent in the social and physical sciences Applications of the method of linear programming were first seriously linear transformation in mathematics a rule for changing one geometric figure or matrix or vector into another using a formula with a specified format The format must be a linear combination in which the original components e g the x and y coordinates of each point Lions Pierre Louis French mathematician who was awarded the Fields Medal in 1994 for his work on partial differential equations Lions earned a doctorate from the University of Paris VI in 1979 He joined the faculty of the University of Paris IX in 1981 and from 1993 Liouville Joseph French mathematician known for his work in analysis differential geometry and number theory and for his discovery of transcendental numbers i e numbers that are not the roots of algebraic equations having rational coefficients He was also influential Lissajous figure also called Bowditch Curve pattern produced by the intersection of two sinusoidal curves the axes of which are at right angles to each other First studied by the American mathematician Nathaniel Bowditch in 1815 the curves were investigated independently Liu Hui Chinese mathematician All that is known about the life of Liu Hui is that he lived in the northern Wei kingdom see Three Kingdoms during the 3rd century ce His fame rests on the commentary he completed in 263 on Jiuzhang suanshu The Nine Chapters Lobachevsky Nikolay Ivanovich Russian mathematician and founder of non Euclidean geometry which he developed independently of János Bolyai and Carl Gauss Lobachevsky s first publication on this subject was in 1829 Bolyai s in 1832 Gauss never published his ideas on non Euclidean logarithm the exponent or power to which a base must be raised to yield a given number Expressed mathematically x is the logarithm of n to the base b if b x n in which case one writes x log b n For example 2 3 8 therefore 3 is the logarithm logicism school of mathematical thought introduced by the 19th 20th century German mathematician Gottlob Frege and the British mathematician Bertrand Russell which holds that mathematics is actually logic Logicists contend that all of mathematics can be deduced Lorenz Edward American meteorologist and discoverer of the underlying mechanism of deterministic chaos one of the principles of complexity After receiving degrees from Dartmouth College and Harvard University in mathematics Lorenz turned to weather forecasting Love Augustus Edward Hough British geophysicist and mathematician who discovered a major type of seismic wave that was subsequently named for him Love held the Sedleian professorship of natural philosophy at the University of Oxford from 1899 to 1940 In his analysis of earthquake Lovelace Ada King countess of English mathematician an associate of Charles Babbage for whose prototype of a digital computer she created a program She has been called the first computer programmer She was the daughter of famed poet Lord Byron and Annabella Milbanke Byron who loxodrome curve cutting the meridians of a sphere at a constant nonright angle Thus it may be seen as the path of a ship sailing always oblique to the meridian and directed always to the same point of the compass Pedro Nunes who first conceived the curve 1550 Mac Lane Saunders American mathematician who was a cocreator of category theory an architect of homological algebra and an advocate of categorical foundations for mathematics Mac Lane graduated from Yale University in 1930 and then began graduate work at the University Machin John English mathematician notable for studies in finding the area of a circle In 1706 he was the first to compute the value of the constant π to 100 decimal places Machin s formula for π was adapted by others including Euler to extend his result Machin Maclaurin Colin Scottish mathematician who developed and extended Sir Isaac Newton s work in calculus geometry and gravitation A child prodigy he entered the University of Glasgow at age 11 At the age of 19 he was elected a professor of mathematics at Marischal magic square square matrix often divided into cells filled with numbers or letters in particular arrangements that were once thought to have special magical properties Originally used as religious symbols they later became protective charms or tools for divination Mahavira Indian mathematician who made significant contributions to the development of algebra All that is known about Mahavira s life is that he was a Jain he perhaps took his name to honour the great Jainism reformer Mahavira c 599 527 bce and that he Mandelbrot Benoit Polish born French American mathematician universally known as the father of fractals Fractals have been employed to describe diverse behaviour in economics finance the stock market astronomy and computer science Mandelbrot was educated at the manifold in mathematics a generalization and abstraction of the notion of a curved surface a manifold is a topological space that is modeled closely on Euclidean space locally but may vary widely in global properties Each manifold is equipped with a family mapping any prescribed way of assigning to each object in one set a particular object in another or the same set Mapping applies to any set a collection of objects such as all whole numbers all the points on a line or all those inside a circle For example Margulis Gregori Aleksandrovich Russian mathematician who was awarded the Fields Medal in 1978 for his contributions to the theory of Lie groups Margulis attended Moscow State University Ph D 1970 In 1978 Margulis was awarded the Fields Medal

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  • Mathematics - 7 | Britannica.com
    complexity a subfield of theoretical computer science and mathematics the question of whether all so called NP problems are actually P problems A P problem is one that can be solved in polynomial time which means that an algorithm packing in mathematics a type of problem in combinatorial geometry that involves placement of figures of a given size or shape within another given figure with greatest economy or subject to some other restriction The problem of placement of a given number Painlevé Paul French politician mathematician and patron of aviation who was prime minister at a crucial period of World War I and again during the 1925 financial crisis Painlevé was educated at the École Normale Supérieure now part of the Universities of Paris Pappus of Alexandria the most important mathematical author writing in Greek during the later Roman Empire known for his Synagoge Collection a voluminous account of the most important work done in ancient Greek mathematics Other than that he was born at Alexandria Pappus s theorem in mathematics theorem named for the 4th century Greek geometer Pappus of Alexandria that describes the volume of a solid obtained by revolving a plane region D about a line L not intersecting D as the product of the area of D and the length of the paraboloid an open surface generated by rotating a parabola about its axis If the axis of the surface is the z axis and the vertex is at the origin the intersections of the surface with planes parallel to the xz and yz planes are parabolas see top The intersections parameter in mathematics a variable for which the range of possible values identifies a collection of distinct cases in a problem Any equation expressed in terms of parameters is a parametric equation The general equation of a straight line in slope intercept parametric equation a type of equation that employs an independent variable called a parameter often denoted by t and in which dependent variables are defined as continuous functions of the parameter and are not dependent on another existing variable More than one parameter partial derivative In differential calculus the derivative of a function of several variables with respect to change in just one of its variables Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential partial differential equation in mathematics equation relating a function of several variables to its partial derivatives A partial derivative of a function of several variables expresses how fast the function changes when one of its variables is changed the others being held partition in mathematics and logic division of a set of objects into a family of subsets that are mutually exclusive and jointly exhaustive that is no element of the original set is present in more than one of the subsets and all the subsets together contain Pascal Blaise French mathematician physicist religious philosopher and master of prose He laid the foundation for the modern theory of probabilities formulated what came to be known as Pascal s law of pressure and propagated a religious doctrine that taught Pascaline the first calculator or adding machine to be produced in any quantity and actually used The Pascaline was designed and built by the French mathematician philosopher Blaise Pascal between 1642 and 1644 It could only do addition and subtraction with Peano axioms in number theory five axioms introduced in 1889 by Italian mathematician Giuseppe Peano Like the axioms for geometry devised by Greek mathematician Euclid c 300 bce the Peano axioms were meant to provide a rigorous foundation for the natural numbers Peano Giuseppe Italian mathematician and a founder of symbolic logic whose interests centred on the foundations of mathematics and on the development of a formal logical language Peano became a lecturer of infinitesimal calculus at the University of Turin in 1884 Pearson Karl British statistician leading founder of the modern field of statistics prominent proponent of eugenics and influential interpreter of the philosophy and social role of science Pearson was descended on both sides of his family from Yorkshire Quakers Peirce Benjamin American mathematician astronomer and educator who computed the general perturbations of the planets Uranus and Neptune Peirce graduated from Harvard University in 1829 and accepted a teaching position with George Bancroft at his Round Hill School Penrose Sir Roger British mathematician and relativist who in the 1960s calculated many of the basic features of black holes After obtaining a Ph D in algebraic geometry from the University of Cambridge in 1957 Penrose held temporary posts at a number of universities Perelman Grigori Russian mathematician who was awarded and declined the Fields Medal in 2006 for his work on the Poincaré conjecture and Fields medalist William Thurston s geometrization conjecture In 2003 Perelman had left academia and apparently had abandoned mathematics perfect number a positive integer that is equal to the sum of its proper divisors The smallest perfect number is 6 which is the sum of 1 2 and 3 Other perfect numbers are 28 496 and 8 128 The discovery of such numbers is lost in prehistory It is known however permutation the various ways in which objects from a set may be selected generally without replacement to form subsets This selection of subsets is called a permutation when the order of selection is a factor a combination when order is not a factor By considering perturbation in mathematics method for solving a problem by comparing it with a similar one for which the solution is known Usually the solution found in this way is only approximate Perturbation is used to find the roots of an algebraic equation that differs Peuerbach Georg von Austrian mathematician and astronomer instrumental in the European revival of the technical understanding of the astronomical ideas of Ptolemy fl c ad 140 and the early use of sines in Europe Nothing is known of Peuerbach s life before 1446 when Pfaff Johann Friedrich German mathematician who proposed the first general method

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  • Mathematics - 8 | Britannica.com
    his work in functional analysis Schwartz received his early education at the École Normale Supérieure now part of the Universities of Paris and the Faculty of Science both located Scott Dana Stewart American mathematician logician and computer scientist and cowinner of the 1976 A M Turing Award the highest honour in computer science Scott and the Israeli American mathematician and computer scientist Michael O Rabin were cited in the award Segner Johann Andreas von German physicist and mathematician who in 1751 introduced the concept of the surface tension of liquids likening it to a stretched membrane His view that minute and imperceptible attractive forces maintain surface tension laid the foundation for the Seki Takakazu the most important figure of the wasan Japanese calculation tradition see mathematics East Asian Japan in the 17th century that flourished from the early 17th century until the opening of Japan to the West in the mid 19th century Seki was instrumental Selberg Atle Norwegian born American mathematician who was awarded the Fields Medal in 1950 for his work in number theory In 1986 he shared with Samuel Eilenberg the Wolf Prize Selberg attended the University of Oslo Ph D 1943 and remained there as a research Selten Reinhard German mathematician who shared the 1994 Nobel Prize for Economics with John F Nash and John C Harsanyi for their development of game theory a branch of mathematics that examines rivalries among competitors with mixed interests The son of a bookseller Serre Jean Pierre French mathematician who was awarded the Fields Medal in 1954 for his work in algebraic topology In 2003 he was awarded the first Abel Prize by the Norwegian Academy of Science and Letters Serre attended the École Normale Supérieure 1945 48 and the set In mathematics and logic any collection of objects elements which may be mathematical e g numbers functions or not The intuitive idea of a set is probably even older than that of number Members of a herd of animals for example could be matched set theory branch of mathematics that deals with the properties of well defined collections of objects which may or may not be of a mathematical nature such as numbers or functions The theory is less valuable in direct application to ordinary experience than Shafer Helen Almira American educator noted for the improvements she made in the curriculum of Wellesley College both as mathematics chair and as school president Shafer graduated in 1863 from Oberlin Ohio College After two years of teaching in New Jersey she joined Shannon Claude American mathematician and electrical engineer who laid the theoretical foundations for digital circuits and information theory a mathematical communication model After graduating from the University of Michigan in 1936 with bachelor s degrees in mathematics Shen Kuo Chinese astronomer mathematician and high official whose famous work Mengxi bitan Brush Talks from Dream Brook Dream Brook was the name of his estate in Jingkou contains the first reference to the magnetic compass the first description of movable Shridhara highly esteemed Hindu mathematician who wrote several treatises on the two major fields of Indian mathematics pati ganita mathematics of procedures or algorithms and bija ganita mathematics of seeds or equations Very little is known about Shripati Indian astronomer astrologer and mathematician whose astrological writings were particularly influential Shripati wrote various works in the first two of the three branches of astral science jyotihshastra namely mathematics including astronomy Sierpiński Wacław leading figure in point set topology and one of the founding fathers of the Polish school of mathematics which flourished between World Wars I and II Sierpiński graduated from Warsaw University in 1904 and in 1908 he became the first person anywhere Singer Isadore Manuel American mathematician awarded together with the British mathematician Sir Michael Francis Atiyah the 2004 Abel Prize by the Norwegian Academy of Sciences and Letters for their discovery and proof of the index theorem bringing together topology singularity of a function of the complex variable z is a point at which it is not analytic that is the function cannot be expressed as an infinite series in powers of z although at points arbitrarily close to the singularity the function may be analytic in Sitter Willem de Dutch mathematician astronomer and cosmologist who developed theoretical models of the universe based on Albert Einstein s general theory of relativity De Sitter studied mathematics at the State University of Groningen and then joined the astronomical Slepian Joseph American electrical engineer and mathematician credited with important developments in electrical apparatus and theory Slepian studied at Harvard University earning the Ph D in 1913 After a postdoctoral year in Europe he taught mathematics at Cornell slide rule a device consisting of graduated scales capable of relative movement by means of which simple calculations may be carried out mechanically Typical slide rules contain scales for multiplying dividing and extracting square roots and some also contain slope Numerical measure of a line s inclination relative to the horizontal In analytic geometry the slope of any line ray or line segment is the ratio of the vertical to the horizontal distance between any two points on it slope equals rise over run Smale Stephen American mathematician who was awarded the Fields Medal in 1966 for his work on topology in higher dimensions Smale grew up in a rural area near Flint From 1948 to 1956 he attended the University of Michigan obtaining B S M S and Ph D degrees Smirnov Stanislav Russian mathematician who was awarded the Fields Medal in 2010 for his work in mathematical physics Smirnov graduated with a degree in mathematics in 1992 from St Petersburg State University in St Petersburg Russia He received a doctorate in mathematics Snell Willebrord van Roijen astronomer and mathematician who discovered the law of refraction which relates the degree of the bending of light to the properties of the refractive material This law is basic to modern geometrical optics In 1613 he succeeded his father Rudolph Somerville Mary British science writer whose influential

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  • Mathematics - 9 | Britannica.com
    Bowdoin College Brunswick Maine Princeton University and with Hermann von Helmholtz Berlin Van Vleck John H American physicist and mathematician who shared the Nobel Prize for Physics in 1977 with Philip W Anderson and Sir Nevill F Mott The prize honoured Van Vleck s contributions to the understanding of the behaviour of electrons in magnetic noncrystalline Varadhan S R Srinivasa Indian mathematician awarded the 2007 Abel Prize by the Norwegian Academy of Sciences and Letters for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations Varadhan received a bachelor s Varahamihira Indian philosopher astronomer and mathematician author of the Pancha siddhantika Five Treatises a compendium of Greek Egyptian Roman and Indian astronomy Varahamihira s knowledge of Western astronomy was thorough In five sections his monumental variable In algebra a symbol usually a letter standing in for an unknown numerical value in an equation Commonly used variables include x and y real number unknowns z complex number unknowns t time r radius and s arc length Variables should Veblen Oswald American mathematician who made important contributions to differential geometry and the early development of topology Many of his contributions found application in atomic physics and the theory of relativity Veblen graduated from the University of vector in mathematics a quantity that has both magnitude and direction but not position Examples of such quantities are velocity and acceleration In their modern form vectors appeared late in the 19th century when Josiah Willard Gibbs and Oliver Heaviside vector in physics a quantity that has both magnitude and direction It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity s magnitude Although a vector has magnitude and vector analysis a branch of mathematics that deals with quantities that have both magnitude and direction Some physical and geometric quantities called scalars can be fully defined by specifying their magnitude in suitable units of measure Thus mass can be expressed vector space a set of multidimensional quantities known as vectors together with a set of one dimensional quantities known as scalars such that vectors can be added together and vectors can be multiplied by scalars while preserving the ordinary arithmetic properties Venn diagram graphical method of representing categorical propositions and testing the validity of categorical syllogisms devised by the English logician and philosopher John Venn 1834 1923 Long recognized for their pedagogical value Venn diagrams have been Vernier Pierre French mathematician and government official who is best remembered for his invention of the vernier caliper an instrument for making accurate linear measurements Taught by his scientist father Claude Vernier he developed an early interest in measuring Viète François seigneur de la Bigotiere mathematician who introduced the first systematic algebraic notation and contributed to the theory of equations Viète a Huguenot sympathizer solved a complex cipher of more than 500 characters used by King Philip II of Spain in his war to defend Roman Villani Cédric French mathematician who was awarded the Fields Medal in 2010 for his work in mathematical physics Villani studied mathematics at the École Normale Supériere in Paris He received a master s degree in numerical analysis from Pierre and Marie Curie University Vinogradov Ivan Matveyevich Russian mathematician known for his contributions to analytic number theory especially his partial solution of the Goldbach conjecture proposed in 1742 that every integer greater than two can be expressed as the sum of three prime numbers In 1914 Vinogradov s theorem in number theory theorem that all sufficiently large odd integers can be expressed as the sum of three prime numbers As a corollary all sufficiently large even integers can be expressed as the sum of three primes plus 3 The theorem was proved in Voevodsky Vladimir Russian mathematician who won the Fields Medal in 2002 for having made one of the most outstanding advances in algebraic geometry in several decades Voevodsky attended Moscow State University 1983 89 before earning a Ph D from Harvard University Volterra Vito Italian mathematician who strongly influenced the modern development of calculus Volterra s later work in analysis and mathematical physics was influenced by Enrico Betti while the former attended the University of Pisa 1878 82 Volterra was appointed von Neumann John Hungarian born American mathematician As an adult he appended von to his surname the hereditary title had been granted his father in 1913 Von Neumann grew from child prodigy to one of the world s foremost mathematicians by his mid twenties Important Wallis John English mathematician who contributed substantially to the origins of the calculus and was the most influential English mathematician before Isaac Newton Wallis learned Latin Greek Hebrew logic and arithmetic during his early school years In 1632 Wang Xiaotong Chinese mathematician who made important advances in the solution of problems involving cubic equations During the reign of Li Yuan 618 626 Wang was a suanxue boshi arithmetic officer In 626 he took part in the revision of the Wuying calendar Waring Edward English mathematician whose primary research interests were in algebra and number theory Waring attended Magdalene College University of Cambridge graduating in 1757 as senior wrangler first place in the annual Mathematical Tripos contest He was Waring s problem in number theory conjecture that every positive integer is the sum of a fixed number f n of n th powers that depends only on n The conjecture was first published by the English mathematician Edward Waring in Meditationes Algebraicae 1770 Thoughts Weaver Warren U S mathematician He studied at the University of Wisconsin taught there 1920 32 and directed the Rockefeller Foundation s Natural Science Division 1932 55 He is considered the first person to propose using electronic computers for the translation Weierstrass Karl German mathematician one of the founders of the modern theory of functions His domineering father sent him to the University of Bonn at age 19 to study law and finance in preparation for a position in the

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  • Aesthetics | Britannica.com
    philosopher and critic of aesthetics and literature who after embracing the philosophical school of Immanuel Kant later criticized it while using its analytic method he also deeply influenced German and Italian idealism the view that reality Downey June Etta American psychologist and educator whose studies centred on the psychology of aesthetics and related philosophical issues Downey graduated from the University of Wyoming in 1895 After a year of teaching school in Laramie she resumed her education Emerson Peter Henry English photographer who promoted photography as an independent art form and created an aesthetic theory called naturalistic photography Trained as a physician Emerson first began to photograph as a part of an anthropological study of the peasants Hegel Georg Wilhelm Friedrich German philosopher who developed a dialectical scheme that emphasized the progress of history and of ideas from thesis to antithesis and thence to a synthesis Hegel was the last of the great philosophical system builders of modern times His work following Hemsterhuis Franciscus Dutch philosopher and aesthetician whose works influenced the German Romantic thinkers Johann Gottfried von Herder Friedrich Heinrich Jacobi and Friedrich Holderlin He sought to coordinate Rationalism and sensationalism holding that all things in Herder Johann Gottfried von German critic theologian and philosopher who was the leading figure of the Sturm und Drang literary movement and an innovator in the philosophy of history and culture His influence augmented by his contacts with the young J W von Goethe made him Hildebrand Adolf von German artist and one of the first sculptors of the 19th century to insist upon the aesthetic autonomy of sculpture from painting a doctrine he most effectively promulgated in Das Problem der Form in der bildenden Kunst 1893 which helped establish Hogarth William the first great English born artist to attract admiration abroad best known

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  • Ancient and Medieval Schools | Britannica.com
    noted for his investigations into physics He studied at Prague and then at the University of Paris where he was a master of arts from 1351 to 1362 and rector in 1353 Most probably he is to be identified with Albertus Magnus Saint Dominican bishop and philosopher best known as a teacher of St Thomas Aquinas and as a proponent of Aristotelianism at the University of Paris He established the study of nature as a legitimate science within the Christian tradition By papal decree Albinus Greek philosopher a pupil of Gaius and a teacher of Galen and a forerunner of Neoplatonism Albinus integrated the ideas of various schools of philosophy in order to shed light on the Platonic system of thought One of his major works the Epitome Anselm of Canterbury Saint Italian born theologian and philosopher known as the father of Scholasticism a philosophical school of thought that dominated the Middle Ages He was recognized in modern times as the originator of the ontological argument for the existence of God Anselm of Laon theologian who became eminent in early Scholasticism Anselm apparently studied at Bec Fr under St Anselm of Canterbury In the final quarter of the 11th century he taught with distinction at Paris where with William of Champeaux he supported realism Antiochus of Ascalon Greek philosopher who followed Philo of Larissa as the head of the Academy charting a new course for Platonism He built up his philosophical system on a foundation of three schools Platonism Peripateticism and Stoicism Stoic ideas played the most apathy in Stoic philosophy condition of being totally free from the pathē which roughly are the emotions and passions notably pain fear desire and pleasure Although remote origins of the doctrine can probably be found in the Cynics second half of the Apuleius Lucius Platonic philosopher rhetorician and author remembered for The Golden Ass a prose narrative that proved influential long after his death The work called Metamorphoses by its author narrates the adventures of a young man changed by magic into an Aquinas Thomas Saint Italian Dominican theologian the foremost medieval Scholastic He developed his own conclusions from Aristotelian premises notably in the metaphysics of personality creation and Providence As a theologian he was responsible in his two masterpieces Arcesilaus philosopher who succeeded Crates as head of the Greek Academy he introduced a skepticism derived either from Socrates or from Pyrrhon and Timon Refusing to accept or deny the possibility of certainty in knowing Arcesilaus advocated a skeptical suspension Ariston of Chios Greek philosopher who studied under Zeno the founder of the Stoic school of philosophy he combined Stoic and Cynic ideas in shaping his own beliefs Ariston believed that the only topic of genuine value in philosophy is the study of ethics and went Aristotelianism the philosophy of Aristotle and of those later philosophical movements based on his thought Assessment and nature of Aristotelianism The extent to which Aristotelian thought has become a component of civilization can hardly be overestimated To

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  • Epistemology | Britannica.com
    the study of the nature origin and limits of human knowledge The term is derived from the Greek epistēmē knowledge and logos reason and accordingly the field is sometimes referred to as the theory of knowledge Epistemology has a long history Ferrier James Frederick Scottish metaphysician distinguished for his theory of agnoiology or theory of ignorance Educated at Edinburgh and Oxford Ferrier qualified as a barrister in 1832 but he came under the influence of the Scottish philosopher Sir William Hamilton who Fichte Johann Gottlieb German philosopher and patriot one of the great transcendental idealists Early life and career Fichte was the son of a ribbon weaver Educated at the Pforta school 1774 80 and at the universities of Jena 1780 and of Leipzig 1781 84 he started Grene Marjorie American philosopher who is considered the founder of the philosophy of biology Grene was known for her innovative theories on the nature of the scientific study of life which she addressed in several works on Existentialism including Dreadful Freedom Hegel Georg Wilhelm Friedrich German philosopher who developed a dialectical scheme that emphasized the progress of history and of ideas from thesis to antithesis and thence to a synthesis Hegel was the last of the great philosophical system builders of modern times His work following Hume David Scottish philosopher historian economist and essayist known especially for his philosophical empiricism and skepticism Hume conceived of philosophy as the inductive experimental science of human nature Taking the scientific method of the English innate idea in philosophy an idea allegedly inborn in the human mind as contrasted with those received or compiled from experience The doctrine that at least certain ideas e g those of God infinity substance must be innate because no satisfactory empirical Kant Immanuel German philosopher whose comprehensive and systematic work

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  • Ethics | Britannica.com
    ibn Pakuda dayyan i e judge of a rabbinical court in Muslim Spain and author of a highly influential and popular work of ethical guidance About 1080 Bahya wrote in Arabic Al Hidāyah ilā farāʾ id al qulūb Duties of the Heart In a rather inaccurate 12th century Barbour Ian American theologian and scientist who attempted to reconcile science and religion Barbour was born in Beijing where his Scottish father and American mother both taught at Yanjing University His family moved between the United States and England before Barnes Albert U S Presbyterian clergyman and writer Of Methodist parentage he intended to study law but while at Hamilton College decided to enter the Presbyterian ministry He attended Princeton Theological Seminary and became a pastor in Morristown N J In bioethics branch of applied ethics that studies the philosophical social and legal issues arising in medicine and the life sciences It is chiefly concerned with human life and well being though it sometimes also treats ethical questions relating to the nonhuman Bosanquet Bernard philosopher who helped revive in England the idealism of G W F Hegel and sought to apply its principles to social and political problems Made a fellow of University College Oxford in 1870 Bosanquet was a tutor there until 1881 when he moved to Braithwaite R B British philosopher best known for his theories in the philosophy of science and in moral and religious philosophy Braithwaite was educated at the University of Cambridge in physics and mathematics before switching to the study of philosophy In 1924 Bronowski Jacob Polish born British mathematician and man of letters who eloquently presented the case for the humanistic aspects of science While Bronowski was still a child his family immigrated to Germany and then to England where he became a naturalized British Bruno Giordano Italian philosopher astronomer mathematician and occultist whose theories anticipated modern science The most notable of these were his theories of the infinite universe and the multiplicity of worlds in which he rejected the traditional geocentric Butler Joseph Church of England bishop moral philosopher preacher to the royal court and influential author who defended revealed religion against the rationalists of his time Ordained in 1718 Butler became preacher at the Rolls Chapel in London where he delivered comparative ethics the empirical observational study of the moral beliefs and practices of different peoples and cultures in various places and times It aims not only to elaborate such beliefs and practices but also to understand them insofar as they are causally conditioned consequentialism In ethics the doctrine that actions should be judged right or wrong on the basis of their consequences The simplest form of consequentialism is classical or hedonistic utilitarianism which asserts that an action is right or wrong according to whether Cudworth Ralph English theologian and philosopher of ethics who became the leading systematic exponent of Cambridge Platonism Reared as a Puritan Cudworth eventually adopted such Nonconformist views as the notion that church government and religious practice should Cumberland

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