analysis Field of mathematics that incorporates the methods of algebra and calculus —specifically of limits, continuity, and infinite series —to ana¬ lyze classes of functions and equations having general properties (e.g., differentiability). Analysis builds on the work of G.W. Leibniz and Isaac Newton by exploring the applications of the derivative and the integral. Several distinct but related subfields have developed, including the cal¬ culus of variations, differential equations, Fourier analysis (see Fourier transform), complex analysis, vector and tensor analysis, real analysis, and FUNCTIONAL ANALYSIS. See also NUMERICAL ANALYSIS.
analytic geometry Investigation of geometric objects using coordi¬ nate systems. Because Rene Descartes was the first to apply algebra to geometry, it is also known as Cartesian geometry. It springs from the idea that any point in two-dimensional space can be represented by two num¬ bers and any point in three-dimensional space by three. Because lines, circles, spheres, and other figures can be thought of as collections of points in space that satisfy certain equations, they can be explored via equations and formulas rather than graphs. Most of analytic geometry deals with the conic sections. Because these are defined using the notion of fixed dis¬ tance, each section can be represented by a general equation derived from the DISTANCE FORMULA.
analytic philosophy Philosophical tradition that emphasizes the logi¬ cal analysis of concepts and the study of the language in which they are expressed. It has been the dominant approach in philosophy in the English-speaking world from the early 20th century. With respect to its problems, methods, and style, it is often contrasted with Continental phi¬ losophy, though the significance of the opposition has been widely chal¬ lenged. Analytic philosophers have differed regarding the nature of so-called “ordinary” language and the methodological value of appeals to ordinary usage in the logical analysis of concepts. Those known as for¬ malists hold that, because ordinary language is potentially a source of conceptual confusion, philosophy and science should be conducted in a logically transparent formal language based on modern mathematical, or symbolic, logic. Those known as informalists reject this view, arguing that attempts to “improve” ordinary language in this way inevitably oversim¬ plify or falsify it, thereby creating conceptual confusion of just the sort that the formalists are concerned to avoid. Three figures conventionally recognized as founders of the tradition are Gottlob Frege, G.E. Moore, and Bertrand Russell. Other major figures include Ludwig Wittgenstein, A.J. Ayer, Rudolf Carnap, J.L. Austin, W.V.O. Quine, and David Lewis (1941— 2001). See also logical positivism; Vienna Circle.
analytic psychology Psychoanalytic method of Carl Jung as he dis¬ tinguished it from that of Sigmund Freud. Jung attached less importance than did Freud to the role of childhood sexual conflicts in the develop¬ ment of neurosis. Moreover, he defined the unconscious to include both the individual’s own unconscious and that inherited, partly in the form of archetypes, from his or her ancestors (the “collective unconscious”). He classified people into introvert and extravert types and further distin¬ guished them according to four primary functions of the mind—thinking, feeling, sensation, and intuition—one or more of which predominated in any given person.
analytic-synthetic distinction In both logic and epistemology, the distinction (derived from Immanuel Kant) between statements whose predi¬ cate is included in the subject (analytic statements) and statements whose predicate is not included in the subject (synthetic statements). Some phi¬
losophers prefer to define as analytic all statements whose denial would be self-contradictory, and to define the term synthetic as meaning “not analytic.” The distinction, introduced by Kant in The Critique of Pure Reason , aroused extensive debate in the mid-20th century, particularly in view of objections raised by W.V.O. Quine.
anamorphosis \ l an-9-mor- , fo-sis\ Drawing or painting technique that gives a distorted image of the subject when seen from the usual view¬ point, but when viewed from a particular angle or reflected in a curved mirror shows it in true proportion. Its purpose is to amuse or mystify. It was a curious by-product of the discovery of perspective in the 14th-15th century and was regarded as a display of technical virtuosity. The first examples appear in Leonardo da Vinci’s notebooks.
Ananda Va-non-doV (fl. 6th century bc, India) First cousin and disciple of the Buddha. A monk who served as the Buddha’s personal attendant, he became known as the “beloved disciple.” It was Ananda who persuaded the Buddha to allow women to become nuns. By tradition, he was the only intimate disciple of the Buddha who had not attained enlightenment before his master’s death; he attained that state just before the first Bud¬ dhist Council (c. 544 or 480 bc), when he recited from memory the Sutta Pitaka. He is represented as the author of several Buddhist discourses.
anaphylaxis \,an-9-fi-'lak-sis\ Severe, immediate, potentially fatal bodily reaction to contact with a substance (antigen) to which the indi¬ vidual has previously been exposed. Often triggered by antiserum, anti¬ biotics, or insect stings, the reaction’s symptoms include skin flushing, bronchial swelling (with difficulty breathing), and loss of consciousness. Shock may follow. Milder cases may involve hives and severe headache. Treatment, consisting of injection of epinephrine, followed by antihistamines, cortisone, or similar drugs, must begin within minutes. Anaphylaxis may be caused by extremely small amounts of antigen.
anarchism Van-or-.kiz-omV Political theory holding all forms of govern¬ ment authority to be unnecessary and undesirable and advocating a society based on voluntary cooperation and free association of individuals and groups. The word was used only pejoratively until Pierre-Joseph Proudhon, now regarded as the founder of anarchism, adopted it in What is Property? (1840). The anarchist Mikhail Bakunin clashed with Karl Marx at the First International; when it was dissolved in 1872, Bakunin’s followers retained control of workers’ organizations in Latin countries such as Spain and Italy. Even anarchists who believed that the transition to a government-free soci¬ ety required violent revolution disagreed on the nature of the transition. Anarcho-SYNDICAUSM, which developed in the late 1880s, emphasized labour unions ( syndicats ) and called for general strikes to paralyze the state. In the 19th and 20th centuries, anarchism also inspired experimental communities, including New Lanark in Britain and Brook Farm in the U.S. During the early months of the Spanish Civil War, anarchist militias were in virtual control of much of eastern Spain, where they established hundreds of anarchist collectives. Suppressed as an organized movement by fascism in the 1930s, anarchism reemerged in the 1950s and ’60s through its influ¬ ence on the civil rights movement and the student movements in the U.S. and Europe. The radical ECOLOGY movement in the 1970s also was inspired by anarchist ideas. Beginning in 1999, anarchist-led street demonstrations against the World Bank and International Monetary Fund received unprec¬ edented publicity and inspired new anarchist groups, periodicals, and Inter¬ net sites. Anarchist themes are reflected in the work of many 20th-century artists, writers, and musicians, including Pablo Picasso, the American poets of the Beat movement, the Spanish Surrealist filmmaker Luis Bunuel, and the American composer John Cage.