Branches of Mathematics

Pure mathematics

• Algebra
• Calculus
• Geometry
• Topology
• Trignometry
• Combinatorics
• Logic
• Number theory

Applied mathematics

• Dynamical systems and differential equations
• Mathematical physics
• Computing
• Information theory and signal processing
• Probability and statistics
• Game theory
• Operations research

Related Topics

• Electromagnetic Spectrum
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• Applied Physics
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• Computational Mathematics
• Gamma Ray Optics
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• Relative Velocity
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• Name
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• Opinion
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• P. F. Strawson
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• Patricia Churchland
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• Platonic realism
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• Portal Epistemology
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• Raymond Smullyan
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• Relevance logic
• Reliabilism
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• Robert Audi
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• Rudolf Carnap
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• Stephen Cole Kleene
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• Sufi philosophy
• Sugar pill
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• Syllogism
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• Søren Kierkegaard
• Taoism
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• Theorem
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• Truth table
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• Vagueness
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• Virtue Ethics
• Well-formed formula
• Western philosophy
• William James
• William of Ockham
• WP Books
• Yangism
• Zen
• Zoroastrianism
• Zurvanism
• 
  

• Alan Ross Anderson
  Alan Ross Anderson (1925 – 1973) was an American logician and professor of philosophy at Yale University and the University of Pittsburgh.
  A frequent collaborator with Nuel Belnap, Anderson was instrumental in the development of relevance logic and deontic logic.
  Anderson died of cancer in 1973.
  Anderson believed that the conclusion of a valid inference ought to have something to do with (i.e. be relevant to) the premises. Formally, he captured this "relevance condition" with the principle that
  As simple as this idea appears, implementing it in a formal system requires a radical departure from the semantics of classical logic. Anderson and Belnap (with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer, Anil Gupta (logician), and others) explored the formal consequences of the relevance condition in great detail in their influential Entailment books (see references below), which are the most frequently cited works in the field of relevance logic.
  Anderson and Belnap were quick to observe that the concept of relevance had been central to logic since Aristotle, but had been unduly neglected since Gottlob Frege and George Boole laid the foundations for what would come to be known, somewhat ironically, as "classical" logic. (For an example of classical logic's failure to satisfy the relevance condition, see the article on the principle of explosion.)
  
  wn.com/Alan_Ross_Anderson
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  Alan Turing
  Alan Mathison Turing, OBE, FRS ( /ˈtjʊərɪŋ/ TEWR-ing; 23 June 1912 – 7 June 1954), was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a significant role in the creation of the modern computer. Turing is widely considered to be the father of computer science and artificial intelligence.
  During the Second World War, Turing worked for the Government Code and Cypher School (GCCS) at Bletchley Park, Britain's codebreaking centre. For a time he was head of Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine that could find settings for the Enigma machine.
  After the war he worked at the National Physical Laboratory, where he created one of the first designs for a stored-program computer, the ACE. In 1948 Turing joined Max Newman's Computing Laboratory at Manchester University, where he assisted in the development of the Manchester computers and became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis, and he predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, which were first observed in the 1960s.
  
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  Albert Einstein
  Albert Einstein ( /ˈælbərt ˈaɪnstaɪn/; German: [ˈalbɐt ˈaɪnʃtaɪn] ( listen); 14 March 1879 – 18 April 1955) was a German theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics. While best known for his mass–energy equivalence formula E = mc² (which has been dubbed "the world's most famous equation"), he received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect". The latter was pivotal in establishing quantum theory within physics.
  Near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led to the development of his special theory of relativity. He realized, however, that the principle of relativity could also be extended to gravitational fields, and with his subsequent theory of gravitation in 1916, he published a paper on the general theory of relativity. He continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the thermal properties of light which laid the foundation of the photon theory of light. In 1917, Einstein applied the general theory of relativity to model the structure of the universe as a whole.
  
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  Alexander Bain
  Alexander Bain (11 June 1818 – 18 September 1903) was a Scottish philosopher and educationalist in the British school of empiricism who was a prominent and innovative figure in the fields of psychology, linguistics, logic, moral philosophy and education reform. He founded Mind, the first ever journal of psychology and analytical philosophy, and was the leading figure in establishing and applying applying the scientific method to psychology. Bain was the inaugural Regius Chair in Logic and Professor of Logic at the University of Aberdeen, where he also held Professorships in Moral Philosophy and English Literature and was twice elected Lord Rector of the University of Aberdeen.
  Alexander Bain was born in Aberdeen, Scotland to George Bain, a weaver and veteran soldier, and Margaret Paul. At age eleven he left school to work as a weaver hence the description of him as Weevir, rex philosophorum. He also took to lectures at the Mechanics' Institutes of Aberdeen and the Aberdeen Public Library.
  
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  Alfred Tarski
  Alfred Tarski (January 14, 1901 – October 26, 1983) was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death.
  A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.
  His biographers Anita and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models."
  Alfred Tarski was born Alfred Teitelbaum (Polish spelling: "Tajtelbaum"), to parents who were Polish Jews in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw's Szkoła Mazowiecka. Nevertheless, he entered the University of Warsaw in 1918 intending to study biology.
  
  wn.com/Alfred_Tarski
• Alonzo Church
  Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, solving the Entscheidungsproblem, Frege–Church ontology, and the Church–Rosser theorem.
  Alonzo Church was born on June 14, 1903 in Washington, D.C. where his father, Samuel Robbins Church, was the judge of the Municipal Court for the District of Columbia. The family later moved to Virginia after his father lost this position because of failing eyesight. With help from his uncle, also named Alonzo Church, he was able to attend the Ridgefield School for Boys in Ridgefield, Connecticut. After graduating from Ridgefield in 1920, Church attended Princeton University where he was an exceptional student, publishing his first paper, on Lorentz transformations, and graduating in 1924 with a degree in mathematics. He stayed at Princeton, earning a Ph.D. in mathematics in three years under Oswald Veblen.
  
  wn.com/Alonzo_Church
• Alvin Goldman
  Alvin Ira Goldman (born 1938) is an American professor of Philosophy at Rutgers University in New Jersey. He previously taught at the University of Michigan and at the University of Arizona. He earned his PhD from Princeton University and is married to Holly Smith, a well known ethicist, former administrator, and current professor at Rutgers University. He has done influential work on a wide range of philosophical topics, but his principal areas of research are epistemology, philosophy of mind, and cognitive science.
  Goldman's early book, A Theory of Human Action (a revised version of his Ph.D. thesis), presents a systematic way of classifying and relating the many actions we perform at any time. Its influence was broad and can be found in, among other writings, John Rawls' book A Theory of Justice. Goldman's early work in action theory soon gave way to work in other branches of philosophy, most influentially epistemology.
  
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  Alvin Plantinga
  Alvin Carl Plantinga (born November 15, 1932) is an American analytic philosopher, the emeritus John A. O'Brien Professor of Philosophy at the University of Notre Dame and the inaugural holder of the Jellema Chair in Philosophy at Calvin College.
  He is known for his work in philosophy of religion, epistemology, metaphysics and Christian apologetics. Plantinga is the author of a number of books including God and Other Minds (1967), The Nature of Necessity (1974), and the "warrant" series culminating in Warranted Christian Belief (2000). He has delivered the Gifford Lectures three times, and was described by Time magazine as "America's leading orthodox Protestant philosopher of God".
  Plantinga was born on November 15, 1932 in Ann Arbor, Michigan to Cornelius A. Plantinga (1908–1994) and Lettie G. Bossenbroek (1908–2007). Plantinga's father was a first generation immigrant, born in the Netherlands. His family is from the Dutch province of Friesland. Plantinga’s father earned a Ph.D. in philosophy from Duke University and a Master's Degree in psychology, and taught several academic subjects at different colleges over the years. One of Plantinga's brothers, Cornelius "Neal" Plantinga, Jr., is a theologian and the former president of Calvin Theological Seminary. Another of his brothers, Leon, is an emeritus professor of musicology at Yale University. His brother Terrell worked for CBS News.
  
  wn.com/Alvin_Plantinga
• Andrew Gelman
  Andrew Gelman is an American statistician, professor of statistics and political science, and director of the Applied Statistics Center at Columbia University. He earned an S.B. in mathematics and in physics from MIT in 1986 and a Ph.D. in statistics from Harvard in 1990 under the supervision of Donald Rubin.
  He has received the Outstanding Statistical Application award from the American Statistical Association three times. He is an elected fellow of the American Statistical Association and the Institute of Mathematical Statistics.
  His sister is the psychologist Susan Gelman.
  
  
  
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  Aristotle
  Aristotle (Ancient Greek: Ἀριστοτέλης, Aristotélēs) (384 BC – 322 BC) was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology. Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most important founding figures in Western philosophy. Aristotle's writings were the first to create a comprehensive system of Western philosophy, encompassing morality, aesthetics, logic, science, politics, and metaphysics.
  Aristotle's views on the physical sciences profoundly shaped medieval scholarship, and their influence extended well into the Renaissance, although they were ultimately replaced by Newtonian physics. In the zoological sciences, some of his observations were confirmed to be accurate only in the 19th century. His works contain the earliest known formal study of logic, which was incorporated in the late 19th century into modern formal logic. In metaphysics, Aristotelianism had a profound influence on philosophical and theological thinking in the Islamic and Jewish traditions in the Middle Ages, and it continues to influence Christian theology, especially the scholastic tradition of the Catholic Church. His ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotle's philosophy continue to be the object of active academic study today. Though Aristotle wrote many elegant treatises and dialogues (Cicero described his literary style as "a river of gold"), it is thought that the majority of his writings are now lost and only about one-third of the original works have survived.
  
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  Augustus De Morgan
  Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him.
  Augustus De Morgan was born in 1806. His father was Col. Augustus De Morgan, who held various appointments in the service of the East India Company. His mother descended from James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached", using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.
  When De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England, and her son received his elementary education at various schools of no great account. His mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. She explained the aim of Euclid to Augustus, and gave him an initiation into demonstration.
  
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  Averroes
  ʾAbū l-Walīd Muḥammad bin ʾAḥmad bin Rušd (Arabic: أبو الوليد محمد بن احمد بن رشد‎), better known just as Ibn Rushd (Arabic: ابن رشد‎), and in European literature as Averroes ( /əˈvɛroʊ.iːz/; April 14, 1126 – December 10, 1198), was an Andalusian Muslim polymath; a master of Aristotelian philosophy, Islamic philosophy, Islamic theology, Maliki law and jurisprudence, logic, psychology, politics, Arabic music theory, and the sciences of medicine, astronomy, geography, mathematics, physics and celestial mechanics. He was born in Córdoba, Al Andalus, modern-day Spain, and died in Marrakesh, Morocco. His school of philosophy is known as Averroism.
  Ibn Rushd was a defender of Aristotelian philosophy against claims from the influential Islamic theologian Ghazali who attacked philosophy so it would not become an affront to the teachings of Islam.
  Averroes' name is also seen as Averroës, Averroès or Averrhoës, indicating that the "o" and the "e" form separate syllables. Averroes is a Latinisation of a Hebrew transcription of the Arab name Ibn Rushd.
  
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  Averroism
  Averroism is the term applied to either of two philosophical trends among scholastics in the late 13th century: (a) the Arab philosopher Averroës or Ibn Rushd's interpretations of Aristotle and his reconciliation of Aristotelianism with Islamic faith; and (b) the application of these ideas in the Latin Christian and Jewish intellectual traditions, such as Siger of Brabant, Boetius of Dacia, and Maimonides. The term was originally coined by Ernst Renan.
  Averroes at the time of Dante Alighieri was probably the most widely condemned thinker in the medieval Christian world... Averroism became virtually synonymous with atheism in the late Middle Ages and early Renaissance . As an historiographical category it was first developed by Ernest Renan in Averroès et l'averroïsme (1852). Later historians applied the name to significant uses and expressions of elements of Averroes' doctrines and to the exegetical tradition of Aristotle by Jewish and Christian thinkers as it had come down through the Latin and Hebrew traditions. "Averroist" can therefore describe a range of positions, up to and including scholars interested in the historical phenomenon of Averroes' influence^{[citation needed]}.
  
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  Avicenna
  Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā (Persian پورسينا Pur Sina [ˈpuːr ˈsiːnɑː] "son of Sina"; c. 980, Afshana near Bukhara– 1037, Hamadan, Iran), commonly known as Ibn Sīnā or by his Latinized name Avicenna, was a Persianpolymath, who wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived. In particular, 150 of his surviving treatises concentrate on philosophy and 40 of them concentrate on medicine.
  His most famous works are The Book of Healing, a vast philosophical and scientific encyclopaedia, and The Canon of Medicine, which was a standard medical text at many medieval universities. The Canon of Medicine was used as a text-book in the universities of Montpellier and Leuven as late as 1650. Ibn Sīnā's Canon of Medicine provides a complete system of medicine according to the principles of Galen (and Hippocrates).
  His corpus also includes writing on philosophy, astronomy, alchemy, geology, psychology, Islamic theology, logic, mathematics, physics, as well as poetry. He is regarded as the most famous and influential polymath of the Islamic Golden Age.
  
  wn.com/Avicenna
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  Bertrand Russell
  Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these in any profound sense. He was born in Monmouthshire, into one of the most prominent aristocratic families in Britain.
  Russell led the British "revolt against idealism" in the early 20th century. He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege and his friend Ludwig Wittgenstein, and is widely held to be one of the 20th century's premier logicians. He co-authored, with A. N. Whitehead, Principia Mathematica, an attempt to ground mathematics on logic. His philosophical essay "On Denoting" has been considered a "paradigm of philosophy." His work has had a considerable influence on logic, mathematics, set theory, linguistics, computer science (see type theory and type system), and philosophy, especially philosophy of language, epistemology, and metaphysics.
  
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  Chrysippus
  Chrysippus of Soli (Ancient Greek: Χρύσιππος ὁ Σολεύς, Chrysippos ho Soleus; c. 279 BC – c. 206 BC) was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of Cleanthes in the Stoic school. When Cleanthes died, around 230 BC, Chrysippus became the third head of the school. A prolific writer, Chrysippus expanded the fundamental doctrines of Zeno of Citium, the founder of the school, which earned him the title of Second Founder of Stoicism.
  Chrysippus excelled in logic, the theory of knowledge, ethics and physics. He created an original system of propositional logic in order to better understand the workings of the universe and role of humanity within it. He adhered to a deterministic view of fate, but nevertheless sought a role for personal freedom in thought and action. Ethics, he taught, depended on understanding the nature of the universe, and he taught a therapy of extirpating the unruly passions which depress and crush the soul. He initiated the success of Stoicism as one of the most influential philosophical movements for centuries in the Greek and Roman world.
  
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  Daniel Dennett
  Daniel Clement Dennett (born March 28, 1942) is an American philosopher, writer and cognitive scientist whose research centers on the philosophy of mind, philosophy of science and philosophy of biology, particularly as those fields relate to evolutionary biology and cognitive science. He is currently the Co-director of the Center for Cognitive Studies, the Austin B. Fletcher Professor of Philosophy, and a University Professor at Tufts University. Dennett is a firm atheist and secularist, a member of the Secular Coalition for America advisory board, as well as an outspoken supporter of the Brights movement. Dennett is referred to as one of the "Four Horsemen of New Atheism," along with Richard Dawkins, Sam Harris, and the late Christopher Hitchens.
  Born in Boston, Massachusetts, Dennett spent part of his childhood in Lebanon, where, during World War II, his father was a covert counter-intelligence agent with the Office of Strategic Services posing as a cultural attaché to the American Embassy in Beirut. When he was five, his mother took him back to Massachusetts after his father died in an unexplained plane crash. His sister is the investigative journalist Charlotte Dennett.
  
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  David Hilbert
  David Hilbert, FRS (German pronunciation: [ˈdaːvɪt ˈhɪlbɐt]; January 23, 1862 – February 14, 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.
  Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.
  Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.
  
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  David Hume
  David Hume (7 May [O.S. 26 April] 1711 – 25 August 1776) was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and scepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment. Hume is often grouped with John Locke, George Berkeley, and a handful of others as a British Empiricist.
  Beginning with his A Treatise of Human Nature (1739), Hume strove to create a total naturalistic "science of man" that examined the psychological basis of human nature. In stark opposition to the rationalists who preceded him, most notably Descartes, he concluded that desire rather than reason governed human behaviour, saying: "Reason is, and ought only to be the slave of the passions." A prominent figure in the sceptical philosophical tradition and a strong empiricist, he argued against the existence of innate ideas, concluding instead that humans have knowledge only of things they directly experience. Thus he divides perceptions between strong and lively "impressions" or direct sensations and fainter "ideas", which are copied from impressions. He developed the position that mental behaviour is governed by "custom"; our use of induction, for example, is justified only by our idea of the "constant conjunction" of causes and effects. Without direct impressions of a metaphysical "self", he concluded that humans have no actual conception of the self, only of a bundle of sensations associated with the self.
  
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  Duns Scotus
  John (Johannes, Ioannes) Duns Scotus, O.F.M. (c. 1256 – November 18, 1308) was one of the more important theologians and philosophers of the High Middle Ages. He was given the medieval accolade Doctor Subtilis (Subtle Doctor) for his penetrating and subtle manner of thought.
  Scotus has had considerable influence on Roman Catholic thought. The doctrines for which he is best known are the "univocity of being," that existence is the most abstract concept we have, applicable to everything that exists; the formal distinction, a way of distinguishing between different aspects of the same thing; and the idea of haecceity, the property supposed to be in each individual thing that makes it an individual. Scotus also developed a complex argument for the existence of God, and argued for the Immaculate conception of Mary.
  Little is known of Scotus' life. He was probably born in 1265, probably at Duns, in Berwickshire, Scotland. In 1291 he was ordained as a priest in Northampton, England. A note in Codex 66 of Merton College, Oxford, records that Scotus "flourished at Cambridge, Oxford and Paris. He began lecturing on Peter Lombard's Sentences at the prestigious University of Paris in the Autumn of 1302. Later in that academic year, however, he was expelled from the University of Paris for siding with then Pope Boniface VIII in his feud with Philip the Fair of France, over the taxation of church property.
  
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• Edmund Gettier
  Edmund L. Gettier III (born 1927 in Baltimore, Maryland) is an American philosopher and Professor Emeritus at the University of Massachusetts Amherst, known best for his 1963 paper, "Is Justified True Belief Knowledge?".
  Gettier was educated at Cornell University, where his mentors included Max Black and Norman Malcolm. Gettier, himself, was originally attracted to the views of the later Ludwig Wittgenstein. His first teaching job was at Wayne State University in Detroit, Michigan, where his colleagues included Keith Lehrer, R. C. Sleigh, and Alvin Plantinga. Because he was short on publications, his colleagues urged him to write up any ideas he had just to satisfy the administration. The result was a three-page paper that remains one of the most famous in recent philosophical history. According to anecdotal comments that Plantinga has given in lectures, Gettier was originally so unenthused about the paper that he wrote it, had someone translate it into Spanish, and published in in an obscure South American journal. The paper was later published in the United States. Gettier has since published nothing, but he has invented and taught to his graduate students new methods for finding and illustrating countermodels in modal logic, as well as simplified semantics for various modal logics.
  
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  Ernst Schröder
  Ernst Schröder (25 November 1841, Mannheim, Baden, Germany – 16 June 1902, Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic (a term he may have invented), by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce. He is best known for his monumental Vorlesungen über die Algebra der Logik (Lectures on the algebra of logic), in 3 volumes, which prepared the way for the emergence of mathematical logic as a separate discipline in the twentieth century by systematizing the various systems of formal logic of the day.
  Schröder learned mathematics at Heidelberg, Königsberg, and Zürich, under Hesse, Kirchhoff, and Franz Neumann. After teaching school for a few years, he moved to the Technische Hochschule Darmstadt in 1874. Two years later, he took up a chair in mathematics at the Polytechnische Schule in Karlsruhe, where he spent the remainder of his life. He never married.
  
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  Ernst Zermelo
  Ernst Friedrich Ferdinand Zermelo (German: [ʦɛrˈmeːlo]; 1871–1953) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy.^{[citation needed]} He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.
  He graduated from Berlin's Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose guidance he began to study hydrodynamics. In 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899.
  In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at Zurich University, which he resigned in 1916. He was appointed to an honorary chair at Freiburg im Breisgau in 1926, which he resigned in 1935 because he disapproved of Hitler's regime. At the end of World War II and at his request, Zermelo was reinstated to his honorary position in Freiburg.
  
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  Georg Cantor
  Georg Ferdinand Ludwig Philipp Cantor ( /ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
  
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  George Berkeley
  George Berkeley ( /ˈbɑrkliː/; 12 March 1685 – 14 January 1753), also known as Bishop Berkeley (Bishop of Cloyne), was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism" (later referred to as "subjective idealism" by others). This theory denies the existence of material substance and instead contends that familiar objects like tables and chairs are only ideas in the minds of perceivers, and as a result cannot exist without being perceived. Thus, as Berkeley famously put it, for physical objects "esse est percipi" ("to be is to be perceived"). Berkeley is also known for his critique of abstraction, an important premise in his argument for immaterialism.
  In 1709, Berkeley published his first major work, An Essay towards a New Theory of Vision, in which he discussed the limitations of human vision and advanced the theory that the proper objects of sight are not material objects, but light and color. This foreshadowed his chief philosophical work A Treatise Concerning the Principles of Human Knowledge in 1710 which, after its poor reception, he rewrote in dialogue form and published under the title Three Dialogues between Hylas and Philonous in 1713.
  In this book, Berkeley's views were represented by Philonous (Greek: 'lover of mind'), while Hylas (Greek: 'matter') embodies the Irish thinker’s opponents, in particular John Locke. Berkeley argued against Sir Isaac Newton's doctrine of absolute space, time and motion in De Motu (on Motion), published 1721. His arguments were a precursor to the views of Mach and Einstein. In 1732, he published Alciphron, a Christian apologetic against the free-thinkers, and in 1734, he published The Analyst, an empiricist critique of the foundations of infinitesimal calculus, which was influential in the development of mathematics.
  
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  George Boole
  George Boole ( /ˈbuːl/; 2 November 1815 – 8 December 1864) was an English-born mathematician and logician. His work was in the fields of differential equations and algebraic logic, and he is now best known as the author of The Laws of Thought. As the inventor of the prototype of what is now called Boolean logic, which became the basis of the modern digital computer, Boole is regarded in hindsight as a founder of the field of computer science. Boole said,
  ... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning ...
  George Boole's father, John Boole (1779–1848), was a tradesman in Lincoln and gave him lessons. He had an elementary school education, but little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin; which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern languages. At age 16 Boole took up a junior teaching position in Doncaster, at Heigham's School, being at this point the breadwinner for his parents and three younger siblings. He taught also in Liverpool, briefly.
  
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• George Boolos
  George Stephen Boolos (September 4, 1940, New York City – May 27, 1996) was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
  Boolos graduated from Princeton University in 1961 with an A.B. in mathematics. Oxford University awarded him the B.Phil in 1963. In 1966, he obtained the first Ph.D. in philosophy ever awarded by the Massachusetts Institute of Technology, under the direction of Hilary Putnam. After teaching three years at Columbia University, he returned to MIT in 1969, where he spent the rest of his career until his death from cancer.
  A charismatic speaker well known for his clarity and wit, he once delivered a lecture (1994b) giving an account of Gödel's second incompleteness theorem, employing only words of one syllable. At the end of his viva, Hilary Putnam asked him, "And tell us, Mr. Boolos, what does the analytical hierarchy have to do with the real world?" Without hesitating Boolos replied, "It's part of it".
  An expert on puzzles of all kinds, in 1993 Boolos reached the London Regional Final of The Times crossword competition. His score was one of the highest ever recorded by an American. He wrote a paper on "the hardest logic puzzle ever"—one of many puzzles created by Raymond Smullyan.
  
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  Gerhard Gentzen
  Gerhard Karl Erich Gentzen (November 24, 1909, Greifswald, Germany – August 4, 1945, Prague, Czechoslovakia) was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague.
  Gentzen was a student of Paul Bernays at the University of Göttingen. Bernays was fired as "non-Aryan" in April 1933 and therefore Hermann Weyl formally acted as his supervisor. At great risk to his career Gentzen kept in contact with Bernays until the beginning of the Second World War. In November 1933, he joined the Sturmabteilung to pass the state exam for teachers.^{[citation needed]} In 1935, he corresponded with Abraham Fraenkel from Jerusalem and was implicated by the Nazi teachers' union as one who "keeps contacts to the Chosen People." In 1935 and 1936, Hermann Weyl, head of the Göttingen mathematics department in 1933 until his resignation under Nazi pressure, made strong efforts to bring him to the Institute for Advanced Study in Princeton.
  
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  Giuseppe Peano
  Giuseppe Peano (Italian pronunciation: [dʒuˈzɛppe peˈaːno]; 27 August 1858 – 20 April 1932) was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin.
  Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. He attended the Liceo classico Cavour in Turin, and enrolled at the University of Turin in 1876, graduating in 1880 with high honours, after which the University employed him to assist first Enrico D'Ovidio, and then Angelo Genocchi, the Chair of Infinitesimal calculus. Due to Genocchi's poor health, Peano took over the teaching of the infinitesimal calculus course within 2 years. His first major work, a textbook on calculus, was published in 1884 and was credited to Genocchi. A few years later, Peano published his first book dealing with mathematical logic. Here the modern symbols for the union and intersection of sets appeared for the first time.
  
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  Gottlob Frege
  Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925, pronounced [ˈɡɔtloːp ˈfreːɡə ]) was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.
  Frege was born in 1848 in Wismar, in the state of Mecklenburg-Schwerin (the modern German federal state Mecklenburg-Vorpommern). His father Carl (Karl) Alexander Frege (3 August 1809 – 30 November 1866) was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege (née Bialloblotzky, 12 January 1815 – 14 October 1898).
  
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• Haskell Curry
  Haskell Brooks Curry (September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic; while the initial concept of combinatory logic was based on a single paper by Moses Schönfinkel, much of the development was done by Curry. Curry is also known for Curry's paradox and the Curry–Howard correspondence. There are three programming languages named after him, Haskell, Brooks and Curry, as well as the concept of currying, a technique used for transforming functions in mathematics and computer science.
  Curry was born on September 12, 1900, in Millis, Massachusetts, to Samuel Silas Curry and Anna Baright Curry, who ran a school for elocution. He entered Harvard University in 1916 to study medicine but switched to mathematics before graduating in 1920. After two years of graduate work in electrical engineering at MIT, he returned to Harvard to study physics, earning a MA in 1924. Curry's interest in mathematical logic started during this period when he was introduced to the Principia Mathematica, the attempt by Alfred North Whitehead and Bertrand Russell to ground mathematics in symbolic logic. Remaining at Harvard, Curry pursued a Ph.D. in mathematics. While he was directed by George Birkhoff to work on differential equations, his interests continued to shift to logic. In 1927, while an instructor at Princeton University, he discovered the work of Moses Schönfinkel in combinatory logic. Schönfinkel's work had anticipated much of Curry's own research, and as a consequence, he moved to Göttingen where he could work with Heinrich Behmann and Paul Bernays, who were familiar with Schönfinkel's work. Curry was supervised by David Hilbert and worked closely with Bernays, receiving a Ph.D. in 1930 with a dissertation on combinatory logic.
  
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  Hilary Putnam
  Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science. He is known for his willingness to apply an equal degree of scrutiny to his own philosophical positions as to those of others, subjecting each position to rigorous analysis until he exposes its flaws. As a result, he has acquired a reputation for frequently changing his own position. Putnam is currently Cogan University Professor Emeritus at Harvard University.
  In philosophy of mind, Putnam is known for his argument against the type-identity of mental and physical states based on his hypothesis of the multiple realizability of the mental, and for the concept of functionalism, an influential theory regarding the mind-body problem. In philosophy of language, along with Saul Kripke and others, he developed the causal theory of reference, and formulated an original theory of meaning, inventing the notion of semantic externalism based on a famous thought experiment called Twin Earth.
  
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  Immanuel Kant
  Immanuel Kant (German pronunciation: [ɪˈmaːnu̯eːl ˈkant]; 22 April 1724 – 12 February 1804) was a German philosopher from Königsberg (today Kaliningrad of Russia), who researched, lectured, and wrote on philosophy and anthropology during the Enlightenment at the end of the 18th century. At the time, there were major successes and advances in the sciences (for example, Isaac Newton, Carl Friedrich Gauss, and Robert Boyle) applying reason and logic.
  Kant’s major work, the Critique of Pure Reason (Kritik der reinen Vernunft, 1781), aimed to unite reason with experience to move beyond what he took to be failures of traditional philosophy and metaphysics. He hoped to end an age of speculation where objects outside experience were used to support what he saw as futile theories, while opposing the skepticism of thinkers such as Descartes, Berkeley and Hume.
  He stated:
  Kant proposed a ‘Copernican Revolution’ in reverse, saying that:
  Kant published other important works on ethics, religion, law, aesthetics, astronomy, and history. These included the Critique of Practical Reason (Kritik der praktischen Vernunft, 1788) and the Metaphysics of Morals (Die Metaphysik der Sitten, 1797), which dealt with ethics. And the Critique of Judgment (Kritik der Urteilskraft, 1790), which looks at aesthetics and teleology. He aimed to resolve disputes between empirical and rationalist approaches. The former asserted that all knowledge comes through experience; the latter maintained that reason and innate ideas were prior. Kant argued that experience is purely subjective without first being processed by pure reason. He also said that using reason without applying it to experience only leads to theoretical illusions. The free and proper exercise of reason by the individual was both a theme of the Enlightenment, and of Kant's approaches to the various problems of philosophy.
  
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  Jerry Fodor
  Jerry Alan Fodor (born 1935) is an American philosopher and cognitive scientist. He holds the position of State of New Jersey Professor of Philosophy at Rutgers University and is the author of many works in the fields of philosophy of mind and cognitive science, in which he has laid the groundwork for the modularity of mind and the language of thought hypotheses, among other ideas. He is known for his provocative and sometimes polemical style of argumentation.
  Fodor argues that mental states, such as beliefs and desires, are relations between individuals and mental representations. He maintains that these representations can only be correctly explained in terms of a language of thought (LOT) in the mind. Furthermore, this language of thought itself is an actually existing thing that is codified in the brain and not just a useful explanatory tool. Fodor adheres to a species of functionalism, maintaining that thinking and other mental processes consist primarily of computations operating on the syntax of the representations that make up the language of thought.
  
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  John Locke
  John Locke FRS ( /ˈlɒk/; 29 August 1632 – 28 October 1704), widely known as the Father of Classical Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social contract theory. His work had a great impact upon the development of epistemology and political philosophy. His writings influenced Voltaire and Rousseau, many Scottish Enlightenment thinkers, as well as the American revolutionaries. His contributions to classical republicanism and liberal theory are reflected in the United States Declaration of Independence.
  Locke's theory of mind is often cited as the origin of modern conceptions of identity and the self, figuring prominently in the work of later philosophers such as Hume, Rousseau and Kant. Locke was the first to define the self through a continuity of consciousness. He postulated that the mind was a blank slate or tabula rasa. Contrary to pre-existing Cartesian philosophy, he maintained that we are born without innate ideas, and that knowledge is instead determined only by experience derived from sense perception.
  
  wn.com/John_Locke
• Jon Barwise
  Kenneth Jon Barwise (June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used.
  Born in Independence, Missouri to Kenneth T. and Evelyn, he was a precocious child.
  A pupil of Solomon Feferman at Stanford University, Barwise started his research in infinitary logic. After positions as assistant professor at the Universities of Yale and Wisconsin, during which time his interests turned to natural language, he returned to Stanford in 1983 to direct the Center for the Study of Language and Information. He began teaching at Indiana University in 1990. He was elected a Fellow of the American Academy of Arts and Sciences in 1999.
  Barwise contended that, by being explicit about the context in which a proposition is made, the situation, many problems in the application of logic can be eliminated. He sought ... to understand meaning and inference within a general theory of information, one that takes us outside the realm of sentences and relations between sentences of any language, natural or formal. In particular, he claimed that such an approach resolved the liar paradox. He made use of Peter Aczel's non-well-founded set theory in understanding "vicious circles" of reasoning.
  
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  Kurt Gödel
  Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German pronunciation: [ˈkʊʁt ˈɡøːdəl] ( listen); April 28, 1906 – January 14, 1978) was an Austrian/American logician, mathematician, and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
  Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
  
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• Laurence BonJour
  Laurence BonJour (born 31 August 1943) is an American philosopher and Professor of Philosophy at the University of Washington.
  He received his bachelor's degrees in Philosophy and Political Science from Macalester College and his doctorate in 1969 from Princeton University with a dissertation directed by Richard Rorty. Before moving to UW he taught at the University of Texas at Austin.
  His areas of specialty include epistemology, Kant, and British empiricism.
  BonJour's work, along with that of Roderick Chisholm, Keith Lehrer, and Michael Williams, is at the center of current debates in epistemology. Initially defending coherentism in his anti-foundationalist critique The Structure of Empirical Knowledge, BonJour has since moved to defend Cartesian^{[disambiguation needed ]}foundationalism in such works as Epistemology and In Defense of Pure Reason. The latter book is a sustained defense of a priori justification, strongly criticizing empiricists and pragmatists who dismiss it (such as W. V. O. Quine and Richard Rorty).
  
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• Leopold Löwenheim
  Leopold Löwenheim (26 June 1878 in Krefeld, Germany (also the birthplace of Max Zorn) – 5 May 1957 in Berlin) was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin. Nevertheless, he survived the Second World War, after which he resumed teaching mathematics.
  Löwenheim (1915) gave the first proof of what is now known as the Löwenheim–Skolem theorem, often considered the starting point for model theory.
  Primary:
  Secondary:
  
  
  
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  Lewis Carroll
  Charles Lutwidge Dodgson ( /ˈtʃɑrlz ˈlʌtwɪdʒ ˈdɒdʒsən/ CHARLZ LUDT-wij DOJ-sən; 27 January 1832 – 14 January 1898), better known by the pseudonym Lewis Carroll (/ˈkærəl/ KARR-əl), was an English author, mathematician, logician, Anglican deacon and photographer. His most famous writings are Alice's Adventures in Wonderland and its sequel Through the Looking-Glass, as well as the poems "The Hunting of the Snark" and "Jabberwocky", all examples of the genre of literary nonsense. He is noted for his facility at word play, logic, and fantasy, and there are societies dedicated to the enjoyment and promotion of his works and the investigation of his life in many parts of the world, including the United Kingdom, Japan, the United States, and New Zealand.
  Dodgson's family was predominantly northern English, with Irish connections. Conservative and High Church Anglican, most of Dodgson's ancestors were army officers or Church of England clergy. His great-grandfather, also named Charles Dodgson, had risen through the ranks of the church to become the Bishop of Elphin. His grandfather, another Charles, had been an army captain, killed in action in Ireland in 1803 when his two sons were hardly more than babies. His mother's name was Frances Jane Lutwidge.
  
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  Ludwig Wittgenstein
  Ludwig Josef Johann Wittgenstein (26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947. In his lifetime, he published just one book review, one article, a children's dictionary, and the 75-page Tractatus Logico-Philosophicus (1921). In 1999, his posthumously published Philosophical Investigations (1953) was ranked as the most important book of 20th-century philosophy, standing out as "...the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations".Bertrand Russell described him as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating".
  Born in Vienna into one of Europe's wealthiest families, he gave away his entire inheritance. Three of his brothers committed suicide, with Ludwig contemplating it too. He left academia several times: serving as an officer on the frontline during World War I, where he was decorated a number of times for his courage; teaching in schools in remote Austrian villages, where he encountered controversy for hitting children when they made mistakes in mathematics; and working during World War II as a hospital porter in London, where he told patients not to take the drugs they were prescribed, and where no-one knew he was one of the world's most famous philosophers. He described philosophy, however, as "the only work that gives me real satisfaction."
  
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• Lynne Rudder Baker
  Lynne Rudder Baker (b. 14 February 1944) is an American philosopher and author, currently a "Distinguished Professor" at the University of Massachusetts Amherst. She is a native of Atlanta. She got her Ph.D. in 1972 from Vanderbilt University. She was a fellow of the National Humanities Center (1983–1984) and the Woodrow Wilson International Center for Scholars (1988–1989). She joined the faculty of UMass Amherst in 1989. She is the author of several books, notably Saving Belief: A Critique of Physicalism (1987), Explaining Attitudes: A Practical Approach to the Mind (1995), Persons and Bodies: A Constitution View (2000), and The Metaphysics of Everyday Life: An Essay in Practical Realism (2007). Along with several other scholars, Baker delivered the 2001 Gifford Lectures in Natural Theology at the University of Glasgow, published as The Nature and Limits of Human Understanding (ed. Anthony Sanford, T & T Clark, 2003). She is a member of the Amherst Grace Episcopal Church.
  Baker imputes to scientists generally the view that human beings are just another species rather than a special creation of God:
  
  wn.com/Lynne_Rudder_Baker
• Mazdak
  Mazdak (in Persian مزدک) (died c. 524 or 528) was a proto-socialistPersian reformer and religious activist who gained influence under the reign of the Sassanian Shahanshah Kavadh I. He claimed to be a prophet of God, and instituted communal possessions and social welfare programs.
  Mazdak was the chief representative of a religious and philosophical teaching called Mazdakism, which he viewed as a reformed and purified version of Zoroastrianism, although his teaching has been argued to display influences from Manichaeism as well. Zoroastrianism was the dominant religion of Sassanid Persia, and Mazdak himself was a Zoroastrian priest, or mobed, but most of the Zoroastrian clergy regarded his teaching as a heresy. Information about it is scarce and little is known beyond what is listed below; some further details may be inferred from the later doctrine of Khurramism, which has been seen as a continuation of Mazdakism.
  Some sources claim that the original founders of this sect lived earlier than Mazdak. These were another mobed named Zardusht (of the same name, but not the same person as the founder of Zoroastrianism) and/or a Zoroastrian philosopher known as Mazdak the Elder, who taught a combination of altruism and hedonism: "he directed his followers to enjoy the pleasures of life and satisfy their appetite in the highest degree with regard to eating and drinking in the spirit of equality,to aim at good deeds; to abstain from shedding blood and inflicting harm on others; and to practice hospitality without reservation". This doctrine was further developed by the best known Mazdak, Mazdak the Younger, son of Bāmdād. At later stages the opposition to Mazdak blamed them for heresy and sharing women etc. which is not true. They are considered to be the first real socialists of human history who emphasized the community work and benefits for all.
  
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• Nelson Goodman
  Henry Nelson Goodman (7 August 1906, Somerville, Massachusetts – 25 November 1998, Needham, Massachusetts) was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, irrealism and aesthetics.
  Goodman graduated from Harvard University, A.B., magna cum laude (1928). During the 1930s, he ran an art gallery in Boston, Massachusetts while studying for a Harvard Ph.D. in philosophy, which he completed in 1941. His experience as an art dealer helps explain his later turn towards aesthetics, where he became better known than in logic and analytic philosophy. During World War II, he served in the US Army.
  He taught at the University of Pennsylvania, 1946–1964, where his students included Noam Chomsky, Sydney Morgenbesser, and Hilary Putnam. He left Penn because he was not granted the control he desired over the philosophy department. He was a research fellow at the Harvard Center for Cognitive Studies from 1962 to 1963 and was a Professor at several universities from 1964 to 1967, before being appointed Professor of Philosophy at Harvard in 1968.
  
  wn.com/Nelson_Goodman
• 
  Patricia Churchland
  Patricia Smith Churchland (born July 16, 1943) is a Canadian-American philosopher noted for her contributions to neurophilosophy and the philosophy of mind. She has been a Professor at the University of California, San Diego (UCSD) since 1984. Since 1999 she has been UC President's Professor of Philosophy at UCSD, and has held an adjunct professorship at the Salk Institute for Biological Studies since 1989. Educated at the University of British Columbia, the University of Pittsburgh, and the University of Oxford, she taught philosophy at the University of Manitoba from 1969 to 1984 and is the wife of philosopher Paul Churchland.
  Churchland was born Patricia Smith in Oliver, British Columbia,^{[citation needed]} and raised on a farm there in the South Okanagan valley. Both of her parents lacked a high-school education, her father and mother left school after grades 6 and 8 respectively. Her mother was a nurse and her father worked in newspaper publishing in addition to running the family farm. In spite of their limited education, Churchland has described her parents as interested in the sciences, and the worldview they instilled in her as a secular one. She has also described her parents as eager for her to attend college, and though many farmers in their community thought this "hilarious and a grotesque waste of money", they saw to it that she did so. She took her undergraduate degree at the University of British Columbia, graduating with honors in 1965. She received a Woodrow Wilson Fellowship to study at the University of Pittsburgh, where she took an M.A. in 1966. Thereafter she studied at Oxford University as a British Council and Canada Council Fellow, obtaining a B. Phil in 1969.
  
  wn.com/Patricia_Churchland
• Paul Bernays
  Paul Isaac Bernays (17 October 1888, London – 18 September 1977, Zurich) was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator with, David Hilbert.
  Bernays spent his childhood in Berlin. Bernays attended the Köllner Gymnasium, 1895-1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson.
  In 1912, the University of Berlin awarded him a Ph.D. in mathematics, for a thesis, supervised by Landau, on the analytic number theory of binary quadratic forms. That same year, the University of Zurich awarded him the Habilitation for a thesis on function theory and Picard's theorem. The examiner was Ernst Zermelo. Bernays was Privatdozent at the University of Zurich, 1912-17, where he came to know George Pólya.
  
  wn.com/Paul_Bernays
• Paul Churchland
  Paul Churchland (born Oct. 21, 1942, Vancouver, B.C., Canada) is a philosopher noted for his studies in neurophilosophy and the philosophy of mind. He is currently a Professor at the University of California, San Diego, where he holds the Valtz Chair of Philosophy. Churchland holds a joint appointment with the Cognitive Science Faculty and the Institute for Neural Computation. He earned his Ph.D. from the University of Pittsburgh in 1969 under the direction of Wilfrid Sellars. Churchland is the husband of philosopher Patricia Churchland. He is also the father of two children, Mark and Anne Churchland, both of whom are neuroscientists.
  Churchland began his professional career as an instructor at the University of Pittsburgh in 1969; he also lectured at the University of Toronto from 1967-69. In 1969, Churchland took a position at the University of Manitoba, where he would teach for fifteen years: as an assistant professor (69 - 74) and associate professor (74 - 79), and then as a full professor from 1979 - 1984. Professor Churchland joined the Institute for Advanced Study in Princeton in 1982, staying as a member until 1983. He joined the faculty at the University of California, San Diego in 1983, serving as Department Chair from 1986 - 1990.
  
  wn.com/Paul_Churchland
• Paul Grice
  Herbert Paul Grice (March 13, 1913, Birmingham, England – August 28, 1988, Berkeley, California), usually publishing under the name H. P. Grice, H. Paul Grice, or Paul Grice, was a British-educated philosopher of language, who spent the final two decades of his career in the United States.
  Born and raised in Harborne (now a suburb of Birmingham), in the United Kingdom, he was educated at Clifton College and then at Corpus Christi College, Oxford. After a brief period teaching at Rossall School, he went back to Oxford where he taught until 1967. In that year, he moved to the United States to take up a professorship at the University of California, Berkeley, where he taught until his death in 1988. He returned to the UK in 1979 to give the John Locke lectures on Aspects of Reason. He reprinted many of his essays and papers in his valedictory book, Studies in the Way of Words (1989).
  He was married and had two children. He and his wife lived in an old Spanish style house in the Berkeley Hills.^{[citation needed]}
  
  wn.com/Paul_Grice
• Peter Geach
  Peter Thomas Geach, MA, FBA ( /ˈɡiːtʃ/; born 29 March 1916) is a British philosopher. His areas of interest are the history of philosophy, philosophical logic, and the theory of identity.
  Geach was educated at Balliol College, Oxford. He taught at Birmingham University (1951–1966) and from 1966 at the University of Leeds where he was Professor of Logic in the Department of Philosophy. Geach was given the title of Emeritus Professor of Logic on his retirement in 1981.
  Geach was elected a Fellow of the British Academy (FBA) in 1965.
  He has been awarded the papal cross "Pro Ecclesia et Pontifice" by the Holy See for his philosophical work.
  His early work includes the classic texts Mental Acts and Reference and Generality, the latter defending an essentially modern conception of reference against medieval theories of supposition.
  His Catholic perspective is integral to his philosophy. He is perhaps the founder of Analytical Thomism (though the current of thought running through his and Elizabeth Anscombe's work to the present day was only ostensibly so named forty years later by John Haldane), the aim of which is to synthesise Thomistic and Analytic approaches. He defends the Thomistic position that human beings are essentially rational animals, each one miraculously created. He dismisses Darwinistic attempts to regard reason as inessential to humanity, as "mere sophistry, laughable, or pitiable." He repudiates any capacity for language in animals as mere "association of manual signs with things or performances."
  
  wn.com/Peter_Geach
• 
  Plato
  Plato ( /ˈpleɪtoʊ/; Greek: Πλάτων, Plátōn, "broad"; 424/423 BC – 348/347 BC) was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the foundations of Western philosophy and science. In the words of A. N. Whitehead:
  The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. I do not mean the systematic scheme of thought which scholars have doubtfully extracted from his writings. I allude to the wealth of general ideas scattered through them.
  Plato's sophistication as a writer is evident in his Socratic dialogues; thirty-six dialogues and thirteen letters have been ascribed to him. Plato's writings have been published in several fashions; this has led to several conventions regarding the naming and referencing of Plato's texts. Plato's dialogues have been used to teach a range of subjects, including philosophy, logic, ethics, rhetoric, and mathematics.
  
  wn.com/Plato
• Raymond Smullyan
  Raymond Merrill Smullyan (born May 25, 1919) is an American mathematician, concert pianist, logician, Taoist philosopher, and magician.
  Born in Far Rockaway, New York, his first career (like Persi Diaconis a generation later) was stage magic. He then earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959. He is one of many logicians to have studied under Alonzo Church.
  Born in Far Rockaway, New York, he showed musical talent, winning a gold medal in a piano competition when he was aged 12. The following year, his family moved to Manhattan and he attended Theodore Roosevelt High School in The Bronx as this school offered courses suited to his musical talents, but he left to study on his own as the school did not offer similar courses in mathematics.
  He attended several colleges, studying mathematics and music.
  While a Ph.D. student, Smullyan published a paper in the 1957 Journal of Symbolic Logic showing that Gödelian incompleteness held for formal systems considerably more elementary than that of Gödel's 1931 landmark paper. The contemporary understanding of Gödel's theorem dates from this paper. Smullyan later made a compelling case that much of the fascination with Gödel's theorem should be directed at Tarski's theorem, which is much easier to prove and equally disturbing philosophically. The culmination of Smullyan's lifelong reflection on the classic limitative theorems of mathematical logic is quite readable:
  
  wn.com/Raymond_Smullyan
• 
  René Descartes
  René Descartes (French pronunciation: [ʁəne dekaʁt]; Latinized form: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day. In particular, his Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system — allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius.
  
  wn.com/René_Descartes
• Robert Audi
  Robert Audi (born November 1941) is an American philosopher whose major work has focused on epistemology, ethics – especially on ethical intuitionism – and the theory of action. He is O'Brien Professor of Philosophy at the University of Notre Dame, and previously held a Chair in the Business School there. His 2005 book, The Good in the Right, updates and strengthens Rossian intuitionism and develops the epistemology of ethics. He has also written important works of political philosophy, particularly on the relationship between church and state.
  Audi earned his BA from Colgate University and his MA and PhD from the University of Michigan. He taught briefly at the University of Texas at Austin, and for many years as the Charles J. Mach University Professor of Philosophy at the University of Nebraska, Lincoln before moving to University of Notre Dame as Professor of Philosophy, Professor of Management, and the David E. Gallo Chair in Ethics. He has served as General Editor of the First Edition (1995) and Second Edition (1999) of The Cambridge Dictionary of Philosophy. He has also served as the general editor for "Modern Readings in Epistemology", as well as for "Modern readings in Metaphysics".
  
  wn.com/Robert_Audi
• Robert Nozick
  Robert Nozick (November 16, 1938 – January 23, 2002) was an American political philosopher, most prominent in the 1970s and 1980s. He was a professor at Harvard University. He is best known for his book Anarchy, State, and Utopia (1974), a libertarian answer to John Rawls's A Theory of Justice (1971). His other work involved decision theory and epistemology.
  Nozick was born in Brooklyn, the son of a Jewish entrepreneur from the Russian shtetl who had been born with the name of Cohen. Nozick was married to the poet Gjertrud Schnackenberg. He died in 2002 after a prolonged struggle with stomach cancer. He is interred at Mount Auburn Cemetery in Cambridge, Massachusetts.
  Nozick was educated at Columbia (A.B. 1959, summa cum laude), where he studied with Sidney Morgenbesser, and later at Princeton (Ph.D. 1963), and Oxford as a Fulbright Scholar (1963-1964).
  
  wn.com/Robert_Nozick
• 
  Romanian philosophy
  Romanian philosophy is a name covering either a) the philosophy done in Romania or by Romanians, or b) an ethnic philosophy, which expresses at a high level the fundamental features of the Romanian spirituality, or which elevates to a philosophical level the Weltanschauung of the Romanian people, as deposited in language and folklore, traditions, architecture and other linguistic and cultural artifacts.
  The discussion concerning the existence of a Romanian philosophy has known three stages.
  Between the two World Wars, after the emergence of Greater Romania, Romanian national identity went through a crisis. Since it was not centered anymore on immediate political goals (national rights, independence, national unity etc.), the idea now had a more pronounced cultural character. Therefore, discussions of "national specificity" in historiography, literature and philosophy became very common. The first history of Romanian philosophy was published in 1922 by Marin Ştefănescu, proving that philosophical thinking in Romania had reached the level of self-reflexivity; in other words, it had become conscious of itself. The general conclusion of interbellum discussions, which involved almost every notable philosopher, was that there is a Romanian philosophy proper, with a distinct profile among other national philosophies. Constantin Noica, who became one of the most prominent Romanian philosophers, thought early on that Romanian philosophy is characterized by paganism, cosmicism (i.e. no acute separation of the world of the man from, transcendence) and determinism (or rather, "fatalism").
  
  wn.com/Romanian_philosophy
• Rudolf Carnap
  Rudolf Carnap (May 18, 1891 – September 14, 1970) was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism.
  Carnap was born to a west German family that had been humble until his parents' generation. He began his formal education at the Barmen Gymnasium. From 1910 to 1914, he attended the University of Jena, intending to write a thesis in physics. But he also studied carefully Kant's Critique of Pure Reason during a course taught by Bruno Bauch, and was one of very few students to attend Gottlob Frege's courses in mathematical logic. After serving in the German army during World War I for three years, he was given permission to study physics at the University of Berlin, 1917–18, where Albert Einstein was a newly appointed professor. Carnap then attended the University of Jena, where he wrote a thesis defining an axiomatic theory of space and time. The physics department said it was too philosophical, and Bruno Bauch of the philosophy department said it was pure physics. Carnap then wrote another thesis, with Bauch's supervision, on the theory of space in a more orthodox Kantian style, and published as Der Raum (Space) in a supplemental issue of Kant-Studien (1922). In it he makes the clear distinction between formal, physical and perceptual (e.g., visual) spaces.
  
  wn.com/Rudolf_Carnap
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  Saul Kripke
  Saul Aaron Kripke (born November 13, 1940) is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center. Since the 1960s Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory. Much of his work remains unpublished or exists only as tape-recordings and privately circulated manuscripts. Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. A recent poll conducted among philosophers ranked Kripke among the top ten most important philosophers of the past 200 years.
  Kripke has made influential and original contributions to logic, especially modal logic, since he was a teenager. Unusual for a professional philosopher, his only degree is an undergraduate degree from Harvard, in mathematics. His work has profoundly influenced analytic philosophy, with his principal contribution being a semantics for modal logic, involving possible worlds as described in a system now called Kripke semantics. Another of his most important contributions is his argument that necessity is a 'metaphysical' notion, which should be separated from the epistemic notion of a priori, and that there are necessary a posteriori truths, such as "Water is H₂O." He has also contributed an original reading of Wittgenstein, referred to as "Kripkenstein." His most famous work is Naming and Necessity (1980).
  
  wn.com/Saul_Kripke
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  Stephen Cole Kleene
  Stephen Cole Kleene (January 5, 1909, Hartford, Connecticut, United States – January 25, 1994, Madison, Wisconsin) was an American mathematician who helped lay the foundations for theoretical computer science. One of many distinguished students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory. Kleene's work grounds the study of which functions are computable. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixpoint theorem. He also invented regular expressions, and was a leading American advocate of mathematical intuitionism.
  Kleene pronounced his last name /ˈkleɪniː/ KLAY-nee. Commonplace mispronunciations include /ˈkliːniː/ and /ˈkliːn/. (His son, Ken Kleene, wrote: "As far as I am aware this pronunciation is incorrect in all known languages. I believe that this novel pronunciation was invented by my father.")^{[citation needed]}
  
  wn.com/Stephen_Cole_Kleene
• 
  Stephen Stich
  Stephen Stich (born May 9, 1943) is a professor of Philosophy at Rutgers University. He is also currently an Honorary Professor of the department of Philosophy at the University of Sheffield. Stich's main philosophical interests are in the philosophy of mind, cognitive science, epistemology, and moral psychology. He is arguably best known for his contributions to philosophy of mind. His 1983 book, From Folk Psychology to Cognitive Science: The Case Against Belief, received much attention as he argued for a form of eliminative materialism about the mind. However, he has since modified his position, especially in his 1996 book Deconstructing the Mind.
  Stephen Stich attended the University of Pennsylvania from 1960-1964 where he was a member of the Philomathean Society. He received his B.A. in 1964 (Summa Cum Laude with distinction in Philosophy). After attending Princeton University from 1964–1968, he received his Ph.D. in 1968.
  Stich has taught at several universities, including University of Michigan, University of Maryland, College Park, University of California, San Diego, University of Sheffield, and Rutgers University.
  
  wn.com/Stephen_Stich
• 
  Søren Kierkegaard
  Søren Aabye Kierkegaard (/ˈsɔrən ˈkɪərkəɡɑrd/ or /ˈkɪərkəɡɔr/; Danish: [ˈsɶːɐn ˈkiɐ̯ɡəɡɒːˀ] ( listen)) (5 May 1813 – 11 November 1855) was a Danish philosopher, theologian and religious author. He was a critic of idealist intellectuals and philosophers of his time, such as Georg Wilhelm Friedrich Hegel, Friedrich Wilhelm Joseph Schelling and Karl Wilhelm Friedrich Schlegel. He was also critical of the state and practice of Christianity, primarily that of the Church of Denmark. He is widely considered to be the first existentialist philosopher.
  Much of his philosophical work deals with the issues of how one lives as a "single individual", giving priority to concrete human reality over abstract thinking, and highlighting the importance of personal choice and commitment.
  His theological work focuses on Christian ethics, on the institution of the Church, and on the differences between purely objective proofs of Christianity. He wrote of the individual's subjective relationship to Jesus Christ, the God-Man, which came through faith. Much of his work deals with the art of Christian love.
  
  wn.com/Søren_Kierkegaard
• 
  Thomas Aquinas
  Thomas Aquinas, O.P. ( /əˈkwaɪnəs/ ə-KWY-nəs; 1225 – 7 March 1274), also Thomas of Aquin or Aquino, was an Italian Dominican priest of the Roman Catholic Church, and an immensely influential philosopher and theologian in the tradition of scholasticism, known as Doctor Angelicus ([the] Angelic Doctor), Doctor Communis, or Doctor Universalis. "Aquinas" is not a surname, but is a Latin demonym for a resident of Aquino, his place of birth. He was the foremost classical proponent of natural theology, and the father of Thomism. His influence on Western thought is considerable, and much of modern philosophy was conceived as a reaction against, or as an agreement with, his ideas, particularly in the areas of ethics, natural law, metaphysics, and political theory.
  Thomas is held in the Catholic Church to be the model teacher for those studying for the priesthood. The works for which he is best-known are the Summa Theologica and the Summa Contra Gentiles. As one of the 33 Doctors of the Church, he is considered the Church's greatest theologian and philosopher. Pope Benedict XV declared: "This (Dominican) Order ... acquired new luster when the Church declared the teaching of Thomas to be her own and that Doctor, honored with the special praises of the Pontiffs, the master and patron of Catholic schools."
  
  wn.com/Thomas_Aquinas
• Thoralf Skolem
  Thoralf Albert Skolem (23 May 1887 – 23 March 1963) (Norwegian pronunciation: [ˈtuːɾɑlf ˈskuːləm]) was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.
  Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany.
  In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters.
  
  wn.com/Thoralf_Skolem
• 
  William James
  William James (January 11, 1842 – August 26, 1910) was a pioneering American psychologist and philosopher who was trained as a physician. He was the first educator to offer a psychology course in the U.S. He wrote influential books on the young science of psychology, educational psychology, psychology of religious experience and mysticism, and on the philosophy of pragmatism. He was the brother of novelist Henry James and of diarist Alice James. In the summer of 1878, James married Alice Gibbens.
  William James was born at the Astor House in New York City. He was the son of Henry James Sr., a noted and independently wealthy Swedenborgian theologian well acquainted with the literary and intellectual elites of his day. The intellectual brilliance of the James family milieu and the remarkable epistolary talents of several of its members have made them a subject of continuing interest to historians, biographers, and critics.
  James interacted with a wide array of writers and scholars throughout his life, including his godfather Ralph Waldo Emerson, his godson William James Sidis, as well as Charles Sanders Peirce, Bertrand Russell, Josiah Royce, Ernst Mach, John Dewey, Macedonio Fernández, Walter Lippmann, Mark Twain, Horatio Alger, Jr., Henri Bergson and Sigmund Freud.
  
  wn.com/William_James
• 
  William of Ockham
  William of Ockham ( /ˈɒkəm/; also Occam, Hockham, or several other spellings; c. 1288 – c. 1348) was an English Franciscan friar and scholastic philosopher, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of the fourteenth century. Although he is commonly known for Ockham's razor, the methodological principle that bears his name, William of Ockham also produced significant works on logic, physics, and theology. In the Church of England, his day of commemoration is 10 April.
  William of Ockham joined the Franciscan order at an early age. It is believed that he studied theology at the University of Oxford from 1309 to 1321, but never completed his master's degree (the usual undergraduate degree in those times). Because of this, he acquired the byname Venerabilis Inceptor, or "Worthy Beginner" (an inceptor was a student formally admitted to the ranks of teachers by the university authorities). He was also known as the Doctor Invincibilis or "Unconquerable Teacher."
  
  wn.com/William_of_Ockham

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