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MIT Press
The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States).
The MIT Press traces its origins back to 1926 when MIT published under its own name a lecture series entitled Problems of Atomic Dynamics given by the visiting German physicist and later Nobel Prize winner, Max Born. Six years later, MIT's publishing operations were first formally instituted by the creation of an imprint called Technology Press in 1932. This imprint was founded by James R. Killian, Jr., at the time editor of MIT's alumni magazine and later to become MIT president.
Technology Press published eight titles independently, then in 1937 entered into an arrangement with John Wiley & Sons in which Wiley took over marketing and editorial responsibilities. In 1962 the association with Wiley came to an end after a further 125 titles had been published. The press acquired its modern name after this separation, and has since functioned as an independent publishing house.
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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.)
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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.
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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.
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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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Auguste Comte
Isidore Auguste Marie François Xavier Comte (19 January 1798 – 5 September 1857), better known as Auguste Comte [oɡyst kɔ̃t], was a French philosopher. He was a founder of the discipline of sociology and of the doctrine of positivism. He may be regarded as the first philosopher of science in the modern sense of the term.
Strongly influenced by the Utopian socialist Henri Saint-Simon, Comte developed the positive philosophy in an attempt to remedy the social malaise of the French revolution, calling for a new social doctrine based on the sciences. Comte was a major influence on 19th century thought, impacting the work of social thinkers such as Karl Marx, John Stuart Mill, and George Eliot. His concept of sociologie and social evolutionism, though now outdated, set the tone for early social theorists and anthropologists such as Harriet Martineau and Herbert Spencer, evolving into Modern academic sociology presented by Émile Durkheim as practical and objective social research.
Comte's social theories culminated in the "religion of humanity", which was influential to the development of religious humanist and Secular Humanist organizations in the 19th century. Comte likewise coined the word "altruism".
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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.
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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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Carl Gustav Hempel
Carl Gustav "Peter" Hempel (January 8, 1905 in Oranienburg, Germany - November 9, 1997 in Princeton, New Jersey) was a philosopher of science and a major figure in 20th-century logical empiricism. He is especially well known for his articulation of the Deductive-nomological model of scientific explanation, which was considered the "standard model" of scientific explanation during the 1950s and 1960s. He is also known for the Raven paradox, which highlights the problem of induction.
Hempel studied mathematics, physics, and philosophy at the University of Göttingen, Heidelberg and Berlin. In Göttingen he encountered David Hilbert and was impressed by his attempt to base all of mathematics on solid logical foundations derived from a limited number of axioms (Hilbert's Program). Having moved to Berlin he participated in a congress on scientific philosophy in 1929, where he met Rudolf Carnap and became involved in the Berlin Circle of philosophers that was associated with the Vienna Circle. In 1934 he received his doctoral degree from the University of Berlin with a dissertation on probability theory.
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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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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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Francis Bacon
Francis Bacon, 1st Viscount St Alban(s),KC (22 January 1561 – 9 April 1626) was an English philosopher, statesman, scientist, jurist and author. He served both as Attorney General and Lord Chancellor of England. Although his political career ended in disgrace, he remained extremely influential through his works, especially as philosophical advocate and practitioner of the scientific method during the scientific revolution.
Bacon has been called the creator of empiricism. His works established and popularised inductive methodologies for scientific inquiry, often called the Baconian method, or simply the scientific method. His demand for a planned procedure of investigating all things natural marked a new turn in the rhetorical and theoretical framework for science, much of which still surrounds conceptions of proper methodology today.
Bacon was knighted in 1603, and created both the Baron Verulam in 1618, and the Viscount St Alban in 1621; as he died without heirs both peerages became extinct upon his death. He famously died by contracting pneumonia while studying the effects of freezing on the preservation of meat.
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Galileo Galilei
Galileo Galilei (Italian pronunciation: [ɡaliˈlɛːo ɡaliˈlɛi]; 15 February 1564 – 8 January 1642), was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism. Galileo has been called the "father of modern observational astronomy", the "father of modern physics", the "father of science", and "the Father of Modern Science".
His contributions to observational astronomy include the telescopic confirmation of the phases of Venus, the discovery of the four largest satellites of Jupiter (named the Galilean moons in his honour), and the observation and analysis of sunspots. Galileo also worked in applied science and technology, inventing an improved military compass and other instruments.
Galileo's championing of heliocentrism was controversial within his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism due to the absence of an observed stellar parallax. The matter was investigated by the Roman Inquisition in 1615, and they concluded that it could only be supported as a possibility, not as an established fact. Galileo later defended his views in Dialogue Concerning the Two Chief World Systems, which appeared to attack Pope Urban VIII and thus alienated him and the Jesuits, who had both supported Galileo up until this point. He was tried by the Inquisition, found "vehemently suspect of heresy", forced to recant, and spent the rest of his life under house arrest. It was while Galileo was under house arrest that he wrote one of his finest works, Two New Sciences. Here he summarized the work he had done some forty years earlier, on the two sciences now called kinematics and strength of materials.
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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 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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Henri Poincaré
Jules Henri Poincaré (29 April 1854 – 17 July 1912) (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃nkaˈʁe]) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.
Poincaré made clear importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.
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Herbert Spencer
Herbert Spencer (27 April 1820 – 8 December 1903) was an English philosopher, biologist, sociologist, and prominent classical liberal political theorist of the Victorian era.
Spencer developed an all-embracing conception of evolution as the progressive development of the physical world, biological organisms, the human mind, and human culture and societies. He was "an enthusiastic exponent of evolution" and even "wrote about evolution before Darwin did." As a polymath, he contributed to a wide range of subjects, including ethics, religion, anthropology, economics, political theory, philosophy, biology, sociology, and psychology. During his lifetime he achieved tremendous authority, mainly in English-speaking academia. "The only other English philosopher to have achieved anything like such widespread popularity was Bertrand Russell, and that was in the 20th century." Spencer was "the single most famous European intellectual in the closing decades of the nineteenth century" but his influence declined sharply after 1900; "Who now reads Spencer?" asked Talcott Parsons in 1937.
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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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Hugh of Saint Victor
Hugh of Saint Victor (c. 1096 – 11 February 1141) was born perhaps in France, or more probably in Saxony. His origins and early life are rather obscure. He studied and taught at the Augustinian Abbey of Saint Victor in Paris after which he is named. His writings include works of theology, mysticism, philosophy and the arts. Appointed a canon of the Victorines around the turn of the twelfth century.
The early Didascalion was an elementary, encyclopedic approach to God and Christ, in which Hugh avoided controversial subjects and focused on what he took to be commonplaces of Catholic Christianity. In it he outlined three types of philosophy or "science" [scientia] that can help mortals improve themselves and advance toward God: theoretical philosophy (theology, mathematics, physics) provides them with truth, practical philosophy (ethics, economics, politics) aids them in becoming virtuous and prudent, and "mechanical" or "illiberal" philosophy (e.g., carpentry, agriculture, medicine) yields physical benefits. A fourth philosophy, logic, is preparatory to the others and exists to ensure clear and proper conclusions in them. Hugh's deeply mystical bent did not prevent him from seeing philosophy as a useful tool for understanding the divine, or from using it to argue on behalf of faith.
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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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Imre Lakatos
Imre Lakatos (November 9, 1922 – February 2, 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.
Lakatos was born Imre (Avrum) Lipschitz to a Jewish family in Debrecen, Hungary in 1922. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. He avoided Nazi persecution of Jews by changing his name to Imre Molnár. His mother and grandmother died in Auschwitz. He became an active communist during the Second World War. He changed his last name once again to Lakatos (Locksmith) in honor of Géza Lakatos.
After the war, from 1947 he worked as a senior official in the Hungarian ministry of education. He also continued his education with a PhD at Debrecen University awarded in 1948, and also attended György Lukács's weekly Wednesday afternoon private seminars. He also studied at the Moscow State University under the supervision of Sofya Yanovskaya in 1949. When he returned, however, he found himself on the losing side of internal arguments within the Hungarian communist party and was imprisoned on charges of revisionism from 1950 to 1953. More of Lakatos' activities in Hungary after World War II have recently become known.
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Isaac Newton
Sir Isaac Newton PRS MP (25 December 1642 – 20 March 1727 [NS: 4 January 1643 – 31 March 1727]) was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived." His monograph Philosophiæ Naturalis Principia Mathematica, published in 1687, lays the foundations for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws, by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the Scientific Revolution.
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John Dewey
John Dewey (October 20, 1859 – June 1, 1952) was an American philosopher, psychologist and educational reformer whose ideas have been influential in education and social reform. Dewey was an important early developer of the philosophy of pragmatism and one of the founders of functional psychology. He was a major representative of progressive education and liberalism.
Although Dewey is known best for his publications concerning education, he also wrote about many other topics, including experience, nature, art, logic, inquiry, democracy, and ethics.
In his advocacy of democracy, Dewey considered two fundamental elements—schools and civil society—as being major topics needing attention and reconstruction to encourage experimental intelligence and plurality. Dewey asserted that complete democracy was to be obtained not just by extending voting rights but also by ensuring that there exists a fully formed public opinion, accomplished by effective communication among citizens, experts, and politicians, with the latter being accountable for the policies they adopt.[citation needed]
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John Stuart Mill
John Stuart Mill, FRSE (20 May 1806 – 8 May 1873) was a British philosopher, political economist and civil servant. He was an influential contributor to social theory, political theory, and political economy. He has been called "the most influential English-speaking philosopher of the nineteenth century". Mill's conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of utilitarianism, an ethical theory developed by Jeremy Bentham. Hoping to remedy the problems found in an inductive approach to science, such as confirmation bias, he clearly set forth the premises of falsification as the key component in the scientific method. Mill was also a Member of Parliament and an important figure in liberal political philosophy.
John Stuart Mill was born on Rodney Street in the Pentonville area of London, the eldest son of the Scottish philosopher, historian and economist James Mill, and Harriet Burrow. John Stuart was educated by his father, with the advice and assistance of Jeremy Bentham and Francis Place. He was given an extremely rigorous upbringing, and was deliberately shielded from association with children his own age other than his siblings. His father, a follower of Bentham and an adherent of associationism, had as his explicit aim to create a genius intellect that would carry on the cause of utilitarianism and its implementation after he and Bentham had died.
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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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Jürgen Habermas
Jürgen Habermas ( /ˈjɜrɡən/ or /ˈjʊərɡən ˈhɑːbərmɑːs/;German: [ˈjʏʁɡən ˈhaːbɐmaːs]; born June 18, 1929) is a German sociologist, geographer and philosopher in the tradition of critical theory and pragmatism. He is perhaps best known for his theory on the concepts of 'communicative rationality' and the 'public sphere'. His work focuses on the foundations of social theory and epistemology, the analysis of advanced capitalistic societies and democracy, the rule of law in a critical social-evolutionary context, and contemporary politics, particularly German politics. Habermas's theoretical system is devoted to revealing the possibility of reason, emancipation, and rational-critical communication latent in modern institutions and in the human capacity to deliberate and pursue rational interests. Habermas is known for his work on the concept of modernity, particularly with respect to the discussions of "rationalization" originally set forth by Max Weber. While influenced by American pragmatism, action theory, and even poststructuralism, many of the central tenets of Habermas' thought remain broadly Marxist in nature. Global polls identified him as one of the leading intellectuals of the present day.
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Karl Pearson
Karl Pearson FRS (27 March 1857 – 27 April 1936) was an influential English mathematician who has been credited for establishing the discipline of mathematical statistics.
In 1911 he founded the world's first university statistics department at University College London. He was a proponent of eugenics, and a protégé and biographer of Sir Francis Galton.
A sesquicentenary conference was held in London on 23 March 2007, to celebrate the 150th anniversary of his birth.
Carl Pearson, later known as Karl Pearson (1857–1936), was born to William Pearson and Fanny Smith, who had three children, Aurthur, Carl (Karl) and Amy. William Pearson also sired an illegitimate son, Frederick Mockett.
Pearson's mother, Fanny Pearson née Smith, came from a family of master mariners who sailed their own ships from Hull; his father read law at Edinburgh and was a successful barrister and Queen's Counsel (QC). William Pearson's father's family came from the North Riding of Yorkshire.
"Carl Pearson" inadvertently became "Karl Pearson" when he enrolled at the University of Heidelberg in 1879, which changed the spelling. He used both variants of his name until 1884 when he finally adopted Karl — supposedly also after Karl Marx[citation needed], though some argue otherwise. Eventually he became universally known as "KP".
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Karl Popper
Sir Karl Raimund Popper, CH FRSFBA (28 July 1902 – 17 September 1994) was an Austro-Britishphilosopher and professor at the London School of Economics. He is generally regarded as one of the greatest philosophers of science of the 20th century; he also wrote extensively on social and political philosophy.
Popper is known for his attempt to repudiate the classical observationalist/inductivist form of scientific method in favour of empirical falsification. He is also known for his opposition to the classical justificationist account of knowledge which he replaced with critical rationalism, "the first non justificational philosophy of criticism in the history of philosophy". In political discourse, he is known for his vigorous defence of liberal democracy and the principles of social criticism that he came to believe made a flourishing "open society" possible.
Karl Popper was born in Vienna (then in Austria-Hungary) in 1902, to upper middle-class parents. All of Karl Popper's grandparents were Jewish, but the Popper family converted to Lutheranism before Karl was born. Karl's father Simon Siegmund Carl Popper was a lawyer from Bohemia, and mother Jenny Schiff was of Silesian and Hungarian descent. After establishing themselves in Vienna, the Poppers made a rapid social climb in Viennese society: Simon Siegmund Carl became a legal partner[clarification needed] of Vienna's liberal mayor Raimond Grübl and, after his death in 1898, took over the firm (Karl received his middle name from the mayor).
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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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Larry Laudan
Larry Laudan (b. Austin, Texas, 1941) is a contemporary philosopher of science and epistemologist. He has strongly criticized the traditions of positivism, realism, and relativism, and he has defended a view of science as a privileged and progressive institution against popular challenges. Laudan's philosophical view of "research traditions" is seen as an important alternative to Imre Lakatos's "research programs."
His most important contributions to the philosophy of science can be found in his book Progress and its Problems (1977) while acting as chair of the history and philosophy of science department at the University of Pittsburgh. Laudan charges philosophers of science with paying lip service to the view that “science is fundamentally a problem-solving activity” without taking seriously the view's implications for the history of science and its philosophy, and without questioning certain issues in the historiography and methodology of science. Against empiricism, which is represented by Karl Popper, and "revolutionism," represented by Thomas Kuhn, Laudan maintained in Progress and its Problems that science is an evolving process that accumulates more empirically validated evidence while solving conceptual anomalies at the same time. Mere evidence collecting or empirical confirmation does not constitute the true mechanism of scientific advancement; conceptual resolution and comparison of the solutions of anomalies provided by various theories form an indispensable part of the evolution of science.
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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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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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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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Otto Neurath
Otto Neurath (Vienna, December 10, 1882 - Oxford, December 22, 1945) was an Austrian philosopher of science, sociologist, and political economist. Before he was to flee his native country in 1934, Neurath was one of the leading figures of the Vienna Circle.
Neurath was born in Vienna, the son of Wilhelm Neurath (1840–1901), a well-known political economist at the time. He studied mathematics in Vienna and gained his Ph.D. in the department of Political Science and Statistics at the University of Berlin.
He married Anna Schapire in 1907. She died as a result of childbirth (Paul Neurath) in 1911, and he married a close friend, the mathematician and philosopher Olga Hahn. Perhaps because of Olga's blindness and then because of the outbreak of war, his son, Paul Neurath was sent to a children's home outside Vienna, where Neurath's mother lived, and returned to live with his father and Olga when he was nine years old.
Neurath taught political economy at the Neue Wiener Handelsakademie (New College of Commerce, Vienna) until war broke out. Subsequently he became director of the Deutsches Kriegwirtschaftsmuseum (German Museum of War Economy, later the Deutsches Wirtschaftmuseum) at Leipzig. Wolfgang Schumann, known from the Dürerbund for which Neurath had written many articles, urged him to work out a plan for socialization. Neurath joined the German Social Democratic Party in 1918-19 and ran an office for central economic planning in Munich. When the Bavarian Soviet Republic was defeated, Neurath was imprisoned but returned to Austria after intervention from the Austrian government. While in prison he wrote "Anti-Spengler", a critical attack on Oswald Spengler's "Decline of the West".
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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.
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Paul Feyerabend
Paul Karl Feyerabend (January 13, 1924 – February 11, 1994) was an Austrian-born philosopher of science best known for his work as a professor of philosophy at the University of California, Berkeley, where he worked for three decades (1958–1989). He lived a peripatetic life, living at various times in England, the United States, New Zealand, Italy, Germany, and finally Switzerland. His major works include Against Method (published in 1975), Science in a Free Society (published in 1978) and Farewell to Reason (a collection of papers published in 1987). Feyerabend became famous for his purportedly anarchistic view of science and his rejection of the existence of universal methodological rules. He is an influential figure in the philosophy of science, and also in the sociology of scientific knowledge.
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Paul Häberlin
Paul Haeberlin (February 17, 1878, Kesswil – September 29, 1960, Basel) was a Swiss philosopher who at different times in his career took the standpoint that either religion or theoretical knowledge was the answer to human problems. He always gave philosophy an important role, but religion was to him the only way man could understand his real position in existence. Haeberlin made contributions to characterology and psychotherapeutics, and was especially successful in treating psychopathic youth and teens. Made a full professor of philosophy, psychology and pedagogics at the University of Basel.
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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."
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Pierre Duhem
Pierre Maurice Marie Duhem ([pjˈɛʁ moʁˈis maʁˈi dyˈɛm] ( listen); 9 June 1861 – 14 September 1916) was a French physicist, mathematician and philosopher of science, best known for his writings on the indeterminacy of experimental criteria and on scientific development in the Middle Ages. Duhem also made major contributions to the science of his day, particularly in the fields of hydrodynamics, elasticity, and thermodynamics.
Duhem's views on the philosophy of science are explicated in The Aim and Structure of Physical Theory. In this work, he opposed Newton's statement that the Principia's law of universal mutual gravitation was deduced from 'phenomena', including Kepler's second and third laws. Newton's claims in this regard had already been attacked by critical proof-analyses of the German logician Leibniz and then most famously by Immanuel Kant, following Hume's logical critique of induction. But the novelty of Duhem's work was his proposal that Newton's theory of universal mutual gravity flatly contradicted Kepler's Laws of planetary motion because the interplanetary mutual gravitational perturbations caused deviations from Keplerian orbits. Since no proposition can be validly logically deduced from any it contradicts, according to Duhem, Newton must not have logically deduced his law of gravitation directly from Kepler's Laws.
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Pierre Gassendi
Pierre Gassendi (January 22, 1592 – October 24, 1655) was a French philosopher, priest, scientist, astronomer, and mathematician. With a church position in south-east France, he also spent much time in Paris, where he was a leader of a group of free-thinking intellectuals. He was also an active observational scientist, publishing the first data on the transit of Mercury in 1631. The lunar crater Gassendi is named after him.
He wrote numerous philosophical works, and some of the positions he worked out are considered significant, finding a way between scepticism and dogmatism. Richard Popkin indicates that Gassendi was one of the first thinkers to formulate the modern "scientific outlook", of moderated scepticism and empiricism. He clashed with his contemporary Descartes on the possibility of certain knowledge. His best known intellectual project attempted to reconcile Epicurean atomism with Christianity.
Gassendi was born at Champtercier, near Digne, in France. A youthful prodigy, at a very early age he showed academic potential and attended the college at Digne, where he displayed a particular aptitude for languages and mathematics. Soon afterwards he entered the University of Aix-en-Provence, to study philosophy under Philibert Fesaye. In 1612 the college of Digne called him to lecture on theology. Four years later he received the degree of Doctor of Theology at Avignon, and in 1617 he took holy orders. In the same year he answered a call to the chair of philosophy at Aix-en-Provence University, and seems gradually to have withdrawn from theology.
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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.
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Ray Solomonoff
Ray Solomonoff (July 25, 1926 – December 7, 2009) was the inventor of algorithmic probability, and founder of algorithmic information theory, He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability. He circulated the first report on non-semantic machine learning in 1956.
Solomonoff first described algorithmic probability in 1960, publishing the crucial theorem that launched Kolmogorov complexity and algorithmic information theory. He first described these results at a Conference at Caltech in 1960, and in a report, Feb. 1960, "A Preliminary Report on a General Theory of Inductive Inference." He clarified these ideas more fully in his 1964 publications, "A Formal Theory of Inductive Inference," Part I and Part II.
Algorithmic probability is a mathematically formalized combination of Occam's razor, and the Principle of Multiple Explanations. It is a machine independent method of assigning a probability value to each hypothesis (algorithm/program) that explains a given observation, with the simplest hypothesis (the shortest program) having the highest probability and the increasingly complex hypotheses receiving increasingly small probabilities.
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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:
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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.
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Robert Kilwardby
Robert Kilwardby (c. 1215–11 September 1279) was an Archbishop of Canterbury in England and as well as a cardinal.
Kilwardby studied at the University of Paris, then was a teacher of grammar and logic there. He then joined the Dominican Order and studied theology, and became regent at Oxford University before 1261, probably by 1245. He was named provincial prior of the Dominicans for England in 1261, and in October 1272 Pope Gregory X appointed him as Archbishop of Canterbury to end a disupte over the election. Kilwardby was provided to the archbishopric on 11 October 1272, given the temporalities on 12 December 1272, and consecrated on 26 February 1273.
Kilwardby crowned Edward I and his wife Eleanor as king and queen of England in August 1274, but otherwise took little part in politics. He instead concentrated on his ecclesiastical duties, including charity to the poor and donating to the Dominicans.
In 1278 Pope Nicholas III named Kilwardby Cardinal Bishop of Porto and Santa Rufina. He then resigned Canterbury and left England, taking with him papers, registers and documents belonging to the see. He also left the see deep in debt again, after his predecessor had cleared the debt. He died in Italy in 1279 and was buried in the Dominican convent in Viterbo, Italy. While in theory this was a promotion, probably it was not, as the pope was unhappy with Kilwardy's support of efforts to resist the payment of papal revenues and with the lack of effort towards the reforms demanded at the Second Council of Lyon in 1274.
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Roger Bacon
Roger Bacon, O.F.M. (c. 1214–1294), (scholastic accolade Doctor Mirabilis, meaning "wonderful teacher"), was an English philosopher and Franciscan friar who placed considerable emphasis on the study of nature through empirical methods. He is sometimes credited, mainly starting in the 19th century, as one of the earliest European advocates of the modern scientific method inspired by the works of Aristotle and later pseudo-Aristotelian works, like the works of Muslim scientist Alhazen. However, more recent reevaluations emphasize that he was essentially a medieval thinker, with much of his "experimental" knowledge obtained from books, in the scholastic tradition. A survey of the reception of Bacon's work over centuries found that it often reflects the concerns and controversies central to the receivers.
Roger Bacon was born in Ilchester in Somerset, England, possibly in 1213 or 1214 at the Ilchester Friary. The only source for his date of birth is his statement in the Opus Tertium, written in 1267, that "forty years have passed since I first learned the alphabet". The 1214 birth date assumes he was not being literal, and may have meant 40 years had passed since he matriculated at Oxford at the age of 13. If he had been literal, his birth date was more likely to have been around 1220/1222. In the same passage he reports that for all but two of those forty years he had always been engaged in study. His family appears to have been well-off, but, during the stormy reign of Henry III of England, their property was despoiled and several members of the family were driven into exile.
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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").
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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.
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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).
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Scottish people
The Scottish people (Scots Gaelic: Albannaich), or Scots, are a nation and ethnic group native to Scotland. Historically they emerged from an amalgamation of the Picts and Gaels, incorporating neighbouring Britons to the south as well as invading Germanic peoples such as the Anglo-Saxons and the Norse. Later the Normans also had some influence.
In modern use, "Scottish people" or "Scots" is used to refer to anyone whose linguistic, cultural, family ancestral or genetic origins are from within Scotland. The Latin word Scotti originally applied to a particular, 5th century, Goidelic tribe that inhabited Ireland. Though sometimes considered archaic or pejorative, the term Scotch has also been used for the Scottish people, though this usage is current primarily outside Scotland.
There are people of Scottish descent in many countries other than Scotland. Emigration, influenced by factors such as the Highland and Lowland Clearances, Scottish participation in the British Empire, and latterly industrial decline and unemployment, resulted in Scottish people being found throughout the world. Large populations of Scottish people settled the new-world lands of North and South America, Australia and New Zealand. There is a Scottish presence at a particularly high rate in Canada, which has the second largest population of descended Scots ancestry, after the United States. They took with them their Scottish languages and culture.
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Sextus Empiricus
Sextus Empiricus (c. 160-210 AD), was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism.
In his medical work, tradition maintains that he belonged to the "empiric school", as reflected by his name. However, at least twice in his writings, Sextus seems to place himself closer to the "methodic school", as his philosophical views imply.
Sextus Empiricus's three known works are the Outlines of Pyrrhonism (Πυῤῥώνειοι ὑποτύπωσεις or Pyrrhōneioi hypotypōseis, thus commonly abbreviated PH), and two distinct works preserved under the same title, Against the Mathematicians (Adversus Mathematicos), one of which is probably incomplete.
The first six books of Against the Mathematicians are commonly known as Against the Professors, but each book also has a traditional title (Against the Grammarians (book I), Against the Rhetoricians (book II), Against the Geometricians (book III), Against the Arithmeticians (book IV), Against the Astrologers (book V), Against the Musicians (book VI). It is widely believed, with perhaps the exception of Sextus scholar Richard Bett, that this is Sextus's latest and most mature work.
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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]
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Stephen Toulmin
Stephen Edelston Toulmin (25 March 1922 - 4 December 2009) was a British philosopher, author, and educator. Influenced by Ludwig Wittgenstein, Toulmin devoted his works to the analysis of moral reasoning. Throughout his writings, he sought to develop practical arguments which can be used effectively in evaluating the ethics behind moral issues. His works were later found useful in the field of rhetoric for analyzing rhetorical arguments. The Toulmin Model of Argumentation, a diagram containing six interrelated components used for analyzing arguments, was considered his most influential work, particularly in the field of rhetoric and communication, and in computer science.
Stephen Toulmin was born in London, England, on 25 March 1922 to Geoffrey Edelson Toulmin and Doris Holman Toulmin. He earned his Bachelor of Arts degree from King's College, Cambridge in 1942. Soon after, Toulmin was hired by the Ministry of Aircraft Production as a junior scientific officer, first at the Malvern Radar Research and Development Station and later at the Supreme Headquarters of the Allied Expeditionary Force in Germany. At the end of World War II, he returned to England to earn a Master of Arts degree in 1947 and a PhD in philosophy from Cambridge University, subsequently publishing his dissertation as An Examination of the Place of Reason in Ethics (1950). While at Cambridge, Toulmin came into contact with the Austrian philosopher Ludwig Wittgenstein, whose examination on the relationship between the uses and the meanings of language shaped much of Toulmin’s own work.
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Thomas Hobbes
Thomas Hobbes of Malmesbury (5 April 1588 – 4 December 1679), in some older texts Thomas Hobbs of Malmsbury, was an English philosopher, best known today for his work on political philosophy. His 1651 book Leviathan established the foundation for most of Western political philosophy from the perspective of social contract theory.
Hobbes was a champion of absolutism for the sovereign but he also developed some of the fundamentals of European liberal thought: the right of the individual; the natural equality of all men; the artificial character of the political order (which led to the later distinction between civil society and the state); the view that all legitimate political power must be "representative" and based on the consent of the people; and a liberal interpretation of law which leaves people free to do whatever the law does not explicitly forbid.
He was one of the founders of modern political philosophy. His understanding of humans as being matter and motion, obeying the same physical laws as other matter and motion, remains influential; and his account of human nature as self-interested cooperation, and of political communities as being based upon a "social contract" remains one of the major topics of political philosophy.
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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.
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Wilhelm Windelband
Wilhelm Windelband (May 11, 1848, Potsdam – October 22, 1915, Heidelberg) was a German philosopher of the Baden School.
Windelband is now mainly remembered for the terms nomothetic and idiographic, which he introduced. These have currency in psychology and other areas, though not necessarily in line with his original meanings. Windelband was a neo-Kantian who protested other neo-Kantians of his time and maintained that "to understand Kant rightly means to go beyond him". Against his positivist contemporaries, Windelband argued that philosophy should engage in humanistic dialogue with the natural sciences rather than uncritically appropriating its methodologies. His interests in psychology and cultural sciences represented an opposition to psychologism and historicism schools by a critical philosophic system.
Windelband relied in his effort to reach beyond Kant on such philosophers as Georg Wilhelm Friedrich Hegel, Johann Friedrich Herbart and Hermann Lotze. Closely associated with Windelband was Heinrich Rickert. Windelband's disciples were not only noted philosophers, but sociologists like Max Weber and theologians like Ernst Troeltsch and Albert Schweitzer.
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Wilhelm Wundt
Wilhelm Maximilian Wundt (16 August 1832 - 31 August 1920 in Germany) was a German physician, psychologist, physiologist, philosopher, and professor, known today as one of the founding figures of modern psychology. He is widely regarded as the "father of experimental psychology". In 1879, Wundt founded the first formal laboratory for psychological research at the University of Leipzig.
By creating this laboratory he was able to explore the nature of religious beliefs, identify mental disorders and abnormal behavior, and find damaged parts of the brain. In doing so, he was able to establish psychology as a separate science from other topics. He also formed the first journal for psychological research in 1881.
Asteroids 635 Vundtia and 11040 Wundt are named in his honour.
Wundt was born at Neckarau, Baden on August 16th, 1832 (now part of Mannheim), the fourth child to parents Maximilian Wundt (a Lutheran minister), and his wife Marie Frederike. When about four years of age, Wundt's family moved to Heidelsheim which was known to be a small town. He studied from 1851 to 1856 at the University of Tübingen, University of Heidelberg, and the University of Berlin. After graduating in medicine from Heidelberg (1856), Wundt studied briefly with Johannes Peter Müller, before joining the University's staff, becoming an assistant to the physicist and physiologist Hermann von Helmholtz in 1858. There he wrote Contributions to the Theory of Sense Perception (1858–62). In 1865, he wrote a textbook about human physiology. In 1867 he became a professor in acquainting medical students with the exact physical needs for medical investigation. In 1874, he became a professor of "Inductive Philosophy" in Zurich. In 1875, he moved back to Leipzig.
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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."
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William Whewell
William Whewell ( /ˈhjuːəl/ HEW-əl; 24 May 1794 – 6 March 1866) was an English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science. He was Master of Trinity College, Cambridge.
Whewell was born in Lancaster. His father, a carpenter, wished him to follow his trade, but his success in mathematics at Lancaster and Heversham grammar schools won him an exhibition (a type of scholarship) at Trinity College, Cambridge (1812). In 1814 he was awarded the Chancellor's Gold Medal for poetry. He was Second Wrangler in 1816, President of the Cambridge Union Society in 1817, became fellow and tutor of his college, and, in 1841, succeeded Dr Christopher Wordsworth as master. He was professor of mineralogy from 1828 to 1832 and Knightbridge Professor of Philosophy (then called "moral theology and casuistical divinity") from 1838 to 1855.
Whewell died in Cambridge 1866 as a result of a fall from his horse.
Full bibliographical details are given by Isaac Todhunter, W. Whewell: an Account of his Writings (2 vols., 1876). See also Life of W. Whewell, by Mrs Stair Douglas (1881).
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Abstract Reasoning, Logical Reasoning, Inductive Reasoning - How To Ace Aptitude Tests 5/7
- Order: Reorder
- Duration: 13:55
- Updated: 13 Jan 2013
This segment, introduced by chartered occupational psychologist Ben Williams, discusses abstract reasoning tests that are widely used in career tests and recruitment. Learn about rule-finding, building blocks, series, see how to find the correct figure, get tips for your preparation! www.careergym.com
http://wn.com/Abstract_Reasoning,_Logical_Reasoning,_Inductive_Reasoning_How_To_Ace_Aptitude_Tests_5_7
This segment, introduced by chartered occupational psychologist Ben Williams, discusses abstract reasoning tests that are widely used in career tests and recruitment. Learn about rule-finding, building blocks, series, see how to find the correct figure, get tips for your preparation! www.careergym.com
Geometry: 2-1: Using Inductive Reasoning to Make Conjectures
- Order: Reorder
- Duration: 5:18
- Updated: 07 Jan 2013
www.tylertarver.com Use inductive reasoning to identify patterns and make conjectures. To find counterexamples to disprove conjectures. To bore the life out of people.
http://wn.com/Geometry_2-1_Using_Inductive_Reasoning_to_Make_Conjectures
www.tylertarver.com Use inductive reasoning to identify patterns and make conjectures. To find counterexamples to disprove conjectures. To bore the life out of people.
- published: 26 Jul 2010
- views: 9030
- author: Sir Tyler Tarver
Inductive Reasoning
- Order: Reorder
- Duration: 7:17
- Updated: 13 Jan 2013
mathispower4u.wordpress.com
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Pitfalls of Arguments: Inductive Reasoning
- Order: Reorder
- Duration: 7:49
- Updated: 13 Dec 2012
Adam and Micah explain the value and dangers of using Inductive reasoning. Join the discussion at arguingart.com, leave a comment and come back for more. Thanks for watching.
http://wn.com/Pitfalls_of_Arguments_Inductive_Reasoning
Adam and Micah explain the value and dangers of using Inductive reasoning. Join the discussion at arguingart.com, leave a comment and come back for more. Thanks for watching.
- published: 16 Feb 2010
- views: 2418
- author: arguingart
Deductive and Inductive Reasoning
- Order: Reorder
- Duration: 1:47
- Updated: 17 Jan 2013
ToK video about reasoning - made using www.xtranormal.com
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ToK video about reasoning - made using www.xtranormal.com
- published: 17 Dec 2009
- views: 46356
- author: Itsdude56
Inductive Reasoning
- Order: Reorder
- Duration: 1:13
- Updated: 19 Nov 2012
gscim.com This video is about inductive reasoning in logic. http genesismission.4t.com http
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gscim.com This video is about inductive reasoning in logic. http genesismission.4t.com http
- published: 31 Mar 2010
- views: 2330
- author: chuckpcr
Geometry - Inductive Reasoning
- Order: Reorder
- Duration: 5:05
- Updated: 11 Jan 2013
Logic can be so illogical at times. Join us as we tackly inductive reasoning and take a look at concepts such as the converse, inverse, and contrapositive of statements. Plus I'm sporting my space costume for the first time, so YayMath is now officially out-of-this-world. YAY MATH! Please visit yaymath.org Videos copyright (c) Yay Math
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Logic can be so illogical at times. Join us as we tackly inductive reasoning and take a look at concepts such as the converse, inverse, and contrapositive of statements. Plus I'm sporting my space costume for the first time, so YayMath is now officially out-of-this-world. YAY MATH! Please visit yaymath.org Videos copyright (c) Yay Math
- published: 05 Oct 2009
- views: 7425
- author: yaymath
Induction and Scientific Reasoning
- Order: Reorder
- Duration: 9:41
- Updated: 18 Jan 2013
www.criticalthinkeracademy.com. A sample video from the video tutorial course titled "Basic Concepts in Logic and Argumentation". You can preview the full course at the web link above. Table of Contents Part 1 What is an Argument? 1.1 Definition of an argument 1.2 Definition of a claim, or statement 1.3 Definition of a good argument (I) 1.4 Identifying premises and conclusions Part 2: What is a Good Argument? 2.1 The truth condition 2.2 The logic condition 2.3 Valid vs invalid arguments 2.4 Strong vs weak arguments 2.5 Definition of a good argument (II) Part 3: Deductive versus Inductive Arguments 3.1 Deduction and valid reasoning 3.2 Induction and invalid reasoning 3.3 Induction and scientific reasoning
http://wn.com/Induction_and_Scientific_Reasoning
www.criticalthinkeracademy.com. A sample video from the video tutorial course titled "Basic Concepts in Logic and Argumentation". You can preview the full course at the web link above. Table of Contents Part 1 What is an Argument? 1.1 Definition of an argument 1.2 Definition of a claim, or statement 1.3 Definition of a good argument (I) 1.4 Identifying premises and conclusions Part 2: What is a Good Argument? 2.1 The truth condition 2.2 The logic condition 2.3 Valid vs invalid arguments 2.4 Strong vs weak arguments 2.5 Definition of a good argument (II) Part 3: Deductive versus Inductive Arguments 3.1 Deduction and valid reasoning 3.2 Induction and invalid reasoning 3.3 Induction and scientific reasoning
- published: 16 Nov 2009
- views: 12663
- author: Kevin deLaplante
Deductive Vs Inductive
- Order: Reorder
- Duration: 6:14
- Updated: 16 Jan 2013
Jordan Taylor and Shawn Sparks explain to you the difference between inductive and deductive reasoning!
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Jordan Taylor and Shawn Sparks explain to you the difference between inductive and deductive reasoning!
- published: 05 Jun 2011
- views: 9804
- author: ssparky21
Dewey B. Larson on inductive reasoning - Reciprocal System
- Order: Reorder
- Duration: 66:31
- Updated: 30 Nov 2012
Dewey B. Larson's 1978 speech to the 3rd New Science Advocates conference. Inductive/deductive reasoning & Reciprocal System (Universe of Motion..) Original video from rstheory.org Apologies for the grainy video, tape being 30 years old.
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Dewey B. Larson's 1978 speech to the 3rd New Science Advocates conference. Inductive/deductive reasoning & Reciprocal System (Universe of Motion..) Original video from rstheory.org Apologies for the grainy video, tape being 30 years old.
- published: 24 Jan 2012
- views: 445
- author: Birdland11
141 1.2.2 Inductive Deductive Reasoning M2U00342
- Order: Reorder
- Duration: 7:37
- Updated: 21 Nov 2012
WEBSITE: www.teachertube.com Inductive and Deductive Reasoning
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WEBSITE: www.teachertube.com Inductive and Deductive Reasoning
- published: 31 Aug 2009
- views: 2234
- author: teachertubemath
Inductive Reasoning - YourTeacher.com - Geometry Help
- Order: Reorder
- Duration: 0:56
- Updated: 21 Jun 2012
For a complete lesson on inductive reasoning, go to www.yourteacher.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn the definitions of inductive and deductive reasoning, as well as the common mistakes that can be made when solving problems that involve each type of reasoning.
http://wn.com/Inductive_Reasoning_YourTeachercom_Geometry_Help
For a complete lesson on inductive reasoning, go to www.yourteacher.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn the definitions of inductive and deductive reasoning, as well as the common mistakes that can be made when solving problems that involve each type of reasoning.
- published: 15 Apr 2008
- views: 6432
- author: yourteachermathhelp
U12_L1_T3_we1 Inductive Reasoning 1
- Order: Reorder
- Duration: 2:07
- Updated: 25 Sep 2012
Learn more: www.khanacademy.org U12_L1_T3_we1 Inductive Reasoning 1
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Learn more: www.khanacademy.org U12_L1_T3_we1 Inductive Reasoning 1
- published: 27 Jun 2010
- views: 19368
- author: khanacademy
Inductive Reasoning 2
- Order: Reorder
- Duration: 2:21
- Updated: 25 Sep 2012
Learn more: www.khanacademy.org U12_L1_T3_we2 Inductive Reasoning 2
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Learn more: www.khanacademy.org U12_L1_T3_we2 Inductive Reasoning 2
- published: 27 Jun 2010
- views: 18310
- author: khanacademy
deductive VS inductive reasoning
- Order: Reorder
- Duration: 1:58
- Updated: 27 Feb 2012
connor nick and larissa made this one
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Deductive and Inductive Reasoning with Examples
- Order: Reorder
- Duration: 5:27
- Updated: 01 Jan 2013
Detailed explanation of deductive and inductive method with examples. For Notes www.biologyexams4u.com
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Detailed explanation of deductive and inductive method with examples. For Notes www.biologyexams4u.com
- published: 30 Dec 2012
- views: 40
- author: biologyexams4u
Inductive Reasoning Patterns.avi
- Order: Reorder
- Duration: 2:14
- Updated: 07 Dec 2011
In this video, we will investigate Inductive reasoning using a pattern.
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In this video, we will investigate Inductive reasoning using a pattern.
- published: 13 Aug 2010
- views: 821
- author: mrPSERIS
2.1 (1 of 2) Inductive Reasoning & Conjectures
- Order: Reorder
- Duration: 4:54
- Updated: 18 Sep 2012
Inductive reasoning and conjectures.
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Geometry: Inductive Reasoning
- Order: Reorder
- Duration: 3:42
- Updated: 12 Dec 2012
Watch more at www.educator.com Other subjects include Algebra 1/2, Basic Math, Trigonometry, Pre-Calculus, Pre-Algebra, Calculus, Statistics, Biology, Chemistry, Physics, Organic Chemistry, and Computer Science. -All lectures are broken down by individual topics -No more wasted time -Just search and jump directly to the answer
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Watch more at www.educator.com Other subjects include Algebra 1/2, Basic Math, Trigonometry, Pre-Calculus, Pre-Algebra, Calculus, Statistics, Biology, Chemistry, Physics, Organic Chemistry, and Computer Science. -All lectures are broken down by individual topics -No more wasted time -Just search and jump directly to the answer
- published: 27 Sep 2010
- views: 1908
- author: EducatorVids
Lead Change with Inductive Reasoning
- Order: Reorder
- Duration: 4:09
- Updated: 19 Jul 2012
People don't like to change. How do you bring about positive change in your life and work? Use inductive reasoning to build a case for your intended change. Here's how.
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People don't like to change. How do you bring about positive change in your life and work? Use inductive reasoning to build a case for your intended change. Here's how.
- published: 19 Jan 2012
- views: 90
- author: Soma Jurgensen
Inductive Reasoning #1
- Order: Reorder
- Duration: 2:44
- Updated: 13 Dec 2012
Learn about Inductive Reasoning.
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Inductive Reasoning Counterexample
- Order: Reorder
- Duration: 1:38
- Updated: 14 Jun 2012
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Dembski's inductive argument gets it's ass handed back to it.
- Order: Reorder
- Duration: 6:06
- Updated: 15 May 2012
Yeah, I'm obsessed with Dembski. 1. It's not scientific. 2. It's not a hypothesis 3. It's not a valid inductive argument www.youtube.com
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Yeah, I'm obsessed with Dembski. 1. It's not scientific. 2. It's not a hypothesis 3. It's not a valid inductive argument www.youtube.com
- published: 02 Feb 2009
- views: 853
- author: urbanelf
photo: AP / Sergei Chuzavkov
19 Jan 2013
MOSCOW — The government of President Viktor F. Yanukovich of Ukraine said Friday that it planned to bring murder charges against his political rival, the jailed former prime minister, Yulia V. Tymoshenko, in connection with the 1996 assassination of a member of Parliament. Connect With Us on Twitter. Follow @nytimesworld for international breaking news and headlines. Twitter List. Reporters and Editors ... He said that Ms ... Ms ... Ms ... ... ....(size: 8.0Kb)
photo: AP / Echorouk Elyaoumi
21 Jan 2013
Algerian security forces on Sunday made grisly discoveries of dozens of bodies at a sprawling natural gas facility, a day after a violent showdown put an end to an international hostage crisis. But as Algeria tallied up the losses, a debate was emerging about whether the militant group linked to al-Qaida that seized the plant had been intent on a massacre or whether it had simply been after money ... ....(size: 2.9Kb)
photo: US Army / Sgt. 1st Class Abram Pinnington
20 Jan 2013
WASHINGTON -- The war rages about cities with names such as Goa and Timbuktu, in a sparsely populated, mostly flat, dusty and landlocked country in northwest Africa. The combatants include a nomadic Berber people known as Tuareg, the French Foreign Legion and a coalition of al-Qaida affiliates who identify themselves with the Maghreb, the desert region of Northwestern Africa ... Some U.S ... Afghanistan ... J ... "They go to where the fight is ... ....(size: 6.6Kb)
photo: WN / Imran Nissar
20 Jan 2013
The world's 100 richest people earned enough money last year to end world extreme poverty four times over, according to released by international rights group and charity Oxfam. The $240 billion net income of the world's 100 richest billionaires would have ended poverty four times over, according to the London-based group's report released on Saturday ... 412 Source.. Agencies. Share ... Dear President Obama, please stop being so reasonable ... ....(size: 18.5Kb)
photo: AP / Patrick Semansky
20 Jan 2013
US President Barack Obama has blamed "terrorists" for the death of 23 hostages at a besieged desert gas facility in Algeria. A four-day siege at the In Amenas site was ended on Saturday by an army raid. At least five Britons, five Norwegians and 10 Japanese are dead or missing, while Algeria said its troops had killed all 32 hostage-takers ... Continue reading the main story Foreign citizens involved ... Hostage crisis in Algeria ... ....(size: 5.9Kb)
noodls
17 Jan 2013
When someone reasons to a conclusion by weighing factors ... This seems to be the position of those who consider this type of reasoning different from standard deductive or inductive reasoning, calling it "conductive," says philosophy professor emeritus J. Anthony Blair, a senior fellow of the Centre for Research in Reasoning, Argumentation and Rhetoric....(size: 2.1Kb)
North Jersey
11 Jan 2013
This is the curious, creative, uninhibited side of the brain where ideas are more valuable than facts, inductive reasoning trumps deductive reasoning, interpretation supplants analysis, and recognizing patterns works better than adding and subtracting numbers. ... of stories, all of which, I promise you, will lead inductively to powerful conclusions....(size: 13.1Kb)
New Straits/Business Times
10 Jan 2013
It is governed by a set of rules and customs ... He notes that by playing video games, children gain problem-solving abilities, increase in perseverance and develop skills for pattern recognition, testing hypotheses, estimating, inductive reasoning, resource management and mapping logistics. They also have improved memory, are able to think quickly and can make reasonable judgements ... ....(size: 1.7Kb)
The Examiner
08 Jan 2013
It's called inductive reasoning, and it is not a valid line of reasoning, because generalizations based on specific examples are easily disproved. For example, inductive reasoning would argue that because life-long smoker George Burns died of a heart attack at the age of 100 (the specific example), smoking is not hazardous to your health (the general conclusion)....(size: 5.5Kb)
Vanity Fair
03 Jan 2013
MSNBC reports. [NASA scientists] arrived at their estimate after studying a five-planet system called Kepler-32, which lies about 915 light-years from Earth. ... This is an example of deductive reasonin—what? Did you mumble something, Kepler? Oh, it’s actually an example of inductive reasoning? Is that right? Ah, well ... ....(size: 1.6Kb)
The Providence Journal
14 Dec 2012
This January, such core government functions as scientific research, education, public safety and environmental protection may well face deep cuts under budget sequestration ... Scientific literacy is an important part of a well rounded education ... Science education does more than teach facts; it will also teach critical thinking, deductive and inductive reasoning ... ....(size: 1.5Kb)
Foreign Policy
13 Dec 2012
Instead, the court interpreted Islamic law de novo using its own distinctive, somewhat idiosyncratic, version of modernist reasoning ... Derived through a process of inductive reasoning, these principles (the so-called qawa`id fiqhiyya) were thought to be generally applicable principles of law ... But even when his works used inductive reasoning in new ways, nobody referred to them as explorations of qawa`id fiqhiyya....(size: 17.7Kb)
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24 Nov 2012
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Inductive reasoning, also known as induction, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations of individual instances of members of the same class. Inductive reasoning contrasts with deductive reasoning in that a general conclusion is arrived at by specific examples.
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Definition[link]
The philosophical definition of inductive reasoning is much more nuanced than simple progression from particular/individual instances to wider generalizations. Rather, the premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it. In this manner, there is the possibility of moving from generalizations to individual instances. Inductive reasoning consists of inferring general principles or rules from specific facts. A well-known laboratory example of inductive reasoning works like a guessing game. The participants are shown cards that contain figures differing in several ways, such as shape, number, and color. On each trial, they are given two cards and asked to choose the one that represents a particular concept. After they choose a card, the researcher says "right" or "wrong."[1]
Though many dictionaries define inductive reasoning as reasoning that derives general principles from specific observations, this usage is outdated.[2]
Examples[link]
This is an example of deductive reasoning:
- 90% of humans are right-handed.
- Joe is a human.
- Therefore, the probability that Joe is right-handed is 90%. (See section on Statistical syllogism.)
Probability is employed, for example, in the following argument:
- Every life form we know of depends on liquid water to exist.
- All life depends on liquid water to exist.
However, induction is employed in the following argument:
- Every life form that everyone knows of depends on liquid water to exist.
- Therefore, all known life depends on liquid water to exist.
Inductive vs. deductive reasoning[link]
Inductive reasoning allows for the possibility that the conclusion be false, even where all of the premises are true.[3] The previous deduction was a false assertion of inductive reasoning based on the weak inductive conjecture of John Vickers.
His example is as follows:
- All of the swans we have seen are white.
- All swans are white.
The previous statement is an example of probabilistic reasoning, which is a weak type of induction. It is not an example of Strong Inductive Reasoning.
A proper example of inductive reasoning is as follows:
- All of the swans that all living beings have ever seen are white
- Therefore, all swans are white.
Note that this definition of inductive reasoning excludes mathematical induction, which is considered to be a form of deductive reasoning.
Strong and weak induction[link]
The words 'strong' and 'weak' are sometimes used to praise or demean the quality of an inductive argument. The idea is that you say "this is an example of strong induction" when you would decide to believe the conclusion if presented with the premises. Alternatively, you say "that is weak induction" when your particular world view does not allow you to see that the conclusions are likely given the premises.[citation needed]
Strong induction[link]
The equation
- "the gravitational force between two objects equals the gravitational constant times the product of the masses divided by the distance between them squared,"
has allowed us to describe the rate of fall of all objects we have observed.
Therefore:
- The gravitational force between two objects equals the gravitational constant times the product of the masses divided by the distance between them squared.
The conclusion of this argument is not absolutely certain, even given the premise. At speeds we normally experience, Newtonian mechanics holds quite well. But at speeds approaching that of light, the Newtonian system is not accurate and the conclusion in that case would be false. However, since, in most cases that we experience, the premise as stated would usually lead to the conclusion given, we are logical in calling this argument an instance of strong induction.
Even very strong inductions are potentially flawed interpretations of the truth, however reasonable and logical they might appear.
Weak induction[link]
Consider this example:
- I always hang pictures on nails.
- Therefore:
- All pictures hang from nails.
Here, the link between the premise and the conclusion is very weak. Not only is it possible for the conclusion to be false given the premise, it is even fairly likely that the conclusion is false. Not all pictures are hung from nails; moreover, not all pictures are hung. Thus we say that this argument is an instance of weak induction.
The previous is an example of probabilistic reasoning which employs weak induction. Therefore the previous example is closer to an example of probabilistic reasoning rather than Induction. Weak Induction is merely a type of conjecture, not a proof.
Induction[link]
Main article: Problem of induction
Inductive reasoning has been attacked for millennia by thinkers as diverse as Sextus Empiricus[4] and Karl Popper.[5]
The classic philosophical treatment of the problem of induction was given by the Scottish philosopher David Hume. Hume highlighted the fact that our everyday habits of mind depend on drawing uncertain conclusions from our relatively limited experiences rather than on deductively valid arguments. For example, we believe that bread will nourish us because it has done so in the past, despite no guarantee that it will do so. Hume argued that it is impossible to justify inductive reasoning. Inductive reasoning certainly cannot be justified deductively, and so our only option is to justify it inductively. However, to justify induction inductively is circular. Therefore, it is impossible to justify induction.[6]
However, Hume immediately argued that even if induction were proved unreliable, we would have to rely on it. So he took a middle road. Rather than approach everything with severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.[7]
Bias[link]
Inductive reasoning is also known as hypothesis construction because any conclusions made are based on educated predictions. There are three biases that could distort the proper application of induction[citation needed], thereby preventing the reasoner from forming the best, most logical conclusion based on the clues. These biases include the availability bias, the confirmation bias, and the predictable-world bias.
The availability bias causes the reasoner to depend primarily upon information that is readily available to him/her. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents would choose the causes that have been most prevalent in the media such as terrorism, and murders, and airplane accidents rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around him/her.
The confirmation bias is based on the natural tendency to confirm rather than to deny a current hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is in fact a sociable individual.
The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist. A major aspect of this bias is superstition, which is derived from the inability to acknowledge that coincidences are merely coincidences. Gambling, for example, is one of the most obvious forms of predictable-world bias. Gamblers often begin to think that they see patterns in the outcomes and, therefore, believe that they are able to predict outcomes based upon what they have witnessed. In reality, however, the outcomes of these games are always entirely random. There is no order. Since people constantly seek some type of order to explain human experiences, it is difficult for people to acknowledge that order may be nonexistent.[8]
Types[link]
Generalization[link]
A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.
- The proportion Q of the sample has attribute A.
- Therefore:
- The proportion Q of the population has attribute A.
- Example
There are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. A good inductive generalization would be that there are 15 black, and five white, balls in the urn.
How much the premises support the conclusion depends upon (a) the number in the sample group compared to the number in the population and (b) the degree to which the sample represents the population (which may be achieved by taking a random sample). The hasty generalization and the biased sample are generalization fallacies.
Statistical syllogism[link]
Main article: Statistical syllogism
A statistical syllogism proceeds from a generalization to a conclusion about an individual.
- A proportion Q of population P has attribute A.
- An individual X is a member of P.
- Therefore:
- There is a probability which corresponds to Q that X has A.
The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".
Simple induction[link]
Simple induction proceeds from a premise about a sample group to a conclusion about another individual.
- Proportion Q of the known instances of population P has attribute A.
- Individual I is another member of P.
- Therefore:
- There is a probability corresponding to Q that I has A.
This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.
Argument from analogy[link]
Main article: Argument from Analogy
The process of analogical inference involves noting the shared properties of two or more things, and from this basis inferring that they also share some further property:[9]
- P and Q are similar in respect to properties a, b, and c.
- Object P has been observed to have further property x.
- Therefore, Q probably has property x also.
Analogical reasoning is very frequent in common sense, science, philosophy and the humanities, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning. For more information on inferences by analogy, see Juthe, 2005.
Causal inference[link]
A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.
Prediction[link]
A prediction draws a conclusion about a future individual from a past sample.
- Proportion Q of observed members of group G have had attribute A.
- Therefore:
- There is a probability corresponding to Q that other members of group G will have attribute A when next observed.
Bayesian inference[link]
As a logic of induction rather than a theory of belief, Bayesian inference does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to a prior probability for a hypothesis based on logic or previous experience, and when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.
Inductive inference[link]
Around 1960, Ray Solomonoff founded the theory of universal inductive inference, the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. This is a mathematically formalized Occam's razor. Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity.
See also[link]
- Abductive reasoning
- Analogy
- Deductive reasoning
- Explanation
- Falsifiability
- Inductive reasoning aptitude
- Inductive Logic Programming
- Inferential statistics
- Inquiry
- Lateral thinking
- Logic
- Open world assumption
- Machine learning
- Mathematical induction
- Mill's Methods
- Raven paradox
- Retroduction
- Laurence Jonathan Cohen
- Counterinduction
References[link]
- ^ Carlson, N.R. & Heth, C.D.(2009).Psychology the Science of Behaviour.Toronto:Pearson Education Canada
- ^ "Deductive and Inductive Arguments", Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/ded-ind/, "Some dictionaries define “deduction” as reasoning from the general to specific and “induction” as reasoning from the specific to the general. While this usage is still sometimes found even in philosophical and mathematical contexts, for the most part, it is outdated. For example, [~snip~]"
- ^ John Vickers. The Problem of Induction. The Stanford Encyclopedia of Philosophy.
- ^ Sextus Empiricus, Outlines Of Pyrrhonism. Trans. R.G. Bury, Harvard University Press, Cambridge, Massachusetts, 1933, p. 283.
- ^ Karl R. Popper, David W. Miller. "A proof of the impossibility of inductive probability." Nature 302 (1983), 687–688.
- ^ Vickers, John. "The Problem of Induction" (Section 2). Stanford Encyclopedia of Philosophy. 21 June 2010
- ^ Vickers, John. "The Problem of Induction" (Section 2.1). Stanford Encyclopedia of Philosophy. 21 June 2010.
- ^ Gray, Peter. Psychology. New York: Worth, 2011. Print.
- ^ Baronett, Stan (2008). Logic. Upper Saddle River, NJ: Pearson Prentice Hall. pp. 321–325.
- Herms, D. "Logical Basis of Hypothesis Testing in Scientific Research" (pdf). http://www.dartmouth.edu/~bio125/logic.Giere.pdf.
- Kemerling, G (27 October 2001). "Causal Reasoning". http://www.philosophypages.com/lg/e14.htm.
- Holland, JH; Holyoak KJ; Nisbett RE; Thagard PR (1989). Induction: Processes of Inference, Learning, and Discovery. Cambridge, MA, USA: MIT Press. ISBN 0-262-58096-9.
- Holyoak, K; Morrison R (2005). The Cambridge Handbook of Thinking and Reasoning. New York: Cambridge University Press. ISBN 978-0-521-82417-0.
External links[link]
 |
Look up inductive reasoning in Wiktionary, the free dictionary. |
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Wikisource has the text of a 1920 Encyclopedia Americana article about Inductive reasoning. |
- Four Varieties of Inductive Argument from the Department of Philosophy, University of North Carolina at Greensboro.
- Inductive Logic from the Stanford Encyclopedia of Philosophy.
- Properties of Inductive ReasoningPDF (166 KiB), a psychological review by Evan Heit of the University of California, Merced.
- The Mind, Limber An article which employs the film The Big Lebowski to explain the value of inductive reasoning.
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