Branches of Mathematics

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• 
  

• 
  Alhazen
  Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham (Arabic: أبو علي، الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated)Alhazen) (965 in Basra – c. 1040 in Cairo) was a Muslimscientist and polymath described in various sources as either Arab or Persian. Alhazen made significant contributions to the principles of optics, as well as to physics, astronomy, mathematics, ophthalmology, philosophy, visual perception, and to the scientific method. He also wrote insightful commentaries on works by Aristotle, Ptolemy, and the Greek mathematician Euclid.
  He is frequently referred to as Ibn al-Haytham, and sometimes as al-Basri (Arabic: البصري), after his birthplace in the city of Basra. He was also nicknamed Ptolemaeus Secundus ("Ptolemy the Second") or simply "The Physicist" in medieval Europe.
  Born circa 965, in Basra, present-day Iraq, he lived mainly in Cairo, Egypt, dying there at age 74. Over-confident about practical application of his mathematical knowledge, he assumed that he could regulate the floods of the Nile. After being ordered by Al-Hakim bi-Amr Allah, the sixth ruler of the Fatimid caliphate, to carry out this operation, he quickly perceived the impossibility of what he was attempting to do, and retired from engineering. Fearing for his life, he feigned madness and was placed under house arrest, during and after which he devoted himself to his scientific work until his death.
  
  wn.com/Alhazen
• 
  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.
  
  wn.com/Aristotle
• Bas van Fraassen
  Bastiaan Cornelis van Fraassen (born in Goes, the Netherlands on 5 April 1941) is a Distinguished Professor of Philosophy at San Francisco State University, teaching courses in the philosophy of science, philosophical logic and the role of models in scientific practice. He previously taught at Yale University, the University of Southern California (USC), the University of Toronto and Princeton University. He coined the term "constructive empiricism" in his 1980 book The Scientific Image. Van Fraassen earned his B.A. (1963) from the University of Alberta and his M.A. (1964) and Ph.D. (1966, under the direction of Adolf Grünbaum) from the University of Pittsburgh.
  A philosopher of science, van Fraassen's 1989 book Laws and Symmetry attempted to lay the ground-work for explaining physical phenomena without using the assumption that such phenomena are caused by rules or laws which can be said to cause or govern their behavior. Focusing on the problem of underdetermination, he argued for the possibility that theories could have empirical equivalence but differ in their ontological commitments. He rejects the notion that the aim of science is to produce an account of the physical world that is literally true, but rather that its aim is to produce theories that are empirically adequate. Van Fraassen has also studied the philosophy of quantum mechanics, philosophical logic, and epistemology.
  
  wn.com/Bas_van_Fraassen
• 
  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.
  
  wn.com/Bertrand_Russell
• 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.
  
  wn.com/Carl_Gustav_Hempel
• 
  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.
  
  wn.com/Galileo_Galilei
• Gilbert Ryle
  Gilbert Ryle (19 August 1900, Brighton – 6 October 1976, Oxford), was a British philosopher, a representative of the generation of British ordinary language philosophers that shared Wittgenstein's approach to philosophical problems, and is principally known for his critique of Cartesian dualism, for which he coined the phrase "the ghost in the machine". Some of his ideas in the philosophy of mind have been referred to as "behaviourist." Ryle's best known book is The Concept of Mind (1949), in which he writes that the "general trend of this book will undoubtedly, and harmlessly, be stigmatised as 'behaviourist'." Ryle, having engaged in detailed study of the key works of Bernard Bolzano, Franz Brentano, Alexius Meinong, Edmund Husserl, and Martin Heidegger, himself suggested instead that the book "could be described as a sustained essay in phenomenology, if you are at home with that label."
  Ryle was born in Brighton, England in 1900 and grew up in an environment of learning. His father was a Brighton doctor, a generalist who had interests in philosophy and astronomy, and passed on to his children an impressive library. Ryle was educated at Brighton College, and in 1919, he went up to Queen's College at Oxford, initially to study Classics but was quickly drawn to Philosophy. He would graduate with first class honours in 1924 and was appointed to a lectureship in Philosophy at Christ Church, Oxford. A year later, he was to become a tutor. Ryle remained at Christ Church until World War II.
  
  wn.com/Gilbert_Ryle
• 
  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).
  
  wn.com/Gottlob_Frege
• 
  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.
  
  wn.com/Hilary_Putnam
• 
  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.
  
  wn.com/Isaac_Newton
• 
  Jaakko Hintikka
  Kaarlo Jaakko Juhani Hintikka (born January 12, 1929) is a Finnish philosopher and logician.
  Hintikka was born in Vantaa. After teaching for a number of years at Florida State University, Stanford, University of Helsinki, and the Academy of Finland, he is currently Professor of Philosophy at Boston University. The prolific author or co-author of over 30 books and over 300 scholarly articles has contributed to mathematical logic, philosophical logic, the philosophy of mathematics, epistemology, language theory, and the philosophy of science. His works have appeared in over nine languages.
  Hintikka is regarded as the founder of formal epistemic logic and of game semantics for logic. Early in his career, he devised a semantics of modal logic essentially analogous to Kripke's frame semantics, and discovered the now widely taught semantic tableau, independently of Evert Willem Beth. In recent decades, he has worked mainly on game semantics, and on independence-friendly logic, known for its "branching quantifiers" which he believes do better justice to our intuitions about quantifiers than does conventional first-order logic. He has done important exegetical work on Aristotle, Kant, Wittgenstein, and C.S. Peirce. Hintikka's work can be seen as a continuation of analytic tendency in philosophy founded by Brentano and Peirce, advanced by Frege and Bertrand Russell, and continued by Carnap, Quine, and by Hintikka's teacher Georg Henrik von Wright. For instance, in 1998 he wrote The Principles of Mathematics Revisited which takes an exploratory stance comparable to that Russell made with his The Principles of Mathematics in 1903.
  
  wn.com/Jaakko_Hintikka
• John Wisdom
  Arthur John Terence Dibben Wisdom (John Wisdom; 12 September 1904, Leyton, Essex – 9 December 1993, Cambridge) was a leading British philosopher considered to be an ordinary language philosopher, a philosopher of mind and a metaphysician. He was influenced by G.E. Moore, Ludwig Wittgenstein and Sigmund Freud, and in turn explained and extended their work.
  Wisdom was for most of his career at Trinity College, Cambridge, and became Professor of Philosophy at Cambridge University. Near the end of his career he was Professor of Philosophy at the University of Oregon. He was president of the Aristotelian Society from 1950 to 1951.
  His famous "Parable of the Invisible Gardener" is a dialectic on the existence or absence of God.
  
  wn.com/John_Wisdom
• 
  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."
  
  wn.com/Ludwig_Wittgenstein
• 
  Michael Dummett
  Professor Sir Michael Anthony Eardley Dummett, FBA, D.Litt (27 June 1925 – 27 December 2011) was a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford. He wrote on the history of analytic philosophy, most notably as an interpreter of Frege, and has made original contributions to the subject, particularly in the philosophies of mathematics, logic, language and metaphysics. He was known for his work on truth and meaning and their implications for the debates between realism and anti-realism, a term he helped popularize. He devised the Quota Borda system of proportional voting, based on the Borda count.
  Dummett was the son of a merchant of silks. He studied at Sandroyd School and was a First Scholar at Winchester College, later winning a Major Scholarship to study History at Christ Church, Oxford in 1943. He was called up that year and served, initially as a private in the Royal Artillery before joining the Intelligence Corps in India and Malaya. He was also awarded a fellowship to All Souls College, Oxford.
  
  wn.com/Michael_Dummett
• Paul Grice
  Herbert Paul Grice (March 13, 1913, Birmingham, England – August 28, 1988, Berkeley, California), usually publishing under the name H. P. Grice, H. Paul Grice, or Paul Grice, was a British-educated philosopher of language, who spent the final two decades of his career in the United States.
  Born and raised in Harborne (now a suburb of Birmingham), in the United Kingdom, he was educated at Clifton College and then at Corpus Christi College, Oxford. After a brief period teaching at Rossall School, he went back to Oxford where he taught until 1967. In that year, he moved to the United States to take up a professorship at the University of California, Berkeley, where he taught until his death in 1988. He returned to the UK in 1979 to give the John Locke lectures on Aspects of Reason. He reprinted many of his essays and papers in his valedictory book, Studies in the Way of Words (1989).
  He was married and had two children. He and his wife lived in an old Spanish style house in the Berkeley Hills.^{[citation needed]}
  
  wn.com/Paul_Grice
• 
  Peter Singer
  Peter Albert David Singer (born 6 July 1946) is an Australian philosopher who is the Ira W. DeCamp Professor of Bioethics at Princeton University and Laureate Professor at the Centre for Applied Philosophy and Public Ethics at the University of Melbourne. He specialises in applied ethics and approaches ethical issues from a secular, preference utilitarian perspective.
  He has served, on two occasions, as chair of philosophy at Monash University, where he founded its Centre for Human Bioethics. In 1996, he unsuccessfully stood as a Greens candidate for the Australian Senate. In 2004, he was recognised as the Australian Humanist of the Year by the Council of Australian Humanist Societies. He has been voted one of Australia's ten most influential public intellectuals. Singer serves on the Advisory Board of Incentives for Global Health, the NGO formed to develop the Health Impact Fund proposal.
  Outside academic circles, Singer is best known for his book Animal Liberation, widely regarded as the touchstone of the animal liberation movement. Not all members of the animal liberation movement share this view, and Singer himself has said the media overstates his status. His views on that and other issues in bioethics have attracted attention and a degree of controversy.
  
  wn.com/Peter_Singer
• 
  René Descartes
  René Descartes (French pronunciation: [ʁəne dekaʁt]; Latinized form: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day. In particular, his Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system — allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius.
  
  wn.com/René_Descartes
• Rudolf Carnap
  Rudolf Carnap (May 18, 1891 – September 14, 1970) was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism.
  Carnap was born to a west German family that had been humble until his parents' generation. He began his formal education at the Barmen Gymnasium. From 1910 to 1914, he attended the University of Jena, intending to write a thesis in physics. But he also studied carefully Kant's Critique of Pure Reason during a course taught by Bruno Bauch, and was one of very few students to attend Gottlob Frege's courses in mathematical logic. After serving in the German army during World War I for three years, he was given permission to study physics at the University of Berlin, 1917–18, where Albert Einstein was a newly appointed professor. Carnap then attended the University of Jena, where he wrote a thesis defining an axiomatic theory of space and time. The physics department said it was too philosophical, and Bruno Bauch of the philosophy department said it was pure physics. Carnap then wrote another thesis, with Bauch's supervision, on the theory of space in a more orthodox Kantian style, and published as Der Raum (Space) in a supplemental issue of Kant-Studien (1922). In it he makes the clear distinction between formal, physical and perceptual (e.g., visual) spaces.
  
  wn.com/Rudolf_Carnap
• 
  Saul Kripke
  Saul Aaron Kripke (born November 13, 1940) is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center. Since the 1960s Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory. Much of his work remains unpublished or exists only as tape-recordings and privately circulated manuscripts. Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. A recent poll conducted among philosophers ranked Kripke among the top ten most important philosophers of the past 200 years.
  Kripke has made influential and original contributions to logic, especially modal logic, since he was a teenager. Unusual for a professional philosopher, his only degree is an undergraduate degree from Harvard, in mathematics. His work has profoundly influenced analytic philosophy, with his principal contribution being a semantics for modal logic, involving possible worlds as described in a system now called Kripke semantics. Another of his most important contributions is his argument that necessity is a 'metaphysical' notion, which should be separated from the epistemic notion of a priori, and that there are necessary a posteriori truths, such as "Water is H₂O." He has also contributed an original reading of Wittgenstein, referred to as "Kripkenstein." His most famous work is Naming and Necessity (1980).
  
  wn.com/Saul_Kripke

Failure Analysis (2004)

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