Branches of Mathematics

Pure mathematics

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

Applied mathematics

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

Related Topics

• Electromagnetic Spectrum
• Linear Equations
• Applied Physics
• Nano Technology
• Engineering Sciences
• Mathematical Medicine
• Fluid dynamics
• Computational Mathematics
• Gamma Ray Optics
• Onfrared Optics
• Relative Velocity
• Iciam
• Intelligent Imaging Neuroscience
• Biomedical Engineering
• Neuroscience Research
• Quantum Cosmology
• Brain Research

• Abstract object
• Adolph Stöhr
• Anima mundi
• Anti-realism
• Aristotle
• Arthur Schopenhauer
• Availability
• Axiology
• Baruch Spinoza
• Being
• Bertrand Russell
• Binomial coefficient
• Brian Tracy
• Category Metaphysics
• Category of being
• Causality
• Choice
• Choice architecture
• Choice theory
• Cogito ergo sum
• Cognition
• Collaborative
• Concept
• Convenience
• Cool (aesthetic)
• Cosmology
• David Hume
• Determinism
• Dorothy Emmet
• Duns Scotus
• Economy
• Embodied cognition
• Emotion
• Enactivism
• Entity
• Epistemology
• Essence
• Essentialism
• Ethnic heritage
• Example Choice
• Existence
• Existentialism
• Experience
• Feeling
• Free will
• Friedrich Nietzsche
• G. E. Moore
• George Berkeley
• George Loewenstein
• Gilbert Ryle
• Habit (psychology)
• Henri Bergson
• Hilary Putnam
• Hindu idealism
• Hobson's choice
• Idea
• Idealism
• Identity and change
• Image
• Immanuel Kant
• Information
• Insight
• Intelligence
• Intention
• Isaac Newton
• Jean-Paul Sartre
• John Locke
• Judging
• Kapila
• Liberty
• Linguistic modality
• Ludwig Wittgenstein
• Magic 8-Ball
• Maimonides
• Martin Heidegger
• Materialism
• Matter (philosophy)
• Meaning of life
• Medical conditions
• Memetics
• Mental function
• Meta
• Metaphysics
• Mind
• MIT Press
• Monism
• Motion (physics)
• Necessity
• New Zealand
• Nihilism
• Notion (philosophy)
• Nutrition
• Object (philosophy)
• Ontology
• Ouija
• P. F. Strawson
• Parmenides
• Pattern
• Perception
• Phenomenalism
• Philosophy of mind
• Philosophy of self
• Physical body
• Physicalism
• Pilot in command
• Plato
• Platonic idealism
• Plotinus
• Portal Metaphysics
• Portal thinking
• Preference
• Principle
• Pro-choice
• Psychological
• Public choice theory
• Qualia
• Quality (philosophy)
• R. G. Collingwood
• Reality
• Regret (emotion)
• Relativism
• René Descartes
• Robert Gates
• Route planner
• Rudolf Carnap
• Sankhya
• Saul Kripke
• Scientific realism
• Social choice theory
• Social pressure
• Solipsism
• Soul
• Subject (philosophy)
• Subjectivism
• Substance theory
• Substantial form
• Teleology
• Template Metaphysics
• Theoretical physics
• Thomas Aquinas
• Thought
• Time
• Tradition
• Truth
• Type theory
• United Kingdom
• United States
• Unobservable
• Value (ethics)
• W. V. Quine
• Will (philosophy)
• 
  

• 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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  New Zealand
  New Zealand (Māori: Aotearoa) is an island country located in the southwestern Pacific Ocean. The country geographically comprises two main landmasses ‒ that of the North and South Islands ‒ as well as numerous smaller islands. New Zealand is situated some 1,500 kilometres (900 mi) east of Australia across the Tasman Sea and roughly 1,000 kilometres (600 mi) south of the Pacific island nations of New Caledonia, Fiji, and Tonga. Because of its remoteness, it was one of the last lands to be settled by humans.
  During its long isolation, New Zealand developed a distinctive biodiversity of both animal and plant life. Most notable are the large number of unique bird species, many of which became extinct after the arrival of humans and introduced mammals. With a mild maritime climate, the land was mostly covered in forest. The country's varied topography and its sharp mountain peaks owe much to the tectonic uplift of land and volcanic eruptions caused by the Pacific and Indo-Australian Plates clashing beneath the earth's surface.
  
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  United Kingdom
  The United Kingdom of Great Britain and Northern Ireland (commonly known as the United Kingdom, the UK, or Britain) is a sovereign state located off the north-western coast of continental Europe. The country includes the island of Great Britain, the north-eastern part of the island of Ireland and many smaller islands. Northern Ireland is the only part of the UK that shares a land border with another sovereign state—the Republic of Ireland. Apart from this land border the UK is surrounded by the Atlantic Ocean, the North Sea, the English Channel and the Irish Sea.
  The United Kingdom is a unitary state governed under a constitutional monarchy and a parliamentary system, with its seat of government in the capital city of London. It is a country in its own right and consists of four countries: England, Northern Ireland, Scotland and Wales. There are three devolved administrations, each with varying powers, based in Belfast, Edinburgh and Cardiff, the capitals of Northern Ireland, Scotland, and Wales. Associated with the UK, but not constitutionally part of it, are three Crown Dependencies. The United Kingdom has fourteen overseas territories. These are remnants of the British Empire which, at its height in 1922, encompassed almost a quarter of the world's land surface and was the largest empire in history. British influence can still be observed in the language, culture and legal systems of many of its former territories.
  
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  United States
  The United States of America (commonly abbreviated to the United States, the U.S., the USA, America, and the States) is a federal constitutional republic comprising fifty states and a federal district. The country is situated mostly in central North America, where its forty-eight contiguous states and Washington, D.C., the capital district, lie between the Pacific and Atlantic Oceans, bordered by Canada to the north and Mexico to the south. The state of Alaska is in the northwest of the continent, with Canada to the east and Russia to the west, across the Bering Strait. The state of Hawaii is an archipelago in the mid-Pacific. The country also possesses several territories in the Pacific and Caribbean.
  At 3.79 million square miles (9.83 million km²) and with over 312 million people, the United States is the third or fourth largest country by total area, and the third largest by both land area and population. It is one of the world's most ethnically diverse and multicultural nations, the product of large-scale immigration from many countries. The U.S. economy is the world's largest national economy, with an estimated 2011 GDP of $15.1 trillion (22% of nominal global GDP and over 19% of global GDP at purchasing-power parity). Per capita income is the world's sixth-highest.
  
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• Adolph Stöhr
  Adolph Stöhr (Sankt Pölten, 1855–1921) was professor of philosophy at the University of Vienna. His lectures and publications covered subjects such as logic, metaphysics, philosophy of language, experimental psychology, psychology of perception, and psychology of association.
  
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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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  Arthur Schopenhauer
  Arthur Schopenhauer (22 February 1788 – 21 September 1860) was a German philosopher known for his pessimism and philosophical clarity. At age 25, he published his doctoral dissertation, On the Fourfold Root of the Principle of Sufficient Reason, which examined the four separate manifestations of reason in the phenomenal world.
  Schopenhauer's most influential work, The World as Will and Representation, claimed that the world is fundamentally what humans recognize in themselves as their will. His analysis of will led him to the conclusion that emotional, physical, and sexual desires can never be fully satisfied. The corollary of this is an ultimately painful human condition. Consequently, he considered that a lifestyle of negating desires, similar to the ascetic teachings of Vedanta, Buddhism and the Church Fathers of early Christianity, was the only way to attain liberation.
  Schopenhauer's metaphysical analysis of will, his views on human motivation and desire, and his aphoristic writing style influenced many well-known thinkers, including Friedrich Nietzsche,Richard Wagner, Ludwig Wittgenstein, Erwin Schrödinger, Albert Einstein,Sigmund Freud, Otto Rank, Carl Jung, Joseph Campbell, Leo Tolstoy, Thomas Mann, and Jorge Luis Borges.
  
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  Baruch Spinoza
  Baruch Spinoza and later Benedict de Spinoza (24 November 1632 – 21 February 1677) was a Dutch philosopher. Revealing considerable scientific aptitude, the breadth and importance of Spinoza's work was not fully realized until years after his death. By laying the groundwork for the 18th century Enlightenment^{[citation needed]} and modern biblical criticism, he came to be considered one of the great rationalists of 17th-century philosophy. His magnum opus, the posthumous Ethics, in which he opposed Descartes's mind–body dualism, has earned him recognition as one of Western philosophy's most important contributors. In the Ethics, "Spinoza wrote the last indisputable Latin masterpiece, and one in which the refined conceptions of medieval philosophy are finally turned against themselves and destroyed entirely." Philosopher Georg Wilhelm Friedrich Hegel said of all contemporary philosophers, "You are either a Spinozist or not a philosopher at all."
  Spinoza's name in different languages is Hebrew: ברוך שפינוזה‎ Baruch Spinoza, Portuguese: Benedito or Bento de Espinosa and Latin: Benedictus de Spinoza; in all these languages, the given name means "the Blessed". Spinoza was raised in the Portuguese Jewish community in Amsterdam. He developed highly controversial ideas regarding the authenticity of the Hebrew Bible and the nature of the Divine. The Jewish religious authorities issued a cherem (Hebrew: חרם, a kind of ban, shunning, ostracism, expulsion, or excommunication) against him, effectively excluding him from Jewish society at age 23. His books were also later put on the Catholic Church's Index of Forbidden Books.
  
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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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• Brian Tracy
  Brian Tracy (born January 5, 1944 in Charlottetown, PEI, Canada) is a self-help author and motivational speaker. He serves as Chairman of Brian Tracy International, a human resource company based in Solana Beach, California, with affiliates throughout the United States and thirty-one other countries.
  After dropping out of high school before graduation, Tracy got a job on a tramp steamer and traveled around the world for eight years, eventually visiting more than eighty countries on five continents.
  When he could no longer find a job as a laborer, he went into sales. Struggling at first, he decided to ask other successful salesmen for advice, and then to emulate them. By the end of his first year, he attained the status of top salesman. After his second year, he was made a Vice President in charge of 95 people. At the time he was only 25 years old.
  Beginning in 1981, Tracy assembled his "success system", which was initially called "The Phoenix Seminar." Three years later, in 1985, he released an updated version of this seminar as a self-help audio tape entitled "The Psychology of Achievement".
  
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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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• Dorothy Emmet
  Dorothy Mary Emmet (29 September 1904 – 20 September 2000) was a British philosopher and head of Manchester University's philosophy department for over twenty years. With Margaret Masterman and Richard Braithwaite she was a founder member of the Epiphany Philosophers.
  
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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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  Friedrich Nietzsche
  Friedrich Wilhelm Nietzsche ( /ˈniːtʃə/;German pronunciation: [ˈfʁiːdʁɪç ˈvɪlhɛlm ˈniːtsʃə]; October 15, 1844 – August 25, 1900) was a German philosopher, poet, cultural critic and classical philologist. He wrote critical texts on religion, morality, contemporary culture, philosophy and science, displaying a fondness for metaphor, irony and aphorism.
  Nietzsche's influence remains substantial within and beyond philosophy, notably in existentialism, nihilism and postmodernism. His style and radical questioning of the value and objectivity of truth have resulted in much commentary and interpretation, mostly in the continental tradition. His key ideas include the death of God, the Übermensch, the eternal recurrence, the Apollonian and Dionysian dichotomy, perspectivism and the will to power. Central to his philosophy is the idea of "life-affirmation", which involves an honest questioning of all doctrines that drain life's expansive energies, however socially prevalent and radical those views might be.
  
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  George Berkeley
  George Berkeley ( /ˈbɑrkliː/; 12 March 1685 – 14 January 1753), also known as Bishop Berkeley (Bishop of Cloyne), was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism" (later referred to as "subjective idealism" by others). This theory denies the existence of material substance and instead contends that familiar objects like tables and chairs are only ideas in the minds of perceivers, and as a result cannot exist without being perceived. Thus, as Berkeley famously put it, for physical objects "esse est percipi" ("to be is to be perceived"). Berkeley is also known for his critique of abstraction, an important premise in his argument for immaterialism.
  In 1709, Berkeley published his first major work, An Essay towards a New Theory of Vision, in which he discussed the limitations of human vision and advanced the theory that the proper objects of sight are not material objects, but light and color. This foreshadowed his chief philosophical work A Treatise Concerning the Principles of Human Knowledge in 1710 which, after its poor reception, he rewrote in dialogue form and published under the title Three Dialogues between Hylas and Philonous in 1713.
  In this book, Berkeley's views were represented by Philonous (Greek: 'lover of mind'), while Hylas (Greek: 'matter') embodies the Irish thinker’s opponents, in particular John Locke. Berkeley argued against Sir Isaac Newton's doctrine of absolute space, time and motion in De Motu (on Motion), published 1721. His arguments were a precursor to the views of Mach and Einstein. In 1732, he published Alciphron, a Christian apologetic against the free-thinkers, and in 1734, he published The Analyst, an empiricist critique of the foundations of infinitesimal calculus, which was influential in the development of mathematics.
  
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• George Loewenstein
  George F. Loewenstein (born August 9, 1955) is an American educator and economist. He is the Herbert A. Simon Professor of Economics and Psychology in the Social and Decision Sciences Department at Carnegie Mellon University and director of the Center for Behavioral Decision Research. He is a leader in the fields of behavioral economics and neuroeconomics.^{[citation needed]}
  He received his B.A. in economics magna cum laude from Brandeis University in 1977 and Ph.D. in economics from Yale University in 1985.^{[citation needed]} He taught at the Booth School of Business at the University of Chicago before taking up his present position at Carnegie Mellon University.^{[citation needed]} Loewenstein became a fellow of the American Academy of Arts and Sciences in 2008.^{[citation needed]} He is the great-grandson of Sigmund Freud.^{[citation needed]}
  Loewenstein is especially known for his work regarding intertemporal choice and affective forecasting.^{[citation needed]}
  Hot-cold empathy gaps are one of Loewenstein's major contributions to behavioral economics. The crux of this idea is that human understanding is "state dependent," that is, when one is angry it is difficult to understand what it is like for one to be happy, and vice versa. The implications of this were explored in the realm of sexual decision-making, where young men in an unaroused "cold state" fail to predict that when they are in an aroused "hot state" they will be more likely to make risky sexual decisions, such as not using a condom (2006).
  
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• 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.
  
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• Henri Bergson
  Henri-Louis Bergson (French pronunciation: [bɛʁksɔn] 18 October 1859 – 4 January 1941) was a major French philosopher, influential especially in the first half of the 20th century. Bergson convinced many thinkers that immediate experience and intuition are more significant than rationalism and science for understanding reality.
  He was awarded the 1927 Nobel Prize in Literature "in recognition of his rich and vitalizing ideas and the brilliant skill with which they have been presented".
  Bergson was born in the Rue Lamartine in Paris, not far from the Palais Garnier (the old Paris opera house) in 1859 (the year in which France emerged as a victor in the Second Italian War of Independence, and in the month before the publication of Charles Darwin's On the Origin of Species). His father, the pianist Michał Bergson, was of a Polish Jewish family background (originally bearing the name Bereksohn). His mother, Katherine Levison, daughter of a Yorkshire doctor, was from an English and Irish Jewish background. The Bereksohns were a famous^{[citation needed]} Jewish entrepreneurial family of Polish descent. Henri Bergson's great-great-grandfather, Szmul Jakubowicz Sonnenberg, called Zbytkower, was a prominent banker and a protégé of Stanisław August Poniatowski, King of Poland from 1764 to 1795.
  
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  Hilary Putnam
  Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science. He is known for his willingness to apply an equal degree of scrutiny to his own philosophical positions as to those of others, subjecting each position to rigorous analysis until he exposes its flaws. As a result, he has acquired a reputation for frequently changing his own position. Putnam is currently Cogan University Professor Emeritus at Harvard University.
  In philosophy of mind, Putnam is known for his argument against the type-identity of mental and physical states based on his hypothesis of the multiple realizability of the mental, and for the concept of functionalism, an influential theory regarding the mind-body problem. In philosophy of language, along with Saul Kripke and others, he developed the causal theory of reference, and formulated an original theory of meaning, inventing the notion of semantic externalism based on a famous thought experiment called Twin Earth.
  
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  Immanuel Kant
  Immanuel Kant (German pronunciation: [ɪˈmaːnu̯eːl ˈkant]; 22 April 1724 – 12 February 1804) was a German philosopher from Königsberg (today Kaliningrad of Russia), who researched, lectured, and wrote on philosophy and anthropology during the Enlightenment at the end of the 18th century. At the time, there were major successes and advances in the sciences (for example, Isaac Newton, Carl Friedrich Gauss, and Robert Boyle) applying reason and logic.
  Kant’s major work, the Critique of Pure Reason (Kritik der reinen Vernunft, 1781), aimed to unite reason with experience to move beyond what he took to be failures of traditional philosophy and metaphysics. He hoped to end an age of speculation where objects outside experience were used to support what he saw as futile theories, while opposing the skepticism of thinkers such as Descartes, Berkeley and Hume.
  He stated:
  Kant proposed a ‘Copernican Revolution’ in reverse, saying that:
  Kant published other important works on ethics, religion, law, aesthetics, astronomy, and history. These included the Critique of Practical Reason (Kritik der praktischen Vernunft, 1788) and the Metaphysics of Morals (Die Metaphysik der Sitten, 1797), which dealt with ethics. And the Critique of Judgment (Kritik der Urteilskraft, 1790), which looks at aesthetics and teleology. He aimed to resolve disputes between empirical and rationalist approaches. The former asserted that all knowledge comes through experience; the latter maintained that reason and innate ideas were prior. Kant argued that experience is purely subjective without first being processed by pure reason. He also said that using reason without applying it to experience only leads to theoretical illusions. The free and proper exercise of reason by the individual was both a theme of the Enlightenment, and of Kant's approaches to the various problems of philosophy.
  
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  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 Locke
  John Locke FRS ( /ˈlɒk/; 29 August 1632 – 28 October 1704), widely known as the Father of Classical Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social contract theory. His work had a great impact upon the development of epistemology and political philosophy. His writings influenced Voltaire and Rousseau, many Scottish Enlightenment thinkers, as well as the American revolutionaries. His contributions to classical republicanism and liberal theory are reflected in the United States Declaration of Independence.
  Locke's theory of mind is often cited as the origin of modern conceptions of identity and the self, figuring prominently in the work of later philosophers such as Hume, Rousseau and Kant. Locke was the first to define the self through a continuity of consciousness. He postulated that the mind was a blank slate or tabula rasa. Contrary to pre-existing Cartesian philosophy, he maintained that we are born without innate ideas, and that knowledge is instead determined only by experience derived from sense perception.
  
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  Kapila
  Kapila (Hindi: कपिल ऋषि) was a Vedic sage credited as one of the founders of the Samkhya school of philosophy. He is prominent in the Bhagavata Purana, which features a theistic version of his Samkhya philosophy. Traditional Hindu sources describe him as a descendant of Manu, a grandson of Brahma. The Bhagavad Gita depicts Kapila as a yogi hermit with highly developed siddhis, or spiritual powers.
  Many of the details about sage Kapila's life are described in Book 3 of the Bhagavata Purana, where it is mentioned that his parents were Kardama Muni and Devahuti. Kapila is considered an incarnation of the supreme-being (Narayana) and listed as such in the list of incarnations in Bhagavata Purana .After his father left home, Kapila instructed his mother, Devahuti in the philosophy of yoga and devotional worship of Lord Vishnu, enabling her to achieve liberation (moksha). Kapila's Sankhya is also given by Krishna to Uddhava in Book 11 of the Bhagavata Purana, a passage also known as the "Uddhava Gita".
  
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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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  Maimonides
  Mosheh ben Maimon משה בן מימון‎, called Moses Maimonides and also known as Mūsā ibn Maymūn (Arabic: موسى بن ميمون‎), or RaMBaM (רמב"ם – Hebrew acronym for "Rabbi Mosheh Ben Maimon"), was a preeminent medieval Jewish philosopher and one of the most prolific and followed Torah scholars and physicians of the Middle Ages. He was born in Córdoba, Almoravid Empire (present-day Spain) on Passover Eve, 1135, and died in Egypt (or Tiberias) on 20th Tevet, December 12, 1204. He was a rabbi, physician and philosopher in Morocco and Egypt.
  Although his writings on Jewish law and ethics were met with acclaim and gratitude from most Jews even as far off as Spain, Iraq and Yemen, and he rose to be the revered head of the Jewish community in Egypt, there were also vociferous critics of some of his rulings and other writings particularly in Spain. Nevertheless, he was posthumously acknowledged to be one of the foremost rabbinical arbiters and philosophers in Jewish history, his copious work a cornerstone of Jewish scholarship. His fourteen-volume Mishneh Torah still carries canonical authority as a codification of Talmudic law. In the Yeshiva world he is known as "haNesher haGadol" (the great eagle) in recognition of his outstanding status as a bona fide exponent of the Oral Torah.
  
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  Martin Heidegger
  Martin Heidegger (September 26, 1889 – May 26, 1976); German pronunciation: [ˈmaɐ̯tiːn ˈhaɪdɛɡɐ]) was a German philosopher known for his existential and phenomenological explorations of the "question of Being."
  Heidegger argues that philosophy is preoccupied with what exists and has forgotten the question of the "ground" of being. We find ourselves "always already" fallen into a world that already existed; but he insists that we have forgotten the basic question of what being itself is. This question defines our central nature. He argues that we are practical agents, caring and concerned about our projects in the world, and allowing it to reveal, or "unconceal" itself to us. He also says that our manipulation of reality is often harmful and hides our true being as essentially limited participants, not masters, of the world which we discover.
  Heidegger wrote about these issues in his best-known book, Being and Time (1927), which is considered to be one of the most important philosophical works of the 20th century. Heidegger's influence is far reaching, from philosophy to deconstructionism and literary theory, theology, architecture, and artificial intelligence.
  
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  Parmenides
  Parmenides of Elea ( /pɑrˈmɛnɨdiːz əv ˈɛliə/; Ancient Greek: Παρμενίδης ὁ Ἐλεάτης; fl. early 5th century BC) was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, Parmenides describes two views of reality. In "the way of truth" (a part of the poem), he explains how reality (coined as "what-is") is one, change is impossible, and existence is timeless, uniform, necessary, and unchanging. In "the way of opinion," he explains the world of appearances, in which one's sensory faculties lead to conceptions which are false and deceitful. These ideas strongly influenced the whole of Western philosophy, perhaps most notably through its effect on Plato.
  
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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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  Plotinus
  Plotinus (Greek: Πλωτῖνος) (ca. 204/5–270 CE) was a major philosopher of the ancient world. In his system of theory there are the three principles: the One, the Intellect, and the Soul. His teacher was Ammonius Saccas and he is of the Platonic tradition. Historians of the 19th century invented the term Neoplatonism^{[citation needed]} and applied it to him and his philosophy which was influential in Late Antiquity. Much of the biographical information about Plotinus comes from Porphyry's preface to his edition of Plotinus' Enneads. His metaphysical writings have inspired centuries of Pagan, Christian, Jewish, Islamic and Gnostic metaphysicians and mystics.
  Porphyry reported that Plotinus was 66 years old when he died in 270, the second year of the reign of the emperor Claudius II, thus giving us the year of his teacher's birth as around 205. Eunapius reported that Plotinus was born in the Deltaic Lycopolis; "Lyco-" from the Greek meaning "wolf" (Greek: λύκος, lykos). It is the same root that gave rise to Aristotle's Lyceum (place of the wolf) in Egypt, which has led to speculations that he may have been a native Egyptian of Roman,Greek, or Hellenized Egyptian descent.
  
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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 Gates
  Robert Michael Gates (born September 25, 1943) is the 24th Chancellor of the College of William and Mary and a retired civil servant and university president who served as the 22nd United States Secretary of Defense from 2006 to 2011. Prior to this, Gates served for 26 years in the Central Intelligence Agency and the National Security Council, and under President George H. W. Bush as Director of Central Intelligence. Gates was also an officer in the United States Air Force and during the early part of his military career, he was recruited by the CIA. After leaving the CIA, Gates became president of Texas A&M University and was a member of several corporate boards. Gates also served as a member of the Iraq Study Group, the bipartisan commission co-chaired by James A. Baker III and Lee H. Hamilton, that has studied the Iraq War. He was also the first pick to serve as the first Director of National Intelligence (DNI), but he declined the appointment in order to remain President of Texas A&M University.
  
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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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  Thomas Aquinas
  Thomas Aquinas, O.P. ( /əˈkwaɪnəs/ ə-KWY-nəs; 1225 – 7 March 1274), also Thomas of Aquin or Aquino, was an Italian Dominican priest of the Roman Catholic Church, and an immensely influential philosopher and theologian in the tradition of scholasticism, known as Doctor Angelicus ([the] Angelic Doctor), Doctor Communis, or Doctor Universalis. "Aquinas" is not a surname, but is a Latin demonym for a resident of Aquino, his place of birth. He was the foremost classical proponent of natural theology, and the father of Thomism. His influence on Western thought is considerable, and much of modern philosophy was conceived as a reaction against, or as an agreement with, his ideas, particularly in the areas of ethics, natural law, metaphysics, and political theory.
  Thomas is held in the Catholic Church to be the model teacher for those studying for the priesthood. The works for which he is best-known are the Summa Theologica and the Summa Contra Gentiles. As one of the 33 Doctors of the Church, he is considered the Church's greatest theologian and philosopher. Pope Benedict XV declared: "This (Dominican) Order ... acquired new luster when the Church declared the teaching of Thomas to be her own and that Doctor, honored with the special praises of the Pontiffs, the master and patron of Catholic schools."
  
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