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

• Abstraction
• Alan Ross Anderson
• Alan Turing
• Alexander Bain
• Alfred Tarski
• Alonzo Church
• Ambiguity
• Analytic grammar
• Anatol Rapoport
• Anthony Wilden
• Antinomy
• Argument
• Argumentation theory
• Aristotle
• Augustus De Morgan
• Averroes
• Avicenna
• Axiology
• Axiom
• Axiom schema
• Axiomatic system
• Belief
• Bertrand Russell
• Bias
• Biological system
• Boolean function
• Boolean logic
• Bra-ket notation
• Béla H. Bánáthy
• C. West Churchman
• Cantor's theorem
• Category Logic
• Category Systems
• Chaos theory
• Charles A S Hall
• Chrysippus
• Church–Turing thesis
• Class (set theory)
• Classical logic
• Claude Shannon
• Cogency
• Complete
• Completeness
• Complex system
• Complex systems
• Computability theory
• Computable function
• Computer science
• Conceptual system
• Consistency
• Control theory
• Countable set
• Credibility
• Critical thinking
• Cybernetics
• David Hilbert
• Decidability (logic)
• Decidable set
• Decision problem
• Deductive reasoning
• Deductive system
• Definition
• Degree of truth
• Deontic logic
• Description
• Description logic
• Dialetheism
• Domain of a function
• Donella Meadows
• Douglas Hofstadter
• Doxastic logic
• Duns Scotus
• Dynamical system
• Economic system
• Ecosystem
• Edward Norton Lorenz
• Effective method
• Empty set
• Entailment
• Entscheidungsproblem
• Enumeration
• Ernst Schröder
• Ernst Zermelo
• Ethics (book)
• Euclid's elements
• Euclidean geometry
• Evidence
• Explanation
• Explanatory power
• Extensionality
• Fact
• Fallacy
• Fictionalism
• Finitary relation
• Finite model theory
• Finite set
• Finitism
• First-order logic
• Formal ethics
• Formal grammar
• Formal language
• Formal method
• Formal proof
• Formal science
• Formal system
• Formation rule
• Francisco Varela
• Free variable
• Fuzzy finite element
• Fuzzy logic
• Fuzzy rule
• Fuzzy set
• Fuzzy set operations
• Generative grammar
• Georg Cantor
• George Boole
• George Boolos
• George Dantzig
• George Klir
• Gerhard Gentzen
• Giuseppe Peano
• Gottlob Frege
• Grammar
• Gregory Bateson
• Haskell Curry
• Heinz von Foerster
• Heyting arithmetic
• Hilary Putnam
• History of logic
• Howard T. Odum
• Human anatomy
• Humberto Maturana
• Ilya Prigogine
• Inductive reasoning
• Inference
• Informal logic
• Information systems
• Inquiry
• Intuitionism
• Intuitionistic logic
• James Grier Miller
• Jay Wright Forrester
• Jon Barwise
• Kenneth E. Boulding
• Kevin Warwick
• Kurt Gödel
• Lambda calculus
• Leopold Löwenheim
• Linear logic
• Linguistics
• List of fallacies
• List of logicians
• List of paradoxes
• Living systems
• Logic
• Logical atomism
• Logical connective
• Logical form
• Logical truth
• Logicism
• Ludwig Wittgenstein
• Margaret Mead
• Mathematical logic
• Mathematics
• Metalanguage
• Metalogic
• Metamathematics
• Metatheorem
• Metric system
• Mihajlo D. Mesarovic
• Modal logic
• Modal operator
• Model theory
• Multi-agent system
• Murray Bowen
• Name
• Natural deduction
• Nervous system
• Niklas Luhmann
• Nominalism
• Non-classical logic
• Non-standard model
• Nonlinearity
• Norbert Wiener
• Object language
• Ontology
• Ontology language
• Operating system
• Opinion
• Ordered pair
• Outline of logic
• Paraconsistent logic
• Paradox
• Parsimony
• Paul Bernays
• Paul Dirac
• Peter Geach
• Philosophical logic
• Philosophy
• Philosophy of logic
• Physical system
• Platonic realism
• Political system
• Portal Logic
• Possible world
• Power set
• Pragmatism
• Predicate logic
• Premise
• Presupposition
• Probability
• Proof calculus
• Proof theory
• Propaganda
• Proposition
• Prudence
• QED manifesto
• Qian Xuesen
• Quantification
• Range (mathematics)
• Raymond Smullyan
• Reason
• Recursion
• Recursive set
• Reference
• Relevance
• Relevance logic
• Rhetoric
• Richard E. Bellman
• Rigour
• Rudolf Carnap
• Rule of inference
• Russell L. Ackoff
• Satisfiability
• Saul Kripke
• Second-order logic
• Semantics
• Semantics of logic
• Sensory system
• Set (mathematics)
• Set theory
• Social system
• Solar System
• Soundness
• Square of opposition
• Statement (logic)
• Stephen Cole Kleene
• Strict implication
• Structural rule
• Subset
• Substitution (logic)
• Substructural logic
• Syllogism
• Symbol (formal)
• Syntax
• Syntax (logic)
• System
• System dynamics
• Systems art
• Systems biology
• Systems ecology
• Systems engineering
• Systems neuroscience
• Systems psychology
• Systems science
• Systems theory
• Talcott Parsons
• Template Logic
• Template talk Logic
• Temporal logic
• Term logic
• Theorem
• Thoralf Skolem
• Truth
• Truth table
• Truth value
• Type theory
• Uncountable set
• Vagueness
• Validity
• Venn diagram
• Well defined
• Well-formed formula
• William of Ockham
• William Ross Ashby
• World-systems theory
• 
  

• 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.
  
  wn.com/Alan_Turing
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  Alexander Bain
  Alexander Bain (11 June 1818 – 18 September 1903) was a Scottish philosopher and educationalist in the British school of empiricism who was a prominent and innovative figure in the fields of psychology, linguistics, logic, moral philosophy and education reform. He founded Mind, the first ever journal of psychology and analytical philosophy, and was the leading figure in establishing and applying applying the scientific method to psychology. Bain was the inaugural Regius Chair in Logic and Professor of Logic at the University of Aberdeen, where he also held Professorships in Moral Philosophy and English Literature and was twice elected Lord Rector of the University of Aberdeen.
  Alexander Bain was born in Aberdeen, Scotland to George Bain, a weaver and veteran soldier, and Margaret Paul. At age eleven he left school to work as a weaver hence the description of him as Weevir, rex philosophorum. He also took to lectures at the Mechanics' Institutes of Aberdeen and the Aberdeen Public Library.
  
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  Alfred Tarski
  Alfred Tarski (January 14, 1901 – October 26, 1983) was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death.
  A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.
  His biographers Anita and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models."
  Alfred Tarski was born Alfred Teitelbaum (Polish spelling: "Tajtelbaum"), to parents who were Polish Jews in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw's Szkoła Mazowiecka. Nevertheless, he entered the University of Warsaw in 1918 intending to study biology.
  
  wn.com/Alfred_Tarski
• Alonzo Church
  Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, solving the Entscheidungsproblem, Frege–Church ontology, and the Church–Rosser theorem.
  Alonzo Church was born on June 14, 1903 in Washington, D.C. where his father, Samuel Robbins Church, was the judge of the Municipal Court for the District of Columbia. The family later moved to Virginia after his father lost this position because of failing eyesight. With help from his uncle, also named Alonzo Church, he was able to attend the Ridgefield School for Boys in Ridgefield, Connecticut. After graduating from Ridgefield in 1920, Church attended Princeton University where he was an exceptional student, publishing his first paper, on Lorentz transformations, and graduating in 1924 with a degree in mathematics. He stayed at Princeton, earning a Ph.D. in mathematics in three years under Oswald Veblen.
  
  wn.com/Alonzo_Church
• Anatol Rapoport
  Anatol Rapoport (Russian: Анато́лий Бори́сович Рапопо́рт, born May 22, 1911 – January 20, 2007) was a Russian-born American Jewish mathematical psychologist. He contributed to general systems theory, mathematical biology and to the mathematical modeling of social interaction and stochastic models of contagion.
  Rapoport was born in Lozоvaya, Kharkov Governorate, Russia (in today's Kharkiv Oblast, Ukraine). In 1922, he came to the United States, and in 1928 he became a naturalized citizen. He started studying music in Chicago and continued with piano, conducting and composition at the Vienna Hochschule für Musik where he studied from 1929 to 1934. However, due to the rise of Nazism, he found it impossible to make a career as a pianist.
  He shifted his career into mathematics, getting a Ph.D. degree in mathematics under Otto Schilling and Abraham Adrian Albert at the University of Chicago in 1941 on the thesis Construction of Non-Abelian Fields with Prescribed Arithmetic. According to the Toronto Globe and Mail, he was a member of the American Communist Party for three years, but quit before enlisting in the U.S. Army Air Corps in 1941, serving in Alaska and India during World War II.
  
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• Anthony Wilden
  Anthony Wilden (14 December 1935, London, England) is a writer, social theorist, college lecturer, and consultant. Wilden has published numerous books and articles which intersect a number of fields, including systems theory, film theory, structuralism, cybernetics, psychiatry, anthropological theory, water control projects, urban ecosystems, resource conservation, and communications and social relations.
  Wilden is credited with one of the first significant introductions to the work of Jacques Lacan in the Anglo-Saxon world, particularly in his role as one of Lacan's early English translators. Today and into the 21st century Wilden's work (and consequent reputation) is arguably more influential in the fields of communication theory, ecology and social interaction. These fields of study evolved out of a long scholarly tradition of "interactional semiotics" that originated with Plato's Cratylus. Along with such figures as Gregory Bateson (ie., Steps to an Ecology of Mind), R. D. Laing (ie., Sanity, Madness and the Family), and Walker Percy (ie., Lost in the Cosmos), Wilden is considered one of this tradition's contemporary (modern and postmodern) pioneers.
  
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  Aristotle
  Aristotle (Ancient Greek: Ἀριστοτέλης, Aristotélēs) (384 BC – 322 BC) was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology. Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most important founding figures in Western philosophy. Aristotle's writings were the first to create a comprehensive system of Western philosophy, encompassing morality, aesthetics, logic, science, politics, and metaphysics.
  Aristotle's views on the physical sciences profoundly shaped medieval scholarship, and their influence extended well into the Renaissance, although they were ultimately replaced by Newtonian physics. In the zoological sciences, some of his observations were confirmed to be accurate only in the 19th century. His works contain the earliest known formal study of logic, which was incorporated in the late 19th century into modern formal logic. In metaphysics, Aristotelianism had a profound influence on philosophical and theological thinking in the Islamic and Jewish traditions in the Middle Ages, and it continues to influence Christian theology, especially the scholastic tradition of the Catholic Church. His ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotle's philosophy continue to be the object of active academic study today. Though Aristotle wrote many elegant treatises and dialogues (Cicero described his literary style as "a river of gold"), it is thought that the majority of his writings are now lost and only about one-third of the original works have survived.
  
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  Augustus De Morgan
  Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him.
  Augustus De Morgan was born in 1806. His father was Col. Augustus De Morgan, who held various appointments in the service of the East India Company. His mother descended from James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached", using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.
  When De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England, and her son received his elementary education at various schools of no great account. His mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. She explained the aim of Euclid to Augustus, and gave him an initiation into demonstration.
  
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  Averroes
  ʾAbū l-Walīd Muḥammad bin ʾAḥmad bin Rušd (Arabic: أبو الوليد محمد بن احمد بن رشد‎), better known just as Ibn Rushd (Arabic: ابن رشد‎), and in European literature as Averroes ( /əˈvɛroʊ.iːz/; April 14, 1126 – December 10, 1198), was an Andalusian Muslim polymath; a master of Aristotelian philosophy, Islamic philosophy, Islamic theology, Maliki law and jurisprudence, logic, psychology, politics, Arabic music theory, and the sciences of medicine, astronomy, geography, mathematics, physics and celestial mechanics. He was born in Córdoba, Al Andalus, modern-day Spain, and died in Marrakesh, Morocco. His school of philosophy is known as Averroism.
  Ibn Rushd was a defender of Aristotelian philosophy against claims from the influential Islamic theologian Ghazali who attacked philosophy so it would not become an affront to the teachings of Islam.
  Averroes' name is also seen as Averroës, Averroès or Averrhoës, indicating that the "o" and the "e" form separate syllables. Averroes is a Latinisation of a Hebrew transcription of the Arab name Ibn Rushd.
  
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  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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• Charles A S Hall
  Charles A.S. Hall is ESF Foundation Distinguished Professor at State University of New York in the College of Environmental Science & Forestry. Hall describes himself primarily as a systems ecologist in the field of Systems ecology with strong interests in biophysical economics, and the relation of energy to society.
  Dr. Hall was trained as a systems ecologist by Howard Odum at the University of North Carolina. Since then he has had a diverse career at Brookhaven Laboratory, The Ecosystems Center at the Marine Biological Laboratory, Woods Hole, Cornell University, University of Montana and, for the last 20 years, at the State University of New York College of Environmental Science and Forestry (SUNY ESF). His work has involved streams, estuaries and tropical forests but focused increasingly on human-dominated ecosystems in the US and Latin America. His research reflects his interest in understanding and developing analyses and computer simulation models of the complex systems of nature and humans and their interactions. Halls focus has been on energy as it relates to economics and environment. His focus is studying material and energy flows referred to as Industrial ecology, and applying this perspective, to attempting to understand human economies from a biophysical rather than just social perspective. Dr. Hall teaches a freshman course called The Global Environment and the Evolution of Human Culture and graduate level courses in Systems Ecology, Ecosystems, Energy systems, Tropical Development and Biophysical Economics.
  
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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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• Claude Shannon
  Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electronic engineer, and cryptographer known as "the father of information theory".
  Shannon is famous for having founded information theory with one landmark paper published in 1948. But he is also credited with founding both digital computer and digital circuit design theory in 1937, when, as a 21-year-old master's student at MIT, he wrote a thesis demonstrating that electrical application of Boolean algebra could construct and resolve any logical, numerical relationship. It has been claimed that this was the most important master's thesis of all time. Shannon contributed to the field of cryptanalysis during World War II and afterwards, including basic work on code breaking.
  Shannon was born in Petoskey, Michigan. His father, Claude Sr (1862–1934), a descendant of early New Jersey settlers, was a self-made businessman and for a while, Judge of Probate. His mother, Mabel Wolf Shannon (1890–1945), daughter of German immigrants, was a language teacher and for a number of years principal of Gaylord High School, Michigan. The first 16 years of Shannon's life were spent in Gaylord, Michigan, where he attended public school, graduating from Gaylord High School in 1932. Shannon showed an inclination towards mechanical things. His best subjects were science and mathematics, and at home he constructed such devices as models of planes, a radio-controlled model boat and a wireless telegraph system to a friend's house half a mile away. While growing up, he worked as a messenger for Western Union. His childhood hero was Thomas Edison, who he later learned was a distant cousin. Both were descendants of John Ogden, a colonial leader and an ancestor of many distinguished people.
  
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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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• Donella Meadows
  Donella H. "Dana" Meadows (March 13, 1941 Elgin, Illinois, USA - February 20, 2001, Hanover, New Hampshire) was a pioneering American environmental scientist, teacher and writer. She is best known as lead author of the influential book The Limits to Growth, which made headlines around the world.
  Born in Elgin, Illinois, Meadows was educated in science, receiving a B.A. in chemistry from Carleton College in 1963, and a Ph.D. in biophysics from Harvard in 1968. After a year-long trip with her husband, Dennis Meadows, from England to Sri Lanka and back, she became, along with him, a research fellow at MIT as a member of a team in the department created by Jay Forrester, the inventor of system dynamics as well as the principle of magnetic data storage for computers. She taught at Dartmouth College for 29 years, beginning in 1972.
  Meadows was honored both as a Pew Scholar in Conservation and Environment (1991) and as a MacArthur Fellow (1994). She received the Walter C. Paine Science Education Award in 1990. She also posthumously received the John H. Chafee Excellence in Environmental Affairs Award for 2001 presented by the Conservation Law Foundation.
  
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  Douglas Hofstadter
  Douglas Richard Hofstadter (born February 15, 1945) is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. He is best known for his book Gödel, Escher, Bach: an Eternal Golden Braid, first published in 1979. It won both the Pulitzer Prize for general non-fiction and a National Book Award for Science.
  Hofstadter was born in New York City, the son of Nobel Prize-winning physicist Robert Hofstadter. He grew up on the campus of Stanford University, where his father was a professor, and he attended the International School of Geneva in 1958–1959. He graduated with Distinction in Mathematics from Stanford University in 1965. He continued his education and received his Ph.D. in Physics from the University of Oregon in 1975.
  
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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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• Edward Norton Lorenz
  Edward Norton Lorenz (May 23, 1917 – April 16, 2008) was an American mathematician and meteorologist, and a pioneer of chaos theory. He discovered the strange attractor notion and coined the term butterfly effect.
  Lorenz was born in West Hartford, Connecticut. He studied mathematics at both Dartmouth College in New Hampshire and Harvard University in Cambridge, Massachusetts. From 1942 until 1946, he served as a meteorologist for the United States Army Air Corps. After his return from the war, he decided to study meteorology. Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor for many years. He was a Professor Emeritus at MIT from 1969 until his death.
  During the 1950s, Lorenz became skeptical of the appropriateness of the linear statistical models in meteorology, as most atmospheric phenomena involved in weather forecasting are non-linear. His work on the topic culminated in the publication of his 1963 paper Deterministic Nonperiodic Flow in Journal of the Atmospheric Sciences, and with it, the foundation of chaos theory. His description of the butterfly effect followed in 1969,. He was awarded the Kyoto Prize for basic sciences, in the field of earth and planetary sciences, in 1991, the Buys Ballot Award in 2004, and the Tomassoni Award in 2008.^{[citation needed]} In his later years, he lived in Cambridge, Massachusetts. He was an avid outdoorsman, who enjoyed hiking, climbing, and cross-country skiing. He kept up with these pursuits until very late in his life, and managed to continue most of his regular activities until only a few weeks before his death. According to his daughter, Cheryl Lorenz, Lorenz had "finished a paper a week ago with a colleague." On April 16, 2008, Lorenz died at his home in Cambridge at the age of 90, having suffered from cancer.
  
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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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  Francisco Varela
  Francisco Javier Varela García (September 7, 1946 – May 28, 2001), was a Chilean biologist, philosopher and neuroscientist who, together with his teacher Humberto Maturana, is best known for introducing the concept of autopoiesis to biology, and for co-founding the Mind and Life Institute to promote dialog between science and Buddhism.
  Francisco Varela was born in 1946 in Santiago in Chile. After completing secondary school at the Liceo Aleman de Santiago (1951–1963). Like his mentor Humberto Maturana, Varela first studied medicine then biology at the University of Chile, then did a Ph.D. in biology at Harvard University. His thesis, defended in 1970 and supervised by Torsten Wiesel, was titled Insect Retinas: Information processing in the compound eye.
  After the 1973 military coup led by Augusto Pinochet, Varela and his family spent 7 years in exile in the USA before returning to Chile to become a Professor of biology.
  Varela became a Tibetan Buddhist in the 1970s, initially studying, together with Keun-Tshen Goba, with the meditation master Chögyam Trungpa Rinpoche, founder of Vajradhatu and Shambhala Training, and later with Tulku Urgyen Rinpoche, a Nepalese meditation master of higher tantras.
  
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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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• George Dantzig
  George Bernard Dantzig (November 8, 1914 – May 13, 2005) was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics.
  Dantzig is known for his development of the simplex algorithm, an algorithm for solving linear programming problems, and his work with linear programming, some years after it was invented by the Soviet mathematician & economist Leonid Kantorovich^{[citation needed]}. In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture of Jerzy Neyman.
  Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford.
  Born in Portland, Oregon, George Bernard Dantzig was named after George Bernard Shaw, the Irish writer.. His father, Tobias Dantzig, was a Baltic German mathematician and linguist, and his mother, Anja Dantzig (née Ourisson), was a French linguist. Dantzig's parents met during their study at the Sorbonne University in Paris, where Tobias studied mathematics under Henri Poincaré. The Dantzigs immigrated to the United States, where they settled in Portland, Oregon.
  
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  George Klir
  George Jiri Klir (born 1932 Prague, Czechoslovakia) is a Czech-American computer scientist and professor of systems sciences at the Center for Intelligent Systems at Binghamton University in Binghamton, New York.
  George Klir was born in 1932 in Prague, Czechoslovakia. In 1957 he received a M.S. degree in Electrical engineering at the Czech Technical University in Prague. In the early 1960s he taught at the Institute of Computer Research in Prague. In 1964 he received a doctorate in Computer science from the Czechoslovak Academy of Sciences.
  In the 1960s Klir went to Iraq to teach at the Baghdad University for two years. At the end he managed to immigrate to the U.S. He started teaching computer science at UCLA and at the Fairleigh Dickinson University. In 1969 he came to Binghamton University, where he later became professor of Systems science. One year (1982–1983) he stayed as a fellow in Dutch Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS), where he completed the manuscript of his book Architecture of Systems Problem Solving. In 2007 he retired after 37 years at the University. One of his doctoral students was Richard Sternberg.
  
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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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  Gregory Bateson
  Gregory Bateson (9 May 1904 – 4 July 1980) was an English anthropologist, social scientist, linguist, visual anthropologist, semiotician and cyberneticist whose work intersected that of many other fields. In the 1940s he helped extend systems theory/cybernetics to the social/behavioral sciences, and spent the last decade of his life developing a "meta-science" of epistemology to bring together the various early forms of systems theory developing in various fields of science. Some of his most noted writings are to be found in his books, Steps to an Ecology of Mind (1972) and Mind and Nature (1979). Angels Fear (published posthumously in 1987) was co-authored by his daughter Mary Catherine Bateson.
  Bateson was born in Grantchester in Cambridgeshire, England on 9 May 1904 - the third and youngest son of [Caroline] Beatrice Durham and of the distinguished geneticist William Bateson. The younger Bateson attended Charterhouse School from 1917 to 1921, obtained a BA in biology at St. John's College, Cambridge in 1925, and continued at Cambridge from 1927 to 1929. Bateson lectured in linguistics at the University of Sydney in 1928. From 1931 to 1937 he was a Fellow of St. John's College, Cambridge, spent the years before World War II in the South Pacific in New Guinea and Bali doing anthropology. During 1936-1950 he was married to Margaret Mead. At that time he applied his knowledge to the war effort before moving to the United States.
  
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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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  Heinz von Foerster
  Heinz von Foerster (born Heinz von Förster; November 13, 1911, Vienna – October 2, 2002, Pescadero, California) was an Austrian American scientist combining physics and philosophy. Together with Warren McCulloch, Norbert Wiener, John von Neumann, Lawrence J. Fogel, and others, Heinz von Foerster was an architect of cybernetics.
  Von Foerster was born in 1911 in Vienna, Austria-Hungary, as Heinz von Förster. He studied physics at the Technical University of Vienna and at the University of Breslau, where in 1944 he received a Ph.D. in physics.
  He moved to the USA in 1949, and worked at the University of Illinois at Urbana-Champaign, where he was a professor of electrical engineering from 1951 to 1975. From 1962 to 1975 he also was professor of biophysics and 1958–75 director of the Biological Computer Laboratory. Additionally, in 1956–57 and 1963–64 he was a Guggenheim-Fellow. From 1963 to 1965 he was president of the Wenner-Gren-Foundation for anthropological research.
  He knew well and was in conversation with John von Neumann, Norbert Wiener, Humberto Maturana, Francisco Varela, Gordon Pask, Gregory Bateson, Lawrence J. Fogel and Margaret Mead, among many others. He influenced generations of students as a teacher and inclusive, enthusiastic collaborator.
  
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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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  Humberto Maturana
  Humberto Maturana (born September 14, 1928, in Santiago, Chile) is a Chilean biologist and philosopher. He is considered a member of the second wave of cybernetics, known for developing a theory of autopoiesis about the nature of reflexive feedback control in living systems. His work has been influential in many fields, mainly the field of systems thinking. Overall, his work is concerned with the biology of cognition.
  After completing secondary school at the Liceo Manuel de Salas in 1947, Maturana enrolled at the University of Chile, studying first medicine then biology. In 1954, he obtained a scholarship from the Rockefeller Foundation to study anatomy and neurophysiology at University College London. He obtained a PhD in biology from Harvard University in 1958.
  He works in neuroscience at the University of Chile, in the research center "Biología del Conocer" (Biology of Knowledge). Maturana's work has been developed and integrated into the work on Ontological coaching done by Fernando Flores and Julio Olalla.
  
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• Ilya Prigogine
  Ilya, Viscount Prigogine (Russian: Илья́ Рома́нович Приго́жин, Ilya Romanovich Prigozhin) (25 January 1917 – 28 May 2003) was a Belgian physical chemist and Nobel Laureate noted for his work on dissipative structures, complex systems, and irreversibility.
  Prigogine was born in Moscow a few months before the Russian Revolution of 1917, into a Jewish family. His father, Roman (Ruvim Abramovich) Prigogine, was a chemical engineer at the Moscow Institute of Technology; his mother, Yulia Vikhman, was a pianist. Because the family was critical of the new Soviet system, they left Russia in 1921. They first went to Germany and in 1929, to Belgium, where Prigogine received Belgian citizenship in 1949.
  Prigogine studied chemistry at the Free University of Brussels, where in 1950, he became professor. In 1959, he was appointed director of the International Solvay Institute in Brussels, Belgium. In that year, he also started teaching at the University of Texas at Austin in the United States, where he later was appointed Regental Professor and Ashbel Smith Professor of Physics and Chemical Engineering. From 1961 until 1966 he was affiliated with the Enrico Fermi Institute at the University of Chicago. In Austin, in 1967, he co-founded what is now called The Center for Complex Quantum Systems. In that year, he also returned to Belgium, where he became director of the Center for Statistical Mechanics and Thermodynamics.
  
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• James Grier Miller
  James Grier Miller (1916 – 7 November 2002, California) was an American biologist, a pioneer of systems science, who originated the modern use of the term "behavioral science", founded and directed the multi-disciplinary Mental Health Research Institute at the University of Michigan, and originated the living systems theory.
  Miller received his A.B. summa cum laude in 1937, an A.M. in psychology in 1938, an M.D. cum laude in 1942, and Ph.D. in psychology in 1943 all from Harvard University, where he was also a junior fellow of the Society of Fellows. Following military service in World War II, he served as Chief of the newly formed Clinical Psychology section of the Veteran's Administration central office in Washington.
  He has served on the faculty at Harvard. In 1948, he accepted the position of Chairman of the Department of Psychology at the University of Chicago, which he held until 1955. From 1955 to 1967 he directed the multi-disciplinary Mental Health Research Institute At Michigan. In 1967 to pursue an administrative career, first as provost of the newly founded Cleveland State University, and then in 1973 as president of the University of Louisville, from which position he retired in 1980.
  
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• Jay Wright Forrester
  Jay Wright Forrester (born July 14, 1918-) is a pioneer American computer engineer, systems scientist and was a professor at the MIT Sloan School of Management. Forrester is known as the founder of System Dynamics, which deals with the simulation of interactions between objects in dynamic systems.
  Forrester was born in 1918 on a cattle ranch near Anselmo, Nebraska. His early interest in electricity was possibly sparked by the fact that the ranch had none. While in high school, he built a wind-driven 12-volt electrical system using old car parts—it gave the ranch its first electric power. After finishing high school, he had received a scholarship to go to the Agricultural College. Three weeks before enrolling, he realized a future of herding cattle in Nebraska winter blizzards had never appealed to him. So in 1936 he enrolled in the Engineering College at the University of Nebraska to study electrical engineering. As it turns out this study was about the only academic field with a solid, central core of theoretical dynamics.
  
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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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  Kevin Warwick
  Kevin Warwick (born 9 February 1954 Coventry, UK) is a British scientist and professor of cybernetics at the University of Reading, Reading, Berkshire, United Kingdom. He is known for his studies on direct interfaces between computer systems and the human nervous system, and has also done research in the field of robotics.
  Kevin Warwick was born in 1954 in Coventry in the United Kingdom. He attended Lawrence Sheriff School in Rugby, Warwickshire. He left school in 1970 to join British Telecom, at the age of 16. In 1976 he took his first degree at Aston University, followed by a Ph.D and a research post at Imperial College London.
  He subsequently held positions at Oxford, Newcastle and Warwick universities before being offered the Chair in Cybernetics at the University of Reading in 1987.
  Warwick is a Chartered Engineer, a Fellow of the Institution of Engineering and Technology and a Fellow of the City and Guilds of London Institute. He is Visiting Professor at the Czech Technical University in Prague and the University of Strathclyde and in 2004 was Senior Beckman Fellow at the University of Illinois at Urbana-Champaign, USA. He is also Director of the University of Reading Knowledge Transfer Partnerships Centre, which links the University with Companies and is on the Advisory Boards of the Instinctive Computing Laboratory, Carnegie Mellon University and the Centre for Intermedia, University of Exeter.
  
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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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• 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.
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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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  Margaret Mead
  Margaret Mead (December 16, 1901 – November 15, 1978) was an American cultural anthropologist, who was frequently a featured writer and speaker in the mass media throughout the 1960s and 1970s. She earned her bachelor degree at Barnard College in New York City, and her M.A. and Ph.D. degrees from Columbia University.
  She was both a popularizer of the insights of anthropology into modern American and Western culture and a respected, if controversial, academic anthropologist. Her reports about the attitudes towards sex in South Pacific and Southeast Asian traditional cultures amply informed the 1960s sexual revolution. Mead was a champion of broadened sexual mores within a context of traditional western religious life.
  An Anglican Christian, she played a considerable part in the drafting of the 1979 American Episcopal Book of Common Prayer.
  Mead, the first of five children, was born in Philadelphia, Pennsylvania, but raised in Doylestown. Her father, Edward Sherwood Mead, was a professor of finance at the Wharton School of the University of Pennsylvania, and her mother, Emily (Fogg) Mead, was a sociologist who studied Italian immigrants. Her sister Katharine (1906–1907) died at the age of nine months. This was a traumatic event for Mead, who had named this baby, and thoughts of her lost sister permeated her daydreams for many years. Her family moved frequently, so her early education alternated between home-schooling and traditional schools. Born into a family of varying religious outlooks, she searched for a form of religion that gave an expression of the faith that she had been formally acquainted with, Christianity. In doing so, she found the rituals of the Episcopal Church to fit the expression of religion she was seeking. Margaret studied one year, 1919, at DePauw University, then transferred to Barnard College where she earned her bachelor's degree in 1923.
  
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• Murray Bowen
  Murray Bowen, M.D., (31 January 1913, Waverly, Tennessee - 9 October 1990) was an American psychiatrist and a professor in psychiatry at the Georgetown University. Bowen was among the pioneers of family therapy and founders of systemic therapy. Beginning in the 1950s, he developed a systems theory of the family.
  Murray Bowen was born in 1913 as the oldest of five and grew up in the small town of Waverly, Tennessee, where his father was the mayor for some time. Bowen got his B.S. in 1934 at the University of Tennessee in Knoxville. He received his MD in 1937 at the Medical School of the University of Tennessee Medical School in Memphis. After that, he had internships at the Bellevue Hospital in New York City in 1938 and at the Grasslands Hospital, Valhalla, New York, from 1939 to 1941. From 1941 to 1946, he had his military training followed by five years of active duty with Army in the United States and Europe. During the war, his interest changed from surgery to psychiatry. While observing the soldiers he decided that mental illness was a more pressing and worthwhile goal. After his military service he had been accepted for fellowship in surgery at Mayo Clinic. But in 1946, he started at the Menninger Foundation in Topeka, Kansas, as fellow in psychiatry and personal psychoanalysis. This psychiatric training and experience lasted here until 1954.
  
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  Niklas Luhmann
  Niklas Luhmann (December 8, 1927 - November 6, 1998) was a German sociologist, and a prominent thinker in sociological systems theory.
  Luhmann was born in Lüneburg, Lower Saxony, where his father's family had been running a brewery for several generations. After graduating from the Johanneum school in 1943, he was conscripted as a Luftwaffenhelfer in World War II and served for two years until, at the age of 17, he was taken prisoner of war by American troops in 1945. After the war Luhmann studied law at the University of Freiburg from 1946 to 1949, when he obtained a law degree, and then began a career in Lüneburg's public administration. During a sabbatical in 1961, he went to Harvard, where he met and studied under Talcott Parsons, then the world's most influential social systems theorist.
  In later years, Luhmann dismissed Parsons' theory, developing a rival approach of his own. Leaving the civil service in 1962, he lectured at the national Deutsche Hochschule für Verwaltungswissenschaften (University for Administrative Sciences) in Speyer, Germany, until 1965, when he was offered a position at the Sozialforschungsstelle (Social Research Centre) of the University of Münster, led by Helmut Schelsky. 1965/66 he studied one semester of sociology at the University of Münster.
  
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  Norbert Wiener
  Norbert Wiener (November 26, 1894, Columbia, Missouri – March 18, 1964, Stockholm, Sweden) was an American mathematician. He was Professor of Mathematics at MIT.
  A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.
  Wiener is regarded as the originator of cybernetics, a formalization of the notion of feedback, with many implications for engineering, systems control, computer science, biology, philosophy, and the organization of society.
  Wiener was the first child of Leo Wiener and Bertha Kahn, Jews of Polish and German descent, respectively. Norbert Wiener became a famous child prodigy. Leo had educated Norbert at home until 1903, employing teaching methods of his own invention, except for a brief interlude when Norbert was 7 years of age. Earning his living teaching German and Slavic languages, Leo read widely and accumulated a personal library from which the young Norbert benefited greatly. Leo also had ample ability in mathematics, and tutored his son in the subject until he left home.
  
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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 Dirac
  
  Paul Adrien Maurice Dirac, OM, FRS ( /dɪˈræk/ di-RAK; 8 August 1902 – 20 October 1984) was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. He held the Lucasian Chair of Mathematics at the University of Cambridge, was a member of the Center for Theoretical Studies, University of Miami, and spent the last decade of his life at Florida State University.
  Among other discoveries, he formulated the Dirac equation, which describes the behaviour of fermions, and predicted the existence of antimatter.
  Dirac shared the Nobel Prize in Physics for 1933 with Erwin Schrödinger, "for the discovery of new productive forms of atomic theory."
  Paul Adrien Maurice Dirac was born at his parents' home in Bristol, England, on 8 August 1902, and grew up in the Bishopston area of the city. His father, Charles Adrien Ladislas Dirac, was an immigrant from Saint-Maurice in the Canton of Valais, Switzerland, who worked in Bristol as a French teacher. His mother, Florence Hannah Dirac, née Holten, the daughter of a ship's captain, was born in Cornwall and worked as a librarian at the Bristol Central Library. Paul had a younger sister, Béatrice Isabelle Marguerite, known as Betty, and an older brother, Reginald Charles Félix, known as Felix, who committed suicide in March 1925. Dirac later recalled: "My parents were terribly distressed. I didn't know they cared so much. /.../ I never knew that parents were supposed to care for their children, but from then on I knew."
  
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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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  Qian Xuesen
  Qian Xuesen (simplified Chinese: 钱学森; traditional Chinese: 錢學森; pinyin: Qián Xuésēn; Wade–Giles: Ch'ien Hsüeh-sên) (11 December 1911 – 31 October 2009) was a scientist who made important contributions to the missile and space programs of both the United States and People's Republic of China. Historical documents in the U. S. commonly refer to him with the earlier family-name last spelling, Hsue-Shen Tsien or H.S. Tsien.
  During the 1940s Qian was one of the founders of the Jet Propulsion Laboratory at the California Institute of Technology. During the Second Red Scare of the 1950s, the United States government accused Qian of having communist sympathies, and he was stripped of his security clearance in 1950. Qian then decided to return to China, but instead was detained at Terminal Island near Los Angeles. After spending 5 years under virtual house arrest, Qian was released in 1955, in exchange for the repatriation of American pilots captured during the Korean War. Notified by U.S. authorities that he was free to go, Qian immediately arranged his departure, leaving for China in September 1955, on the passenger liner SS President Cleveland of American President Lines, via Hong Kong. He returned to lead the Chinese rocket program, and became known as the "Father of Chinese Rocketry" (or "King of Rocketry").
  
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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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• 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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  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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  Talcott Parsons
  Talcott Parsons (December 13, 1902 – May 8, 1979) was an American sociologist who served on the faculty of Harvard University from 1927 to 1973.
  Parsons developed a general theory for the study of society called action theory, based on the methodological principle of voluntarism and the epistemological principle of analytical realism. The theory attempted to establish a balance between two major methodological traditions: the utilitarian-positivist and hermeneutic-idealistic traditions. For Parsons, voluntarism established a third alternative between these two. More than a theory of society, Parsons presented a theory of social evolution and a concrete interpretation of the "drives" and directions of world history.
  Parsons analyzed the work of Émile Durkheim and Vilfredo Pareto and evaluated their contributions through the paradigm of voluntaristic action. Parsons was also largely responsible for introducing and interpreting Max Weber's work to American audiences. Although he was generally considered a major structuralist functionalist scholar, in an article late in life, Parsons explicitly wrote that the term "functional" or "structural functionalist" were inappropriate ways to describe the character of his theory. For Parsons, "structural functionalism" was a particular stage in the methodological development of the social science, and "functionalism" was a universal method; neither term was a name for any specific school. In the same way, the concept "grand theory" is a derogatory term, which Parsons himself never used.
  
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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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  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 Ross Ashby
  W. Ross Ashby (London, 6 September 1903 – 15 November 1972) was an English psychiatrist and a pioneer in cybernetics, the study of complex systems. His first name was not used: he was known as Ross Ashby.
  His two books, Design for a brain and An introduction to cybernetics, were landmark works. They introduced exact and logical thinking to the nascent discipline, and were highly influential.
  William Ross Ashby was born in 1903 in London, where his father was working at an advertising agency. From 1917 to 1921 William studied at the Edinburgh Academy in Scotland, and from 1921 at Sidney Sussex College, Cambridge, where he received his B.A. in 1924 and his M.B. and B.Ch. in 1928. From 1924 to 1928 he worked at the St. Bartholomew's Hospital in London. Later on he also received a Diploma in Psychological Medicine in 1931, and an M.A. 1930 and M.D. from Cambridge in 1935.
  Ross Ashby started working in 1930 as a Clinical Psychiatrist in the London County Council. From 1936 until 1947 he was a Research Pathologist in the St Andrew's Hospital in Northampton in England. From 1945 to 1947 he served in India where he was a Major in the Royal Army Medical Corps.
  
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