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Solomon Feferman
Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for his contributions to the history of logic (for instance, via biographical writings on figures such as Kurt Gödel, Alfred Tarski, and Jean van Heijenoort) and as a vocal proponent of the philosophy of mathematics known as predicativism, notably from an anti- platonist stance. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to the United States after World War I and had met and married in New York. Neither parent had any advanced education. The family moved to Los Angeles, where Feferman graduated from high school at age 16. He received his B.S. from the California Institute of Technology in 1948, and in 1957 his Ph.D. in mathematics from the University of California, Berkeley, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Western Philosophy
Western philosophy refers to the Philosophy, philosophical thought, traditions and works of the Western world. Historically, the term refers to the philosophical thinking of Western culture, beginning with the ancient Greek philosophy of the Pre-Socratic philosophy, pre-Socratics. The word ''philosophy'' itself originated from the Ancient Greek (φιλοσοφία), literally, "the love of wisdom" , "to love" and σοφία ''Sophia (wisdom), sophía'', "wisdom". History Ancient The scope of ancient Western philosophy included the problems of philosophy as they are understood today; but it also included many other disciplines, such as pure mathematics and natural sciences such as physics, astronomy, and biology (Aristotle, for example, wrote on all of these topics). Pre-Socratics The pre-Socratic philosophers were interested in cosmology (the nature and origin of the universe), while rejecting unargued fables in place for argued theory, i.e., dogma superseded reason, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tarski
Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. 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, type theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school, Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983.#FefA, Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicativism
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions. The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over a type at one 'level' results in types at a new, higher, level. A prototypical example is intuitionistic type theory, which retains ramification (without the explicit levels) so as to discard impredicativity. The 'levels' here correspond to the number of layers of dependency in a term definition. Russell's paradox is a famous example of an impredicative construction—namely the set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Van Heijenoort
Jean Louis Maxime van Heijenoort ( ; ; ; July 23, 1912 – March 29, 1986) was a historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and an American Trotskyist until 1947. Life Van Heijenoort was born in Creil, France. His parents had immigrated from the Netherlands before his birth. When van Heijenoort was only two years old, his father passed away, leaving his family in financial hardship. Despite these challenges, he pursued his education and became proficient in French. Throughout his life, he maintained strong connections with his extended family and friends in France, making biannual visits after he obtained American citizenship in 1958. Political views In 1932, van Heijenoort was recruited by Yvan Craipeau to join the Trotskyist movement. He joined the Communist League in the same year. After Trotsky was exiled, he hired van Heijenoort as a secretary and bodyguard, primarily because of his fluency in French, Russian, G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurt Gödel
Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell,For instance, in their "Principia Mathematica' (''Stanford Encyclopedia of Philosophy'' edition). Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor. Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Logic
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in Indian logic, India, Logic in China, China, and Greek philosophy, Greece. Greek methods, particularly Aristotelian logic (or term logic) as found in the ''Organon'', found wide application and acceptance in Western science and mathematics for millennia.Boehner p. xiv The Stoicism, Stoics, especially Chrysippus, began the development of predicate logic. Christian philosophy, Christian and Logic in Islamic philosophy, Islamic philosophers such as Boethius (died 524), Avicenna (died 1037), Thomas Aquinas (died 1274) and William of Ockham (died 1347) further developed Aristotle's logic in the Medieval philosophy#High Middle Ages, Middle Ages, reaching a high point in the mid-fourteenth century, with Jean Buridan. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosopher
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational and critical inquiry that reflects on its methods and assumptions. Historically, many of the individual sciences, such as physics and psychology, formed part of philosophy. However, they are considered separate academic disciplines in the modern sense of the term. Influential traditions in the history of philosophy include Western philosophy, Western, Islamic philosophy, Arabic–Persian, Indian philosophy, Indian, and Chinese philosophy. Western philosophy originated in Ancient Greece and covers a wide area of philosophical subfields. A central topic in Arabic–Persian philosophy is the relation between reason and revelation. Indian philosophy combines the Spirituality, spiritual problem of how to reach Enlightenment in Buddhism, enlighten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
