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Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the '' Foundations of Arithmetic'' is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano. Overview Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On Sense And Reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the two ways he believed a singular term may have meaning. The reference (or "referent"; ''Bedeutung'') of a ''proper name'' is the object it means or indicates (''bedeuten''), whereas its sense (''Sinn'') is what the name expresses. The reference of a ''sentence'' is its truth value, whereas its sense is the thought that it expresses."On Sense and Reference" Über Sinn und Bedeutung" '' Zeitschrift für Philosophie und philosophische Kritik'', vol. 100 (1892), pp. 25–50, esp. p. 31. Frege justified the distinction in a number of ways. #Sense is something possessed by a name, whether or not it has a reference. For example, the name "Odysseus" is intelligible, and therefore has a sense, even though there is no individual object (its refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Philosophy
Analytic philosophy is a broad movement within Western philosophy, especially English-speaking world, anglophone philosophy, focused on analysis as a philosophical method; clarity of prose; rigor in arguments; and making use of formal logic, mathematics, and to a lesser degree the natural sciences.Mautner, Thomas (editor) (2005) ''The Penguin Dictionary of Philosophy'', entry for "Analytic philosophy", pp. 22–23 It is further characterized by an interest in language, semantics and Meaning (philosophy), meaning, known as the linguistic turn. It has developed several new branches of philosophy and logic, notably philosophy of language, philosophy of mathematics, philosophy of science, modern predicate logic and mathematical logic. The proliferation of analysis in philosophy began around the turn of the 20th century and has been dominant since the latter half of the 20th century. Central figures in its historical development are Gottlob Frege, Bertrand Russell, G. E. Moore, and L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Sluga
Hans D. Sluga (; born 24 April 1937) is a German philosopher who spent most of his career as professor of philosophy at the University of California, Berkeley. Sluga teaches and writes on topics in the history of analytic philosophy, the history of continental philosophy, as well as on political theory, and ancient philosophy in Greece and China. He has been particularly influenced by the thought of Gottlob Frege, Ludwig Wittgenstein, Martin Heidegger, Friedrich Nietzsche, and Michel Foucault. Education and career Hans Sluga studied at the University of Bonn and the University of Munich. He subsequently obtained a BPhil at Oxford, where he studied under R. M. Hare, Isaiah Berlin, Gilbert Ryle and Michael Dummett. Since 1970, Sluga has been a professor of philosophy at the University of California, Berkeley, serving from 2009 as the William and Trudy Ausfahl Professor of Philosophy until his retirement in 2020. He previously served as a lecturer in philosophy at University Colle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frege%27s Theorem
In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his 1884 ''Die Grundlagen der Arithmetik'' ('' The Foundations of Arithmetic'')Gottlob Frege, ''Die Grundlagen der Arithmetik'', Breslau: Verlag von Wilhelm Koebner, 1884, §63. and proven more formally in his 1893 ''Grundgesetze der Arithmetik'' I (''Basic Laws of Arithmetic'' I).Gottlob Frege, ''Grundgesetze der Arithmetik'' I, Jena: Verlag Hermann Pohle, 1893, §§20 and 47. The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics known as neo-logicism (at least of the Scottish School variety). Overview In '' The Foundations of Arithmetic'' (1884), and later, in ''Basic Laws of Arithmetic'' (vol. 1, 1893; vol. 2, 1903), Frege attempted to de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Language
Philosophy of language refers to the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy), meaning, intentionality, reference, the constitution of sentences, concepts, learning, and thought. Gottlob Frege and Bertrand Russell were pivotal figures in analytic philosophy's "linguistic turn". These writers were followed by Ludwig Wittgenstein (''Tractatus Logico-Philosophicus''), the Vienna Circle, Logical positivism, logical positivists, and Willard Van Orman Quine. History Ancient philosophy In the West, inquiry into language stretches back to the 5th century BC with philosophers such as Socrates, Plato, Aristotle, and the Stoics. Linguistic speculation predated systematic descriptions of grammar which emerged in India and in Greece. In the dialogue ''Cratylus (dialogue), Cratylus'', Plato considered the question of whether ... [...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] |
Philosophy Of Mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major themes that are dealt with in philosophy of mathematics include: *''Reality'': The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. *''Logic and rigor'' *''Relationship with physical reality'' *''Relationship with science'' *''Relationship with applications'' *''Mathematical truth'' *''Nature as human activity'' (science, the arts, art, game, or all together) Major themes Reality Logic and rigor Mathematical reasoning requires Mathematical rigor, rigor. This means that the definitions must be absolutely unambiguous and th ... [...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] |
Michael Resnik .
Michael David Resnik (; born March 20, 1938) is a leading contemporary American philosopher of mathematics. Biography Resnik obtained his B.A. in mathematics and philosophy at Yale University in 1960, and his PhD in Philosophy at Harvard University in 1964. He wrote his thesis on Frege. He was appointed Associate Professor at the University of North Carolina at Chapel Hill in 1967, Professor in 1975, and University Distinguished Professor in 1988. He is Professor Emeritus of University of North Carolina at Chapel Hill and currently resides in rural Chatham County, North Carolina Chatham County ( ) , from the North Carolina Collection's website at the University of North Car ... Publications Books * * * * *Journal articles * * * * * * * ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancestral Relation
In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation ''R'' is its transitive closure, however defined in a different way, see below. Ancestral relations make their first appearance in Frege's ''Begriffsschrift''. Frege later employed them in his ''Grundgesetze'' as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicist foundation of arithmetic. Definition The numbered propositions below are taken from his ''Begriffsschrift'' and recast in contemporary notation. A property ''P'' is called ''R''-hereditary if, whenever ''x'' is ''P'' and ''xRy'' holds, then ''y'' is also ''P'': :(Px \land xRy) \rightarrow Py An individual ''b'' is said to be an ''R''-ancestor of ''a'', written ''aR*b'', if ''b'' has every ''R''-hereditary property that all objects ''x'' such that ''aRx'' have: :\mathbf\ \vdash aR^*b \leftrightarrow \forall F forall x (aRx \to Fx) \land \forall x \forall y ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linguistic Turn
The linguistic turn was a major development in Western philosophy during the early 20th century, the most important characteristic of which is the focusing of philosophy primarily on the relations between language, language users, and the world. Very different intellectual movements were associated with the "linguistic turn", although the term itself is commonly thought to have been popularised by Richard Rorty's 1967 anthology ''The Linguistic Turn'', in which he discusses the turn towards linguistic philosophy. According to Rorty, who later dissociated himself from linguistic philosophy and analytic philosophy generally, the phrase "the linguistic turn" originated with philosopher Gustav Bergmann. Analytic philosophy Traditionally, the linguistic turn is taken to also mean the birth of analytic philosophy. One of the results of the linguistic turn was an increasing focus on logic and philosophy of language, and the cleavage between ideal language philosophy and ordinary lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
