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Foundations Of Arithmetic
''The Foundations of Arithmetic'' (german: Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other theories of number and develops his own theory of numbers. The ''Grundlagen'' also helped to motivate Frege's later works in logicism. The book was not well received and was not read widely when it was published. It did, however, draw the attentions of Bertrand Russell and Ludwig Wittgenstein, who were both heavily influenced by Frege's philosophy. An English translation was published (Oxford, 1950) by J. L. Austin, with a second edition in 1960. Criticisms of predecessors Psychologistic accounts of mathematics Frege objects to any account of mathematics based on psychologism, that is the view that math and numbers are relative to the subjective thoughts of the people who think of them. According to Frege, psychological accounts appeal to what is subjective, while mathematics i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empiricism
In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empiricism emphasizes the central role of empirical evidence in the formation of ideas, rather than innate ideas or traditions. However, empiricists may argue that traditions (or customs) arise due to relations of previous sensory experiences. Historically, empiricism was associated with the "blank slate" concept (''tabula rasa''), according to which the human mind is "blank" at birth and develops its thoughts only through experience. Empiricism in the philosophy of science emphasizes evidence, especially as discovered in experiments. It is a fundamental part of the scientific method that all hypotheses and theories must be tested against observations of the natural world rather than resting solely on ''a priori'' reasoning, intuition, or r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Psychologism Dispute
In logic, anti-psychologism (also logical objectivism or logical realism) is a theory about the nature of logical truth, that it does not depend upon the contents of human ideas but exists independent of human ideas. Overview The anti-psychologistic treatment of logic originated in the works of Immanuel Kant and Bernard Bolzano. The concept of logical objectivism or anti-psychologism was further developed by Johannes Rehmke (founder of Greifswald objectivism) and Gottlob Frege (founder of logicism the most famous anti-psychologist in the philosophy of mathematics), and has been the centre of an important debate in early phenomenology and analytical philosophy. Frege's work was influenced by Bolzano. Elements of anti-psychologism in the historiography of philosophy can be found in the work of the members of the 1830s speculative theist movement and the late work of Hermann Lotze. The psychologism dispute (german: Psychologismusstreit) in 19th-century German-speaking philosophy i ... [...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 and the other humanities 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 philos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundationalism
Foundationalism concerns philosophical theories of knowledge resting upon non-inferential justified belief, or some secure foundation of certainty such as a conclusion inferred from a basis of sound premises.Simon Blackburn, ''The Oxford Dictionary of Philosophy'', 2nd (New York: Oxford University Press, 2005)p 139 The main rival of the foundationalist theory of justification is the coherence theory of justification, whereby a body of knowledge, not requiring a secure foundation, can be established by the interlocking strength of its components, like a puzzle solved without prior certainty that each small region was solved correctly. Identifying the alternatives as either circular reasoning or infinite regress, and thus exhibiting the regress problem, Aristotle made foundationalism his own clear choice, positing basic beliefs underpinning others.Ted Poston"Foundationalism"(Internet Encyclopedia of Philosophy) Descartes, the most famed foundationalist, discovered a foundation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context Principle
In the philosophy of language, the context principle is a form of semantic holism holding that a philosopher should "never ... ask for the meaning of a word in isolation, but only in the context of a proposition" (Frege 884/1980x). Analysis The context principle is one of Gottlob Frege's "three fundamental principles" for philosophical analysis, first discussed in his Introduction to ''The Foundations of Arithmetic'' (''Grundlagen der Arithmetik'', 1884). Frege argued that many philosophical errors, especially those related to psychologism in the philosophy of logic and philosophy of mathematics, could be avoided by adhering carefully to the context principle. The view of meaning expressed by the context principle is sometimes called semantic contextualism. This view need not be contrasted with the view that the meanings of words or expressions can (or must) be determined prior to, and independently of, the meanings of the propositions in which they occur, which is often referred ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Begriffsschrift
''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, for pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his ''calculus ratiocinator'' (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though not impossible—task). Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century. This is the first work in Analytical Philosophy, a field that future British and Anglo philosophers such as Bertr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Law V
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 deri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evanston, Illinois
Evanston ( ) is a city, suburb of Chicago. Located in Cook County, Illinois, United States, it is situated on the North Shore along Lake Michigan. Evanston is north of Downtown Chicago, bordered by Chicago to the south, Skokie to the west, Wilmette to the north, and Lake Michigan to the east. Evanston had a population of 78,110 . Founded by Methodist business leaders in 1857, the city was incorporated in 1863. Evanston is home to Northwestern University, founded in 1851 before the city's incorporation, one of the world's leading research universities. Today known for its socially liberal politics and ethnically diverse population, Evanston was historically a dry city, until 1972. The city uses a council–manager system of government and is a Democratic stronghold. The city is heavily shaped by the influence of Chicago, externally, and Northwestern, internally. The city and the university share a historically complex long-standing relationship. History Prior to the 1830s, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important ''Introduction to the Second Edition'', an ''Appendix A'' that replaced ✸9 and all-new ''Appendix B'' and ''Appendix C''. ''PM'' is not to be confused with Russell's 1903 ''The Principles of Mathematics''. ''PM'' was originally conceived as a sequel volume to Russell's 1903 ''Principles'', but as ''PM'' states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of ''Principles of Mathematics''... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic 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 theory is co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
