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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] |
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] |
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, philosophy of logic, logic, and Philosophy of mathematics, 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 History of logic#Rise of modern logic, 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 ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the ... [...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 published 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. 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 not members of themselves. (This set is sometimes called "the Russell set".) If ''R'' is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : Let R = \. Then R \in R \iff R \not \in R. Russell also showed that a version of the paradox co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Dedekind
Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as ''logicism''. Life Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests at Braunschweig Main Cemetery. He first attended the Collegium Carol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notation, 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. He also created an international auxiliary language, Latino sine flexione ("Latin without inflections"), which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, while others are in Italian. Biography 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 Universi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology. In his early career Whitehead wrote primarily on mathematics, logic, and physics. He wrote the three-volume ''Principia Mathematica'' (1910–1913), with his former student Bertrand Russell. ''Principia Mathematica'' is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library."The Modern Library ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naive Set Theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping stone towards more formal treatments. Method A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. Such theory treats sets as platonic absolute o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Principles Of Mathematics
''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell, in which the author presented Russell's paradox, his famous paradox and argued his thesis that mathematics and logic are identical. The book presents a view of the foundations of mathematics and Alexius Meinong, Meinongianism and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others. In 1905 Louis Couturat published a partial French translation that expanded the book's readership. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were published in 1938, 1951, 1996, and 2009. Contents ''The Principles of Mathematics'' consists of 59 chapters divided into seven parts: indefinables in mathematics, number, quantity, order, infinity and continuity, spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epistemology
Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowledge in the form of skills, and knowledge by acquaintance as a familiarity through experience. Epistemologists study the concepts of belief, truth, and justification to understand the nature of knowledge. To discover how knowledge arises, they investigate sources of justification, such as perception, introspection, memory, reason, and testimony. The school of skepticism questions the human ability to attain knowledge while fallibilism says that knowledge is never certain. Empiricists hold that all knowledge comes from sense experience, whereas rationalists believe that some knowledge does not depend on it. Coherentists argue that a belief is justified if it coheres with other beliefs. Foundationalists, by contrast, maintain th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
