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Intuitionistic Negation
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. Truth and proof The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can ... [...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] |
Michael Dummett
Sir Michael Anthony Eardley Dummett (; 27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He was, until 1992, Wykeham Professor of Logic at the University of Oxford. He wrote on the history of analytic philosophy, notably as an interpreter of Frege, and made original contributions particularly in the philosophies of mathematics, logic, language and metaphysics. He was known for his work on truth and meaning and their implications to debates between realism and anti-realism, a term he helped to popularize. In mathematical logic, he developed an intermediate logic, a logical system intermediate between classical logic and intuitionistic logic that had already been studied by Kurt Gödel: the Gödel–Dummett logic. In voting theory, he devised the Quota Borda system of proportional voting, based on the Borda count, and conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitism
Finitism is a philosophy of mathematics that accepts the existence only of finite set, finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as existing. Main idea The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the ''set'' of all natural numbers is not considered to exist as a mathematical object. Therefore quantifier (logic), quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic. History The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase when Georg Cantor in 1874 introduced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, "'" ("God made the integers, all else is the work of man").The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886". Weber, Heinrich L. 1891–1892Kronecker 2:5-23. (The quote is on p. 19.) Kronecker was a student and life-long friend of Ernst Kummer. Biography Leopold Kronecker was born ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite set, infinite and well-order, well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal number, cardinal and ordinal number, ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Wey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfinite Arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The sum of two well-ordered sets and is the ordinal representing the variant of lexicographical order with least significant position first, on the union of the Cartesian products and . This way, every element of is smaller than every element of , comparisons within keep the order they already have, and likewise for comparisons within . The definition of addition can also be given by transfinite recursion on . When the right ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Esenin-Volpin
Alexander Sergeyevich Esenin-Volpin (also written Ésénine-Volpine and Yessenin-Volpin in his French and English publications; rus, Алекса́ндр Серге́евич Есе́нин-Во́льпин, p=ɐlʲɪˈksandr sʲɪrˈɡʲejɪvʲɪtɕ jɪˈsʲenʲɪn ˈvolʲpʲɪn, a=Alyeksandr Syergyeyevich Yesyenin-Vol'pin.ru.vorb.oga; May 12, 1924March 16, 2016) was a Russian-American poet and mathematician. A dissident, political prisoner and a leader of the Soviet human rights movement, he spent a total of six years incarcerated and repressed by the Soviet authorities in psikhushkas and exile. In mathematics, he is known for his foundational role in ultrafinitism. Life Alexander Volpin was born on May 12, 1924, in the Soviet Union. His mother, Nadezhda Volpin, was a poet and translator from French and English. His father was Sergei Yesenin, a celebrated Russian poet, who never knew his son. Alexander and his mother moved from Leningrad to Moscow in 1933. His first psy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Set Theory
Constructivism may refer to: Art and architecture * Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes * Constructivist architecture, an architectural movement in the Soviet Union in the 1920s and 1930s * British Constructivists, a group of British artists who were active between 1951 and 1955. Education * Constructivism (philosophy of education), a theory about the nature of learning that focuses on how humans make meaning from their experiences * Constructivism in science education * Constructivist teaching methods, based on constructivist learning theory Mathematics * Constructivism (philosophy of mathematics), a logic for founding mathematics that accepts only objects that can be effectively constructed * Constructivist set theory * Constructivist type theory Philosophy * Constructivism (philosophy of mathematics), a philosophical view that asserts the necessity of constructing a mathematical object to p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theorems, proof (mathematics), proofs, algorithms, etc. in particular. This may also include the philosophy of mathematics, philosophical study of the relation of this framework with reality. The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements, Euclid's ''Elements''. A mathematical assertion is considered as truth (mathematics), truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality Of The Continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \bold, \bold\mathbb R\bold, . The real numbers \mathbb R are more numerous than the natural numbers \mathbb N. Moreover, \mathbb R has the same number of elements as the power set of \mathbb N. Symbolically, if the cardinality of \mathbb N is denoted as \aleph_0, the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers ''a'' < ''b'', no matter how close they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
