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Topic: Mathematical intuitionism

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	Philosophy of mathematics - Open Encyclopedia (Site not responding. Last check: 2007-10-20)
		Some philosophers of mathematics view their task as giving an account of mathematics and mathematical practice as it stands, as interpretation rather than criticism.
		Three schools, intuitionism, logicism and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that mathematics (as it stood), and analysis in particular, did not live up to the standards of certainty and rigour with which it was over-credited.
		The capacity to acquire mathematics, and competence in it, called numeracy, is seen as separate from literacy and the acquisition of language.
	open-encyclopedia.com /Philosophy_of_mathematics (3541 words)

	Intuitionism - Wikipedia, the free encyclopedia
		In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans.
		Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction.
		Intuitionism also rejects the abstraction of actual infinity; i.e., it does not consider as given objects infinite entities such as the set of all natural numbers or an arbitrary sequence of rational numbers.
	en.wikipedia.org /wiki/Mathematical_intuitionism (313 words)

	Mathematics - Wikipedia, the free encyclopedia
		Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language.
		This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis).
		Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
	en.wikipedia.org /wiki/Mathematics (2972 words)

	Philosophy of mathematics - Wikipedia, the free encyclopedia
		Mathematical logicians study formal systems but are just as often realists as they are formalists.
		But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current Mathematical Community.
		They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-belief in axiomatic proof and peer review as practices.
	en.wikipedia.org /wiki/Mathematical_realism (3623 words)

	Philosophy of mathematics (Site not responding. Last check: 2007-10-20)
		Some philosophers of mathematics view their task as being to give an account of mathematics and mathematical practice as it stands, as interpretation rather than criticism.
		Criticisms can however have important ramifications for mathematical practice and claims for finished mathematics and so the philosophy of mathematics can be of very direct interest to working mathematicians, particularly in new fields where the process of peer review of mathematical proofs is not firmly established, raising probability of an undetected error.
		Putnam strongly rejected the term "Platonist" as implying an overly-specific ontology that was not necessary to mathematical practice in any real sense - he advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics - a term that he was involved in coining (see below).
	1-free-software.com /en/wikipedia/p/ph/philosophy_of_mathematics.html (3428 words)

	[No title]
		The whole of philosophy of mathematics consists of two areas of studies: The interpretation of mathematical concepts: numbers, infinity, and certainty.1 The nature of mathematical propositions: theorems and proofs.
		Formalism and intuitionism argued that mathematics is an invention of the mind, like backgammon or chess, an enterprise that is disconnected from the world.
		They did have theories for mathematical concepts, but those theories were constructed as necessary to work for their main goal: deduction of mathematical theorems from logical axioms.
	www.usfca.edu /philosophy/discourse/8/bold.doc (4206 words)

	Brouwer's Cambridge Lectures on Intuitionism
		The gradual transformation of the mechanism of mathematical thought is a consequence of the modifications which, in the course of history, have come about in the prevailing philosophical ideas, firstly concerning the origin of mathematical certainty, secondly concerning the delimitation of the object of mathematical science.
		Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognising that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time.
		The belief in the universal validity of the principle of the excluded third in mathematics is considered by the intuitionists as a phenomenon of the history of civilization of the same kind as the former belief in the rationality of pi, or in the rotation of the firmament about the earth.
	www.marxists.org /reference/subject/philosophy/works/ne/brouwer.htm (2524 words)

	Intuitionistic logic (Site not responding. Last check: 2007-10-20)
		Intuitionistic Logic is the logical branch of Mathematical intuitionism.
		Roughly speaking, 'intuitionism' holds that logic and math are 'constructive' mental activities.
		While it may be argued whether such a formal calculus really captures the philosophical aspects of intuitionism, it has properties which are also quite useful from a practical point of view.
	www.termsdefined.net /in/intuitionistic-logic.html (675 words)

	Stephen Cole Kleene (Site not responding. Last check: 2007-10-20)
		Kleene was best known for founding the branch of mathematical logic known as recursion theory together with Alonzo Church, Kurt Gödel;, Alan Turing and others; and for inventing regular expressions.
		From 1930 to 1935, he was a graduate student and research assistant at Princeton University, where he received his doctorate in mathematics in 1934, supervised by Alonzo Church, for a thesis entitled A Theory of Positive Integers in Formal Logic.
		He was chair of mathematics and computer sciences in 1962 and 1963, and dean of the College of Letters and Science from 1969 to 1974.
	www.bidprobe.com /en/wikipedia/s/st/stephen_cole_kleene.html (408 words)

	Inconsistent Mathematics
		Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
		Gödel's contribution to the philosophy of mathematics was to show that the first three of these can be rigorously expressed in arithmetical theories, albeit in theories which are either inconsistent or incomplete.
		Projective geometry is a mathematical theory which is interesting because we are creatures with an eye, since it explains why it is that things look the way they do in perspective.
	www.braungardt.com /Mathematica/inconsistent_mathematics.htm (1793 words)

	Philosophy of mathematics - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-20)
		For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see 's Realism in Mathematics.
		Contributions to this school have been made by Imre Lakatos and, although it is not clear that either would endorse the title.
		Judea Pearl claimed that all of mathematics as presently understood was based on an algebra of seeing - and proposed an to complement it - this is a central concern of the philosophy of action and other studies of how "knowing" relates to "doing", or knowledge to action.
	eastcleveland.us /project/wikipedia/index.php/Philosophy_of_mathematics (3633 words)

	Read This: Gnomes in the Fog
		He is also the founder of mathematical intuitionism, and a key player in the debate on foundations of mathematics that raged for a brief decade in the 1920s, and then subsided.
		But so prevalent within their mathematical training was the dismissal of this debate as a dead end in the philosophy of mathematics that they, too, subscribed to the prevailing wisdom that it was a non-issue, a cul-de-sac not worth exploring.
		The separation between mathematics and mathematical language is the first of the two "acts" in the development of intuitionism.
	www.maa.org /reviews/gnomesfog.html (2372 words)

	Mathematical intuitionism (Site not responding. Last check: 2007-10-20)
		Society for Mathematical Psychology Promotes the advancement and communication of research in mathematical psychology, broadly defined to include work of a theoretical character that uses mathematical methods, formal logic, or computer simulation.
		Mathematical Geology Journal published by the International Association for Mathematical Geology (IAMG).
		International Association of Mathematical Physics The International Association of Mathematical Physics (IAMP) was founded in order to promote research in mathematical physics.
	www.serebella.com /encyclopedia/article-Mathematical_intuitionism.html (261 words)

	Mathematical constructivism (Site not responding. Last check: 2007-10-20)
		In the philosophy of mathematics, mathematical constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.
		Constructivism is often confused with mathematical intuitionism, but in fact, intuitionism is only one kind of constructivism.
		Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.
	www.city-search.org /ma/mathematical-constructivism.html (412 words)

	Intuitionism -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		Any mathematical object is considered to be a product of a construction of a (That which is responsible for one's thoughts and feelings; the seat of the faculty of reason) mind, and therefore, the existence of an object is equivalent to the possibility of its construction.
		As such, intuitionism is a variety of (Click link for more info and facts about mathematical constructivism) mathematical constructivism; but it is not the only kind.
		For example, to say A (A room in a hospital equipped for the performance of surgical operations) or B, to an intuitionist, is to claim that either A or B can be proved.
	www.absoluteastronomy.com /encyclopedia/i/in/intuitionism.htm (402 words)

	Joshua Helston
		Mathematics can refer to any system of objects or members and relationships whose titles can be chosen in such a way to guarantee that all the axioms of pure mathematics are true of the objects, members and those relationships.
		Logic differs from mathematics, so logic may use modus tollens and be consistent with its set of rules, but it may not count as valid mathematics.
		This first wave of intuitionist mathematics versus classical mathematics, personified by the conflict between Cantor and Kronecker, proved to be the first in at least a two-part series.
	www.gustavus.edu /oncampus/academics/philosophy/Helston.html (5499 words)

	Brouwer (Site not responding. Last check: 2007-10-20)
		L E J Brouwer founded the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are governed by self-evident laws.
		His doctoral thesis in 1907 on the foundations of mathematics attacked the logical foundations of mathematics and form the beginning of the Intuitionist School.
		He rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false.
	www.math.uri.edu /~kulenm/mth381pr/fixedpoint/brouwer.html (454 words)

	The Time of Formalization: 1854-1937 (Site not responding. Last check: 2007-10-20)
		Symbol manipulation (calculi)[<>]: Symbolism is the idea that mathematics and logics can be reduced to symbol manipulation [Log], defined as a "calculus" of manipulation rules (in particular sound logical derivations) which depend on the symbols' form, not meaning (hence formal logic/mathematics).
		Frege was not out to eliminate mathematical ontology, since he held that logic itself has an ontology, containing concepts and their extensions» [x].
		«Intuitionism stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms» [ref].
	www.cs.mun.ca /~ulf/csh/formal.html (5570 words)

	1.1.4 Intuitionism -- Prof Scowcroft -- 16 HT (Site not responding. Last check: 2007-10-20)
		[Candidates for Mathematics and Philosophy who are taking option (ii) Mathematics and Philosophy in Part II and wish to offer Intuitionism for Papers c1(P) and c2(P) must apply through their college for approval of this subject to the Joint Committee for Mathematics and Philosophy (see Examination Decrees and Regulations)].
		Intuitionism, the philosophy of mathematics invented by the Dutch topologist L. Brouwer, views mathematics as an elaboration over time of certain mental constructions rather than as a report on a reality existing independently of human mathematical activity.
		Here are some of the topics to be covered in the lectures: the intuitionistic interpretation of the connectives and quantifiers; intuitionistic first-order arithmetic and its relation to classical first-order arithmetic; semantics for intuitionistic predicate logic; intuitionistic analysis and continuity principles.
	www.maths.ox.ac.uk /current-students/undergraduates/handbooks-synopses/2002/html/sect-c-02/node7.html (237 words)

	Learn more about Intuitionism in the online encyclopedia. (Site not responding. Last check: 2007-10-20)
		Learn more about Intuitionism in the online encyclopedia.
		As such, intutionism is a variety of mathematical constructivism; but it is not the only kind.
		This requires the reconstruction of the most part of set theory and calculus, leading to theories highly different from their classical versions.
	www.onlineencyclopedia.org /i/in/intuitionism.html (368 words)

	intuitionism -- Britannica Concise Encyclopedia - Your gateway to all Britannica has to offer!
		School of mathematical thought introduced by the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881–1966).
		In contrast with mathematical Platonism, which holds that mathematical concepts exist independent of any human realization of them, intuitionism holds that only those mathematical concepts that can be demonstrated, or constructed, following a finite number of steps are legitimate.
		Few mathematicians have been willing to abandon the vast realms of mathematics built with nonconstructive proofs.
	concise.britannica.com /ebc/article-9368116 (82 words)

	Intuitionism and Type Theory (Site not responding. Last check: 2007-10-20)
		In his 1907 doctoral thesis [Brouwer 23] Brouwer advanced the view that the basis of mathematics and of logic is found in the capacity of human beings to carry out mental constructions.
		On the method of proof by contradiction he says, ``The words of your mathematical demonstration merely accompany a mathematical construction that is effected without words.
		Bishop's works, and those of his followers [Bishop 67,Bishop and Bridges 85,Bishop and Cheng 72,Bridges 79,Chan 74], have started a modern school of constructive mathematics that we believe is in harmony with the influence of computing in mathematics, an influence seen in many quarters [Nepeivoda 82,Goad 80,Goto 79,Takasu 78].
	www.cs.cornell.edu /Info/Projects/NuPrl/book/node28.html (349 words)

	Inconsistent Mathematics
		This study has been further developed in Mortensen (2002a), where category theory is applied to give a general description of the relationships between the various theories and their consistent cut-downs and incomplete duals.
		For an informal account which highlights the difference between visual "paradoxes" and the philosophically more common paradoxes of language, such as the Liar, see Mortensen (2002b).
		It should be emphasised that these constructions do not in any way challenge or repudiate existing mathematics, but extend our conception of what is mathematically possible.
	plato.stanford.edu /entries/mathematics-inconsistent (2006 words)

	Publisher description for Library of Congress control number 82025257 (Site not responding. Last check: 2007-10-20)
		The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.
		It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography.
		The volume will be welcomed as a major work of reference at this level in the field.
	www.loc.gov /catdir/description/cam022/82025257.html (115 words)

	Find in a Library: Mathematical intuitionism and intersubjectivity : a critical exposition of arguments for intuitionism
		Find in a Library: Mathematical intuitionism and intersubjectivity : a critical exposition of arguments for intuitionism
		Mathematical intuitionism and intersubjectivity : a critical exposition of arguments for intuitionism
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/5a3e140ac297ff9aa19afeb4da09e526.html (66 words)

	References for Brouwer (Site not responding. Last check: 2007-10-20)
		P Mancosu (ed.), From Hilbert to Brouwer : The Debate on the Foundations of Mathematics in the 1920s (Oxford, 1988).
		M I Panov, L E J Brouwer and Soviet mathematics (Russian), Patterns in the development of modern mathematics (Moscow, 1987), 250-278.
		D van Dalen, The War of the Frogs and the Mice, or the Crisis of the Mathematische Annalen, Mathematical Intelligencer 12 (4) (1990), 17-31.
	www-history.mcs.st-and.ac.uk /history/Printref/Brouwer.html (422 words)

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