In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. That is, they are not analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs. To meet Wikipedias quality standards, this article or section may require cleanup. ... In some circles of mathematical philosophy the pre-intuitionists are considered to be a small but influential group who informally shared similar philosophies on the nature of mathematics. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

Truth and proof

The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. As the name suggests, in Brouwer's original intuitionism, the truth of a statement is taken to be equivalent to the mathematician being able to intuit the statement. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, however Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill defined. Regardless of how it is interpreted, intuitionism does not equate the truth of a mathematical statement with its provability. However, because the intuitionistisic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he/she proves is in fact intuitionistically true. This gives rise to intuitionistic logic.

To claim an object with certain properties exists, is, to an intuitionist, to claim to be able to construct a certain object with those properties. 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. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a constructive proof of existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind. For other uses, see Mind (disambiguation). ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

As well, to say A or B, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, A or not A, is disallowed since one cannot assume that it is always possible to either prove the statement A or its negation. OR logic gate. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Negation (i. ...

The interpretation of negation is also different. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable (i.e., that there is a proof that there is no proof of it). The asymmetry between a positive and negative statement becomes apparent. If a statement P is provable, then it is certainly impossible to prove that there is no proof of P; however, just because there is no proof that there is no proof of P, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

Intuitionistic logic substitutes justification for truth in its logical calculus. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has given philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett. Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... In philosophy, the term anti-realism is used to describe any position involving either the denial of the objective reality of entities of a certain type or the insistence that we should be agnostic about their real existence. ... Sir Michael Anthony Eardley Dummett F.B.A., D. Litt, (born 1925) is a leading British philosopher. ...

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. This requires the reconstruction of the foundations of set theory and calculus as constructivist set theory and constructivist analysis respectively. Abstraction is the process of reducing the information content of a concept, typically in order to retain only information which is relevant for a particular purpose. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Calculus [from Latin, literally pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ... Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. ... In mathematics, constructive analysis is mathematical analysis done according to the principles of constructivist mathematics. ...

History of Intuitionism

Intuitionist mathematics originated in part from (i) the strong disagreement between Cantor and his teacher Kronecker — a confirmed finitist — that led to Cantor's hospitalization, and (ii) the failure of Frege's effort to reduce all of mathematics to a logical formulation — in face of the letter from Bertrand Russell received by Frege just as his life's work was about to be published. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and Heijenoort's commentary. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ...

In the early twentieth century the battle was taken up Brouwer the intuitionist versus Hilbert the formalist — see van Heijenoort. Kurt Gödel the Platonist had his opinions (see various sources re Gödel) and Alan Turing considers: Brouwer is the last name of different people. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... The term formalist can have many applications: The Chambers 1994 edition Dictionary indicates a pejorative quality, a person having an exaggerated regard to rules or established usages. In the philosophy of mathematics a formalist is a person who belongs to the school of formalism, a certain mathematical-philosophical doctrine which... [...]I dont believe in natural science. ... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ... Alan Mathison Turing, OBE (June 23, 1912 – June 7, 1954), was an English mathematician, logician, and cryptographer. ...

"non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive" (Turing (1939) Systems of Logic Based on Ordinals in Undecidable, p. 210)

In the middle of the century Kleene brought forth a more rational consideration of intuitionism in his Introduction to Meta-mathematics (1952). Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ...

For the view that there are no paradoxes in Cantorian set theory — thus calling into question the program of intuitionist mathematics, see Alejandro Garciadiego's now-classic Bertrand Russell and the Origins of the Set-Theoretic Paradoxes.

Contributors to intuitionism

• L. E. J. Brouwer
• Arend Heyting
• Stephen Kleene
• Michael Dummett

Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ... Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... Sir Michael Anthony Eardley Dummett F.B.A., D. Litt, (born 1925) is a leading British philosopher. ...

Branches of intuitionistic mathematics

• Intuitionistic logic
• Intuitionistic arithmetic
• Intuitionistic type theory
• Intuitionistic set theory
• Intuitionistic analysis

Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Heyting arithmetic is the basic arithmetic of intuitionism (not to be confused with Heyting algebra). ... Intuitionistic Type Theory, or Constructive Type Theory, or Martin-Löf Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ... Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. ... In mathematics, constructive analysis is mathematical analysis done according to the principles of constructivist mathematics. ...

See also

• Anti-realism
• BHK interpretation
• Computability logic
• Curry-Howard isomorphism
• Foundations of mathematics
• Game semantics
• Intuition (knowledge)
• Ultraintuitionism

In philosophy, the term anti-realism is used to describe any position involving either the denial of the objective reality of entities of a certain type or the insistence that we should be agnostic about their real existence. ... In mathematical logic, the BHK interpretation of intuitionistic predicate logic was proposed by L. E. J. Brouwer, Arendt Heyting and independently by Kolmogorov. ... Computability logic is a formal theory of computability, introduced by Giorgi Japaridze in 2003. ... The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Game semantics (German: dialogische Logik) is an approach to the semantics of logic that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player. ... Intuition is an unconscious form of knowledge. ... In the philosophy of mathematics, ultrafinitism, or ultraintuitionism, is an extreme version of finitism. ...

Further reading

• van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.

* Luitzen Egbertus Jan Brouwer, 1923, On the significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort]

* Andrei Nikolaevich Kolmogorov, 1925, On the princple of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]

* Luitzen Egbertus Jan Brouwer, 1927, On the domains of definitions of functions, [reprinted with commentary, p. 446, van Heijenoort]

Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.

* Luitzen Egbertus Jan Brouwer, 1927(2), Intuitionistic reflections on formalism, [reprinted with commentary, p. 490, van Heijenoort]

* Jacques Herbrand, (1931b), "On the consistency of arithmetic", [reprinted with commentary, p. 618ff, van Heijenoort]

From van Heijenoort's commentary it is unclear whether or not Herbrand was a true "intuitionist"; Gödel (1963) asserted that indeed "...Herbrand was an intuitionist". But van Heijenoort says Herbrand's conception was "on the whole much closer to that of Hilbert's word 'finitary' ('finit') that to "intuitionistic" as applied to Brouwer's doctrine".

• Arend Heyting: Heyting, Arend [1956] (1971). Intuitionism: An Introduction, 3d rev. ed., Amsterdam: North-Holland Pub. Co. ISBN 0-7204-2239-6. 

• Hesseling, Dennis E. (2003). Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s. Birkhäuser. ISBN 3-7643-6536-6. 

• Paul Rosenbloom, The Elements of Mathematical Logic, Dover Publications Inc, Mineola, New York, 1950.

In a style more of Principia Mathematica -- many symbols, some antique, some from German script. Very good discussions of intuitionism in the following locations: pages 51-58 in Section 4 Many Valued Logics, Modal Logics, Intuitionism; pages 69-73 Chapter III The Logic of Propostional Functions Section 1 Informal Introduction; and p. 146-151 Section 7 the Axiom of Choice.

• Stephen Cole Kleene and Richard Eugene Vesley, The Foundations of Intuistionistic Mathematics, North-Holland Publishing Co. Amsterdam, 1965. The lead sentence tells it all "The constructive tendency in mathematics...". A text for specialists, but written in Kleene's wonderfully-clear style.

• Kleene, Stephen C. [1952] (1991). Introduction to Meta-Mathematics, Tenth impression 1991, Amsterdam NY: North-Holland Pub. Co. ISBN 0-7204-2103-9. 

In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.

• "analysis." Encyclopædia Britannica. 2006. Encyclopædia Britannica 2006 Ultimate Reference Suite DVD 15 June 2006, "Constructive analysis" (Ian Stewart, author)

• W. S. Anglin, Mathematics: A Concise history and Philosophy, Springer-Verlag, New York, 1994.

In Chapter 39 Foundations, with respect to the 20th century Anglin gives very precise, short descriptions of Platonism (with respect to Godel), Formalism (with respet to Hilbert), and Intuitionism (with respect to Brouwer).

• Constance Reid, Hilbert, Copernicus - Springer-Verlag, 1st edition 1970, 2nd edition 1996.

Definitive biography of Hilbert places his "Program" in historical context together with the subsequent fighting, sometimes rancorous, between the Intuitionists and the Formalists.

• John W. Dawson Jr., Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley, MA, 1997.

Less readable than Goldstein but, in Chapter III Excursis, Dawson gives an excellent "A Capsule History of the Development of Logic to 1928".

• Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Godel, Atlas Books, W.W. Norton, New York, 2005.

In Chapter II Hilbert and the Formalists Goldstein gives further historical context. As a Platonist Gödel was reticent in the presence of the logical positivism of the Vienna Circle. She discusses Wittgenstein's impact and the impact of the formalists. Goldstein notes that the intuitionists were even more opposed to Platonism than Formalism.

Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ... Brouwer is the last name of different people. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a... Brouwer is the last name of different people. ... Brouwer is the last name of different people. ... Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician. ... Stephen Cole Kleene (January 5, 1909 – January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. ... In mathematics, constructive analysis is mathematical analysis done according to the principles of constructivist mathematics. ... Ian Stewart, FRS (b. ... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ... The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. ... Constance Bowman Reid is the author of several biographies of mathematicians and popular books about mathematics. ... John W. Dawson was Governor of Utah Territory in 1861. ... [...]I dont believe in natural science. ... Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein Rebecca Goldstein (née Newberger, born 1950) is an American novelist, philosopher and teacher. ... Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... Logical positivism is a school of philosophy that combines empiricism—the idea that observational evidence is indispensable for knowledge of the world — with a version of rationalism—the idea that our knowledge includes a component that is not derived from observation. ... Ludwig Wittgenstein (1889-1951), pictured here in 1930, made influential contributions to Logic and the philosophy of language, critically examining the task of conventional philosophy and its relation to the nature of language. ... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ... The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. ...

Secondary References

• A. A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]

A secondary reference for specialists: Markov opined that "The entire significance for mathematics of rendering more precise the concept of algorithm emerges, however, in connection with the problem of a constructive foundation for mathematics....[p. 3, italics added.] Markov believed that further applications of his work "merit a special book, which the author hopes to write in the future" (p. 3). Sadly, said work apparently never appeared.

Andrey Andreyevich Markov (Андрей Андреевич Марков) (June 14, 1856 N.S. _ July 20, 1922) was a Russian mathematician. ...

External links

• Ten Questions about Intuitionism