Dictionary Definition
intuitionism n : (philosophy) the doctrine that
knowledge is acquired primarily by intuition
User Contributed Dictionary
English
Noun
 An approach to mathematics/logic which avoids proof by contradiction, insisting on doing everything constructively
See also
Extensive Definition
 This article is about Intuitionism in mathematics and philosophical logic. For other uses, see Ethical intuitionism.
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, mathematics does
not consist of 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.
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 intuitionistic 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 nonexistence.
For the intuitionist, this is not valid; the refutation of the
nonexistence 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.
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
can construct, via
Gödel's incompleteness theorems, a mathematical statement that
can be neither proven nor disproved.
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 notnotP.
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 Antirealism
of Michael
Dummett.
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.
History of Intuitionism
Intuitionism's history can perhaps be traced to the nineteenth century. Cantor and his teacher Kronecker — a confirmed finitist — disagreed. Frege's effort to reduce all of mathematics to a logical formulation was a scientific breakthrough in the department of logic, and it greatly inspired the younger generation, including a youthful Bertrand Russell.But Frege himself counted it as failure when a
young Bertrand
Russell sent Frege a letter about his hotoffthepresses first
volume, outlining the famous paradox now known as Russell's
Paradox, that showed how one of Frege's rules of selfreference was
selfcontradictory. Frege, the story goes, plunged into depression
and did not publish the second and third volumes of his work as he
had planned. 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.
In the early twentieth century
Brouwer represented the intuitionist position and Hilbert the
formalist — see van
Heijenoort. Kurt
Gödel offered opinions referred to as Platonist. (see various
sources re Gödel). Alan Turing
considers: "nonconstructive 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) Later, Kleene brought forth
a more rational consideration of intuitionism in his Introduction
to Metamathematics (1952). 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
nowclassic Bertrand Russell and the Origins of the SetTheoretic
Paradoxes.
Contributors to intuitionism
Branches of intuitionistic mathematics
See also
Further reading
 "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, SpringerVerlag, New York, 1994.
 Martin Davis (ed.) (1965), The Undecidable, Raven Press, Hewlett, NY. Compilation of original papers by Gödel, Church, Kleene, Turing, Rosser, and Post.
 Engines of Logic: Mathematicians and the origin of the Computer
 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.
 van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 18791931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. The following papers appear in van Heijenoort:

 L.E.J. 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 principle of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]
 L.E.J. 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.
 L.E.J. 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".
 Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s
 Arend Heyting: Intuitionism: An Introduction
 Introduction to MetaMathematics
 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.
 Stephen Cole Kleene and Richard Eugene Vesley, The Foundations of Intuistionistic Mathematics, NorthHolland Publishing Co. Amsterdam, 1965. The lead sentence tells it all "The constructive tendency in mathematics...". A text for specialists, but written in Kleene's wonderfullyclear style.
 Constance Reid, Hilbert, Copernicus  SpringerVerlag, 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.
 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 5158 in Section 4 Many Valued Logics, Modal Logics, Intuitionism; pages 6973 Chapter III The Logic of Propostional Functions Section 1 Informal Introduction; and p. 146151 Section 7 the Axiom of Choice.
Secondary References
 A. A. Markov (1954) Theory of algorithms. [Translated by Jacques J. SchorrKon 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 algorifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 6051085.]
 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.
External links
intuitionism in Czech: Intuicionistická
logika
intuitionism in German: Intuitionismus
intuitionism in Spanish: Intuicionismo
intuitionism in Esperanto: Intuiciismo
intuitionism in Croatian: Intuicionizam
intuitionism in Italian: Intuizionismo
intuitionism in Dutch: Intuïtionisme
intuitionism in Japanese: 数学的直観主義
intuitionism in Polish: Intuicjonizm
(matematyka)
intuitionism in Portuguese: Intuicionismo
intuitionism in Russian: Интуиционизм
intuitionism in Turkish: Sezgici Matematik
intuitionism in Chinese:
数学直觉主义