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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, 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.

Truth and proof

In classical mathematics, mathematical statements assert something about truth. Intuitionism takes the truth of a mathematical statement to be equivalent to its having been proved; what other criteria can there be for truth, an intuitionist would argue, if mathematical objects are merely mental constructions? This means that an intuitionist may not believe that a mathematical statement has the same meaning that a classical mathematician would.

For example, 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.

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.

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.

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

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 on these sad events 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 excellent commentary.

In the early twentieth century the battle was taken up Brouwer the intuitionist versus Hilbert the logicist — see van Heijenoort. Kurt Gödel the Platonist had his opinions (see various sources re Gödel) and even Alan Turing considers:

"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).

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

Branches of intuitionistic mathematics

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

See also

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 definitons 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 0720422396.
  • Hesseling, Dennis E. (2003). Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s, Birkhäuser. ISBN 3764365366.
  • 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 0720421039.
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.

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.

External links

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