Intuitionism


 * 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

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

Branches of intuitionistic mathematics

 * Intuitionistic logic
 * Intuitionistic arithmetic
 * Intuitionistic type theory
 * Intuitionistic set theory
 * Intuitionistic analysis

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.