intuitionist adj : of or relating to intuitionism
- A person who studies intuitionistic mathematics
- 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 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 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 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 IntuitionismIntuitionism'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 hot-off-the-presses first volume, outlining the famous paradox now known as Russell's Paradox, that showed how one of Frege's rules of self-reference was self-contradictory. 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: "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) Later, 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.
Branches of intuitionistic mathematics
- W. S. Anglin, Mathematics: A Concise history and Philosophy, Springer-Verlag, 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, 1879-1931, 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 Meta-Mathematics
- 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.
- 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.
- 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.
- 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 algorifmov. [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.
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