Intuitionism (mathematical)
Intuitionism (mathematical)
The name given to the school (of mathematics) founded by L. E. J. Brouwer (q. v.) and represented also by Hermann Weyl, Hans Freudenthal, Arend Heyting, and others. In some respects a historical forerunner of intuitionism is the mathematician Leopold Kronecker (1823-1891). Views related to intuitionism (but usually not including the rejection of the law of excluded middle) have been expressed by many recent or contemporary mathematicians, among whom are J. Richard, Th. Skolem, and the French semi-intuitionists — as Heyting calls them — E. Borel, H. Lebesgue, R. Baire, N. Lusin. (Lusin is Russian but has been closely associated with the French school.)
For the account given by Brouwerian intuitionism of the nature of mathematics, and the asserted priority of mathematics to logic and philosophy, see the article Mathematics. This account, with its reliance on the intuition of ordinary thinking and on the immediate evidence of mathematical concepts and inferences, and with its insistence on intuitively understandable construction as the only method for mathematical existence proofs, leads to a rejection of certain methods and assumptions of classical mathematics. In consequence, certain parts of classical mathematics have to be abandoned and others have to be reconstructed in different and often more complicated fashion.
Rejected in particular by intuitionism are
the use of impredicative definition (q. v.);
the assumption that all things satisfying a given condition can be united into a set and this set then treated as an individual thing — or even the weakened form of this assumption which is found in Zermelo’s Aussonderungsaxiom or axiom of subset formation (see logic, formal, 9);
the law of excluded middle as applied to propositions whose expression lequires a quantifier for which the variable involved has an infinite range.
As an example of the rejection of the law of excluded middle, consider the proposition, “Either every even number greater than 2 can be expressed as the sum of two prime numbers or else not every even number greater than 2 can be expressed as the sum of two prime numbers.” This proposition is intuitionistically unacceptable, because there are infinitely many even numbers greater than 2 and it is impossible to try them all one by one and decide of each whether or not it is the sum of two prime numbers. An intuitionist would accept the disjunction only after a proof had been given of one or other of the two disjoined propositions — and in the present state of mathematical knowledge it is not certain that this can be done (it is not certain that the mathematical problem involved is solvable). If, however, we replace “greater than 2” by “greater than 2 and less than 1,000,000,000,” the resulting disjunction becomes intuitionistically acceptable, since the number of numbers involved is then finite. The intuitionistic rejection of the law of excluded middle is not to be understood as an assertion of the negation of the law of excluded middle; on the contrary, Brouwer asserts the negation of the negation of the law of excluded middle, i.e., ~~[p ? ~p]. Still less is the intuitionistic rejection of the law of excluded middle to be understood as the assertion of the existence of a third truth-value intermediate between truth and falsehood.
The rejection of the law of excluded middle carries with it the rejection of various other laws of the classical propositional calculus and functional calculus of first order, including the law of double negation (and hence the method of indirect proof). In general the double negation of a proposition is weaker than the proposition itself; but the triple negation of a proposition is equivalent to its single negation. Noteworthy also is the rejection of ~(x)F(x) ? (Ex)~F(x); but the reverse implication is valid. (The sign ? here does not denote material implication, but is a distinct primitive symbol of implication.) — A.C.
L. E. J. Brouwer,
De onbetrnuwhaarheid der logische principes, Tijdsdschrift voor Wijsbegeerte, vol 2 (1908), pp 152-158; reprinted in Brouwer’s Wiskunde. Waarheid, Werketijkhetd. Groningen, 1919.
L. E. J. Brouwer,
Intuitionism and formalism. English translation by A. Dresden. Bulletin of the American Mathematical Society, vol. 20 (1913), pp 81-96.
H. Weyl,
Consistency in mathematics. The Rice Institute Pamphlet, vol 16 (1929), pp 245-265.
A. Heyting,
Mathematische Grundlagen-forschung, Intuitionismus, Beweistheorie, Berlin, 1934.