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Scepticism in mathematics over the new set theory of Georg Cantor (1845 - 1918) (and in particular his work on infinity) led to the formation of the school of mathematics known as intuitionism in the early years of the 20th century. The main thesis of the school was that the basis of mathematics should be in thought rather than in logic or symbols, and in particular, the basic intuitions of the construction of an infinite series of numbers. The chief proponent of this idea was , Luitzen Egbertus Jan Brouwer (1881 - 1966). Intuitionists felt that the idea of the infinite in the work of Cantor was counter-intuitive, and that it should therefore be avoided. Their main argument was that in order to meaningfully assert the existence of something (for example, a number satisfying certain properties), it is necessary to give a definite method of (at least theoretically) constructing this object. They rejected the ideas of formalism and logicism (which were mainly to do with the language in which mathematical thought should be expressed), to say that mathematical concepts have an existence independent of the language used to express them, and therefore that the only important property a language should have is that it should express mathematical ideas clearly.
They argued, to counter Cantor\'s set theory, that it is never actually necessary to look at the whole of an infinite set at once; only finite parts of it are used at any one time. The set of numbers, for example, is only a ‘potential infinity’. The followers of this school attempted to show the results that other mathematicians proved using only finite and constructive methods (that is, by direct construction); many of their proofs were forced to be much longer and more difficult to follow. Intuitionism tended always to follow behind the mainstream of mathematics, rather than to produce its own results. In the end, intuitionism perished under its own weight as mathematicians turned to the increasingly easier methods of mainstream mathematics; professional mathematicians became more confident about handling infinite sets, and so very few mathematicians still have any philosophical or intuitive quarrels with the idea of infinite sets. SMcL |
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