Intuitionism
Intuitionism is a philosophy of mathematics which regards the objects of mathematical discourse to be mental constructions based upon intuitively self-evident ideas. Thus, for the intuitionist, mathematical "objects" are purely mental and have no independent physical existence. Founded by the Dutch mathematician L. E. J. Brouwer (1881-1966), the intuitionst philosophy emerged as a reaction to Georg Cantor's theory of infinite sets and the application of standard logic to such sets. In particular, Brouwer required that all mathematical proof be "constructive." By this, he meant that in order to prove the existence of any mathematical object, one must provide the instructions for mentally constructing the object in a finite number of steps. Traditionally, mathematicians have been satisfied to show that a mathematical entity exists by denying its existence and showing that such a denial leads to a contradiction of some known mathematical truth. This method of proof, called indirect proof, is unacceptable to intuitionists because it provides no finite step-by-step method for constructing the entity in question. Indirect proof is based upon the "law of the excluded middle" from standard logic, which intuitionists reject when it is applied to infinite sets. The law of the excluded middle says that a statement S is either true or false. There is no middle ground, hence the term "excluded middle." Intuitionists, on the other hand, regard any unproved statement about infinite sets to be neither true nor false. The "intuition" that Brouwer and his followers claim to rely on is the intuition we have about the natural numbers. Here is an infinite set that can be constructed by our intuitive understanding of succession. The set of natural numbers can be built up by starting with the number 1, constructing its successor, 2, and then 3, and so on. Now, Brouwer claims, all the rest of what we regard as legitimate mathematics must come from this intuitive set of natural numbers. All the familiar concepts of mathematics must be built constructively from the natural numbers.
Paraphrasing the mathematician Leopold Kronecker (1823-1891), intuitionists seem to say that God created the natural numbers and all the rest of mathematics is the work of humans. While Brouwer regarded the set of natural numbers as a completed infinite set, there are intuitionists who do not believe in the concept of completed infinite sets. They believe only in the concept of "potential infinity," an idea that can be traced back to Aristotle. Other factions of intuitionism deny the existence of infinite sets altogether. To them, infinity is not an intuitive concept.
The appeal of intuitionism is that if we base our mathematics only upon those ideas which are intuitively obvious or self-evident, then we may hold to the conclusions of our mathematics with more certainty than if we use ideas about which we have some uneasiness. If all of our proofs are purely constructive, then the evidence for their truth is spelled out in a finite step-by-step sequence for all to see. It is not unreasonable to suggest that a completely constructive proof is more convincing than an indirect proof. Nevertheless, intuitionists make up a very small subset of working mathematicians. The reason for this is that to be a thoroughgoing intuitionist one must give up much of classical mathematics or reconstruct it from scratch. Constructivist proofs are longer and more difficult than proofs which allow the law of excluded middle and Cantor's theory of infinite sets. The more radical forms of intuitionism that deny the existence of any infinite sets, are even more crippling to the actual practice of mathematics. As a practical matter, most working mathematicians choose to ignore intuitionism even if they think that it has some merit philosophically. It is simply too much trouble to sacrifice the toolkit of classical mathematics for the sake of a little more certainty. The great German mathematician, David Hilbert (1862-1943) said that taking away the law of the excluded middle from the mathematician would be like forbidding the astronomer to use a telescope. This is probably the view of most modern mathematicians, even though, in the latter half of the 20th century, intuitionists have managed to revise much of classical mathematics using constructivist methods. One such example is the 1967 work Foundations of Constructive Analysis by Errett Bishop in which the author develops a large part of 20th century real analysis using intuitionist principles. It remains to be seen whether intuitionist mathematics will gain an equal footing or surpass classical mathematics in the 21st century. A case can be made that intuitionism is more compatible with the algorithmic approach of computer science than is traditional mathematics. If so, then perhaps intuitionism has not yet seen its heyday.
This is the complete article, containing 776 words
(approx. 3 pages at 300 words per page).
