# Mathematical Intuitionism

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## Intuitionism, Mathematical

a trend in the philosophy of mathematics that rejects the set-theoretic treatment of mathematics and considers intuition to be the only source of mathematics and the principal criterion of the rigor of its constructions.

The intuitionist view, which can be traced back to ancient mathematics, was shared by such scientists as K. F. Gauss, L. Kronecker, H. Poincare, H. Lebesgue, E. Borel, and H. Weyl. At the beginning of the 20th century, L. E. J. Brouwer advanced a comprehensive critique of classical mathematics and proposed a radical program for an intuitionist reconstruction of mathematics. The program, now called intuitionism (Brouwer himself used the term “neointuitionism”), developed amid a heated controversy with mathematical formalism against the background of a crisis brought about in the foundations of mathematics by antinomies (paradoxes) in set theory. Brouwer decisively rejected both belief in the actual existence of infinite sets and in the legitimacy of extending the laws of traditional logic developed for finite sets to the realm of infinite sets. From the intuitionist standpoint, mental constructions, which can be considered as such, “regardless of such questions about the nature of the constructed objects as whether these objects exist independently of our knowledge of them” (A. Heyting, the Netherlands), are the subject matter of mathematical investigation. Thus, mathematical statements are a form of information about completed constructions. The use of mental constructions requires a special logic, called intuitionist logic, which, in particular, does not accept the law of the excluded middle in anything like its full scope.

In a series of articles beginning in 1918, Brouwer and his followers systematically constructed the principal branches of intuitionist mathematics—set theory, mathematical analysis, topology, and geometry. Intuitionist mathematics is now (1970’s) a quite thoroughly worked out trend. As a result of the requirements of the intuitionist program for the foundations of mathematics, some branches of traditional mathematics have acquired a highly unusual form. This is due to the refusal to consider actually infinite sets as an object of study and to the insistence that every construction meet the requirement of effectiveness. Highly unique are the concept of a freely formed sequence (in different terminology, choice sequence)—the basic tool of intuitionism—and the related new treatment of the number continuum as a “medium of formation” of a sequence of partitioned rational intervals (in contrast to the traditional construction of the continuum out of separate points). In its simplest form, a freely-formed sequence is a function that transforms natural numbers into natural numbers such that any of its value can be effectively calculated. Precise studies have shown that there are several types of sequences depending on the amount of information known about the sequence.

In opposition to formalism and believing intuitition to be the principal criterion for the truth of constructions, Brouwer objected to attempts to formalize intuitionist mathematics and, in particular, intuitionist logic. But the “intuition” of intuitionism, independently of Brouwer’s and Weyl’s philosophical positions and views on it, in essence constitutes a clear intellectual persuasiveness of the simplest constructive processes that take place in people’s minds in the process of social development and education and, as such, is completely open to investigation by precise methods.

Significant advances were achieved in the study of intuitionist logic after its basic laws were precisely formulated in the form of calculi to which precise methods of mathematical logic could be applied. For example, we may mention the well-known interpretation of the intuitionist predicate calculus proposed by A. N. Kolmogorov, the embedment of classical formal arithmetic in intuitionist arithmetic (K. Godel), the proof of the independence of the logical connectives and the impossibility of representing the intuitionist predicate calculus in the form of a finite-valued logic (K. Godel), the theory of models for the intuitionist logic, and many other results that elucidate the importance and distinctive features of the intuitionist logic in comparison with classical logic, which could not in principle be obtained without a preliminary precise formulation. A precise formulation of the laws of intuitionist logic and of intuitionist arithmetic was proposed as early as the 1930’s by Heyting. A satisfactory construction of the theory of sequences of free formation and higher branches of intuitionist mathematics was not completed until the 1970’s by S. C. Kleene and other mathematicians.

Mathematical intuitionism is still being intensively developed. The stress placed by intuitionism on the effectiveness of its results agrees well with the computational tendency in modern mathematics and has drawn a great number of creative mathematicians to the study of intuitionist logic. In the USSR a group of mathematical logicians headed by A. A. Markov has undertaken the development of a constructive mathematics, a tendency that is close to mathematical intuitionism.

### REFERENCES

Weyl, H.*O filosofii matematiki*. Moscow-Leningrad, 1934. (Collection of articles translated from German.)

Heyting, A.

*Intuitsionizm*. Moscow, 1965. (Translated from English.)

Fraenkel, A. A., and Y. Bar-Hillel.

*Osnovaniia teorii mnozhestv*. Moscow, 1966. (Translated from English.)

A. G. DRAGALIN and B. A. KUSHNER