One of the principal trends in the foundations of mathematics whose representatives, followers of D. Hilbert, believe that every branch of mathematics can (and, at a sufficiently advanced stage in its construction, should) be completely formalized, that is, set forth in the form of a calculus (formal system) developed according to certain well-defined rules. Furthermore, the legitimacy of the existence and study of a given branch of mathematics should be based exclusively on its consistency and not on the possibility of its interpretation in terms of any reality external to it. These assumptions, particularly the second, have far-reaching consequences only for those branches of mathematics that involve some form of the concept of infinity.
Systematic formulation of the concept of mathematical formalism arose directly as a reaction to the paradoxes discovered within set theory, which studies the concept of infinity. Briefly, mathematical formalism asserts that “finitary” (that is, meaningfully interpretable, without the use of the concept of infinity) conclusions from a mathematical theory have meaningful validity only if the consistency of this formalized theory is proved by finitary methods.
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Kleene, S. K. Vvedenie v metamatematiku. (With bibliography.) Moscow, 1957. Chapters 8, 14, 15, 42, 79. (Translated from English.)
Novikov, P. S. Elementy matematicheskoi logiki. (Introduction.) Moscow, 1959.
Church, A. Vvedenie v metamatematicheskuiu logiku, vol 1. (Introduction.) Moscow, 1960. (Translated from English.)
Gentzen, G. “Neprotivorechivost’ chistoi teorii chisel.” In Matematicheskaia teoriia logicheskogo vyvoda. Moscow, 1967. Pages 77-163. (Translated from German.)
Curry, H. B. Osnovaniia matematicheskoi logiki. Moscow, 1969. Chapters 1-4. (Translated from English.)