calculus

1. a branch of mathematics, developed independently by Newton and Leibniz. Both differential calculus and integral calculus are concerned with the effect on a function of an infinitesimal change in the independent variable as it tends to zero.

2. any mathematical system of calculation involving the use of symbols

3. Logic an uninterpreted formal system

4. Pathol a stonelike concretion of minerals and salts found in ducts or hollow organs of the body

Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

calculus

[′kal·kyə·ləs]

(anatomy)

A small and cuplike structure.

(mathematics)

The branch of mathematics dealing with differentiation and integration and related topics.

(pathology)

An abnormal, solid concretion of minerals and salts formed around organic materials and found chiefly in ducts, hollow organs, and cysts.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Calculus

a formal apparatus, based on clearly formulated rules, for operating with symbols of a specified type, which permits an exhaustively exact description of a certain class of problems as well as solution algorithms for certain subclasses of this class. The subclasses in question coincide with the whole class only in the case of the simplest calculi. Examples of calculi are the set of arithmetic rules for operating with numbers (that is, numerical symbols), the literal calculus of elementary algebra, differential calculus, integral calculus, the calculus of variations, and other branches of mathematical analysis and the theory of functions.

Despite its early origin, the term “calculus” was used in mathematics without a rigorous general definition until very recently. With the development of mathematical logic a demand arose for a general theory of calculus as well as for the refinement of the concept of “calculus” itself, which underwent a more systematic formalization. In most cases, however, the following conception of a calculus (originating from D. Hilbert) proves to be sufficient. Consider a certain alphabet (generally speaking, infinite, although also possibly given by means of a finite number of symbols), whose elements, called letters, are used to construct formulas of the calculus under consideration (sometimes called words or expressions) with the help of clearly stated formation rules. Some of these (“well-formed”) formulas are declared to be axioms, and from these, with the help of transformation rules (or rules of deduction), new formulas are “deduced,” which are called theorems of the given calculus. Sometimes the term “calculus” is applied only to the “dictionary” (“expression”) part of the structure described, and it is said that joining the “deductive” part to it (that is, adding both the rules and axioms of formation to the rules of deduction and to the alphabet) produces a formal system. Besides, these terms are often also considered synonymous (and the terms “logistic system,” “formal theory,” “formalism,” and many others are also used as synonyms for this). If such a noninterpreted (“meaningless”) calculus is juxtaposed to a certain interpretation (or, as is said, a purely syntactic treatment is supplemented by semantics), then a formalized language is obtained. The representation of substantive logical (and logico-mathematical) theories in the form of formalized languages is a characteristic feature of mathematical logic.

REFERENCES

Kleene, S. C. Vvedenie v metamatematiku, sections 14–20. Moscow, 1957. (Translated from English.)
Markov, A. A. “Teoriia algorifmov.” Moscow-Leningrad, 1954. (Tr. Matematicheskogo in-ta im. V. A. Steklova, vol. 42.)
Curry, H. B. Osnovaniia matemalicheskoi logiki, chap. 2. Moscow, 1969. (Translated from English.)
Matematicheskaia teoriia logicheskogo vyvoda.Collection of translations. Edited by A. V. Idel’son and G. E. Mints. Moscow, 1967.
Logicheskie i logiko-matematicheskie ischisleniia, 1.Collection of works. Edited by V. P. Orevkov. Leningrad, 1968.

IU. A. GASTEV