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Formalism

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A program for the foundations of mathematics initiated by D. Hilbert. The aim of this program was to prove the consistency of mathematics by precise mathematical means. Hilbert's program envisaged making precise the concept of a proof, so that these latter could become the object of a mathematical theory — proof theory.

In order to make it possible to study proofs precisely, one gives them a single precisely-defined form. This is carried out with the aid of a formalization (cf. Formalization method) of the theory: the assertions of the theory are changed to finite sequences of definite signs, and the logical methods of inference — to formal rules for generating new formally-represented statements from those already proved. Thus, a mathematical theory is turned into a formal system.

Despite the fact that Hilbert's program did not achieve its ultimate goal (it could not because of the Gödel incompleteness theorem, [3]), the research done within this program has had great significance for the development of many areas of mathematical logic.

The term "formalism" is often used as a synonym for a formal system, and generally for the notion of a calculus enabling one to change operations with objects to operations with symbols corresponding to them. In the philosophy of mathematics formalism means a view of the nature of mathematics according to which mathematics is characterized by its methods rather than by the objects it studies; its objects have no meaning other than the one derived from their formal definition (a possible "underlying nature" is regarded as irrelevant).

References

[1] D. Hilbert, "Grundlagen der Geometrie" , Springer (1913)
[2] G. Gentzen, "Collected papers" , North-Holland (1969)
[3] K. Gödel, "Ueber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" Monatsh. Math. Physik , 38 (1931) pp. 178–198
[4] P.S. Novikov, "Elements of mathematical logic" , Oliver & Boyd (1964) (Translated from Russian)
[5] S.C. Kleene, "Mathematical logic" , Wiley (1967)
[6] H.B. Curry, "Foundations of mathematical logic" , McGraw-Hill (1963)
How to Cite This Entry:
Formalism. V.E. Plisko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Formalism&oldid=16237
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098