Mathematical Formalism

Formalism, Mathematical


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.


Hilbert, D. Osnovaniia geometrii. Moscow-Leningrad, 1948. Appendices 6-10. (Translated from German.)
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.)
References in periodicals archive ?
(iii) students translated the motion into mathematical formalism by converting their life-sized sketch into velocity and acceleration vectors at a number of points along the path;
The birth and development of quantum mechanics in the 1920's and 1930's brought about various problems that questioned our comprehension of nature, and specifically the interpretation of high mathematical formalism that seemingly leads to a probabilistic theory.
The paper is arranged as follows: in Section 2, a mathematical formalism of an anisotropic DE universe is presented along with the relevant physical parameters.
Koch, "Petri nets - A mathematical formalism to analyze chemical reaction networks," Molecular Informatics, vol.
After a brief scene-setting summary of the ideas of the great classical economists Adam Smith, John Stuart Mill and Karl Marx, Bookstaber's account starts in earnest with the rise of neoclassical economics and mathematical formalism in the late nineteenth century.
The realisation of the project will validate and enrich the mathematical formalism and also promises a next generation of physical effects related to the conjunction of both gravity and quantum noncommutativy, which will stimulate original and creative approaches to quantum gravity across several EU institutes.
Chapter Eight shifts gears again to focus on mathematical notation, and the idea that progress in mathematics can sometimes be chalked up to revolutions in mathematical formalism. One striking example of this, discussed by Colyvan, is the shift from the Roman numerals to the Arabic numerals.
In this paper, we have developed a mathematical formalism to predict speckle-like interferometric out-of-focus patterns created by ellipsoidal rough scattering objects.
In this book, the authors provide a novel general relativistic theory of the internal constitution of liquid stars, using a mathematical formalism first introduced by Abraham Zelmanov for calculating physically observable quantities in a four-dimensional pseudo-Riemannian space, known as the theory of chronometric invariants.
Though written primarily for the scientific-minded layman there are two short appendices that provide a more detailed and mathematical formalism than the main text.
Hilbert's argument grounding Spitzer's past-time proof is clearly a mathematical demonstration, yet its application to finite cosmological structures is a sort of mixed demonstration that relies on the use of a mathematical formalism as analogy for a material reality.
The facts are represented using the conventional knowledge representation and the adequate mathematical formalism.

Full browser ?