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 ?
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.
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.
and] also had a contribution to make to the philosophical tradition in subjecting Aristotelian physics, and the ontology whose vehicle it was, to an orderly criticism in aid of a mathematical formalism that took its inspiration from Plato" (p.
He has sought to primarily present only simple theoretical ideas and phenomenological models that are well supported by experimental data without going too deep into the underlying mathematical formalism, although he does include a chapter devoted to quantum chromodynamics and also includes discussion of recent ideas concerning the quark-gluon plasma and predictions concerning the results of collisions planned at the Large Hadron Collider in Europe.
The lectures went on telling about mathematical formalism and abstract set theory starting with Eider's diagrams or circles from 1761 (Sandifer 2007) and George Boole's logical system (Boolean algebra) from 1854 (Gowers 2008, Wolff 1963), then Georg Cantor's theory of sets and a little about countable sets and transfinite cardinals (Gowers 2008), and after this ending up with Bertrand Russell's famous paradox from 1904 (Gowers 2008, Olesen (13) 2007).
The traditional approaches to construction project modeling, such as ACDs, however, rely on a representation that lacks an underlying mathematical formalism.
At the turn of the twentieth century, German physicist Max Planck proposed a mathematical formalism involving the notion that continuous electromagnetic waves were actually tiny, individual particles.
The aim of this paper was to develop the mathematical formalism that allows later study of nonlinear control systems on time scales.

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