Domination, Discretization, and Scalarization

This is an overview of a few possibilities that are open by model theory in applied mathematics. The most attention is paid to the present state and frontiers of the Cauchy method of majorants, approximation of operator equations with finite-dimensional analogs, and the Lagrange multiplier principle in multiobjective decision making. DOI: 10.1134/S1990478909010116 The union of functional analysis and applied mathematics celebrates its sixtieth anniversary this year. The present article focuses on the trends of interaction between model theory and the methods of domination, discretization, and scalarization. 1. PURE AND APPLIED MATHEMATICS Provable counting is the art of calculus which is mathematics in modern parlance. Mathematics exists as a science more than two and a half millennia, and we can never mixed it with history or chemistry. In this respect our views of what is mathematics are independent of time. The objects of mathematics are the quantitative forms of human reasoning. Mathematics functions as the science of convincing calculations. Once-demonstrated, the facts of mathematics will never vanish. Of course, mathematics renews itself constantly, while the stock increases of mathematical notions and constructions and the understanding changes of the rigor and technologies of proof and demonstration. The frontier we draw between pure and applied mathematics is also time-dependent. Francis Bacon wrote in his celebrated book “The Advancement of Learning”: The Mathematics are either pure or mixed. To the Pure Mathematics are those sciences belonging which handle quantity determinate, merely severed from any axioms of natural philosophy; and these are two, Geometry and Arithmetic; the one handling quantity continued, and the other dissevered. Mixed hath for subject some axioms or parts of natural philosophy, and considereth quantity determined, as it is auxiliary and incident unto them. For many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the mathematics; of which sort are perspective, music, astronomy, cosmography, architecture, enginery, and divers others. In the Mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the Pure Mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the Mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended. . . . And as for the Mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed. E-mail: [email protected] 1 Cp. [1, 2]. 2 The complete title was as follows: “The tvvoo bookes of Francis Bacon, of the proficience and aduancement of learning, diuine and humane. To the King. At London: Printed for Henrie Tomes, 1605.”