Nonstandard Models and Optimization

This is an overview of a few possibilities that are open by model theory in applied mathematics. 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. 1. Agenda The union of functional analysis and applied mathematics celebrates its sixtieth anniversary this year. This talk focuses on the trends of interaction between model theory and the methods of domination, discretization, and scalarization. 2. The Art of Calculus 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 well never vanish. Of course, mathematics renews itself constantly, while the stock increases of mathematical notions and construction 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.