Semantic Models in the History of Mathematics

Abstract

It is generally assumed that semantic models did not appear until the mathematics of the 19th century (models of algebraic structures, models of non-Euclidean geometries) and the theory of models appeared only in the 20th century (Tarski). This claim is a simplification, because it can be shown that Descartes used geometrical and arithmetic models of his algebra as early as the 17th century. For Eudoxos's axiomatic theory of sizes it was possible to show mathematical models as early as the 4th century before the birth of Christ. Since the beginning of epistemology and the philosophy of mathematics (Aristotle), some assertions from the area of the theory of models have also been expressed.