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Préhistoire de la géométrie algébrique réelle: De Descartes à Tarski. (Prehistory of real algebraic geometry: From Descartes to Tarski). (French) Zbl 0733.14022
Cah. Sémin. Hist. Math., 2. Sér. 1, 1-17 (1991).
The author points out that the criteria for existence of roots or a polynomial in an interval, culminating in Sturm’s theorem, inspired A. Tarski to generalize them to a general logical process, the “elimination of quantifiers”, which establishes the equivalence of a proposition implying quantifiers (in Sturm’s case, the existence of a real number satisfying a property) with a proposition containing no quantifiers (in Sturm’s case, a system of inequalities). She then shows how Tarski went on to consider general real closed fields instead of \({\mathbb{R}}\), and formulated his “transfer principle” according to which any proposition about real numbers, formulated in the language of the first order, is valid in any real closed field. Finally, she shows that the concept of “elementary set” introduced by Tarski is the first example of the semi- algebraic sets, which is the fundamental notion in real algebraic geometry, now actively investigated by many mathematicians.

14P10 Semialgebraic sets and related spaces
01A60 History of mathematics in the 20th century
14-03 History of algebraic geometry
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
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