
06619300
j
2016e.00602
Anatriello, Giuseppina
Tortoriello, Francesco Saverio
Vincenzi, Giovanni
On an assumption of geometric foundation of numbers.
Int. J. Math. Educ. Sci. Technol. 47, No. 3, 395407 (2016).
2016
Taylor \& Francis, Abingdon, Oxfordshire
EN
F50
G40
Euclidean geometries
geometric constructions
plane geometry
complex numbers
Pythagorean theorem
doi:10.1080/0020739X.2015.1078004
Summary: In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the {\it Elements} of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in {\it Ausdehnungslehre} in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.