
05495465
j
2009e.00371
Griesel, Heinz
Reform of the construction of the number system with reference to Gottlob Frege.
ZDM, Int. J. Math. Educ. 39, No. 12, 3138 (2007).
2007
Springer, Berlin/Heidelberg
EN
F50
F10
E20
D20
number concepts
construction of the system of real numbers
foundations of mathematics
mathematics and philosophy
ontological commitments
H.G. Steiner
doi:10.1007/s1185800600032
Summary: Due to missing ontological commitments Frege rejected Hilbert's Fundamentals of Geometry as well as the construction of the system of real numbers by Dedekind and Cantor. Almost all of school mathematics is ontologically committed. Therefore, H.G. Steiner considered Frege's viewpoint of mathematics fundamentals, refined by Tarski's semantics, as suitable for math education. Frege committed numbers ontologically by using measurement to define numbers. He invented the concept of quantitative domain (Gr\"o{\ss}engebiet), which  it is now known by reconstruction of that concept by the NewFregean Movement  agrees with the concept of quantity domain (Gr\"o{\ss}enbereich) as established in the German reform of the applicationoriented construction of the system of real numbers. Concepts of quantity (ratioscale) and intervalscale in comparative measurement theory  going beyond Frege  show the way how the negative numbers can be ontologically committed and the operations of addition and multiplication can be included. In this work it is shown how Frege's viewpoint of mathematics fundamentals, as propagated by H.G. Steiner, can be better implemented in the current construction of the system of real numbers in school.