From Objects to Structures
From antiquity to the nineteenth century and even up to now, two theses are being among the most debated subjects in philosophy of mathematics:
This paper draws on two talks given at the AlPS-conference at A Coruna and at the University of Toulouse.
The work was carried out as part of the ANR/DFG project Mathematics: Objectivity by Representation.G. Heinzmann (s)
Universite de Lorraine/CNRS & Archives Henri-Poincare, Nancy, France
e-mail: gerhard.heinzmann@univ-lorraine.fr
© Springer International Publishing AG 2017 385
E. Agazzi (ed.), Varieties of Scientific Realism,
DOI 10.1007/978-3-319-51608-0_21
(a) According to the Aristotelian tradition, mathematical objects such as numbers, quantities and figures, are thought of as entities belonging to different categories.
(b) Mathematical objects are extra-linguistic entities existing independently from our representations of them in an abstract world. They are conceived by analogy with the physical world and designated by singular terms of the mathematical language.
Although the Aristotelian thesis and the ontological ‘Platonism’ were countered by nominalism and early tendencies of algebraic formalization, they become more problematic when mathematicians like Niels Abel thought the relations before the relata or, like Hermann Hankel, maintained that mathematics is a pure theory of forms whose purpose is not that of treating of quantities or combinations of numbers (see: Bourbaki 1968, 317).
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