Title:
DedekindianAbstraction and Its Philosophical Background
Abstract:
Dedekind is known to have made fundamentalcontributions to the modern axiomatization of number theory usingvarious abstraction procedures in a series of writings from1872-1888. However, his way of proceeding has been interpreted indifferent ways. Whereas some saw it as offering a clear example ofstructural or axiomatic definitions of abstract mathematical objects(regardless – and sometimes despite – his language of“abstraction” and “creation”), others deemed it to show apsychological understanding of abstraction conflating genetic withaxiomatic methods. This paper aims to gain insight into the notion ofabstraction at work in these writings by placing it in the contextnineteenth-century attempts to conceptualize ideal objects such asMoritz Wilhelm Drobisch’s and HermannLotze’s. Dedekind’s connections to nineteenth-century logic openthe door a non-psychologistic rendering of his procedure as firstproposed by the neo-Kantian philosopher Ernst Cassirer, and morerecently by Erich Reck and Audrey Yap. It will be suggested that thisline of interpretation is worth exploring further, not only becauseit sheds light on the philosophical background of Dedekind’snotion, but also because it brings out the premises for a specificversion of mathematical structuralism.