Course Description

Structuralist views have become a widely discussed topic in the philosophy of mathe­matics. Richard Dedekind (1831-1916) is often mentioned as a precursor of, and an influence on, current proponents of structuralism; arguably, his position also constitutes an attractive alternative to theirs. In addition, Dedekind made several technical contribu­tions to the foundations of mathematics that have to be taken into account by any position in the philosophy of mathematics, structuralist or not. Dedekind's works are thus relevant to contemporary research both in a historical and a systematic sense. This class has three goals: First, to clarify the sense in which Dedekind held structuralist views, by studying his contributions to the foundations of arithmetic, analysis, and set theory in some detail; second, to discuss the strengths and weaknesses of his position, relative to contemporary alternatives; and third, to explore the relationship of his foundational contributions with his other mathematical works, another topics that has started to attract serious attention in the literature recently. Along the way a number of issues central to the history and philo­sophy of mathematics, also beyond Dedekind, will be discussed.

Background knowledge in three areas will be helpful: set theory (naïve set theory is enough); number theory (basic notions and techniques, especially in algebraic number theory); the philosophy of mathematics (logical and metaphysical issues in arithmetic, analysis, and set theory). However, none of this is a strict prerequisite; the material will be discussed in such a way that the relevant concepts and techniques are introduced as we go along. Consequently, the first three quarters of the class can be viewed as a general introduction to the philosophy of mathematics, presented from a historical point of view.

 
(1) R. Dedekind: Essays on the Theory of Numbers, W. Beman, ed. and ts., Dover, 1963.

(2) R. Dedekind: Theory of Algebraic Integers, J. Stillwell, ed. and ts., Cambridge Uni­versity Press, 1996.

(3) A selection of secondary literature, in the form of Xerox copies (see the syllabus).