The History of Algebra’s Impact on the Philosophy of Mathematics

Abstract

Structuralism in the philosophy of mathematics encompasses a range of views, many of which see structures, such as the natural numbers, as the proper objects of mathematics, rather than objects like individual numbers. This position is relatively recent—many see a paper by Paul Benacerraf in 1965 as one of its earliest articulations, though others have written about Richard Dedekind as an earlier precursor. This paper will consider how Emmy Noether, extending Dedekind’s work and general approach, contributed to a transition in mathematics enabling a fuller range of structuralist positions. Her work in ideal theory is a perfect illustration of the kind of conceptual step that permits seeing structures themselves as objects, which is required for certain contemporary views even to make sense.