35 - Learning to Move with Dedekind

Summary

Es steht schon bei Dedekind.

— Emmy Noether

The history of mathematics sometimes calls our attention to intellectual hurdles that our students must face, showing that ideas and conceptual moves that have become second nature to us are in fact quite daring and difficult to learn. This article focuses on a particular conceptual move, which we call “the Dedekind move” because it was so characteristic of Richard Dedekind's work. Briefly put, the idea is to define a mathematical object as a set of other mathematical objects. We then treat the whole set as a single thing, and do our best to forget its original plural nature.

Students typically meet this idea for the first time in an “introduction to abstraction” class, when they learn about equivalence classes. It really comes into its own, however, when the quotient construction is introduced in abstract algebra. This is a notorious stumbling block for students. A little history can help us understand why, and suggests some ideas for helping students over the hump.

Historical Background: What Dedekind Did

Suppose we are confronted with the need to come up with a definition of some mathematical entity. There are many ways to go about this. Some mathematical definitions, for example, are entirely functional: we explain what it does, and ignore completely the issue of what it is. But this is rarely completely satisfying. How do we even know that the object in question exists? Some construction is usually wanted.

Richard Dedekind was faced with this situation more than once. His approach was fairly consistent.