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# Examples vs Definitions

One tradition in mathematics is to make a definition and then give examples. I’m going to argue that this is the wrong way round.

Consider how new mathematics is developed. A definition is introduced to help put a name to a concept that has been useful — for example, a group. We don’t often hear of an important definition that was made on a whim, explored and found to be useful. Are there any? (Suggestions in the comments, please!) Thus examples precede definitions and are the result of seeing a unifying feature.

In lectures we often give the definition, in other words the unifying feature, and then show the examples and the consequences following from it. In doing so we miss providing the students with a motivation for the definition. Certainly, it can be argued that there isn’t time to motivate everything (and there isn’t — there is never enough time to tell the students everything we think that they should know!). It can also be argued that mathematics research is a messy business with lots of dead ends and false starts. By putting definitions first we are tidying up the mess.

That’s true but next time, if you don’t already do so, maybe try as an experiment to put the definition after the examples. This was successful for me when teaching topology. By giving a diagram with lots of examples at the start of the very first lecture I was able to clearly motivate definitions such as homotopy and homeomorphism.

Let’s return to the definition of group. Considering examples first means we can have non-commutative examples such as matrix multiplication. When write try to write a definition that brings together all the examples and extracts their important features it is then clearly a choice whether or not we include commutativity of the binary operation in our axioms. By considering that it would be good to include matrices we motivate the exclusion of commutativity from the definition.