Definitions come from choices

Posted by Kevin Houston on Mar 9, 2019 in How To Think Like A Mathematician, Mathematical thinking, Mathematics education | 0 comments

In yesterday’s post I touched on the need to stress the importance of definitions and gave a way to help students learn them. Today’s post is about something more subtle. It is often said that we can define a word to mean whatever we want it to mean. Of course, that does not mean we can redefine continuous function in our elementary analysis courses.

Instead, it is important to let students know that a definition arises from choices. Any definition was likely a choice amongst a number of possibilities. The first place this appears is in the definition of natural number. Should we include as a natural number? Some say yes, some say no and I tell my students that they will meet different lecturers with different ideas as to whether is natural or not.*

One of the first definitions met by students is that of prime number and there is an argument to be had here on what a good definition is. One has the standard discussion about why counts as a prime — it just makes statements nicer. (See One is the loneliest number for a children’s book author defending their decision to have as a prime.)

Leaving aside that discussion we can argue about good taste in definitions. One can define a prime as “a natural number greater than that is divisible only by and itself” or as “A natural number with only two factors”. For the latter, is automatically excluded as a prime since it has only one factor — itself. One can then argue as to which is the better definition. They’re equivalent of course but is one more elegant than the other?

There are plenty of examples to get students talking about the importance of agreeing on a definition. Does a cylinder count as a prism? Is an even or odd number? Why is ? (If we define to be , then it is not clear what should be. Students are often told to take it as as that makes for better formulae. It’s clearly , however, if we define as the number of ways of writing the set with elements.)

Polyhedra are great for generating discussion of definitions. What should a polyhedron be? This can then lead on to the idea that we choose the definition that gives us “good looking” theorems. See Polyhedra by Cromwell or Proofs and Refutations by Lakatos for excellent resources on discussing definitions.

* My opinion is that is not a natural number as it took a long time to be accepted as a number in its own right. That does not satisfy the concept of “natural” in my opinion. And if natural numbers are the counting numbers (think definition of countable), then we certainly don’t start counting with zero. Nobody says “Zero, one, two, three, …”.

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