In the beginning there was… something. That something was you. The concept of the number one is integral to your existence. You are one thing. When you first learned of one, you didn’t have language to describe it, it just was.
Any attempt at defining one seems destined to either involve concepts that are far more complex than what we are after or fall into a circular trap. My Collins Canadian English Dictionary defines “one” as follows:
one [wun] a. lowest cardinal number; single; united; only, without others; identical –n. number or figure 1; unity; single specimen
I feel like I knew “one” before I knew what cardinal numbers were. And defining single without “one” seems futile. The “without others” definition is interesting. Can you have a concept of “one” without knowing the possibility of other numbers?
It might be illuminating to consider the case of the Pirahã people in the Amazon. It is tempting to call their culture “primitive,” though that is merely us enforcing our own value system on them. According to Dan Everett they have no number words at all. As can be seen in the following video, the Pirahã operate using only words for few, and many.
So maybe my assertions that one is a pre-language concept aren’t true. Maybe one is a concept that we had to learn. We will be coming back to the Pirahã many times at this blog, as they have a fascinating culture that has developed without anything that outsiders would call Mathematics. (The fact that such people can lead full and interesting lives gives this mathematician a small sense of existential dread.)
Here is an activity for you or your students:
Write down a list of things that you do in a day that don’t make use of the concept of one or numbers in general. In a classroom setting this should lead to some interesting discussion. This amounts to putting the common assertion that “math is everywhere” to the test.