All mathematics begins as measurement

In order to thrive when doing mathematics we need to understand what it is that we are doing. The title of this post is a place to begin. When you read it, do you agree or disagree? Are you mentally exploring the implications of the statement? I encourage you to. My purpose is to find the beginning of a thread that can be woven into a tapestry.

The title is an answer to the question, Where does mathematics come from? Often the answers have revolved around whether mathematics is discovered or invented. On the side of discovery is the “unreasonable effectiveness” of mathematics in explaining the universe. The case for invention comes from the observation that mathematics is done in our minds. Certainly the language and symbols that we associate with mathematics are creations of the human mind. The relationships they describe though, seem to function independent of our intervention. It is most likely that such a dichotomous argument is spurious and there are elements of both views that are true.

What I have in mind with the current topic though is more about where our interaction with what we now know as mathematics began. It is known that humans are not the only species who can count, or even perform arithmetic. Bees construct hives made up of hexagons.

Measurement is the act of comparing quantities. When the quantity is measured in discrete amounts, sheep in a fold, or grapes in a bunch, we call it counting. When the quantity is continuous like length or volume we simply say we measure it. In both instances the reference unit, the one, needs to be clearly defined and accepted.

Geometry literally means ‘to measure the earth’ and this takes in length and area but also angle and shape. We distinguish one shape from another by their properties, one of the most basic being the number of sides a figure has. Sophistication is added when we further distinguish types of the same shape from other examples. Triangles can be equilateral, a sides the same length, scalene, all sides of different length, or isosceles, two sides of equal length and the third being different. We determine the type of triangle by both counting and comparing measurements.

Chance and statistics, the third region of mathematics taught in schools around the globe, has its roots in fractions of the whole, which instead of measuring out from the unit, measures inward, where the one is the entirety of some population or amount, and what is being measured is some part of that entirety that conforms to a particular characteristic.

From these beginnings grow the rest of mathematical construction.