In the diagram below, I have collected some of the examples introduced in the prior post in the series on multiplication. Each example is presented in two ways: on top, as a box with an input and an output, and on the bottom, as a graph.
Each shows a multiplicative structure. The graphs have something striking in common with each other, but they are not identical. The same: they are straight lines; they start at the corner. Different: some of the red lines are steeper than others; their axes are labeled differently.
I could imagine a world in which these boxes are cheap and plentiful, so that I’d have a whole bunch of them in my drawer. They are all labeled, but except for the label they all look kind of the same. Some get used a lot: the “plus sales tax” one is worn smooth from use, while the ‘after 35% discount” one has hardly been used at all. We could call these boxes special purpose tools, consistent with the advantages and disadvantages of all special purpose tools.
What would the general purpose version of these tools look like? What would be the crescent wrench to this multi-tool wrench set? In he diagram below, the top row is like before; the bottom row suggest what the crescent wrench might look like. Before you can use it, the crescent wrench must be adjusted to the particular use. This is indicated through the line on the left, which can be set to a number, e.g. like a thermostat.
Note that you can’t tell anymore from the label on the box what is going on. The whole point of a general purpose tool is that it can be used for many different things. Also note that this general purpose tool embodies ideas about multiplication and yet is still basically one value in / one value out. More generalization is needed before we can say we have a good model for multiplication as a binary operation. We’ll come back to this.