Representations II: Dry Bones

In my abstract math website here I wrote about “two levels of images and metaphors” in math, the rich and the rigorous. There are several things wrong with that presentation and I intend to rewrite it. This post is a first attempt to get things straight.

When we are trying to understand or explain math, we may use various kinds of images and metaphors about the subject matter to construct a colorful and rich representation of the mathematical objects and processes involved. I described some of these briefly in the previous post on representations.

When we set out to prove some math statement, we go into what I called “rigorous mode”. We feel that we have to forget about all the color and excitement of the rich view. We must think of math objects as totally inert and static. The don’t move or change over time and they don’t interact with other objects or the real world. In other words, pretend that all math objects are dead.

We don’t always go all the way into this rigorous mode, but if we use an image or metaphor in a proof and someone challenges us about it, we may rewrite that part to get rid of the colorful representation and replace it by a calculation or line of reasoning that refers to the math objects as if they were inert and static – dead.

I now think that “rigorous mode” is a misleading description. The description of math objects as inert and static is just another representation. We need a name for this representation; I thought about using “the dead representation” and “the leached out representation” (the name comes from a remark by Steven Pinker), but my working name in this post is the dry bones representation (from the book of Ezekiel).

Well, there is a sense in which the dry bones representation is not just another representation. It is unusual because it is a representation of every mathematical object. Most representations, images, metaphors, models of math objects apply only to some objects. You can say that the function $y = 25 – t^2$ “rises and then falls” but you can’t say the monster group rises and falls. The dry bones representation applies to all objects. Its representation of that function, or of the monster group, is that it is one object, all there all at once, not changing, not affecting anything, a kind of