Why do we call something a “number”?: Well, perhaps because it has a “direct” relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name.
And we extend our concept of number as in spinning a thread we twist fiber on fiber. And the strength of the thread does not reside in the fact that some one fiber runs through its whole length, but in the overlapping of the fibers.
Ludwig Wittgenstein, Philosophical Investigations EEM
Always have loved them. Always have obsessed about them. I treasure my favorites and revisit them again and again — I could barely think without them. I have a tag for them in this blog (here) and I almost started “a collection of beautiful definitions” to go with my eemadges website (“a collection of beautiful descriptions”). A good definition more than justifies a whole book. A good book always has many good definitions in it. Good people always carry several good definitions with them — you just have to know how to tease them out.
And yet I seem to get into all kinds of tiresome, silly discussions when I try to share them with friends. Besides my not to be belittled incompetence as an explainer and my fabled monomanias, I believe a basic misunderstanding regarding their nature is at the heart of the matter.
You see, most people seem to never have moved over the idea of a definition as distilled truth — the one true essence which both captures everything that should be captured and leaves nothing that shouldn’t be left out. Definitions as platonic ideals — the perfect divine forms of which we only see shadows. The one golden fiber that runs trough all the thread.
The problem with this view, of course, is that it is crippling in its obsession with perfection. It intimidates and nurtures ridiculous expectations. If we had had to delay mathematics until we had a “perfect” definition of number we would still be waiting.
In their supposed perfection, definitions only become cages. And we easily get to the point when not only it isn’t believed that things like “love”, “mind”, “conscience”, or “happiness” could ever be defined (again, as if there was one true definition to rule them all), but the very possibility is viewed with dread. Dread that what once was magic and alive is cramped and crippled into a cage.
A much more interesting view of definitions, in my opinion, is to regard them as tools for thought, and as such, to value them on their usefulness and pick the one appropriate for the task at hand — platonic truth is only one of the many, many things we can ask of them. Most importantly, we ought to recognize that we need them — a brain unaided can do only so much. Thinking without them is like hammering with your bare fists — it’s painful and ineffectual. Yes, they are only one (verbal) kind of tool and we run the risk of starting to see everything as a nail, but they are still one of the most basic and powerful tools we have and they have so far been needlessly feared and vilified.
Definitions are semantic flashlights, casting light on some meaning corners, shadow on some others. That everything be alight is only one criteria (ultimately impossible; only emptiness can be shadelessly illuminated), there are others — that it be bright, that it be dim, that it illuminate (or obscure!) a particular patch, that it be pristinely white, that it tint its subjects with its color, that it be diffuse, that it be focused, that it be favorable, that it be unfavorable… We say, teasingly, that an American is a “man with two hands and four wheels” not because we believe that it happens to be a perfect embodiment of what it means to be an American, but because we believe it casts them in an interesting light.
So the effort to define “play” or “capital” or “freedom” is not to pin the butterfly down and put it in formaldehyde, it’s to find new ways to look at it, new sources of joy and understanding. Definitions do not diminish their subjects, they reveal them.