How to Define Concepts

Let us say you are the sort of thoughtful (or idle) person who occasionally wonders about the meaning of everyday concepts. So there you are, at the fair, laughing at yourself in a concave mirror, when suddenly it hits you. You don't really know what "concave" means. You just recall vague ideas of concave and convex lenses and mirrors from high school and using the term in general conversation to describe certain shapes. So you decide to figure out a definition. What do you? How do you make up a definition? Let's get you into some trouble. So the first thought you have is: concavity has something to do with "indentations" or inward curvature of shapes. You quickly abandon open-ended curves, of the sort you see in graphs of population growth and the like. Being smart, you realize that the concavity there is not fundamental -- you could turn a concave graph upside down, and make it convex with respect to your preferred visual orientation of "up is against gravity." So you decide that the notion of concavity probably only makes sense for closed curves: things with an inside and an outside (congrats, you just found a use for the Jordan curve theorem!). You draw yourself a prototypical closed curve, like the one on the left below, and stare at it:

 

You think, "hmm... really what is going on with concavity is that I can sort of take a short cut across some parts by going outside it." That leads to your first stab at a definition: a figure is concave if there exists a pair of points in it such that the straight line between them is not contained entirely within it. You draw a couple of lines and convince yourself, like on the right. At this point, if you like to rush to math, you might even write down an equation like this one: [tex] \bar{x}(\theta)= \theta\bar{x}_1+(1-\theta)\bar{x}_2 [/tex] and go, "Aha! a closed curve is concave if and only if you can find a pair of points like so, and for some theta, the point on the line given by my clever equation isn't inside the figure!" You've just found an attribute of convexity that you think is necessary and sufficient to define it. But then, suddenly a thought occurs to you. You sketch:

and go, "Uh Oh!" What just happened here? Why does this bother you? You've found a way to draw a line across two closed figures that don't "look" concave to you, and satisfied a formal notion of concavity. You really want to say that concavity is a notion that only applies to single figures. So what do you mean by "single"? Is a Figure 8 single? Is it concave? You are in trouble. At this point, if you really cared enough, you'd go on to reinvent a good deal of topology, invent the notion of "simply connected," figure out that you need the notion of closed and open sets and interiors and boundaries (to handle the Figure 8 case) and so forth. But let's not go down that road. Let's ask the more interesting question, why didn't you just define concavity to be anything that satisfies your original "straight line" test? (For many purposes in math, that is in fact exactly what you do, use the definition without worrying about connectedness -- that's the impatient, technical, "let's get on with it" aspect of mathematics, but you and I like to fuss over what we mean instead of getting somewhere). The interesting thing about the way our minds work is that math and formalism is subservient to a fuzzier notion of "what I want to get at." As we refine technical definitions (or natural language definitions of entities like "culture") we tend to move the definition to get at an understood but inexpressible concept. We practically never reduce the concept itself to the definition we are working with. This sort of thing is an example of the operation of what philosophers like to call intension (with an 's'). Intension, roughly speaking is the "true" meaning of a concept we are after. The difference between definition and meaning is what philosophers like to characterize as primary (or a priori) and secondary (or a posteriori) intension. The primary intension of "water" is "watery stuff." That is why a sentence like "Ammonia is the water of Titan" makes sense to us -- we imagine ammonia oceans. By contrast, "Water is H20" is a secondary intension. David Chalmers has a beautiful discussion of intension in The Conscious Mind: In Search of a Fundamental Theory. Does this apply to this example? Concavity, unlike "water" is an abstraction of real-world things like inkblots, bays, dents, holes and so forth. It references too many things in the real world for us to useful say something like "concavity is sort of like bays or dents." Despite this, however, our brains seem to work with a primary intension of concavity that draws efforts at definition spiraling towards itself. We grope towards what we mean through attribute-based tests expressed in terms of simpler concepts (like "straight line" in our case). What makes math special is that starting with a few prototypes that suggest a useful notion, we can often converge in a finite number of steps to a watertight characterization of an abstract concept within a useful closed domain. Brouwer, was perhaps the only major mathematician who tried to articulate this fundamental aspect of the structure of mathematical thought -- that technicalities follow from trying to capture intuition. Leaky abstractions like "culture" and "war" though, are another matter. I don't yet have a good handle on how to think about the process of achieving clarity with such concepts. Until then, all I can offer is my own rule of thumb, "Seek to capture the intension!"