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!”

Hi Venkat,

Don’t want to make it an “Orkut” but Hi.

At the risk of being ridiculed, I would like to share the perspective from the other side. Most of the world is a heavy and unaware consumer of these concepts. ( One man discovered the bulb and rest of them used it, most of them without ever knowing the word tungsten ). Its as if the world is giant cloud of particles and the people who understand and define these “concepts” are magicians waving their wands guiding the particles at will.

Trishant (TK)

Hey TK,

Yes, it is interesting that most of us don’t question the definitions of very basic things, so the people who manage to own the definitions can exercise a lot of influence. I like the giant cloud of particles metaphor.

Not quite magic though, since an unsound definition usually loses ground over time. Sort of like darwinian evolution among concepts, similar to evolutionary competition among living things or technologies (like VHS/Beta Max). Many math definitions have also gone in and out of fashion because of this type of evolution in concepts.

The interesting thing about concepts that makes them useful for fiction/fantasy writers is that they can be just about anything. But this very thing makes them useful for mathematicians too. :-) It would be a very boring world for mathematicians if they couldn’t invent/discover new concepts and call them axioms, within which they could creatively find new theorems to play with and prove.

“The search for proper concepts and definitions is one of the main features of doing great mathematics.” — Richard Hamming, “The Unreasonable Effectiveness of Mathematics”.

Clearly concavity can be a concept. So can the curling up of a fourth space dimension. So can poisonous glasses that affect the wearer’s feet.

But some concepts may be more interesting than others, no doubt. And these are probably the primary ones. You may be able to deduce poisonous glasses from concavity, and perhaps glasses can be concave only if you have a fourth curled up space dimension. I don’t know.

But here is a foundational question nevertheless.

What are the axiomatic concepts of modern science?

— Viraje

I do not get the meaning of concept. Could you please help me get another word for concept. Thanks