# Concepts and Prototypes

We think about abstract concepts in terms of prototypical instances. These prototypical instances inform how we construct arguments using these concepts. At a more basic level, they determine how we go about constructing definitions themselves. Prototypes pop up in all sorts of conceptual domains, ranging from “war” to “airplane” to “bird.” So how do prototypes work in our thinking? Let’s start with an apparently simple example — the concept of triangle — that can get tricky really quickly.

If I asked you to draw a triangle, you would probably draw one that looked something like the one below, a scalene triangle, almost certainly drawn with the longest side as base and obtuse angle, if there is one, on top. Call this a prototypical triangle, understood as the sort of instance most people would draw. Why we draw such instances is the question of interest here. Let’s exercise this instance in a simple argument to see what role “prototypicity” plays in thinking. We will convince ourselves of the validity of the formula for the area of a triangle — half base times height — through mental visual manipulations.

Imagine a line dropping from the top vertex vertically down to the base. This line enables you to visualize two right triangles. Now imagine a copy of the triangle on the left being rotated clockwise 180 degrees. Position this imaginary triangle so that you now have a complete rectangle on the left. Repeat the process for the right. The two imaginary rectangles now form a larger rectangle. The area of this rectangle is the product of the base and height of the original triangle. Since you constructed this rectangle by copying, rotating and pasting two triangles that exactly covered the original triangle, the original triangle must have an area given by half the product of the base and height.

You used imaginary visual manipulations to convince yourself of the formula for the area of a triangle. Now ask yourself, why did you not start with any of the following set of perfectly legal triangles:

We have here an isoceles right triangle, an isoceles triangle, an equilateral triangle, a long, skinny triangle, a straight line segment and a point. The first 3 cases possess symmetries and the last two contain degeneracies.

Now work through the visual proof of the area of the triangle for each of these cases. The last two are the easiest: the answer is zero by examination, so the formula is trivially correct. More importantly, the answer does not tell us much — any constant instead of 1/2 would work, so we have validated a non-unique candidate. The first three, I assert, are also degenerate. Not in terms of their explicit geometric structure, but because the visual proof of their satisfying a particular asserted property — the area formula in our example — collapses to a simpler case than in the scalene triangle case. Recall your mental manipulations for the scalene triangle. Can you see how there are fewer steps in convincing yourself of the truth of the area formula? In each case, a single cut-and-paste visualization suffices. The scalene triangle proof visualization works (if inefficiently) for the symmetric cases, but the reverse is not true.

So one answer to “why do we choose as prototypes the instances we do?” is an information-theoretic one. A prototypical instance of a concept is one that contains the maximum information that an entity satisfying the definition possibly could. We are naturally inclined to work with the richest-information-structure case. In the case of the explicitly degenerate cases, the information poverty showed up in the non-uniqueness of the formula that was satisfiable. In the more subtle cases with symmetries, it showed up in terms of the degeneracy in the proof construction which would not work in the general case.

That doesn’t explain it all. What about our “long skinny triangle” which is close to degenerate, but not strictly so? Why didn’t we draw something like that? I suspect this has to do with the precision of comparisons we need to make when we mentally manipulate geometric figures: we want enough asymmetry and non-degeneracy to clearly illustrate the information capacity of our concept, but not so much that the precision required of the representation is too high. I am not completely happy with this hand-wavy account, so I’ll revisit this when I come up with something better. If you have a better account right now, post a comment.

A final point: why did we choose to draw our original scalene triangle with the longest side as a visual base? One proximal reason is that the necessary manipulations require more effort if we were to draw it with, say, the obtuse angle at the base (try it). A less obvious reason is that we operate with orientational metaphors that determine notions like “up” and “down” when dealing with abstractions. These metaphors inform both our language (“base” and “height”) and probably explain why non-standard orientations are mentally harder to work with, even though the explicit visual-proof steps are orientation agnostic. These conceptual framing metaphors will come up later when I talk about George Lakoff and his work on metaphor, so I’ll defer discussion of this aspect of prototypicity to later.

When we move from instantiations of abstractions to sets of entities (real or imagined) that we want to define, we run into problems with other methods of picking out elements, such as archetypes and stereotypes, that get in the way. We also run into issues of intension, with an ‘s’. That’s for later.

I first encountered this notion of prototypicity in a biology class when I was about 13. The teacher asked the class clown to come up to the blackboard and draw an amoeba. He drew a neat block ‘L’ shape, and the class burst out laughing. The teacher got mad and told him to stop clowning around and draw a proper amoeba. He countered that since we’d been taught that amoeba proteas could take on any shape, a regular ‘L’ was as much an ‘any’ shape as an irregular blob.

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