Even Oddness, Part 3

In parts 1 and 2 of the "even oddness" tale, I introduced prototypes and idealized cognitive models (ICMs). The goal? To lay the groundwork for explaining the mysterious experimental results of a 1983 study which showed that people judge some even numbers as "better" than others. Today we tie it all together.

How does George Lakoff’s theory of ICMs explain the results of Armstrong, Gleitman and Gleitman’s experiment? In order to understand, we first need to draw a clear distinction between “folk” and “expert” models.

“Folk models” — also called “folk theories” — are mental models held by “regular old folks.” These are the models that make up almost all of Audie’s interpretations of events around us. “Expert models,” or “expert theories,” are Connie’s consciously-constructed models of how the world works. For instance, the “camcorder” model of vision — the idea that they eyes are like camcorders, recording what is "actually out there" — is a commonly held folk model of how vision works. Expert models of vision are created by neuroscientists who understand how the eyes and brain work together to generate the images we perceive as “real.”

In the case of even number, the definition we learn in school is an expert definition: an even number is a multiple of 2. It is consciously accessible, and we know, via Connie, that any given number either is or isn’t even. There is no gray at all.

At the same time, though, Audie, unbeknownst to us, has created folk models of numbers, including even numbers, which affect our judgments unconsciously.[1] The first model, underlying all others, is what we might call the “name equals number” model. Most people who aren’t mathematicians don’t normally distinguish between numbers and their names. Take the number 26: in our familiar base-10 system, 26 is both a number and a name. However, the number 26 can be represented by many different names: 11010 in binary, 1A in hexadecimal, and so on. It just happens that the base-10 system is what we have all grown up with, and is the one we use unconsciously to understand numbers — so much so that we equate numbers with their base-10 names.

Another folk model is the “single-digit numbers are primary” model. Because numbers are infinite, but systems of representing numbers are finite, we have to start somewhere. Our folk starting point is the positive numbers near zero, which in base 10 are represented with single numerals: 1 through 9. Every number in base 10 is represented using some combination of these numerals, plus 0. This gives 0–9 a privileged cognitive status: we view any number, no matter how large, in terms of these numerals.

Consider 106, which was one of the stimuli in the experiment. The name-equals-number model equates the quantity “one hundred six” with the string of numerals, 1 — 0 — 6. This string, three digits long, is more complex than any single numeral. So, via the single-digit-numbers-are-primary model, we understand 106/“one hundred six” in terms of the numbers/digits it’s made up of.

These two models are in us so deeply that it can be hard to see that they’re completely made up. For name-equals-number, it just seems so obvious that the quantity denoted by “106” just is 106. Its hexadecimal name, 6A, or its binary name, 1101010, seems fanciful, false — but 106 seems real.

As for the single-digit-numbers-are-primary model, this stems from the fact that we can imagine numbers much larger than we can count on our ten fingers and ten toes. Human creativity and imagination have outstripped our limited bodies, so we have invented a system of representation that starts with something basic and easy to understand — quantities less than ten — and we’ve figured out a way to represent everything else in terms of this simple concept.

Given these two models, it’s no longer surprising that Armstrong, Gleitman and Gleitman’s experiments gave the results they did. Subjects were asked to make rapid, unconscious judgments about some even numbers. Given the rapidity, Audie was in charge, and drew on the only models it could draw on under such stress: the folk models just discussed. This is what yielded the strange experimental results.

Importantly, when asked explicitly whether even number is a graded category, they said no. Able to draw on Connie to answer the question, they gave the logically obvious and truthful response. Audie, though, left to their own automated, time-pressured devices, begged to differ.

The two models discussed here suffice to explain Armstrong, Gleitman and Gleitman’s results. We also use many other simplifying models to help us understand numbers. For example, multiples of 10, 100, 1,000, etc., are more cognitively basic and provide key reference points. We could expect that the same kinds of experiments Armstrong, Gleitman and Gleitman did would show 1,001 to be more prototypical than 1,173.

George Lakoff’s contributions to category theory have been immense, and aren’t limited to prototype theory. Just within this sub-field, though, he accomplished what many who went before him had tried but failed to accomplish: he provided a highly plausible theory of human cognition that offered to explain the origins of prototype effects, while being completely in line with how psychologists and neuroscientists understand the brain and body to work.

The moral to our story: Audie is a genius at simplifying — so much so that they reach conclusions that are obviously logically absurd. In how many other ways is Audie fooling us, moment to moment?

[1] This discussion is a simplified and modified version of Lakoff’s own discussion in Women, Fire and Dangerous Things (Kindle Locations 2099–2125).