For a long time now, I’ve been writing journal articles and blog posts about large number innumeracy. A basic take-away from my research has been that untrained members of the general population have a wide range of skills for dealing with large numbers. Basically, people are very competent at arithmetic up to about 1,000; they can order the basic scale words thousand, million, billion, and trillion, and can generally even write them correctly. What people struggle to do is to relate across orders of magnitude. In one study, for instance, 65 of 67 college undergraduates represented one thousand, one million, and one billion correctly as numerals, even as one third of them made huge errors estimating their relative magnitudes. This week, a popular internet meme has made my point beautifully.
I have often argued that the idea of an ‘abstraction’, as usually conceived by Cognitive Science, is a myth. The specific notion of abstraction that I’m accusing of mythological status about is one in which concepts can be divided into those which primarily encode specific features or instances or particulars (concrete concepts), and those which do not (abstract concepts); it’s epitomized in the ‘schema abstraction’ notion of category learning. On this notion, initial encounters with an abstract concept involve encoding lots of features that are fundamentally irrelevant to some (abstract, relational) concept. Later, you prune away those irrelevancies, and are left with just the pure relational entity.
Of course, something can be a myth either because it is an unattainable ideal, or because it is fundamentally wrongheaded. Some people have suggested that sure, it might be that even abstract ideas still retain some concrete features, but have fewer of them, somehow. So ‘abstraction’ is, itself, an abstract principle which maybe never happens purely in practice, but is still the limit of real abstractions in the same way that a Platonic circle is the limit of real circle-like shapes. In contrast I’ve usually argued that the myth of ‘abstraction’ is the latter–fundamentally wrong-headed, at least an an explanation of interesting complex reasoning in supposedly abstract fields like math or physics. Maybe this feature-stripping thing happens, but it’s not what we call "abstraction" in mathematics.