Our cognitive skills are sometimes more a product of our culture than of nature, and that cultural aspect can lead to some funny thinking in our “common sense” and what we think we know about the world. Counting, for instance, is a cognitive skill, but innate as it may seem to be, it is one that is definitely acquired from other people. Many indigenous people don’t count (or don’t even have words for numbers higher than five). It takes years for toddlers to pass through the stages that Liz Spelke, a prominent Harvard cognitive development researcher, calls “one-knowers”, “two-knowers”, etc. Counting is not an immediate process, but rather a protracted one that requires several stages.It is only after years of practice that children are able to align their sense of numerosity with the “number chant” that they had been rehersing long before they understand the significance. There is even good MRI evidence showing changes in the parietal lobe as it undergoes this counting training. (Piazza and Izard 2009).
So, what the current story about counting tells us is that we have to work our way into counting in order to open up a world of relationships between our numbers and how they can interact, a world called Mathematics. But mathematics is so useful, why doesn’t it line up with the way that individuals appear to naturally count or perceive quantity? Well, while we may learn a different way to count, it turns out that our more innate sense of number may be a very rational adaptation to a natural world that tends more to behave logarithmically. While the digits we count with (i.e. 1, 2, 3 on through 9) might seem uniformly distributed—that is, they would seem to all show up equally as often—there is an interesting mathematical law that should disabuse us of that “common sense” notion.
That neat mathematical law is called Benford’s Law and it dictates that if you take a collection of numbers, say the heights of the 1000 tallest buildings in the world or the Fibonacci sequence, and you count the first digits of the collection, you will find that there are more 1’s than 2’s, more 3’s than 4’s, etc. It might sound like numerology at first blush, but it can actually be used very accurately in forensic accounting to check and see if a company has cooked its books. Given Benford’s Law and how much it can describe the natural occurrence of numbers, that our numerosity sense would seem to be susceptible to logarithmic distinctions strikes me as a rational adaptation.