What's Halfway Between 1 and 9? Kids and Scientists Say 3

Counting woman hands (0 to 5) isolated on white background
Counting woman hands (0 to 5) isolated on white background

Quick: What's halfway between 1 and 9?

If you're like me, you answered "5," which is (approximately) the right answer. But if you've got a 6-year-old kid on hand, try asking him or her. According to a recent study, most little kids will answer "3," as will many people in non-literate cultures. And as it turns out, there's a solid evolutionary reason why that guess makes sense.

2012-10-18-NautilusCutawayLogarithmicSpiral.jpgThe reason, strange as it may seem, is that our neural circuits are evolved to count logarithmically rather than linearly. This strikes me as ironic, because logarithms always drove me nuts with their anti-intuitiveness back in high school math class. The basic idea, if you remember, is that a logarithm is the inverse of an exponential, so if 102 = 100, then the log10(100) = 2. Unlike me, many scientists and artists are in love with logarithms, because these functions elegantly describe all sorts of ever-tightening curves and spirals, such as fractal patterns, ratios between musical octaves and, most famously, the golden ratio itself.

It's likely that we love logarithmic shapes so much because logarithms are hardwired into our sensory pathways. Stevens' power law explains that we perceive logarithmic increases in light, sound and heat (among lots of other stimuli) as if they were linear increases. For example, as heat rises along a tightening logarithmic curve -- say, from 80º to 84º to 86º to 87º -- our sense of touch is growing correspondingly more sensitive to each increase, causing us to perceive smaller and smaller changes as if they were equal. In short, we might describe that logarithmic temperature change as a linear rise from 80º to 84º to 88º to 92º.

From a survival perspective, this relativistic estimation is actually really practical, because it provides reasonable guesses while saving on energy. Imagine that you're a stone-age warrior stalking through the forest with your tribe, when suddenly a pack of wolves leap out of the shadows. If you can instantly guess whether it's more like a three-wolf pack or a nine-wolf pack, you can make a snap decision about whether to stand and fight or turn and run. On the other hand, distinguishing between a nine-wolf pack and, say, a 10- or 11-wolf pack isn't nearly as helpful. In other words, your brain is evolved to estimate in a way that minimizes relative errors rather than absolute ones; the larger the sample size, the less individual variations matter. (As you might've noticed, this logarithmic curve is the reverse of the temperature curve above, where smaller and smaller changes became more and more significant as they progressed.)

In fact, the differences between logarithmic and linear counting only leap into sharp contrast when we compare our brains' estimates with the precision of mathematics, which brings us back to my original point about numbers between 1 and 9. Because 30 = 1, 31 = 3, and 32 = 9, the number 3 is logarithmically halfway between 1 and 9, and young children somehow intuitively know that. This gives new meaning to the term "baby genius," doesn't it? The folks at Radiolab certainly thought so; they devoted a whole podcast to infant intuition about numbers.

This all points back to one of the stranger ideas in the philosophy of science: Numbers themselves don't actually exist in the same way forests and wolves do; we made them up to count and classify what we observe in the world around us. Math may be the "language of the universe," but it's still a language -- a map, but not the territory itself. So what's "really" halfway between 1 and 9? In the end, the answer depends on how you're counting.