The mind is unable," wrote W. Stanley Jevons, “to estimate any large number of objects without counting them successively." He was right. Think, for example, of what you’d do if I asked you to look out of the window at a traffic jam on the road below and tell me how many cars are involved.
You’d start counting.
But Jevons went on: “A small number, for instance three or four, [the mind] can certainly comprehend and count by an instantaneous and apparently single act of mental attention." Right again. Think of reporting how many fingers of one hand I hold up. Definitely five or less, of course. One quick glance and you’ll be able to tell me “three", or “four". You certainly won’t count the fingers.
Jevons wrote these lines in a scientific paper reporting on a protracted experiment he conducted. First, he would grab a number of beans in his fist. Second, he’d throw them into a box. Third, he’d take a quick glance at the box and write down a guess at how many beans were in there. Fourth, he’d actually count the beans in the box and write down that number, too.
Jevons was a patient, diligent experimenter: He did this bean and box exercise 1,027 times. He put the results into a table where the rows represented his estimates and the columns the actual numbers. According to the table, the smallest number of beans he threw into the box on a given trial was three; the largest 15. Presumably he didn’t want to waste his time on a grab of just one or two beans, and he never grabbed more than 15.
But Jevons found something fascinating with his trials. When there were only three or four beans in the box, he guessed right every time—all 23 and 65 trials, respectively. But with five and more, he started making mistakes. The more numerous the set of beans, the more his guesses were off the mark.
Thus with five beans, he guessed right 102 times, but also guessed “six" four times and “seven" once. With six beans, he guessed “six" 120 times, but also “seven" 20 times and “five" seven times. With 10, it was like this: “eight", six; “nine", 37; “10", 46; “11", 16; and “12", two. With 15, like this: “12", two; “13", one; “14", six; “15", two.
Partly, what this table suggests is that though he did make mistakes, Jevons’s guesses even with larger numbers were mostly correct, or close. I’m not sure that faced with 10 beans, I’d guess correctly 46 times out of 107; or with an error of just 1 bean 99 times out of 107. Or maybe I would, and maybe that’s one point Jevons’s data makes—that we can make pretty good estimates of smallish sets of objects.
But the more intriguing point his data makes is about the number four. It looks like a limit on our ability to quickly estimate the number of objects in a set. Four or less, and we get it right every time. More than four, our accuracy falls off.
This may seem obvious and unremarkable to you. Of course we can guess small numbers precisely, and larger numbers not so precisely. Yet think about this: why is the threshold four? Why not three, or six, or 10? Does it have something to do with our five fingers? Something else?
Jevons published his paper all the way back in 1871 (The Power of Numerical Discrimination, Nature, 9 February 1871), and ever since, there are scientists who have wondered about this threshold of 4. If Jevons wondered about it, too, he didn’t have access to the tools of modern neuroscience to help him find an explanation. But a recent study does have a possible explanation (Distinct neuronal representation of small and large numbers in the human medial temporal lobe, Esther F. Kutter et al., Nature Human Behaviour, 2 October 2023).
The idea was to monitor how the brain behaves when faced with such an estimation task. To do this, naturally we cannot simply implant electrodes into the brains of living humans. Instead, these scientists found a set of people who already have such implants—epileptics who have the electrodes for medical reasons—and were willing to be studied.
These patients were asked to do simple mental arithmetic while their brains were monitored. What they found is, to me, simply startling.
For each number, individual neurons lit up, or became active. That is, a “three"-tuned neuron fires up only when offered the number three; a “six"-tuned one, only the number six. Such “number neurons" had been identified in animals before, but never in humans.
But after she analysed 801 of these neuron firings, Kutter was able to tease out two different patterns. As you might guess, one was for small numbers, the other for large ones. The neurons tuned for four and below were nearly error-free. That is, they lit up only for their preferred numbers and not for others. In contrast, the neurons tuned to steadily higher numbers than four were steadily less precise in their firing. Sometimes they didn’t fire when expected; sometimes they fired when fed a slightly different number.
“We found a boundary in neuronal coding," the paper remarks, “around number 4 that correlates with the behavioural transition from subitizing [perceiving a number correctly at a glance] to estimation."
There was one more facet to this. The neurological mechanism at work here isn’t just that an individual neuron responds to individual numbers. There’s also evidence that the number neurons for four and below are actually prevented from firing when offered the wrong number, thus improving their accuracy. These neurons, Kutter and colleagues explained, “showed superior tuning selectivity accompanied by suppression effects suggestive of surround inhibition as a selectivity-increasing mechanism."
So yes, this is the possible explanation for Jevons’s 150-year-old findings. “Possible", because remember that this is based on work with patients who already have electrodes in their brains. They are implanted in the part of the brain that deals more with memory, not so much with numbers. Still, there are these number neurons there, too. That is why the speculation that this small-number neuron mechanism may underlie the way we estimate numbers.
All this makes me wonder if I can train myself to accurately guess larger sets of objects. I’m aiming for seven.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.
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