This post considering the evolutionary origins of numerical cognition, specifically in terms of the approximation of large numbers, is meant as a companion to this week's series on the developmental origins of numerical cognition and developmental dyscalculia, at Child's Play.

What are the origins of number representation in the mind? Are there any innate building blocks that contribute to our understanding of mathematics and number, or must everything be learned?

Number is an important domain of human knowledge. Many decisions in life are based on quantitative evidence, sometimes with life or death consequences.

Figure 1: Fight or flight?

By now you probably have come to expect that I'll be arguing that there are several innate "building blocks" of cognition that give rise to more complex mathematics. To start with, what are some of the arguments proposed by the empiricists?

(1) Number knowledge is entirely conceptual - it requires seeing objects as belonging to sets;

(2) Number knowledge is abstract. You need to understand the similarity between 3 people, 3 objects, 3 sounds, 3 smells, 3 dollars, 3 seconds, 3 hours, and 3 years;

(3) It doesn't appear to be cross-culturally universal. Some cultures have more advanced mathematics than others; and

(4) babies and monkeys can't do long division.

Surely, humans have something unique that allows us to do things like multivariate regression and construct geometric proofs, however, but let's start at the beginning. I will hopefully convince you that there is an evolutionarily-ancient non-verbal representational system that computes the number of individuals in a set. That knowledge system is available to human adults and infants (even in cultures that don't have a count list), as well as to monkeys, rats, pigeons, and so forth.

In the first study (of human adults) that we'll consider, participants were presented with arrays of dots on a computer screen that were presented only for a fraction of a second. In that time, the participants had to determine if the second array had more or fewer dots than the first array. They controlled or counterbalanced dot size, density, shape, and things like that. How did they do in this task?

Figure 2: Results. They were at chance on trials comparing 32 and 34 dots (center). All other comparisons, adults demonstrated above-chance discrimination.

Here's the important point. If they were counting, then they should have been able to discriminate 32 versus 34 dots just as easily as they were able to discriminate 8 versus 10 dots. Also, if they were counting, it should have taken then longer to count 32 versus 64 dots, than for 8 versus 16 dots. Since the dot arrays were displayed for the same amount of time, and they were able to discriminate both of those sets of numerosities equally well (and perfectly), it follows that they weren't counting.

It turns out that the discriminability of two numerosities depends not on the total number of objects, but on the ratio between the two numerosities. Notice that 32 vs. 64 and 8 vs. 16 have the same 1:2 ratio. This also implies that the mental representations of these numerosities are inexact - they are approximations, not exact numbers.

Figure 3: Where the ratio was 1:2, responses were perfect. Success rate decreased as the ratio decreased. When the ratio was 1:1.1, success was basically at chance, but considerable success at 1:1.15.

Is this limited to the visual domain? In the next experiment, participants saw a similar array of dots, and then they heard a set of sounds. This cross-modal comparison (73% accuracy) was almost as accurate as the visual comparison alone (76% accuracy). The participants were also asked to add sets of objects. They were shown two arrays sequentially, and asked to add mentally approximate the total number of dots on both arrays, and shown a third array. They were asked if the sum of the dots in the first two arrays more or less than the total amount of dots in the third array. This resulted in 72% correct responses. Last, this was done cross-modally. They were shown a dot array, then given a sequence of sounds, and asked to add them. Was the sum of the dots and the sounds more or less than the total amount of dots in a new, third, array? Accuracy was 74%. Accuracy was roughly equal across all four of these conditions, suggesting that numerical representations are abstract. It also means that these approximate abstract representations contribute to our ability to add.

Figure 4: Results. Equal success for each condition.

"But," the empiricist says, "these representations have been mapped onto verbal numerals. Even though you might have an approximate estimation of the number of dots, you still use language. You might think 'that's about 50 dots' or 'it seems like there's around 300 dots.' These people have spent years learning and using formal arithmetic." And in response to the empiricist, the nativist says: "Fine. Have it your way. Bring on the babies and animals."

So we gather a group of six month old babies who don't have language yet. We can't ask babies which array has more dots, so we use a habituation paradigm. We show the babies arrays with the same number of dots (say, 8 dots) until they get bored of it and spend less time looking at it. Then we show them a new array (say, 16 dots). Do they look longer at the new array of 16 dots than at a new array of 10 dots? If so, that means that they discriminate 8 versus 16, but not 8 versus 10.

And so it was. Infants successfully discriminated 8 vs. 16, 16 vs. 32, and 4 vs 8. They failed to discriminate 8 vs. 12, 16 vs. 24, or 4 vs. 6 dots. So infants, too, show a ratio limit, though the critical ratio is higher for infants (1:2) than for adults (1:1.15).

How general is this ability? Does it work for sounds as well? A new group of six-month-old and nine-month-old infants were placed between two speakers, and were habituated to a number of sounds coming from a certain direction. Instead of looking time, the measurement was whether or not the infant turned his or her head to orient toward the source of the sounds. They were familiarized to, for example, 8 sounds or 16 sounds, and then tested with 8 and 16 sounds. The findings from this task were similar to the findings in the dot task.

Six-month-olds discriminated 8 vs. 16 and 4 vs. 8 sounds, but failed at 8 vs. 12 and 4 vs. 6. The nine-month-olds discriminated 8 vs. 12 and 4 vs. 6 sounds, but failed at 8 vs. 10 and 4 vs. 5 sounds. Again, discrimination of numerosities showed a ratio limit, though discrimination gets sharper with age.

Let's push the question a bit farther. Approximate numerical representation exists for visual objects as well as auditory sounds. What about for actions?

Infants were habituated to a cartoon in which a rabbit jumped either 4 times or 8 times. The total distance of the movement of the rabbit was equivalent, such that the each of the 4 jumps were twice the distance of each of the 8 jumps. This allowed the rabbit to end up at the same location at the end of each set of jumps - so the infants couldn't rely on physical displacement as a correlate of number of jumps. The findings were the same as the dots and sounds. Representation of number is truly abstract, even in infants.

To recap: Before learning to count or learning arithmetic, infants represent and discriminate large numerosities. These representations are approximate, and subject to a ratio limit. These representations are also abstract, and the same ratio limits applies to objects, sounds, and actions. This capability is present in infancy, though it increases in precision through development.

On to the animals.

In the 1950s and 1960s, Dr. Francis Mechner did a series of conditioning experiments with rats. In one such study, rats were trained to press a lever 4, 8, 12, or 16 times in order to receive a reward. Tension on the lever was controlled, so that the rats couldn't rely on total effort as a correlate of the number of level presses.

Figure 5: Presses on the lever by rats.

These data indicate that rats also have inexact, approximate representations of large numerosities, and that there is also a ratio limit. Accuracy decreases as the target number of lever presses increases.

Some years ago, a group from Harvard investigated this question in cotton-top tamarin monkeys as well. They did the same auditory discrimination experiment with the monkeys as had been done previously with infants. The monkeys were habituated to a sequence of sounds coming from the right or left, and then presented with a new number of sounds. Again, the turn of the head to orient towards the sound was used as an indication of discrimination. Their performance was similar to the 9-month-old infants: they discriminated 4 vs. 6 and 8 vs. 12, but not 4 vs. 5 or 8 vs. 10. They could discriminate 2:3 ratios, but not 4:5 ratios.

How evolutionarily-ancient is this cognitive capacity? Our common ancestor with tamarins is relatively recent, compared to say, with fish. So let's look some Italian fish.

Figure 6: Eastern mosquitofish (Gambusia holbrooki). This one is about 4cm long.

Female mosquitofish like to hang out with groups of other females as protection from sexually harassing males.

So you take a female mosquitofish, and you let her habituate to the fishtank. On either side of a long fishtank there are two additional groups of females of differing sizes, in their own tanks.

Figure 7: Something like this.

Meanwhile, you take a male mosquitofish and you deprive him of any females for a whole week. Sucks to be him. You introduce him into the tank with the female, and its game on. He desperately wants the feel of her cold, wet, slimey, scaley body. He can't wait to make sweet fishy love. He sees her from across the tank. He works up his nerve and says, (channeling his best Walter Matthau impression) "Maria, there may be lots of fish in the sea, but you're the only one I want to mount over my fireplace."

If he makes at least 10 attempts to have his way with the female in the first five minutes, then you record which group of fish the female tries to join. She should prefer to hang out with the larger group of females.

When the two groups of female fish were different according a 1:2 ratio, she always chose the larger group, but when the ratio was 2:3, she chose randomly. Just like the monkeys, and just like the human infants.

Figure 8: Success for 1:2 ratios, but not for 2:3 ratios.

What can we conclude from this series of studies?

Animals and humans spontaneously represent large (greater than 4), abstract, approximate numerosities. Animals, human infants, and human adults, show the same ratio signatures. Adult tamarins are on par with 9-month-old human infants. With age or training, discrimination ability becomes more precise, and the the critical ratio is reduced a bit.

The large number cognitive system is evolutionarily-ancient and non-verbal, and is likely innate.

Next up, later this week: small numbers.

Barth H, Kanwisher N, & Spelke E (2003). The construction of large number representations in adults. Cognition, 86 (3), 201-21. PMID: 12485738

Lipton JS, & Spelke ES (2003). Origins of number sense. Large-number discrimination in human infants. Psychological science : a journal of the American Psychological Society / APS, 14 (5), 396-401. PMID: 12930467

Mechner F (1958). Probability Relations within Response Sequences under Ratio Reinforcement. Journal of the experimental analysis of behavior, 1 (2), 109-21. PMID: 16811206

Hauser, M., Tsao, F., Garcia, P., & Spelke, E. (2003). Evolutionary foundations of number: spontaneous representation of numerical magnitudes by cotton-top tamarins. Proceedings of the Royal Society B: Biological Sciences, 270 (1523), 1441-1446. DOI: 10.1098/rspb.2003.2414

Agrillo, C., Dadda, M., & Bisazza, A. (2006). Quantity discrimination in female mosquitofish. Animal Cognition, 10 (1), 63-70. DOI: 10.1007/s10071-006-0036-5