4 Numerical cognition
In this section I outline the phylogenetic basis of mathematical cognition. That basis is in our shared sense of quantity and our ability to estimate the size of small sets by making approximate judgements of the size of the set. This ancient endowment is the basis for our mathematical competence, but it is not all there is to mathematical cognition. This is because precise mathematics depends upon a very recent and acquired public system of exact and discrete mathematical thinking. The ancient system is analogue and approximate, but mathematics requires digital and discrete representations and exact operations. These are, of course, recent additions to inherited cognitive capital. I shall show why mathematical cognition requires our ancient capacity for numerosity and how it is constituted by cognitive practices—which transform our cognitive abilities, resulting in novel and unique modern human cognitive capacities. However, this transformation results in two partially overlapping systems—the approximate number system and the discrete number system—with the latter having unique properties acquired from cultural innovation. One of the puzzles is how it is possible to move from an inherited approximate system to an acquired exact system. The process of enculturation provides the mechanisms by which such a move takes place, from the ancient capacity for numerosity to development in a socio-cultural niche, and the orchestrating role of practices in the assembly of the cognitive systems responsible for mathematical cognition.
4.1 Numerosity in animals and humans
There is strong evidence to suggest that we have a basic analogical and non-linguistic capacity to recognise quantity and number. I think that there is overwhelming evidence for an ancient evolutionary capacity to discriminate cardinality, and to determine in an approximate way the quantity of membership of sets. It is obvious how this capacity, for only very small sets, would be beneficial for activities such as foraging, hunting, and so on.
Recent studies have revealed that the neural populations that code for number are distributed in the intraparietal sulcus (Dehaene & Cohen 2007). A growing number of studies show that both animals and humans possess a rudimentary numerical competence, which is an evolutionary endowment. For example, red-backed salamanders have been shown to choose the larger of two groups of live prey (Uller et al. 2003). Single neuron activation studies in rhesus monkeys (Nieder et al. 2006) discovered that individual neurons respond to changes in number when presented visually (and non-symbollically). These neurons are also located in the intraparietal sulci, indicating a probable cross-species homology. The neurons peak at the presentation of a specific quantity of dots, but then decrease as the numbers presented differ from the original. So a neuron that peaks at the presentation of two dots responds less to three or four dots. The further the numerical distance of the array of dots is from the magnitude to which the neuron is tuned, the lower the firing rate of the neuron. Therefore, the ancient capacity for numerosity is an approximate function, not a discrete one (DeCruz 2008).
This is not yet counting; counting is exact enumeration. Subitizing is the ability to immediately recognise the size, or number, of a small set—usually <4. Most animals subitize, rather than count. Infant humans also appear to be able to subitize (Rouselle & Noël 2008). This ancient or approximate number system (ANS) is a non-linguistic continuous representation[27] of quantities above 4; Dehaene calls it the number sense (1997). Take the following example. Whilst it is easy enough to determine which of the following two boxes contains the larger number of dots without having to count them:
Figure 1: Figure 1: Subitizing or counting?
It is less easy to do so for the following (you will probably need to resort to counting):
Figure 2: Figure 2: Subitizing or counting?
It is also possible to make estimations or approximate judgements of scale for numbers. Most people can quickly identify that 7 is larger than 3. Even for more complicated exact operations we can do this:
34 + 47 = 268 (is this right?)
We readily reject this result, because the proposed quantity is too distant from the operands of the addition (Dehaene 2001, p. 28).
34 x 47 = 1598 (is this right?)
Approximation involving proximity and distance will not help here (unless you are very practised at mental multiplication), but you might resort to a multiplication algorithm (which might be routinized). It is clear that we have an ancient sense of quantity and are good at making judgements about more than and less than, but when it comes to precise and discrete quantities (particularly larger numbers) we need new capacities to be able to make judgements about operations on discrete numbers.
4.2 Two overlapping systems
The approximate numerical system is an analogue and approximate system for discriminating non-symbolic numerosities greater than 4, but the “representations” are approximate and noisy. The second system is acquired and concerns discrete symbolic and linguistic representation of individual numbers from our numeral system, including individual words for numbers. This system works with discrete, exact, symbolic representations of quantity and allows for the exact operations of arithmetic and mathematics. I will call this the discrete numerical system (DNS). There is disagreement about how much the two systems overlap. However, what is clear is that the internalisation of the public numeral system allows us to perform the kind of digital mathematical operations that are required for most arithmetic and mathematical operations (Nieder & Dehaene 2009, p. 197).
Dehaene and colleagues produced a series of experiments that demonstrate the separate functioning of the two systems. Russian–English bilinguals were taught a set of exact and approximate sums of two digit numbers in one of their languages (Dehaene et al. 1999, p. 970). Their tasks were split into giving exact answers to additions and giving an approximate answer to the addition task. The interesting result was that:
[w]hen tested on trained exact addition problems, subjects performed faster in the teaching language than in the untrained language, whether they were trained in Russian or English. (Dehaene et al. 1999, p. 971)
This provided evidence that knowledge of arithmetic was being stored in a linguistic format, and that there was a switching cost between the trained and untrained languages. By contrast, there was equivalent performance in the approximation task, and no switching cost between the trained and untrained languages. Dehaene et al. conclude that this provides “evidence that the knowledge acquired by exposure to approximate problems was stored in a language-independent form” (1999, p. 971).
This leads us to the conclusion that there are two overlapping, but not identical, systems for mathematical cognition. The first is the ancient and approximate system, the second is a relatively new and acquired system for discrete and digital representations and operations. As Dehaene & Cohen put it:
The model that emerges suggests that we all possess an intuition about numbers and a sense of quantities and of their additive nature. Upon this central kernel of understanding are grafted the arbitrary cultural symbols of words and numbers […]. The arithmetic intuition that we inherit through evolution is continuous and approximate. The learning of words and numbers makes it digital and precise. Symbols give us access to sequential algorithms for exact calculations. (2007, p. 41)
The two systems are overlapping but not identical because they have quite different properties. First, the ancient system is part of our phylogeny, whereas the discrete system is an acquired set of capacities in ontogeny. Second, the ancient system is analogue and approximate, whereas the discrete system is digital and exact. Third, the discrete system operates on symbols that don’t map directly on to the ancient system.
When we consider very large numbers, such as 10,000,000, there is no obvious analogue in the ANS. Consequently, large or exotic numbers and operations on them do not map onto existing cortical circuitry for numerosity. Lyons et al. (2012) call this phenomenon “symbolic estrangement”. Symbols become estranged through a process of symbol-to-symbol mappings, rather than symbol-to-approximate-quantity mappings (Lyons et al. 2012, p. 635).
However, there appears to be a point of contention here: Dehaene expects there to be a more or less direct mapping of symbols to quantities (e.g., the mental number line). If symbolic estrangement does happen, then this would appear to be mistaken. Lyons, Ansari and Beilock propose a developmental resolution of this apparent disagreement. Children may start out in the acquisition of discrete number systems by a mapping to an existing approximate neural coding of quantity, but as the system matures and symbols become abstracted from the ancient system, the mature system splits into two (related but not entirely overlapping) systems: neural circuitry in the DNS tunes for discrete symbols,[28] whereas circuitry in the ANS tunes for approximate quantities, such that discrete symbols do not map directly onto approximate quantities. E.g., 10,000,000. The DNS has properties that are unique.
In the next section I return to the question of the role of practices in assembling the DNS.