The DNS is dependent upon mathematical practices, systems of number and algorithms for performing mathematical operations, complex mathematical concepts such as sets, functions, and so on. None of these practices, representations, or concepts are innate, and no one seriously thinks that they are. They are culturally inherited and acquired in the right learning niche with experts willing to teach. These new abilities are continuous with our cognitive phylogeny. How, though, can we put the whole package together? This section does that job.
Mathematics and writing systems are examples of culturally evolved symbol systems that are deployed to complete complex cognitive tasks. These systems are structured by rules and norms, but they are deployed as practices: patterns of action spread out across a population. In this case cognitive agents must gain mastery over the symbols, including numerals and operators, as well as the rules for their combination. However, they must also learn how to write and manipulate the symbols according to those rules in order to produce the right products—and this is proceduralised.
There may be more than one way of achieving a solution to the task. One can multiply by the partial products algorithm, or one can use the lattice/grid method or a number of others that have been developed by different cultures using different numerical systems. However, they all involve the same set of features: symbols, rules, operators, spatial configuration, and products, and they jointly constitute a practice for manipulating the symbols to complete mathematical problems. The practices are novel and unique to humans.
The methods apply equally to their off-line equivalents, so in the page-based version of the partial products algorithm we perform the multiplications from right to left and write down their products in rows, carrying numbers where necessary. In the off-line version we can perform the same operations on imagined numerals, multiplying numbers along the line and carrying any numbers as required. It is cognitively taxing to hold the products of the multiplications constant in working memory, though some people can train themselves to become quite good at it. Most people learn off-line multiplication by performing shortcuts; if I want to work out what 25 x 7 is, I just add 25 together 7 times.
On-line methods can change even within the same arithmetical systems, so the partial products algorithm works like this:
23
x 11
23 (1x3 and 1x2)
+ 230 (carry 0, 1x2 and 1x3)
253 (add products together)
However there is an equivalent algorithm that works like this:
23
x 11
200 (10 x 20)
30 (10 x 3)
+ 23 (1 x23)
253 (add products together)
The algorithms may differ, but they still involve the practice of spatially arranging the numerals, and performing operations on them and deriving a product, by performing the staged manipulations on the page. It appears then to matter how we manipulate symbols in public space, but is there any empirical evidence for this conclusion?
CI predicts that it matters how symbols are spatially arranged when they are being manipulated. Landy & Goldstone (2007) found that college-level algebraists could be induced to make errors by altering the layout of numbers that they were to manipulate. They did this by altering the spacing of the equations:
F+z * t+b = z+f * b+t
Although minor, the extra spacing was enough to induce errors. It matters how the symbols are spatially laid out, for this layout is the basis of how we manipulate those symbols. In this case the artificial visual groups created by the irregular spacing affected the judgement of the validity of the equation. If the visual groupings were inconsistent with valid operator precedence then they negatively affected the judgement.[29]
Landy & Goldstone’s work provides evidence that expert algebraists are practised at symbolic reasoning achieved via the perception and manipulation of physical notations (2007; Landy et al. 2014). Rather than an internal system of abstract symbols and rules for their combination (i.e., a language of thought), the system is composed of perceptual-motor systems and the manipulations of numerals. They are careful to say that the manipulations must conform to the abstract norms of algebra. Dutilh Novaes (2013) takes this to be evidence that mathematical competence is constituted by the capacity to manipulate inscriptions of mathematical equations. This fits very well with the CI approach.
Despite some interesting lacunae (savants and blind mathematicians), most mathematicians learn to manipulate numerals and other mathematical symbols on the page, and they continue to do so throughout their mature cognitive lives. Landy and Goldstone’s evidence supports the thesis that mathematical competence is constituted, in part, by our capacity to manipulate symbols in public space; that competence is, properly, a matter of interaction.
We have seen that there is an ancient evolutionary endowment for numerosity—an analogue and approximate system. This system is found in other primates and other species. It provides both the phylogenetic basis of mathematical cognition and the initial constraints for the development of the DNS. The DNS did not spring sui generis into the world. It did so because of a heady mixture of socio-cultural pressures, phenotypic and neural plasticity, social learning strategies, and cultural inheritance. These are the conditions for the scaffolding of the ANS, transforming our basic biological capacities into the DNS.
New cultural functions, discrete mathematical functions, and the practices for manipulating inscriptions transform existing circuitry in the brain. Once we learn how to recognise, understand, and manipulate mathematical symbols our brains undergo a profound transformation. There is a reproducible circuit for mathematical cognition involving a bi-lateral parietal based approximate estimation; a left lateralised verbal framework for arithmetic concepts (e.g., number words); and a occipito-temporal based symbol recognition system (e.g., Arabic numerals). The system also incorporates visual-motor systems for writing (manipulating, or pushing) symbols in public space.
A further important aspect of transformation is symbolic estrangement. As the DNS matures it becomes more abstract and less directly mapped onto the approximate functions of the ANS. Interestingly, at the same time expert mathematicians become reliant upon visual-motor capacities for manipulating inscriptions. Transformation depends upon the novelty and uniqueness of mathematical symbols and practices.
Symbolic number systems and sequential algorithms allow for mathematical and cognitive novelty. Once we have a public system, all manner of exotic numbers and operations can be discovered:[30] negative numbers, square roots, zero, sets, and so on. Its importance lies in the ability to perform computations that cannot be performed by ancient neural functions for numerosity. For example, the neural circuits responsible for numerosity cannot (on their own) represent -3 or √54, and yet this is simply represented in terms of public mathematical symbols (DeCruz 2008). This is because the symbolic representations are novel and unique. Initially, novelty results from the pressures of increasing social and economic complexity. Small roaming bands of foragers do not need to develop symbolic number systems; post-agricultural Neolithic societies settled in villages and towns do. A further issue is how novelty comes about from the ability to abstractly combine symbols and functions that apply to the symbols. I don’t propose to try to answer that question here; however, we might think of this as a curiosity- and creativity-driven processes. Given uniquely human behavioural and neural plasticity and socio-cultural complexity we might expect an increasing drive towards cognitive innovation. This has certainly been the story of recent cultural evolution in modern human societies.
This concludes the discussion of mathematical cognition as enculturation. Now I turn to the objections.