An emergentist perspective on the origin of number sense

1. Introduction

It is widely believed that mathematical learning is rooted into a phylogenetically ancient ‘number sense’ that humans share with many animal species [1,2]. Perceiving the number of objects is a key aspect of the number sense and it is highly adaptive for survival [3,4]. Visual numerosity appears to be extracted directly and spontaneously from vision [5,6] by a specialized mechanism that yields an approximate representation of numerical quantity, the approximate number system (ANS) [7]. The ANS representation can be conceived as a distribution of activation on a putative ‘mental number line’, where the overlap between distributions of activation increases with numerical magnitude due to either scalar variability or compression of the scale [8,9]. Accordingly, discrimination between two numerosities is modulated by their numerical ratio, thereby obeying Weber's Law, and this ratio-dependent effect in numerosity comparison is considered to be a primary signature of the ANS. The striking similarity in performance between human and non-human primates has suggested phylogenetic continuity of the ANS [10], as also indicated by the shared neural correlates found in the intraparietal sulcus of the primate brain [11,12].

The ability to discriminate between numerosities, known as number acuity, improves throughout childhood [13,14]. Human babies seem to be able, since their first hours of life, to discriminate the numerosity of object sets if the ratio is at least 1 : 3 [15]. Dramatic changes in number acuity have been observed within the first years of life: for instance, six-month-old infants can reliably discriminate between sets with a ratio of 1 : 2 but fail with a 2 : 3 ratio, which is instead discriminated by 10-month-olds [16,17]. The 2.5-year-old toddlers discriminate a ratio of 3 : 4 [18]. Changes in number acuity are thought to index the representational precision of the ANS, and it is widely believed that the latter is foundational to the subsequent acquisition of formal numerical competences. Indeed, individual number acuity has been linked to mathematical achievement [19–21] (for a meta-analysis, see [22]) and it has been showed to be impaired in dyscalculia [14,23].

In the present article, we discuss the origin of number sense within the emergentist framework provided by artificial neural network models [24]. The emergentist approach to cognition [25] assumes that the structure seen in overt behaviour and its patterns of change (e.g. during development) reflect the operation of subcognitive processes, such as propagation of activation and inhibition among neurons and adjustment of strengths of connections between them. In the context of the visual number sense (i.e. numerosity perception), the crucial question is what kind of biological constraints lead to its emergence (for a broader discussion, also see [26]). Is the number sense innate? We show below that connectionist simulations with artificial neural networks can provide a fresh perspective on this debate. Elman et al. [27] have thoroughly discussed the issue of innateness in the context of connectionism. Here, we focus on their observation that there are two different ways to think about innateness, which also readily apply to the discussion on the origin of numerical abilities.

2. Number sense emerges from evolutionary pressure

As noted above, the finding that human infants and many other animal species are sensitive to numerical quantity supports the hypothesis that at least some aspects of number sense might be genetically determined. Though learning and experience clearly play a role in the development and refinement of the number sense [1], this evidence has suggested that these abilities are supported by an ‘evolutionary start-up kit’ [30]. This possibility was explored by Hope et al. [40] using artificial life simulations, which exploit behaviour-based selection to capture the impact of evolution. The simulation was built on the hypothesis that quantity comparison emerges from selective pressure to forage effectively [9]: evolving quantity-sensitive foragers would imply the ability to judge quantity and therefore a tendency to ‘go for more’ [37].

The artificial ecosystem used in the simulations is depicted in figure 1a. It consisted of a two-dimensional (2D) grid of 100 × 100 cells, each containing a certain amount of ‘food’. Food was randomly distributed and it could take any value between 0 and 9 in each cell. The ecosystem had a fixed population of 200 agents, each controlled by a recurrent, asymmetrically connected neural network (figure 1b). The ecosystem evolved by iterative update: each update allowed every agent the opportunity to sense its environment and act. Sensory input to a given agent was defined by its ‘field of view’, which included the current cell and the three cells directly ahead. The information gathered from each visible cell was the food quantity n, which consisted in a binary vector with n randomly chosen active neurons. A similar coding scheme has been used by Verguts & Fias [41]; however, it is important to emphasize that this coding strategy dispenses with the problem of normalizing object size [24,42]. Finally, each agent possessed a basic repertoire of actions, encoded at the level of the effector neurons: that is, they could turn left or right, move forward or eat.

Importantly, the agents' neural network was shaped by evolution rather than by lifelong learning. The evolutionary process was controlled by a genetic algorithm, which included crossover and mutation operators (for details, see [40]). The agent's genome determined the connection weights and the size of the hidden layer, which were initially set to random values. The goal of the evolutionary process was to promote the emergence of agents that forage in a quantity-sensitive manner, which can be achieved by defining as ‘fitness function’ the rate at which the agent collects food. At each iteration of the ecosystem, two ‘parents’ were randomly selected from the population and the weaker of the two (in terms of fitness value) was replaced by their ‘child’, which was defined by mixing the parents' genomes and applying a small random mutation. As typical in evolutionary simulations, the average efficiency of food collection increased over time and it became high after several million iterations. At that point, the agents' behaviour was tested systematically in terms of their quantity comparison performance. Each agent was removed from the ecosystem and placed into a 3 × 3 test environment (figure 1c), where the top left and top right cells contained food of varying quantity. Each trial started with the agent in the centre position, where it could ‘sense’ both quantities, and it was allowed to move until it selected one of the two food cells. This method mimicked that used by Uller et al. [37] to capture quantity comparison performance in salamanders. Accordingly, a correct choice was defined as the selection of the larger of the two food values. Each agent in the final population was tested with every combination of food quantities for 50 repetitions, for a total of 3600 trials per agent.

Though many agents performed above the chance level, few agents performed at chance (approx. 20%). The persistence of non-discriminating agents in the population is interesting because it shows that relatively high rates of food collection can be achieved without quantity discrimination, for example, by trading off decision quality in favour of decision speed. Nevertheless, the population included agents that were highly accurate in quantity discrimination.

Notably, accuracy decreased as the two numbers became larger (i.e. size effect) and it was strongly modulated by numerical distance as typically observed in behavioural studies on animals and humans [43]. The distance effect on both accuracy and response time (number of time-steps until selection of the food cell) is shown in figure 1d. Moreover, analyses of the internal dynamics of the neural network revealed that the emergent internal representation of quantity had the form of ‘summation coding’ [42,44], whereby neuronal activity increases or decreases monotonically as a function of numerical magnitude (see examples in figure 1e). Numerosity-sensitive neurons with this type of tuning property have been found in the lateral intraparietal area of the monkey brain [45]. This format is broadly consistent with the accumulator model of Meck & Church [46], as well as with the ‘summation clusters’ in the neural network model of Dehaene & Changeux [42]. As we will discuss below, the same type of coding also emerges from unsupervised deep learning on images of object sets.

In summary, the simulations reviewed above show that representational innatism is viable in the context of evolutionary pressure. However, these results have two important limitations. First, as noted above, the simulations dispense with the non-trivial problem of extracting numerosity information in a way that is invariant from covarying continuous visual properties (cumulative area, object size, density, etc.). Whether evolutionary simulations embedding a more realistic sensory input would still show the emergence of numerosity representations remains an issue that requires further investigation. Second, these simulations do not account for the ontogeny of numerical abilities. Indeed, the improvement of number acuity observed during early development in both humans [7] and animal species (e.g. in fish, see [47]) suggests that other mechanisms are also at play. The simulations reported in the following section investigate the role of experience-dependent learning in determining the development of number acuity.

3. Number sense emerges from architectural and learning constraints

The nature of the mechanisms underlying numerosity perception has been debated for decades [5,6,42,48,49]. However, the advent of a new generation of artificial neural networks, known as deep learning models [50], has provided modellers the unique opportunity to investigate the emergence of high-level visual skills using realistic sensory input [51]. This framework has been exploited by Stoianov & Zorzi [24] to investigate the emergence of a visual number sense. Their simulations, as well as novel simulations described below, are characterized by two key ingredients: generative learning and hierarchical processing. Intuitively, generative learning corresponds to learning by observation; unlike discriminative learning, there is no supervision or reward because the objective of learning is simply to build an internal model by discovering features or latent causes of the sensory information [52]. In other words, there is no task and the neural network does not receive any information about what is in the input (i.e. there is no feedback/supervision signal: all training data are unlabelled). Generative learning becomes particularly powerful when embedded into a hierarchical architecture, where many layers of neurons form a deep neural network [51,53], also known as ‘deep belief network’ [52].

Unsupervised learning in deep neural networks has provided a state-of-the-art and neurobiologically plausible account of how visual numerosity is extracted from real images of object sets [24]. Numerosity emerged as a high-order statistical property of images in a deep network, which learned a hierarchical generative model of the sensory input. The key idea was that numerosity is a statistical invariant of highly variable visual input, and for this reason, it might be encoded as a high-order visual feature (summary statistics) in a deep neural network that simply ‘observes’ images of object sets with variable numerosity. The hypothesis was therefore that numerosity is a latent factor that explains variability in the images of sets of objects. As a result of this unsupervised learning, numerosity-sensitive neurons emerged in the deepest layer of the network, with tuning functions resembling summation coding as observed in the lateral intraparietal area of the monkey brain [45]. The population code provided by number-sensitive neurons in the model was found to be largely invariant to continuous visual properties, and it supported numerosity estimation with the same behavioural signature (i.e. Weber's Law for numbers) and accuracy level (i.e. number acuity) of human adults. Analyses of the emergent computations in the model showed that numerosity was abstracted from lower-level visual primitives through a simple two-level hierarchical process, which exploited cumulative surface area as a normalization signal (also see [54,55]).

In the simulations reported below, we addressed more directly the question of how much sensory experience is necessary for observing number-sensitive behaviour in a deep network. In particular, we pursued the hypothesis that sensitivity to numerosity might emerge in a hierarchical architecture before any sensory experience of object sets, provided that it is endowed with basic visuospatial processing mechanisms. We then investigated how subsequent experience-dependent learning would further shape numerosity representations, thereby leading to the progressive improvement in numerosity discrimination performance.

4. Conclusion and future directions

During the last decades, an impressive amount of empirical research has shown that non-verbal numerical abilities are widespread within the animal kingdom. These findings have been usually interpreted as evidence for evolved, biologically determined numerical capacities across unrelated species, thereby supporting a ‘nativist’ stance on the origin of number sense. In this article, we have framed the problem of the origin of number sense within an ‘emergentist’ perspective [25], whereby the neural processing systems that implement perception and cognition might be shaped by a variety of constraints, which include evolutionary, architectural and learning biases.

We have shown that, although numerical abilities can be supported by domain-specific representations emerging from evolutionary pressure (as in the simulations with the artificial ecosystem), numerical representations need not be genetically pre-determined as such. Indeed, they can also emerge from the interplay between innate constraints (e.g. simple visuospatial processing embedded into a hierarchical architecture) and domain-general learning mechanisms (e.g. unsupervised learning of an internal model of the environment). Our computer simulations demonstrated that multi-layer (deep) neural networks endowed with only basic high-frequency spatial filters exhibit a remarkable performance in numerosity discrimination. Moreover, following exposure to sets of visual objects, the network gradually refined its internal representation of numerosity, thereby improving discrimination performance up to the level of adult human observers. Thus, our simulations are the proof-of-concept that a form of innatism based on architectural and learning biases is a viable approach to understanding the origin and development of number sense across species.

It should be noted that one possible issue with connectionist models is that several modelling choices, such as the format of the input and output representations [85] or the particular choice of learning algorithm [86], may have a crucial impact on the model's behaviour. One strength of the simulations presented in this article is that we made virtually no assumptions about the input/output representation format (e.g. the visual stimuli were real images, encoded at the pixel level) and the learning algorithm (e.g. internal representations in the model emerged from probabilistic, generative learning on the sensory signals, in line with modern theories of cortical learning [87–89]). Notably, a visual number sense emerges even when the deep network is exposed to a more ecological set of images, that is, when the size and the displacement of the items is obtained by segmenting objects in natural scenes (WY Zou, A Testolin, JL McClelland 2017, submitted). However, it is still to be shown whether a further refinement of numerical representations could be boosted by ‘recycling’ [90] generic visual features learned from natural images, as recently shown in the domain of visual letter recognition [91].

Key issues remain to be addressed in future research. For example, it should be stressed that our behavioural task was implemented by training a supervised read-out layer on the internal representations developed by the deep network. Though this is compatible with common experimental procedures, where explicit feedback is provided to the subjects [67–70,72], many studies carried out with newborns and infants are instead based on habituation paradigms [15–17]. There have been concrete proposals about how to simulate habituation tasks using artificial neural networks [92,93], but they have not yet been exploited in the field of numerical cognition.

A promising research direction would also be to more carefully investigate how the basic visuospatial filtering implemented in the first hidden layer of our model relates to spatial acuity in newborns. In our simulations, this early processing stage was fixed for simplicity, but it would be more realistic if spatial acuity also could change during development [57]. Although in principle the refinement of early processing stages should still be supported by unsupervised learning (i.e. it should happen independently from explicit numerical training), simulating the joint development of all hidden layers of a deep neural network is challenging because it requires a progressive learning algorithm (WY Zou, A Testolin, JL McClelland 2017, submitted).

Another interesting question relates to the computational properties of random matrices. How is it possible that random projections, such as those implemented at the deepest layer of our ‘initial’ network, create internal states that can be meaningfully read out by a supervised classifier? Although the advantages of transforming input data using random mappings have been explored in several machine learning algorithms [94,95], a parallel with neurocognitive models has not yet been clearly established. A mathematical motivation for the surprising effectiveness of random projections is based on the Johnson–Lindenstrauss theorem, which suggests that good representations for classification and discrimination of visual objects can indeed be obtained by dot products of the image with random templates, because the latter provide a quasi-isometric embedding of images [96]. However, understanding how this theory extends to the case of visual numerosity, which implies a different type of variability in the sensory input with respect to the case of object recognition, is still an uncharted territory.

Finally, one of the most pressing questions to be addressed in future research is whether the generic processing and learning constraints incorporated in our model would suffice even for developing more sophisticated types of numerical abilities, such as those underlying symbolic quantification and arithmetic, which likely require cultural mediation [97] and whose acquisition profoundly reshapes our brain [98]. Mathematical thinking is a hallmark of human intelligence and one of the most impressive achievements of human cultural evolution, as well as a major target of educational efforts; a deeper understanding of its neurocomputational foundations is therefore the key to the possibility of formally assessing the impact of different learning strategies both for normal children and in remedial treatment of mathematical learning disorders.

Data accessibility

The data and complete source code of our model is available for download (http://ccnl.psy.unipd.it/research/deeplearning).

Authors' contributions

A.T. and M.Z. conceived the simulation, discussed the results and wrote the paper. A.T. ran the simulations and analysed the data.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by grants from the European Research Council (no. 210922) and by the University of Padova (Strategic Grant NEURAT) to M.Z.