A Number Sense as an Emergent Property
of the Manipulating Brain

Mathematics, one of the most distinctive expressions of human intelligence, is founded on the ability to reason about abstract entities. We are interested in the question of how humans develop an intuitive facility with numbers and quantities, and how they come to recognize numbers as an abstract property of sets of objects. There is wide agreement that innate mechanisms play a strong role in developing a number sense [1, 2, 3], that development and learning also play an important role [2], that naming numbers is not necessary for the perception of quantities [4, 5], and a number of brain areas are involved in processing numbers [6, 7]. Quantity-tuned units have been described in physiology  experiments [3, 8, 9, 10] as well as in computational studies [11, 12, 13, 14].

The role of learning in developing abilities that relate to the natural numbers and estimation has been recently explored using computational models. Fang et al. [15] trained a recurrent neural network to count sequentially and Sabathiel et al. [16] showed that a neural network can be trained to anticipate the actions of a teacher on three counting-related tasks – they find that specific patterns of activity in the network’s units correlate with quantities. The ability to perceive numerosity, i.e. a rough estimate of the number of objects in a set, was explored by Stoianov, Zorzi and Testolin [11, 12], who trained a deep network encoder to efficiently reconstruct patterns composed of dots, and found that the network developed units or “neurons” that were coarsely tuned to quantity, and by Nasr et al. [13], who found the same effect in a deep neural network that was trained on visual object classification, an unrelated task. In these models quantity-sensitive units are an emergent property. In a recent study, Kim et al. [14] observed that a random network with no training will exhibit quantity-sensitive units. After identifying these units, [11, 12, 13, 14] train a supervised classifier on a two-set comparison task to assess numerosity properties encoded by the deep networks. These works showed that training a classifier with supervision, in which the classifier is trained and evaluated on the same task and data distribution, is sufficient for recruiting quantity-tuned units for relative numerosity comparison. Our work focuses on this supervised second stage. Can more be learned with less supervision? We show that a representation for numerosity, that generalizes to several tasks and extrapolates to large quntities, may arise through a simple, supervised pre-training task. In contrast to prior work, our pre-training task only contains scenes with up to 3 objects, and our model generalizes to scenes with up to 30 objects.

We focus on the interplay of action and perception as a possible avenue for this to happen. More specifically, we explore whether perception, as it is naturally trained during object manipulation, may develop representations that support a number sense. In order to test this hypothesis we propose a model where perception learns how specific actions modify the world. The model shows that perception develops a representation of the scene which, as an emergent property, can enable the ability to perceive numbers and estimate quantities at a glance [17, 18].

In order to ground intuition, consider a child who has learned to pick up objects, one at a time, and let them go at a chosen location. Imagine the child sitting comfortably and playing with small toys (acorns, Legos, sea shells) which may be dropped into a bowl. We will assume that the child has already learned to perform at will, and tell apart, three distinct operations (Fig. 2A). The put (P) operation consists of picking up an object from the surrounding space and dropping it into the bowl. The take (T) operation consists in doing the opposite: picking up an object from the bowl and discarding it. The shake (S) operation consists of agitating the bowl so that the objects inside change their position randomly without falling out. Objects in the bowl may be randomly moved during put and take as well.

We hypothesize that the visual system of the learner is engaged in observing the scene, and its goal is predicting the action that has taken place [19]as a result of manipulation. By comparing its prediction with a copy of the action signal from the motor system it may correct its perception, and improve the accuracy of its predictions over time. Thus, by performing P, T, and S actions in a random sequence, manipulation generates a sequence of labeled two-set comparisons to learn from.

We assume two trainable modules in the visual system: a “perception” module that produces a representation of the scene, and a “classification” module that compares representations and guesses the action (Fig. 2).

During development, perceptual maps emerge, capable of processing various scene properties. These range from basic elements like orientation [20] and boundaries [21] to more complex features such as faces [22] and objects [23, 24]. We propose that, while the child is playing, the visual system is being trained to use one or more such maps to build a representation that facilitates the comparison of the pair of images that are seen before and after a manipulation. These representations are often called embeddings in machine learning.

A classifier network is simultaneously trained to predict the action (P, T, S) from the representation of the pair of images (see Fig. 2). As a result, the visual system is progressively trained through spontaneous play to predict (or, more accurately, post-dict) which operation took place that changed the appearance of the bowl.

We postulate that signals from the motor system are available to the visual system and are used as a supervisory signal (Fig. 2B). Such signals provide information regarding the three actions of put, take and shake and, accordingly, perception may be trained to predict these three actions. Importantly, no explicit signal indicating the number of objects in the scene is available to the visual system at any time.

Using a simple model of this putative mechanism, we find that the image representation that is being learned for classifying actions, simultaneously learns to represent and perceive the first few natural numbers, to place them in the correct order, from zero to one and beyond, as well as estimate the number of objects in the scene.

We use a standard deep learning model of perception [26, 27, 28]: a feature extraction stage is followed by a classifier (Fig. 2). The feature extraction stage maps the image $x$ to an internal representation $z$, often called an embedding. It is implemented by a deep network [27] composed of convolutional layers (CNN) followed by fully connected layers (FCN 1). The classifier, implemented with a simple fully connected network (FCN 2), compares the representations $z_{t}$ and $z_{t+1}$ of the before and after images to predict which action took place. Feature extraction and classification are trained jointly by minimizing the prediction error. We find that the embedding dimension makes little difference to the performance of the network (Fig. S3). Thus, for ease of visualization, we settled on two dimensions.

We carried out train-test experiments using sequences of synthetic images containing a small number of randomly arranged objects (Fig. 2). When training we limited the top number of objects to three (an arbitrary choice), and each pair of subsequent images was consistent with one of the manipulations (put, take, shake). We ran our experiments twice with different object statistics. In the first dataset the objects were identical squares, in the second they had variable size and contrast. In the following we refer to the model trained on the first dataset as Model A and the model trained on the second dataset as Model B.

The network we train is a standard deep network [28] composed of two stages. First, a feature extraction network maps the original image of the scene into an embedding space (Fig. 2A). Second, a classification network takes the embedding of two sequential images and predicts the action that modified the first into the second (Fig. 2B). Given the fact that the classification network takes the embedding of two distinct images as its input, each computed by identical copies of the feature extraction network, the latter is trained in a Siamese configuration [25].

The feature extraction network is a 9-layer CNN followed by two fully connected layers (details in Fig. S15A). The first 3 layers of the feature extraction network are from AlexNet [27] pre-trained on ImageNet [51] and are not updated during training. The remaining four convolutional layers and two fully connected layers are trained in our action prediction task.

The dimension of the output of the final layer is a free parameter (it corresponds to the number of features and to the dimension of the embedding space). In a control experiment we varied this dimension from one to 256, and found little difference in the action classification error rates (Fig. S3). We settled for a two-dimensional output for the experiments that are reported here.

The classification network is a two-layer fully connected network that outputs a three-dimensional one-hot-encoding vector indicating a put, take or shake action (details in Fig. S15B).

To visualize how well the model was able to perform the action classification task, we predict actions between pairs of images in our first test set. The error, calculated by comparing the ground truth actions to the predicted actions, is plotted with respect to the number of objects in the visual scene at $x_{t}$. 95% Bayesian confidence intervals with a uniform prior were computed for each data point, and a lower bound on the number of samples is provided in the figure captions (Figs. 3, S3, S6).

We first explored the structure of the embedding space by visualizing the image embeddings in two dimensions. The points, each one of which corresponds to one image, are not scattered across the embedding. Rather, they are organized into a structure that exhibits five salient features: (a) the images are arranged along a one-dimensional structure, (b) the ordering of the points along the line is (almost) monotonic with respect to the number of objects in the corresponding images, (c) images are separated into groups at one end of the embedding, and these groups are discovered by unsupervised learning, (d) these first few clusters are in one-to-one correspondence with the first few natural numbers, (e) there is a limit to how many number-specific clusters are discovered (Fig. 4).

To verify that the clusters can be recovered by unsupervised learning we applied a standard clustering algorithm, and found almost perfect correspondence between the clusters and the first few natural numbers (Fig. 4). The clustering algorithm used was the default Python implementation of HDBSCAN [53]. HDBSCAN is a hierarchical, density based clustering algorithm, and we used the euclidean distance as an underlying metric [54]. HDBSCAN has one main free parameter, the minimum cluster size, which was set to 90 in Figure 4. All other free parameters were left at their default values. Varying the minimum cluster size between 5 and 95 does not have an effect on the first few clusters, although it does create variation in the number and size of the later clusters. Beyond 95, the algorithm finds only three clusters corresponding to 0, 1 and greater than 1.

One additional structure is not evident from the the embedding and may be recovered from the action classifier: the connections between pairs of clusters. For any pair of images that are related by a manipulation, two computations will be simultaneously carried out; first, the supervised action classifier in the model will classify the action as either P, T, or S (Fig. 3) and, at the same time, the unsupervised subitization classifier (Fig. S5A) will assign each image in the pair to the corresponding number-specific cluster. As a result, each pair of images that is related by a P action provides a directed link between a pair of clusters (Fig. S5A, red arrows), and following such links one may traverse the sequence of numbers in an ascending order. The T actions provide the same ordering in reverse (blue arrows). Thus, the clusters corresponding to the first few natural numbers are strung together like the beads in a necklace, providing an unambiguous ordering that starts from zero and proceeds through one, two etc. (Fig. S5 A, B). The numbers may be visited both in ascending and descending order. As we pointed out earlier, the same organization may be be obtained more simply by recognizing that the clusters are spontaneously arranged along a line, which also supports the natural ordering of the numbers [55, 56, 40]. However, the connection between the order of the number concepts, and the actions of put and take, will support counting, sum and subtraction.

To estimate whether the embedding structure is approximately one dimensional and linear in higher dimensions we computed the one-dimensional linear approximation to the embedding line, and measured the average distortion of using such approximation for representing the points. More in detail, we first defined a mean-centered embedding matrix with M points and N dimensions, each point corresponding to the embedding of an image. We then computed the best rank 1 approximation to the data matrix by computing its singular value decomposition (SVD) and zeroing all the singular values beyond the first one. If the embedding is near linear, this rank 1 approximation should be quite similar to the original matrix. To quantify the difference between the original matrix and the approximation, we calculated the element-wise residual (the Frobenius norm of the difference between the original matrix and the approximation), then computed the ratio of the Frobenius norm of the residual matrix and the Frobenius norm of the original matrix. The nearer the ratio is to 0, the smaller the residual, and the better the rank 1 approximation. We call this ratio the linear approximation error, we show this error compared to some embeddings in Figure S7. We computed the embedding for dimensions 8, 16, 64, and 256, (one experiment each) and found ratios of 0.702%, 2.23%, 2.77%, and 2.24%, suggesting that they are close to linear.

We can use the perceived numerosity to reproduce a common task performed in human psychophysics. Subjects are asked to compare a reference image to a test image and respond in a two-alternative forced choice paradigm with “more” or “less”. We perform the same task using the magnitude of the embedding as the fiducial signal. The model responds with more if the embedding of the test image has a larger perceived numerosity than the reference image. The psychometric curves generated by our model are presented in Figure 5A and match qualitatively the available psychophysics [31, 34].

As described above, the clusters are spaced regularly along a line and the points in the embedding are ordered by the number of objects in the corresponding images (Fig. S5). We postulate that the number of objects in an image is proportional to the distance of that image’s embedding from the embedding of the empty image. Given the linear structure, any one of the embedding features, or their sum, may be used to estimate the position along the embedding line. In order to produce an estimate we use the embedding of the “zero” cluster as the origin. The zero cluster is special, and may be detected as such without supervision, because all its images are identical and thus it collapses to a point. The distance between “zero” and “one”, computed as the pairwise distance between points belonging to the corresponding clusters, provides a natural yardstick. This value, also learned without further supervision, can be used as a unit distance to to interpret the signal between 0 and n. This estimate of numerosity is shown in Figure 5B against the actual number of objects in the image. We draw two conclusions from this plot. First, our unsupervised model allows an estimate of numerosity that is quite accurate, within 10-15% of the actual number of objects. Second, the model produces a systematic underestimate, similar to what is observed psychophysically in human subjects [33].

Do image properties, other than the abstraction of “object number”, drive the quantity estimate of our model? Many potential confound variables, such as the count of pixels that are not black, are correlated with object number and might play a role in the model’s ability to estimate the number of objects in the scene. If that were the case, one might argue that our model is not learning the abstraction of “number”, but rather learning to measure image properties that are correlated with number.

We controlled for this hypothesis by exploiting the natural variability of our test set images. We explored three image properties that correlate with the number of objects and might thus be exploited to estimate the number of objects: (a) overall image brightness, (b) the area of the envelope of the objects in the image, and (c) the total number of pixels that differ from the background. Since objects in training set B vary both in size and in contrast, these three variables are not deterministically related to object number and thus, we reason, counfound variable fluctuations ought to affect error rates independently of the number of objects.

We focused on close-call relative estimate tasks (e.g. 16 vs 18 objects), where errors are frequent both for our model and for human subjects, and, while holding the number of objects constant in each of the two scenes being compared, we studied the behavior of error rates as a function of fluctuations in the confound variables. One would expect more errors when comparing image pairs where quantities that typically correlate with the number of objects are anticorrelated in the specific example (Fig. S1). Conversely, one would expect lower error rates when the confound variables are positively correlated with number.

In Fig. S2 error rates are plotted vs each one of the confound variables when the n. of objects is held constant. We could not find large systematic biases even for extreme variations in the confound variables. In conclusion, we do not find support for the argument that any of the confound variables we studied is implicated significantly in the estimate of quantity.

Does the dimension of the embedding space influence the action classification error? We wondered what is the effect of this free parameter on the model’s performance. We explored this question by training our model repeatedly with the same training images, and varying the dimension of the embedding (Fig. 2). Figure S3 shows that the effect of the embedding dimension is negligible. This was initially surprising to us. An explanation may be found in the fact that learning produces an embedding that is organized as a line (see Fig 4 and Sec. A.4).

Next, we explored the structure of the embedding space in the region where images containing 0-3 objects (the training range) are represented. As discussed in the main text we find that the embedding is organized into clusters (Fig. S5 (A,B)). Each cluster contains embeddings of images with the same number of objects. For each pair of images that were generated by a put action we drew a red arrow connecting the corresponding embeddings. We used blue arrows for take pairs. It is clear from the figure that by following the red arrows one may visit numbers in increasing order: 0-1-2-3 and vice-versa for blue arrows, i.e. the embedding that is produced by our model supports counting up and down.

In our main experiment we trained our model to classify actions with scenes containing from zero to three objects. Does this choice influence qualitatively or quantitatively our observations?

To explore this question we re-trained our model using images that were generated with a total number of three, five and eight objects. As expected, we find that adding more objects to the training images reduces the action classification error for image pairs with corresponding number of objects (Fig. S6). We find no change in the linearity of the embeddings, however, the number of clusters seems to increase with the training limit (Figs. S7A,B). This increase in clusters that corresponds with training limit likely explains the improvement in action classification performance.

The line-like organization of our embedding space is a striking feature. Is this the result of chance, or is this a robust feature that may be reproduced reliably?

We explored this question by repeating our experiments, varying each the random seed used to generate the training images, as well as the random seed used to initialize the model perception network’s weights. We show all the embeddings we obtained in Fig. S7. Each time we measured how line-like are the embeddings and we report the deviation from an exact line as a percent error below each embedding. We found that the deviations from a perfect line are very small, and most look perfectly linear with a few exceptions where we see slight kinks in the line.

In our main experiment the arrangement of the objects in the scene varied randomly between put, take and shake actions. The size and contrast were varied as well. This was because we did not wish to presume that the agent (a child) playing with the objects would have to be careful with their motions. Furthermore we did not wish to presume that lighting conditions, and thus image contrast, and object pose, and thus their apparent size, would be preserved during the play session. However, one may suspect that scene randomness could help the model abstract the concept of “number” without being distracted by other factors such as object placement, contrast and size.

We explored the effect of randomness by modifying the process that generates data for Model B. In dataset B, object properties (area, intensity) are completely randomized before and after an action (Fig. 2B). We thus constructed a new dataset (Fig. S8), where we restricted the randomness before and after an action by reducing the amount of change in an object’s area and intensity to a small amount of jitter. However, we still randomize object position, which we find is fundamental to learning a generalizable model of numerosity. We find that even after reducing object variation, the model has learned has the same properties as Model B (Fig. S9). However, learning is more sensitive to the initial seed (Fig. S10). We refer to this dataset as the jitter dataset and model’s trained by this dataset as Jitter Models.

Will our model learn the abstraction of “number” even when the put and take actions will place or remove an unpredictable random number of objects?

We explored this question by randomizing the number of objects that each action affects in the range 0-3, as opposed to exactly 1 as in the main experiment. We capped the maximum number of objects to 3, like previous experiments. We find that while precise actions help in building distinct clusters in the subitization range, it is not necessary to retain the important properties of the generalizable number line. We refer to this dataset as the imprecise actions dataset (Fig. S11) and model’s trained by this dataset as Imprecise Action Models. We find that all the properties of the original model retained (Fig. S12) and that the model is reproducible (Fig. S13).