Both humans and non-human animals exhibit sensitivity to the approximate number of items in a visual array, as indexed by their performance in numerosity discrimination tasks, and even neonates can detect changes in numerosity. These findings are often interpreted as evidence for an innate 'number sense'. However, recent simulation work has challenged this view by showing that human-like sensitivity to numerosity can emerge in deep neural networks that build an internal model of the sensory data. This emergentist perspective posits a central role for experience in shaping our number sense and might explain why numerical acuity progressively increases over the course of development. Here we substantiate this hypothesis by introducing a progressive unsupervised deep learning algorithm, which allows us to model the development of numerical acuity through experience. We also investigate how the statistical distribution of numerical and non-numerical features in natural environments affects the emergence of numerosity representations in the computational model. Our simulations show that deep networks can exhibit numerosity sensitivity prior to any training, as well as a progressive developmental refinement that is modulated by the statistical structure of the learning environment. To validate our simulations, we offer a refinement to the quantitative characterization of the developmental patterns observed in human children. Overall, our findings suggest that it may not be necessary to assume that animals are endowed with a dedicated system for processing numerosity, since domain-general learning mechanisms can capture key characteristics others have attributed to an evolutionarily specialized number system.