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Many species of animals exhibit an intuitive sense of number, suggesting a fundamental neural mechanism for representing numerosity in a visual scene. Recent empirical studies demonstrate that early feedforward visual responses are sensitive to numerosity of a dot array but substantially less so to continuous dimensions orthogonal to numerosity, such as size and spacing of the dots. However, the mechanisms that extract numerosity are unknown. Here, we identified the core neurocomputational principles underlying these effects: (1) center-surround contrast filters; (2) at different spatial scales; with (3) divisive normalization across network units. In an untrained computational model, these principles eliminated sensitivity to size and spacing, making numerosity the main determinant of the neuronal response magnitude. Moreover, a model implementation of these principles explained both well-known and relatively novel illusions of numerosity perception across space and time. This supports the conclusion that the neural structures and feedforward processes that encode numerosity naturally produce visual illusions of numerosity. Taken together, these results identify a set of neurocomputational properties that gives rise to the ubiquity of the number sense in the animal kingdom.