Understanding the Countably Infinite: Neural Network Models of the Successor Function and its Acquisition

Abstract: As children enter elementary school, their understanding of the ordinal structure of numbers transitions from a memorized count list of the first 50-100 numbers to knowing the successor function and understanding the countably infinite. We investigate this developmental change in two neural network models that learn the successor function on the pairs (N, N+1) for N in (0, 98). The first uses a one-hot encoding of the input and output values and corresponds to children memorizing a count list, while the second model uses a place-value encoding and corresponds to children learning the language rules for naming numbers. The place-value model showed a predicted drop in representational similarity across tens boundaries. Counting across a tens boundary can be understood as a vector operation in 2D space, where the numbers with the same tens place are organized in a linearly separable manner, whereas those with the same ones place are grouped together. A curriculum learning simulation shows that, in the expanding numerical environment of the developing child, representations of smaller numbers continue to be sharpened even as larger numbers begin to be learned. These models set the stage for future work using recurrent architectures to move beyond learning the successor function to simulating the counting process more generally, and point towards a deeper understanding of what it means to understand the countably infinite.