  The ability to understand and manipulate numbers and quantities emerges
during childhood, but the mechanism through which humans acquire and develop
this ability is still poorly understood. We explore this question through a
model, assuming that the learner is able to pick up and place small objects
from, and to, locations of its choosing, and will spontaneously engage in such
undirected manipulation. We further assume that the learner's visual system
will monitor the changing arrangements of objects in the scene and will learn
to predict the effects of each action by comparing perception with a
supervisory signal from the motor system. We model perception using standard
deep networks for feature extraction and classification, and gradient descent
learning. Our main finding is that, from learning the task of action
prediction, an unexpected image representation emerges exhibiting regularities
that foreshadow the perception and representation of numbers and quantity.
These include distinct categories for zero and the first few natural numbers, a
strict ordering of the numbers, and a one-dimensional signal that correlates
with numerical quantity. As a result, our model acquires the ability to
estimate numerosity, i.e. the number of objects in the scene, as well as
subitization, i.e. the ability to recognize at a glance the exact number of
objects in small scenes. Remarkably, subitization and numerosity estimation
extrapolate to scenes containing many objects, far beyond the three objects
used during training. We conclude that important aspects of a facility with
numbers and quantities may be learned with supervision from a simple
pre-training task. Our observations suggest that cross-modal learning is a
powerful learning mechanism that may be harnessed in artificial intelligence.
