  The unprecedented performance achieved by deep convolutional neural networks
for image classification is linked primarily to their ability of capturing rich
structural features at various layers within networks. Here we design a series
of experiments, inspired by children's learning of the arithmetic addition of
two integers, to showcase that such deep networks can go beyond the structural
features to learn deeper knowledge. In our experiments, a set of images is
constructed, each image containing an arithmetic addition $n+m$ in its central
area, and several classification networks are then trained over a subset of
images, using the sum as the label. Tests on the excluded images show that, as
the image set gets larger, the networks have well learnt the law of arithmetic
additions so as to build up their autonomous reasoning ability strongly. For
instance, networks trained over a small percentage of images can classify a big
majority of the remaining images correctly, and many arithmetic additions
involving some integers that have never been seen during the training can also
be solved correctly by the trained networks.
