 should not obscure that the validity of each move is established by induction. We establish that two plus one is equal to three by generalization from specific instances: that two pebbles and one pebble together make three pebbles, that two horses and one horse together make three horses, and so on. So too other such arithmetic laws. Of course, as is the case with our discovery of physical laws, the laws which we learn first are, as the “Science of Number” grows more sophisticated, subsumed as instances of more fundamental laws—and can eventually be shown to follow from the axioms of mathematics (System, VII: 253, 608ff.).