From an early age, humans learn that objects can be sorted into meaningful orderings. This might be on the basis of an obvious physical feature (like "size") or something more subtle and subjective (like "cost" or "preference"). A particular cognitive skill arises from this understanding: the ability to perform a transitive inference. Although long studied as a facet of human intelligence, many other species are also able to make this inference. When considering several different experimental manipulations, the best explanation for transitive inference in these species is a cognitive representation of order that can be reasoned about and manipulated as though it possesses spatial properties. This cognitive faculty has deep evolutionary roots that predate the talent for mathematical and linguistic abstraction that humans often take for granted. Simulations using a computational model that embodies this theory suggest new directions for the study of this fundamental cognitive mechanism.