The argument from design stands as one of the most intuitively compelling arguments for the existence of a divine Creator. Yet, for many scientists and philosophers, Hume's critique and Darwin's theory of natural selection have definitely undermined the idea that we can draw any analogy from design in artifacts to design in nature. Here, we examine empirical studies from developmental and experimental psychology to investigate the cognitive basis of the design argument. From this it becomes clear that humans spontaneously discern purpose in nature. When constructed theologically and philosophically correctly, the design argument is not presented as conclusive evidence for God's existence but rather as an abductive, probabilistic argument. We examine the cognitive basis of probabilistic judgments in relationship to natural theology. Placing emphasis on how people assess improbable events, we clarify the intuitive appeal of Paley's watch analogy. We conclude that the reason why some scientists find the design argument compelling and others do not lies not in any intrinsic differences in assessing design in nature but rather in the prior probability they place on complexity being produced by chance events or by a Creator. This difference provides atheists and theists with a rational basis for disagreement.