This paper traces the reception of Babylonian astronomy into the history of science, beginning in early to mid twentieth century when cuneiform astronomical sources became available to the scholarly public. The dominant positivism in philosophy of science of this time influenced criteria employed in defining and demarcating science by historians, resulting in a persistently negative assessment of the nature of knowledge evidenced in cuneiform sources. Ancient Near Eastern astronomy (and astrology) was deemed pre- or non-scientific, and even taken to reflect a stage in the evolution of thought before the emergence of science (in ancient Greece). Two principal objections are examined: first, that the Near East produced merely practical as opposed to theoretical knowledge and, second, that astronomy was in the service of astrology and religion. As the notion of a universal scientific method has been dismantled by post-positivists and constructivists of the second half of the twentieth century, an interest in varieties of intellectual and cultural contexts for science has provided a new ground for the re-consideration of Babylonian astronomical texts as science developed here.