Scholars tend to assume that the mathematical sciences and philosophy were distinct disciplines in antiquity, as they are today. From the fourth century B.C.E. onward, mathematicians and philosophers did distinguish themselves. They criticized each other’s work and, in some areas of the Greek world, strong rivalries developed between philosophers and mathematicians. I argue, however, that the distinction between philosophers and mathematicians did not entail that their fields of inquiry were distinct. This chapter examines the relationship between the mathematical sciences and philosophy from the perspective of the practitioners of the mathematical sciences, in particular, Archytas of Tarentum, Hero of Alexandria, and Claudius Ptolemy. I argue that these practitioners viewed the relationship between the mathematical sciences and philosophy as more complex, where the mathematical sciences are not only in relation to philosophy but, even stronger, forms of philosophy.  Moreover, the mathematical sciences answer some of the most fundamental questions of philosophy, e.g., how to obtain knowledge, how to form a just society, and how to attain the good life.

The idea that the universe is infinite in size is central to the Epicurean system. Infinity, however, is also a concept that, in the history of philosophical, scientific and artistic discussion before and after Lucretius, has defied explanation, engendered paradox, and stimulated the romantic sensibility. This chapter looks at the strategies, philosophical and literary, deployed by Lucretius to achieve closure on this inherently open topic. Co-ordinate with the relationships of analogy and complementarity between the text of the DRN and the nature of the universe it describes, the poem’s poetics of closure and enclosure, on the one hand, and of non-closure or ‘false closure’, on the other, express and enact the infinity of the universe conceived both as all-encompassing and as open-ended. These rival conceptions of infinity are modeled throughout the poem, and especially in Epicurus’ triumph of the mind (1.62–79) and Lucretius’ reworking of a thought-experiment attributed to Archytas of Tarentum (1.951–83). Taken together, they bring out the tension, or complementarity, in Lucretius between the totalizing scientist who prescribes an intellectual panacea and the sublime poet who reaches into the beyond.