 Philolaus’ most significant innovation, however, is to argue that the cosmos cannot be adequately explained in terms of unlimiteds alone. There appear to be two related arguments for this thesis. First, Philolaus takes it as self-evident that the world is full of limiters such as shapes. If this is so, however, we must suppose that limiters were among the basic principles from which the world arose, since it is impossible for what limits to arise from what is unlimited (Fr. 2). Second, in Fragment 3, he argues that if all things are unlimited then “there will not be anything that is going to know.” This argument makes most sense if directed against Anaxagoras, who generates the world from elements that are all unlimited (Fr. 1), but who claims that mind (nous), which is also said to be unlimited, knows all the things that are separated off from the original mixture (Fr. 12). Philolaus seems to regard the act of knowing as an act of limitation and hence as impossible, if reality is composed just of unlimiteds (Huffman 1993, 119–120). Parmenides is the only Presocratic before Philolaus to emphasize the role of limit in his account of reality. Parmenides likens reality to a sphere and describes it as held in the bonds of a limit (DK 28 B8). Philolaus’ insistence that the ultimate elements from which the cosmos developed must have included not just unlimiteds but also limiters is best understood as an attempt to integrate Parmenides’ insights about the nature of reality with the earlier Presocratic emphasis on material elements which are unlimited (Huffman 1999). Earlier Presocratics, such as Heraclitus (DK B30, B31), had recognized that there was measure and structure in the world but no Presocratic had made structure an element or principle in the world with the same status as material principles. It is Philolaus who first insists that structure is equal in importance to material constitution (Huffman 2013b; Graham 2014: 53). Philolaus’ explanation of the cosmos in terms of limiters and unlimiteds cannot be fully understood, however, without discussing his third basic principle, harmonia.