 On the other hand, Aristotle always presents the philosophical system of limiters and unlimiteds, which placed an emphasis on the role of number in understanding the cosmos and which included the astronomical system built around the central fire, as the work of “the so-called Pythagoreans,” whom he dates to the second half of the fifth century, at the same time as or a little earlier than the atomists (e.g. Metaph. 985b23 ff.). This is the philosophical system that we find in the fragments of Philolaus, and Aristotle’s dating fits Philolaus exactly so there can be little doubt that Philolaus was the leading figure among those labeled “the so-called Pythagoreans” by Aristotle. Although some scholars continue to argue that most of what Aristotle ascribes to “the so-called Pythagoreans”should be regarded as Pythagoras’ own teaching (Riedweg 2005, 77–80), Aristotle’s testimony doubly distances Pythagoras himself from the Philolaic system. First, Aristotle is emphatic that the system dates after the time of Pythagoras, and second, his expression “so-called Pythagoreans” is most reasonably understood to suggest that, although he knows that the name “Pythagorean” is often given to Philolaus and his contemporaries, he can see little reason to connect the philosophical system they espouse to Pythagoras himself. The later tradition also emphasizes Philolaus’ originality in the area of cosmology by asserting that he was the first Pythagorean to publish a work on nature (D. L. 8.85). The testimony of Aristotle and the later tradition is largely confirmed by the content of Philolaus’ philosophy. Although the contrast between limit and unlimited is implicit from the beginning of the Presocratic tradition, the first explicit champion of limit is Parmenides, and the attempt to explain the world in terms of a combination of limiters and unlimiteds thus makes most sense as arising after Parmenides has championed limit (Huffman 1999). Moreover, the style of argumentation in Fragments 1–6, and particularly Fr. 2, is reminiscent of Parmenides and his successor Melissus (Kahn 2001, 34), so that the central metaphysical system and the arguments for it look very much as if they are the original contribution of Philolaus. The same is true of the astronomical system built around the central fire, which the later tradition assigns to Philolaus and which Aristotle (Metaph. 986a8 ff.) and Fr. 7 explicitly tie to the philosophy of limiters and unlimiteds. Kahn is inclined to regard parts of the cosmogony as going back to Pythagoras. He argues that the idea of the cosmos breathing is archaic and that it may have been attacked long before Philolaus by Xenophanes, who pointedly describes his god as not breathing; Kahn also argues that the equivalence between air and void suggests a date before Anaxagoras (2001, 36–7). In section 4.1 above, however, reasons have been given to show that the Philolaic system, in fact, distinguishes between air and void and hence once again conforms very well to Philolaus’ date after Anaxagoras. The breathing of the cosmos need not be seen as archaic, since biological analogies are still found in the cosmogony of the atomists (DK 67 A1). Xenophanes’ remark is reported in only one testimonium (DK 21 A1) and is not found in any of the extant fragments of his book; if it is his, it need not be directed at anything other than general Greek anthropomorphism.