  If Pythagoras was primarily a figure of religious and ethical significance, who left behind an influential way of life and for whom number and cosmology primarily had significance in this religious and moral context, how are we to explain the prominence of rigorous mathematics and mathematical cosmology in later Pythagoreans such as Philolaus and Archytas? It is important to note that this is not just a question asked by modern scholars but was already a central question in the fourth century BCE. What is the connection between Pythagoras and fifth-century Pythagoreans? The question is implicit in Aristotle’s description of the fifth-century Pythagoreans such as Philolaus as “the so-called Pythagoreans.” This expression is most easily understood as expressing Aristotle’s recognition that these people were called Pythagoreans and at the same time his puzzlement as to what connection there could be between the wonder-worker who promulgated the acusmata, which his researches show Pythagoras to have been, and the philosophy of limiters and unlimiteds put forth in fifth-century Pythagoreanism. The tradition of a split between two groups of Pythagoreans in the fifth century, the mathematici and the acusmatici, points to the same puzzlement. The evidence for this split is quite confused in the later tradition, but Burkert (1972a, 192 ff.) has shown that the original and most objective account of the split is found in a passage of Aristotle’s book on the Pythagoreans, which is preserved in Iamblichus (On Common Mathematical Science, 76.19 ff). The acusmatici, who are clearly connected by their name to the acusmata, are recognized by the other group, the mathematici, as genuine Pythagoreans, but the acusmatici do not regard the philosophy of the mathematici as deriving from Pythagoras but rather from Hippasus. The mathematici appear to have argued that, while the acusmatici were indeed Pythagoreans, it was the mathematici who were the true Pythagoreans; Pythagoras gave the acusmata to those who did not have the time to study the mathematical sciences, so that they would at least have moral guidance, while to those who had the time to fully devote themselves to Pythagoreanism he gave training in the mathematical sciences, which explained the reasons for this guidance. This tradition thus shows that all agreed that the acusmata represented the teaching of Pythagoras, but that some regarded the mathematical work associated with the mathematici as not deriving from Pythagoras himself, but rather from Hippasus (on the controversy about the evidence for this split into two groups of Pythagoreans see the fifth paragraph of section 4.3 above). For fourth-century Greeks as for modern scholars, the question is whether the mathematical and scientific side of later Pythagoreanism derived from Pythagoras or not. If there were no intelligible way to understand how later Pythagoreanism could have arisen out of the Pythagoreanism of the acusmata, the puzzle of Pythagoras’ relation to the later tradition would be insoluble. The cosmos of the acusmata, however, clearly shows a belief in a world structured according to mathematics, and some of the evidence for this belief may have been drawn from genuine mathematical truths such as those embodied in the “Pythagorean” theorem and the relation of whole number ratios to musical concords. Even if Pythagoras’ cosmos was of primarily moral and symbolic significance, these strands of mathematical truth, which were woven into it, would provide the seeds from which later Pythagoreanism grew. Philolaus’ cosmos and his metaphysical system, in which all things arise from limiters and unlimiteds and are known through numbers, are not stolen from Pythagoras. They embody a conception of mathematics, which owes much to the more rigorous mathematics of Hippocrates of Chios in the middle of the fifth century; the contrast between limiter and unlimited makes most sense after Parmenides’ emphasis on the role of limit in the first part of the fifth century. Philolaus’ system is nonetheless an intelligible development of the reverence for mathematical truth found in Pythagoras’ own cosmological scheme, which is embodied in the acusmata.