 Burkert argues that Philolaus’ appeal to number has little to do with the rigorous mathematics that was being developed in the work of his contemporaries Hippocrates of Chios and Theodorus of Cyrene. It is instead a manifestation of the Pythagorean number mysticism, which can be found in the acusmata, the oral apothegms, many of which probably go back to Pythagoras himself (1972a, 465 ff.). Burkert sees evidence that Philolaus was not interested in rigorous mathematics in the mathematically nonsensical account of music in Fr. 6b and testimonium A26 (1972a, 394–400) as well as in what might be the use of mathematics for astrological purposes in A14 (1972a, 349–350).  There are, however, independent reasons for doubting the authenticity of these texts (Huffman 1993, 364–374 and 381–391). Moreover, the mathematics of the surely authentic Fr. 6a is above reproach, and Aristotle clearly presents the interest in number displayed by fifth-century Pythagoreans as the result of their training in rigorous mathematics (Metaph. 985b23 ff.; cf. Iamblichus, Comm. Math. Sc. 78.8–18). Fragments 4 and 5 suggest that Philolaus’ program was to discover the numbers and numerical relations that govern the phenomena which we observe, and Fr. 6a suggests that the whole-number ratios which govern musical scales served as the model of the kind of mathematical account which should be supplied for all phenomena. In identifying numerical relations as the basis for knowledge Philolaus may in part be responding to Parmenides’ strictures on what constitutes a proper object of thought. Mathematical relationships certainly fit a number of the characteristics of a proper object of thought as set out by Parmenides in Fr. 8, e.g., they are ungenerated, imperishable, and unchangeable. Philolaus does not provide many further specific examples of mathematical relations controlling phenomena, which is not surprising given the scientific capabilities of the time. Indeed, the fact that the fragments and testimonia are not full of facile equations between numbers and phenomena once again suggests that Philolaus was not simply interested in symbolic connections between numbers and things and that Philolaus was looking for connections that were scientifically verifiable, as in the case of the ratios that govern musical intervals. Moreover, Greek usage shows that “having number” need not indicate just a simple one to one relation between a number and things but can also have a general sense indicating that something is constituted by an ordered plurality or relations between ordered pluralities (Huffman 1993: 175). The musical scale is one example but the alphabet may be another so that in Philolaus’ epistemology science may involve a classification and systematization of a subject that has similarities to Plato’s collection and division (Graham 2014: 55). His successor in the Pythagorean tradition, Archytas of Tarentum, appears to have adopted and been even more successful in carrying out Philolaus’ project of accounting for things in the phenomenal world in terms of number. In some cases, however, Philolaus may have been content to accept similarities between the properties of number and the properties of things. Aristotle reports that the Pythagoreans identified individual numbers with certain concepts, e.g., the number four is associated with justice (Metaph. 985b29), evidently because it is the first square number (cf. the English expression “a square deal”). It is not certain that this is a reference to Philolaus, but, when Aristotle says that the counter-earth was introduced into the Pythagorean astronomical system in order to bring the number of bodies orbiting the central fire up to the perfect number ten, this must apply to Philolaus, since the astronomical system described belongs to him (see 4.2 below). Since the world of numbers is ordered in tens, the cosmos too must parallel this order. (For further analysis of Philolaus’ use of mathematics see Huffman 1988 and 1993, 54–77.) Zhmud (2016) argues that, while Philolaus and the early Pythagoreans had some interest in the number symbolism that is found in many cultures, they did not engage in arithmology, which focuses on the special properties of the first ten numbers. Certainly it is true that there is no evidence for anything as elaborate as the the later arithmological treatises such as the Theologumena Arithmeticae. However, in order to deny any special significance for the numbers from one to ten in Philolaus one has to dismiss clear Aristotelian evidence and Philolaus’ own use of four psychic faculties and ten bodies rotating around the central fire. 