 The biggest puzzle, however, concerns the philosophical system that Aristotle assigns to the Pythagoreans. For the purposes of his discussion in the Metaphysics, he treats most Pythagoreans as adopting a mainstream system in contrast to another group of Pythagoreans whose system is based on the table of opposites (see section 2.4). The central thesis of the mainstream system is stated in two basic ways: the Pythagoreans say that things are numbers or that they are made out of numbers. In his most extended account of the system in Metphysics 1.5, Aristotle says that the Pythagoreans were led to this view by noticing more similarities between things and numbers than between things and the elements, such as fire and water, adopted by earlier thinkers. The Pythagoreans thus concluded that things were or were made of numbers and that the principles of numbers, the odd and the even, are principles of all things. The odd is limited and the even unlimited. Aristotle criticizes the Pythagoreans for being so enamored of numerical order that they imposed it on the world even where it was not suggested by the phenomena. Thus appearances suggested that there were nine heavenly bodies orbiting in the heavens but, since they regarded ten as the perfect number, they supposed that there must be a tenth heavenly body, the counter-earth, which we cannot see. Later, Aristotle is also critical of the Pythagoreans for employing principles that do not derive from the sensible world, i.e., mathematical principles, even though all their efforts were directed at explaining the physical world (Metaphysics 989b29). How can they explain features of physical bodies such as weight or motion using principles which have no weight and do not move (990a8–990a16)? Indeed, it becomes clear that Aristotle interpreted the Pythagorean cosmogony as starting out by constructing the number one. The one then draws in the unlimited and produces the rest of the number series and evidently the cosmos at the same time. The number one and the other numbers from 1 to 10 are conceived of as physical entities (Metaphysics 1091a13–18). The puzzle is that Aristotle’s description makes clear that he is basically describing Philolaus’ system (e.g., the counter-earth, limit and unlimited, the generation of a one), yet a number of his central assertions are flatly contradicted by the surviving fragments of Philolaus. Most importantly, Philolaus never says that things are numbers or are made out of numbers. For Philolaus things are composed of limiters and unlimiteds held together by harmony (Frs. 1, 2 and 6) and unlimiteds appear to include physical things like fire and breath (Fr. 7, Aristotle Fr. 201). Numbers and the odd and the even do play a prominent role in Philolaus (Frs. 4–5), but there is no hint that they are understood as physical entites. Instead number has an epistemological role: all things are known through number (Fr. 4). How are we to explain this tension between what Aristotle reports and the fragments of Philolaus? One approach is to recognize that Aristotle is not giving a historical report of what the Pythagoreans said but an interpretation of what he found in Philolaus and others. He does not in fact know of any text in which the Pythagoreans said that things were numbers or were made of numbers. Instead this is a conclusion drawn by Aristotle; it is his summary statement of what the Pythagorean system amounts to. That this is what Aristotle is doing is suggested by another passage in the Metaphysics where he starts out by flatly stating that the Pythagoreans say that all things are numbers but then goes on to add “at least they apply mathematical theories to bodies as if they (the bodies) consisted of those numbers”(Metaphysics 1083b16). The “at least” and “as if” show that Aristotle is drawing an inference rather than referring to any explicit statement by the Pythagoreans that things are numbers. Thus for Philolaus there are analogies between numbers and things and numbers give us knowledge of things but Aristotle mistakenly takes this to be equivalent to saying that things are numbers or are made of numbers. Another approach is to argue that Aristotle was right that Philolaus and other Pythagoreans thought of the number one and other numbers as physical entities. The one constructed in Philolaus Fr. 7 is not just the primal physical unity but also the number one (Schofield 2012). At the opposite extreme, Zhmud argues that Aristotle has essentially invented this Pythagorean system with little regard for what any actual Pythagoreans said in order to serve as background for his account of Plato’s theory of principles (2012a, 438, 394–414). Another approach tries to mitigate the differences between Philolaus and Aristotle and suggests that Aristotle’s emphasis on number was derived from Pythagorean numerology that was independent of Philolaus but that was combined with material from Philolaus as a result of Aristotle’s decision to present one mainstream Pythagorean system (Primavesi 2012). 