¹ See , Zeller-Nestle, Die Philosophie der Griechen I(Leipzig, 1923 ⁷), pp. 446ff.;Google ScholarBurnet, J, Early Greek Philosophy (London, 1930 ⁴), pp. 286ff.;Google ScholarKranz, W, Die griechische Philosophie (Birsfelden-Basel, 1955), pp. 41f., 48;Google ScholarGuthrie, W. K. C, A History of Greek Philosophy I (Cambridge, 1962), pp. 229ff.;Google ScholarBarnes, J, The Presocratic Philosophers (London, 1982, rev. ed.), pp. 380ff. These and the following works will subsequently be referred to by the name of the author alone:Google ScholarBecker, O, Das mathematische Denken der Antike (Gottingen, 1966 ²);Google ScholarBoeckh, A, Philolaos (Berlin, 1819);Google ScholarBurkert, W, Lore and Science in Ancient Pythagoreanism, Minar, E. L, Jr. (trans.) (Cambridge, MA, 1972);Google ScholarCherniss, H, Aristotle's Criticism of Presocratic Philosophy (Baltimore, 1935);Google ScholarFrank, E, Plato unddie sogenannten Pythagoreer (Halle, 1923);Google ScholarKahn, C. H, ‘Pythagorean Philosophy before Plato’ in Mourelatos, A. P. D. (ed), The Pre-Socratics (Garden City, NY, 1974), pp. 161–85;Google ScholarKirk, G. S, Raven, J. E and M. Schofield, The Presocratic Philosophers (Cambridge, 1983 ²);Google ScholarNussbaum, M. C, ‘Eleatic Conventionalism and Philolaus on the Conditions of Thought’, HSCP 83 (1979), 63–108;Google ScholarPhilip, J.A, Pythagoras and Early Pythagoreanism (Toronto, 1966);Google ScholarRoss, W. D, Aristotle's Metaphysics (Oxford, 1924);Google ScholarStenzel, J, Zahl und Gestalt bei Platon und Aristoteles (Darmstadt, 1959 ³);Google ScholarStokes, M. C, One and Many in Presocratic Philosophy (Washington, DC, 1971);Google ScholarZhmud', L. Ya, ‘“AH is number?” Basic Doctrine of Pythagoreanism reconsidered’, Phronesis 34 (1989), 270–92.CrossRefGoogle Scholar

² See Huffman, C. A, ‘The role of number in Philolaus' philosophy’, Phronesis 33 (1988), 1–30, incorporated in his recent book, Philolaus of Croton (Cambridge, 1993) (references to Huffman from now on will be to his book). For a similar perspective, but arrived at independently from Huffman, see L. Ya. Zhmud', 270–92. Much of the confidence for the critique of Aristotle is derived from Cherniss' book.CrossRefGoogle Scholar

³ In this, however, the critics of Aristotle differ from Cherniss (p. 37, n. 140, 386), who considered the fragments of Philolaus spurious and of no bearing on Aristotle.

⁴ Stob. 1.21.8 = DK 44B7. With Burkert, p. 255, n. 83, I am assuming that is not dittography with the following , mainly because of Aristotle's testimony about the construction of the Pythagorean one— (Met. 1080b20, cf. 1091a15). Huffman, p. 228, also accepts though not, as we will see, its identification with number.

⁵ Huffman admits that fire and ‘middle’ might seem ‘features of the cosmos in radically different senses’, but his mitigating point that Philolaus was a Presocratic author like Anaxagoras and Empedocles who could write about Mind and Love/Strife as cosmic components has some force.

⁶ So also Kirk, Raven and Schofield, p. 326. This possibility and others—empty spaces and atoms, or material components and their defining shapes—are briefly reviewed and dismissed by Huffman, pp. 37rf.

⁷ On the Greek notion of number, see n. 14 below; on the special status of the one, see further, n. 20 below.

⁸ Similarly, L. Ya. Zhmud', 276, n. 22.

⁹ The opinio communis holds there were no Pythagorean writings before the appearance of Philolaus' book; cf. Guthrie, p. 155; Burkert, pp. 223–38, 239; Huffman, p. 15. From this it is often inferred that when Aristotle makes some definite statement about Pythagoreanism, he is relying on a written source, which therefore must have been Philolaus' book (Archytas is left aside in this reckoning, since Aristotle appears to have devoted special treatises to him); see, for example, Kahn, pp. 170ff. Although there are indisputably important points of contact between Aristotle' testimonies and what we know of Philolau' book, it is hazardous to assume, especially in the light of what we do not know, that there could not have been other Pythagorean sources of information, if not written then oral, available to Aristotle; cf. Philip, p. 120. (I do not place much stock in Diogenes Laertiu' assertion (8.15), probably based on Aristoxenus, that until Philolaus nothing could be known of Pythagorean teaching; the most this tells, in the context of the surrounding claims, is that Pythagorean teachings first become expressed in writing with Philolaus.) Thereby the oral reports need not have been confined to acusmata, i.e. Pythagorean lore and wisdom (pace Burkert, p. 240), but may well have contained something about Pythagorean number philosophy. And because Aristotle, in his surviving works, was not concerned to give a systematic account and critique of Pythagorean philosophy (most likely reserving such an account for his separate treatise on the Pythagoreans; on the pointers thereto, see Guthrie, p. 215, n. 1) but adduces the Pythagoreans to illustrate their views on certain topics and, more often than not, quickly dismisses them again, it is feasible that these incidental reports contain some conflated material. The difficulty then arises in trying to sort out what pertains to Philolaus and what to other Pythagoreans. Relatively certain traces of Philolaus in Aristotle are succinctly summarized by Burkert, p. 234, n. 83; cf. Kirk, Raven and Schofield, pp. 330f, with criticism by Huffman, pp. 58f.

¹⁰ Whatever the ordering of the fragments, I call fr. 4 programmatic because it is instructive for any assertion about knowledge in Philolaus.

¹¹ Huffman: When Philolaus says that of the two proper kinds of number there are many forms that are signified by each individual thing, he implies that there are not many forms of the third kind (this seems to me an obvious implication; cf. Burkert, p. 264, n. 24). In fact, as I will go on to argue, there is only one form of the even-odd and hence only one thing which ‘signifies’ it. The various possibilities of numerical interpretation offered by fr. 5 are are summarized by Becker, pp. 45f.

¹² Huffman, p. 178, adduces this passage (except for the last clause, ‘numbers… are the whole universe’) to vouch for the authenticity of Philolaus fr. 5.

¹³ The objections of Barnes, p. 390, to the connection between unlimiteds/limits and even/odd in Philolaus have been adequately met by Huffman, pp. 182f.

¹⁴ The clearest discussion of is that of Nussbaum, 89ff. The Pythagoreans, as we will discuss at the end of this paper, never divorced number from that which was counted.

¹⁵ As Nussbaum, 90, notes parenthetically: ‘Thus zero is not an arithmos, even in Greek mathematics, since there is no such thing as a null group to be counted; there is a great deal of debate even about one, since we do not count the unitary.’

¹⁶ Furthermore, Huffman's sole emphasis on the epistemological role of number for recognizing the structure of things is to put number on a simple, predicative level, which would invalidate Aristotle's criticisms of the Pythagorean understanding of number as substance and in effect obliterate an important difference between him and the Pythagoreans. Cf. Gloy, K, ‘Aristoteles’ Theorie des Einen auf der Basis des Buches I der “Metaphysik”’, in K., Gloy and E., Rudolph (edd.), Einheit als Grundfrage der Philosophie (Darmstadt, 1985), p. 92: ‘Dadurch, daB das Eine nicht wie nach pythagoreischer Ansicht das Wesen des Seienden ausmacht, sondern dessen Struktur, begleitet es alles Seiende, was Aristoteles auch so ausdrckt, daB die Zahl Zahl von etwas ist.’Google Scholar

¹⁷ Cf. p. 182: ‘Things give signs of numbers which gives us knowledge of those things, but things are not therefore said to be numbers, nor are unlimiteds said to be even numbers or the limiters said to be odd numbers, although we may come to know them through seeing the even and odd numbers to which they point.’ Cf. also p. 56. Similarly, L. Ya. Zhmud', 275.

¹⁸ Huffman's ordering of certain fragments in his book under the separate rubrics of ‘Basic Principles’, ‘Epistemology’, and ‘Cosmogony’ has its organizational usefulness, though it does tend to detract from seeing the close interrelatedness of the topics treated in those fragments.

¹⁹ This is the standard interpretation of the even-odd in early Pythagoreanism, also followed by Huffman (pp. 186f.), in preference to the alternative but weaker intepretation that the evenodd refers to even numbers whose halves are odd (thereto Zeller-Nestle, p. 455, n. 1; cf. Becker, p. 46)

²⁰ Cf. Guthrie, p. 240. (On t he absence of zero, which gave the one a ‘ganz besondere metaphysische Stellung’, see Stenzel, pp. 28, 34.) This explanation is preferable to another ancient explanation that says that when the unit is added to an even number, an odd number results, and when added to an odd number it makes it even. As Guthrie, p. 244, points out, this is unsatisfactory ‘since it applies to every odd number as much as to the unit’ (so also Huffman, p. 186). Moreover, since this explanation holds for every odd number, it would contradict the implication made by Philolaus in fr. 5, to wit, that there are not many forms of the even-odd kind of number.

²¹ A more weighty objection is pointed out to me by Lawrence Schrenk: if the one is a mixture of odd and even, this seems to imply that the one is ontologically posterior to its components, which would contradict its being the starting-point of the number series. Schrenk therefore suggests that, if evens and odds are unlimiteds and limiters, the even-odd refers to the whole mixed class of things (fr. 2), e.g. the world and its contents harmonized from unlimiteds and limiters (fr. 1). While Schrenk's suggestion has its attractiveness (see also now his article, ‘World as Structure: The Ontology of Philolaus of Croton’, Apeiron 27 [1994], 171–90), it still leaves me with the question what place and function to assign to the individual things that give signs of the many forms of odd and even (fr. 5), if the world and the things in it are wholly constituted of the mixed class (hence Huffman terms the one as even-odd a ‘symbol of unity’ for the mixed class of things; see n. 28 below). But given the existence of odds and evens in the world, Philolaus may have introduced them a posteriori to explain the composition of the one. I am not certain that the implied ontological posteriority of the one, and the logical problem this might entail, was at all an issue with him.

²² To accept more would appear to go against the grain of Huffman's thesis, seeing that the numbers resulting from the one are equated by Aristotle with the physical existents of the world (‘numbers… are the whole universe’, cf. Met. 985b32, 987b28, 1090a20). This has been a line of interpretation followed by most scholars; see, e.g. Zeller-Nestle, p. 478 (quoted in n. 49 below), i Philip, pp. 51ff., 61.

²³ ‘Unter dem ist entweder das Eins zu verstehen, welches von den Pythagoreern so genannt wurde…, von dem man allerdings kaum erwarten sollte, daB es als eigene Gattung bezeichnet wiirde, oder diejenigen geraden Zahlen, die durch zwei geteilt ungerade ergeben’, Zeller-Nestle, p. 455 n. 1; against the latter alternative, see n. 19 above.

²⁴ This conjecture has the merit of drawing its examples from something discussed by Philolaus (fr. 6a). With the octave (2:1) Huffman notes a problem ‘since 1 is not simply an odd number, but it does contain the principle of the odd in it according to Philolaus’. But, as I will argue below, the Pythagoreans in actual practice gave a delimiting function to the one, effectively treating it as odd. Philolaus was no exception.

²⁵ The emphasis is thus not on the proportion signified by the ratios but on the use of number as such. Cf. Burkert, , p. 400: ‘Not from Philolaus alone does it become clear that the important thing in Pythagorean musical theory was not the function of the proportion but the meaningful numbers.’ Similarly, West, M. L, Ancient Greek Music (Oxford, 1992), p. 236. See also n. 43 below.Google Scholar

²⁶ Whereby, as already noted, the unit has to function simply as odd.

²⁷ There is no compelling philological reason against this use of (before Aristotle takes it up) to refer to a single kind of thing.

²⁸ Huffman, p. 189, speaks of the one as ‘the symbol of unity’ and adds ‘…granted that the one as even-odd is an excellent symbol for the whole mixed class of things, it cannot serve to give us knowledge of the great variety of things in the mixed class of things; they cannot all be known through the same number one or they would all be the same’ (hence Huffman goes on to give as examples the ratios of the musical concords). While I will grant that the one can be loosely seen as a ‘symbol’ for all kinds of unities, this does not commit me to the proposition that the whole mixed class of things can be known through the one, since in my view there is strictly speaking only one form of the even and odd and therefore, in a strict sense again, only one thing known through the one.

²⁹ ‘as Aristotle takes it to be’ in this quotation refers to Met. 1091a13ff., which Huffman sees as a commentary on fr. 7 (see n. 34 below). I will return to this passage in Aristotle.

³⁰ Cf n. 28 above. When we assign the one as number to the central fire, its symbolic value may be clear to us but not necessarily to the Pythagoreans. As will be discussed at the end of the paper, for the Pythagoreans numbers were not merely symbols but actually identified with that for, which they stood.

³¹ I do not mean to be unkind to Huffman's reasoning, the motivations for which I understand, but it does seem to me contorted to try to garner support for an understanding of the even-odd as the number one by pointing to a fragment in which something is described as the one but then to forbid this to be a reference to the number one, presenting it instead as a unity merely symbolized by the number, while emphasizing that the dissimilar elements that combine to form both (the central fire and the unity) can be closely associated (limits with odd, and unlimiteds with even).

³² Cf. Alexander, in Met. 74.6, where describing how the Pythagoreans gave a proper number to each region of the universe as well as to certain concepts, Alexander says that they assigned the one to the centre of the cosmos, for it was the first thing there—. This would seem to agree with both test. A17 and fr. 7 of Philolaus, but there are certain difficulties with Alexander's report; see Huffman, pp. 285ff.

³³ According to Huffman, the interpretation of the one as the mathematical unit seems natural only after Plato and Aristotle and has come about as follows: although Aristotle carefully notes differences between Plato and the Pythagoreans, particularly in regard to Plato's separation of mathematicals from things, nonetheless in those treatises not specially devoted to the Pythagoreans he is interested in them mainly in connection to Plato and therefore ‘overinterpreted the role of “the one” in light of its importance in Plato’ (p. 209). Against this and Huffman's ongoing discussion in a similar vein (pp. 209ff.) I raise two elemental objections. First, while no one will deny that the Academy took the one to heights not reached in fifthcentury Pythagoreanism, the construction of the number one and its role in the generation of the cosmos (Arist. Met. 1091a12ff.) were not Platonic inventions; the doctrine of comes from pre-Platonic Pythagoreanism (Arist. Met. 1053b12f., van der Waerden, B. L, ‘Pythagoreer’, RE 24, 1 [1963], 247). My second point has been stated well enough by Guthrie; Aristotle was ‘perfectly capable of distinguishing non-Platonic Pythagoreanism from the teaching of his master’ (p. 256, cf. p. 241). This is valid even if, as Huffman supposes (pp. 63f.), Aristotle wanted to bring the Pythagoreans into a debate with Plato.Google Scholar

³⁴ Huffman, pp. 62, 203f., 227, 228; cf. Frank, p. 327; Ross, ad loc. (II 484).

³⁵ Cf. n. 9 above.

³⁶ See the collection of texts by Guthrie, pp. 276f.

³⁷ Frank, pp. 327f., Baldry, H. C, ‘Embryological Analogies in Pre-Socratic Cosmogony’, CQ 26 (1932), 27–34 at 33, Guthrie, pp. 278f., Burkert, p. 37 and n. 47, Kahn, pp. 174f., Huffinan, pp. 43f.CrossRefGoogle Scholar For Lloyd, G. E. R, Polarity and Analogy (Cambridge, 1966), p. 238, n. 2, Philolaus' embryology ‘roughly’ parallels the cosmogonic inhalation.Google Scholar

³⁸ Cf. Burkert, p. 255.

³⁹ Here Huffman indirectly supports the point I made earlier and sought to illustrate from fr. 6, that it will not do to banish Philolaus' epistemological concerns, which apparently could be resolved by the use of number, from his discussions of the generation and composition of things.

⁴⁰ So already Cherniss, p. 39, in regard to Met. 1091a13–22: ‘the One here mentioned need not be considered the numerical unit; it is rather the universe itself…’ in agreement, L. Ya. Zhmud', 288. But cf. Philip, p. 76; ‘The number theory of the Pythagoreans derives from their cosmology and, in its principal aspects, is cosmology’ (his emphasis). Sim. Burkert, p. 36: ‘To the Pythagoreans, number philosophy is cosmogony.’

⁴¹ It is not clear whether the odd and even forms of number were applied respectively to limited and unlimited things or just certain whole numbers to various unities; it is clear only that all known things have number and that the central fire had the number one.

⁴² Cf. Stokes, p. 247: ‘If, as Aristotle says (Metaphysics 986a 19), there was a school of Pythagorean thought holding the unit to be a compound of Limit and Unlimited, then at least these same thinkers believed that number came from the unit, . In so believing they were obviously ignoring the veto—if there ever was a veto—on a “one” producing a “many.” A clearer case of derivation of a plurality from a unity it would be hard to find.’

⁴³ This double status of the one may explain why in another Pythagorean text, the table of opposites (Met. 986a22), it appears with the odd as a manifestation of limit. I do not mean hereby to revive older theories about a chronological development in Pythagoreanism from an older teaching of the one as odd to a newer, post-Parmenidean version of the even-odd, nor to posit different schools of thought within contemporaneous Pythagoreans. On all this see Stokes, pp. 244ff., who himself doubts ‘whether the Table of Opposites implies a different view of “one” from the usual even-odd designation’ (p. 246). It is my belief that the Pythagoreans could view the one as essentially a unity of odd/limit and even/unlimited and, without causing themselves much headache, treat it as odd and assign it a limiting function when required (not much is gained by following Stenzel, p. 6, who ascribes to the Pythagoreans ‘neben der Eins ein anderes einheitliches Prinzip, das Unbegrenzte’). This appears to be the case not only in Philolaus' cosmogony but also in Pythagorean pebble arithmetic (see Arist. Phys. 203a13; thereto, Guthrie, p. 243 and Burkert, p. 33, n. 27) and musical theory—in the ratio of the octave, 2:1. Huffman, in his discussion of Philolaus' fragment on harmonics (pp. 149ff., 161f.), does well to try to clarify the terms by which the octave is designated ( as the concord and harmonia as covering the scale, ‘an attunement an octave long’), but he does not convince me that earlier scholars (Boeckh and Frank; see Huffman, pp. 159f.) were wholly wrong in identifying the principle of limit with the 1 in the octave and the unlimited with the 2 (for which the ‘indefinite dyad’ is of course a later Platonic term). One should not lose sight of the forest for the trees and forget the essential Pythagorean discovery (whatever its empirical bases) of simple numerical ratios to express harmonic intervals, which ‘made it appear that kosmos—order and beauty—was imposed on the chaotic range of sound by means of the first four integers 1,2,3,4’ (Guthrie, p. 224). However Philolaus conceived of the ‘size of harmonia’ (fr. 6a init.), he was primarily interested in the whole numbers of the ratios (pace Huffman, p. 160; cf. n. 25 above), numbers that corresponded to limiting and unlimited principles.

⁴⁴ It is noteworthy that fire and limit seem to be (spatially) associated in Aristotle's account of why certain thinkers, in agreement with the Pythagoreans, make fire central: ‘They think that the most honoured place properly belongs to the most honoured thing, and that fire is more honourable than earth, and limit more honourable than what lies in between, and that the outermost boundary and the centre are a limit’ (de Caelo 293a30). That the priority of fire over earth (or other elements) obtains for Philolaus is suggested by his embryology, in which the hot as the original vital element of the body can be compared to the central fire; and the association of fire with limit fits his system as well, not only because he assigned a limiting action to the central fire but also because, according to test. A16, he located ‘another fire’ at the uppermost boundary surrounding the spherical universe; cf. Huffman, pp. 244ff. A further link between fire and limit may again be sought in Philolaus' embryology. Since Huffman is inclined to believe that the hot, at least at the embryonic stage, is the sole element of the body (in effect, a monistic theory of the body's composition, see p. 294), he allows me to make this, albeit tentative, parallel. The body is originally a ‘one’, in the sense of being solely constituted of the hot; it then draws in the cold air from outside and emits it again, thus starting the process of respiration (which is vital for cooling the body). If in this process the external air may be seen as the unlimited element, the hot is somehow the limiting element (but cf. Huffman, pp. 45f.), perhaps in the measured intake and discharge of the cold air in breathing. Similarly the central fire, the one, takes on a limiting function by drawing in breath (and void and time) from the unlimited, thereby initiating the further development of the cosmos.

⁴⁵ Cf. Boeckh, p. 95, who sees the central fire as ‘die Einheit… in welcher die Welt ihren Halt hat, und welche zugleich als die Einheit Allem das MaB und die Begrenzung giebt.’ I differ from Boeckh, however, in t h a t he does not recognize the dual aspect of the one and treats it solely and consistently as a limiting principle (pp. 54ff); cf. Nussbaum, 97, n. 85. It is precisely the interplay of ‘duality and singleness’, of unlimited and limited, represented in nuce in the one, that makes up the all-pervading harmonia of the cosmos. As Walter Burkert observes (in a letter dated 5.3.94): ‘Die Frage, warum das “Eine”, obschon aus Begrenzendem und Unbegrenztem bestehend, gegeniiber dem unbegrenzten “Hauch” dann de facto als Grenze wirkt, konnte man vielleicht auch von seinem Charakter als Harmonia herleiten: Eben als solche muB es “gegensteuern”, bis das grofie Unbegrenzte seine weitgespannte Grenze in der Kugel des Himmels erhält, so daB dann mit dem “Zwei” gegen “Eines” die “durch alles hindurchgehende” Harmonia, , 2:1 [fr. 6a], gesichert ist.’

⁴⁶ The one is thus more than a paradigm, since the numbers that could apparently render the order and structure of physical bodies intelligible derive from the one.

⁴⁷ The manner in which Iamblichus refers to Philolaus is noteworthy: … Iamblichus characterizes the monad in a participial clause ‘in so far as it is’ (or: ‘would be’—note ) ‘the principle of all things according to Philolaus' and asks parenthetically, ‘for does he not say [the] one is principle of all things?’ (punctuated as a question after Boeckh, p. 150). The parenthesis looks as if it were Iamblichus' way of seeking to confirm what he has just ascribed to Philolaus and thus has a ring of authenticity. (According to test. A10 = Theo Sm. 20.19, Archytas and Philolaus referred indiscriminately to the one and the monad. Since it is probable that Philolaus only spoke of the one, as in fr. 7 and perhaps as here in Iamblichus' parenthesis, and not of a monad, Theon's testimony is best taken to mean that Philolaus did not distinguish between the one and the monad in the Platonic fashion; cf. Huffman, p. 340.) Iamblichus, I might add, is not an inherently unreliable witness, though of course he often needs to be shorn of his Neoplatonic trappings. In Nic. 7.8 = fr. 3 appears to be a genuine nugget (accepted as authentic by Huffman, pp. 113ff.) whereas In Nic. 10.22 = fr. 23 consists wholly of later interpretation (Huffman, pp. 354f.) and allows us at most to infer that number played an important role in Philolaus' cosmology.

⁴⁸ Huffman brings the same objections against Syrianus, in Met. 165.33 (= fr. 8a). Syrianus posits for Philolaus a god who established limit and unlimited and a transcendent unitary cause. Though this is patently Platonic, we should not be too quick to assume that his concluding assertion about the one in Philolaus, …, is alo erroneous, evenif he is merely echoing Iamblichus.

⁴⁹ Cf. Zeller-Nestle, p. 478: ‘Philolaos nennt zwar das Eins den Anfang von allem, aber damit will er schwerlich etwas anderes ausdriicken, als was auch Aristoteles sagt, daB die Zahl Eins die Wurzel aller Zahlen und somit, da alles aus Zahlen besteht, auch der Grund aller Dinge sei.’

⁵⁰ Nielsen, H in The Concept of Matter in Greek and Medieval Philosophy, ed. E., McMullin (Notre Dame, 1963), p. 255.Google Scholar

⁵¹ This would certainly be borne out if we still had Aristotle's lost work on the Pythagoreans; thereto, see Guthrie, p. 215, n. 1.

⁵² The larger context of the passage is quoted above (III init). As Aristotle's inquiry has to do with unchangeable and immovable principles, he cannot accept the Pythagorean ascription of a physical generation to numbers if numbers are to be understood as eternal. But the Pythagoreans, as we will see, failed to conceive of numbers as abstract mathematical principles. Hence it is not the case that ‘Aristotle is confusing… the cosmogony with the number theory’, as Cherniss says (p. 39; cf. n. 40 above). Aristotle wanted to keep mathematicals and physical cosmology in separate categories; the confusion of the two rests squarely with the Pythagoreans.

⁵³ Somewhat analogous to the way the atomists thought of bodies as consisting of invisible atomic units; but even though Aristotle mentions the Pythagoreans in conjunction with the atomists (Met. 985b23) there are important differences between the two (mainly, the indefinite number of atoms supposed by the atomists to constitute bodies as opposed to the limited amount of numbers that sufficed for the Pythagoreans to account for a thing), so that one should not interpret the Pythagorean doctrine as a kind of ‘number atomism’ see Burkert, pp. 4If.

⁵⁴ Ps.-Alexander, Met. 827.9 (DK 45A3), elaborating on Arist. Met. 1092b8ff. Theophrastus, Met. 6a19ff., gives Archytas as the source of the story. On Eurytus' practice, see further Guthrie, pp. 273ff., Barnes, pp. 390f.

⁵⁵ Cf. Guthrie, pp. 237f., Burkert, p. 46, Ross, ad Met. 1080b19 (II 428–9).

⁵⁶ As far as I can see, the closest Philolaus comes to anticipating a symbolic use of number is when he says, in fr. 5, that each thing signifies () the many forms () of number. This allows us to say, in regard to fr. 7, that the central fire is the one thing that signifies the form of the one, yet it still appears, from our examination of fr. 7, that Philolaus identified the first cosmic body with the number signified by it. Even Huffman, who insists on maintaining a strict parallelism between things and numbers in Philolaus, admits, on the basis of Aristotle's evidence, that ‘the Pythagoreans… do not think of the numbers that are pointed to as separate from the things which do the pointing, but rather as in some way part of them’ (p. 193).

⁵⁷ Cf. Aristotle, Met. 989b33ff., chiding the Pythagoreans for using their discovery of nonsensible, immovable mathematicals as supplying the principles for the physical world: ‘their discussions and concerns are yet all about nature (), for they generate the heaven, and they observe what happens in regard to its parts, and attributes, and functions, and use up the principles and the causes for these studies, as though they agreed with the physical philosophers that only that is real which is perceptible and contained by the so-called heaven.’ Huffman, while he admits that Aristotle's words here find an exact target in Philolaus, whose ‘thinking is still very much in the Presocratic mode’ (p. 52), nonetheless does not allow Aristotle's essential criticism of the Pythagorean confusion between mathematicals and substances to apply to Philolaus.

⁵⁸ Ross comments ad loc. (II 147), ‘We are not to suppose that they deliberately rejected the notion that numbers are not spatial. Like all the pre-Socratics, they had not reached the notion of non-spatial reality.’