Early Pythagorean principles. Peras and apeiron

ATTO arretrare ee hes a tl = CEA rs Nis, Early Pythagorean principles. Peras and apeiron : lonian [4483 ici ine $. Studies 1 lonian philosophy, ed. by Boupouris Konstantine J. : Studies in Gree k philos. 453 p. N° 1 Alimos Intern. assoc. for Greek philosophy & Athens Kardamitsa 1989 [13205 ill. index (dépouillé dans le présent vol.). "Early Pythagorean Principles: and apeiron.” lonian Philosophy. Boudouris. Athens: International Philosophy, 1989, pp. 304-311. Edited by Association K. peras J. for Greek Questions the commonly assumed idea that early Greek philosophy was characterized by a dichotomy between the philosophers, like the Milesian cosmologists, who introduced a philosophy of matter, and the Pythagoreans, who advocated a philosophy of form. Regards the opposition between matter and form, at that stage of the development of ideas, as anachronistic, both too primitive and too modern, which is mostly the result of Aristotle’s interpretation of his predecessors. Argues that recent developments in physics, specifically related to new mathematical conceptions of reality (e.g., Einsteins and Heisenberg’s notion of fields in space) the absolute circumstance have cast a shadow of doubt on distinction between matter and form, a which calls for a reappraisal of the ancient Pythagorean principles.

EARLY PYTHAGOREAN PRINCIPLES ble to us: exhalation (= soul?) is “very much disembodied” DA 405a25-7). This does not appear to be fair as a general comment on the Presocratics, but Aristotle gives us the reason too: the earlier thinkers assert the identity of understanding and perceiving, supposing thinking to be, like perceiving, something bodily and that both are of like by like (Met. 1009b12-15, DA 427a21-8).” One may feel tempted to take this as no more than Aristotle’s interpretation of the PresocratERIK NIS OSTENFELD ics (“say” = “must say”, cf. GC 314a8-10). However, even if they did not explicitly say that knowledge is senseperception, they undoubtedly assimilated the former to the latter (cf. also DA 410a25-6 and Hicks’ note). II. Two Traditions of Pythagorean Principles Aristotle’s main account (Met. A5) of the Pythagoreans contemporary with EARLY PYTHAGOREAN PRINCIPLES: PERAS AND APEIRON and before the atomists, after the general doctrine of number/thing likenesses, is two-fold: (a) the Pythagoreans think that number is the principle (arche) both as matter for things and as their modification and states, that the elements of number are I. Pythagoreans in the History of Philosophy It is commonly assumed that the Pythagoreans, the Italian school of philosophy, advocated and introduced a philosophy of form or structure in contrast to earlier Ionian philosophy of matter.’ Presumably by form/structure something like the Aristotelian form-matter distinction is presupposed. Now judging from the best extant evidence, Aristotle’s report in Metaphysics A, it appears that (a) the Pythagoreans are thought different from the physiologoi by their use of stranger elements: non-sensible, immovable, mathematical objects (989b29-32, cf. 1090a32-5).* However, (b) they agree with the physical philosophers that the real is just all that which is perceptible (989b33-990a5). Plato complains that the Pythagoreans are too empirical (Rep. 531c). As for (a), the Pythagoreans seem to speak of another heaven and other bodies, not of the sensible (1090a31-5, cf. 1083b17-19, 990a6-8). The Eleatics somewhat similarly, but more madly (Pythagoreans distinguished from physiologoi (989b29ff, 1053b10-16) but grouped with them against Eleatics (986b8ff)), the even and the odd (unlimited and limited) and that the One proceeds from both of these and number from the One (986a16-21). (b) Other members of the same school say there are ten principles arranged in two columns of cognates: limit/unlimited, odd/even, one/plurality, right/left, male/female, resting/moving, straight/curved, light/darkness, good/bad, square/oblong (986a22-b2). (b) is early Pythagoreanism for the following reasons: Connection with Alcmaeon and the fact that limit/unlimited and thus, one assumes, the closely related odd/even are not his but special to the Pythagoreans (987a13ff, cf. 990a8-10). Moreover, we may take it that the widest and most comprehensive, that from which the others have a chance of being deducted, limit/unlimited, heading the list is also most basic (thus good/bad belong to limit/unlimited (EN 1106b29-30). Cf. also Heidel’s ethical argument and Burkert 32-3 n. 25). Also, in Pythagorean cosmogony, which appears to be archaic (air = empty space), this pair is used (1091a12-18). The same cosmogony seems also to point to the originality of male/female (1091a16) and the anyway ancient pair light/darkness (seed) 1091a16, 1092a32) and central fire (Cael. 293a19)/air or go beyond senseperception in postulating a higher, unitary and immovable realimist). The same applies to one/plurality (On Pyth. frg. 10 Ross, Phys. 213b22). With odd/even goes the pair square/oblong, the graphic representation of ty (demanded if knowledge is to be possible), but yet, as the Pythagoreans number being an ancient form of thinking in relatively concrete terms (1092b11- (989b29), transfer the description of that to sensible things (Cael. 298b20-5, GC 325a2-23). Cf. also Heraclitus’ hidden reality (frg. 123). As for (b), that the real is identical with the perceptible, this is common belief among the Presocratics according to Aristotle (Empedocles, Democritus, Par- 12, Phys. 203a10ff). Cf. Burkert 427ff, Heath Gk. Math I, 76-84. It cannot of course be proved that the whole decalogue is early, but it seems apriori likely that the number of antitheses was complete (ten) from the start (On Pyth. frg. 10 Ross) and there seems no conclusive evidence against. Cf. the rather optimistic menides, Anaxagoras 1009b12-1010a3; cf. Cael. 298b22-3 for Eleatics and DA 404a27ff for Democritus. Cf. also the general opinion that reality is in motion view of Guthrie (I 232-3), with the more cautious Vlastos 167-8. The following characteristics of the two traditions may be noted: (a) is con- (DA 405a28-9) and Heraclitus’ everliving fire which is material but not percepticerned with ontology (or numerology/cosmology) and involves certain positions

that seem peculiar to it: 1) the elements of number are even/odd (On Pyth. 13 Ross, 986a17-18), 2) the one is generated (986a19-20, 1091a15, On Pyth. 13 Ross), 3) use of peperasmenon for limit, 4) good/bad are not principles (1075a36), but effects of a cause (1072b30). (b) is a list of values (On Pyth. 10 Ross, EN 1096b6). In the sequel we shall be mainly dealing with tradition (a). EARLY PYTHAGOREAN PRINCIPLES 307 principles, but he is still concerned with the substance (of things), aas under the definable aspect (987a20-3). How do we reconcile this very Aristotelian report with the material contrary principles we set out with above? IV. The Nature of the Unlimited It may be suggested that there actually was a certain asymmetry between III. The Nature of the Principles peras and apeiron in that only the latter is fully material (hyle 988a24-27). Cf. think . Now, how are these Pythagorean principles to be conceived? One might Howeventities. abstract that being the elements of number they would be fairly 3, 32, er, numbers, being identical with things themselves (987b28, 1090a22- his 990a23), are not abstract (see further below), and Aristotle looking through princiy contrar rean Pythago own spectacles of form/matter concludes that the as ples seem to have been ranged under the head of matter: “for out of these the Platonic development (987b20ff) of the Pythagorean unlimited from being one to a duality/the great and the small (= material principle, whereas the One is or This suggests an asymmetry in the conception of the principles as material. However, in the Physics.203a Aristotle contrasts the Pythagoreans and Plato with all the physicists in that they make the unlimited a principle in the sense of ). By a self-subsistent (material) substance just as the One (peras) was argued to be a in the “substance” we are here to understand “individual things” (substance are ed” “mould and sed” sense defined 1017b10-14, cf. 24; 1028b8-16). “Compo as concrete as possible in their implications. However, in contrast to the earlier materialists the two principles of finiself-subsistent substance (cf. above). So the two principles seem to be on a par. The unlimited does indeed seem passive and limit active in the cosmogony: immanent parts they say substance is composed and moulded” (986b4-8 other natures tude (peperasmenon) and unlimited {and the one] are not certain (ousia) of the ce substan the ves (physeis) like fire, earth, etc., but they are themsel 19).° The (987a15ted predica things of which they (sc. the one, unlimited) are point of this obscure saying appears from several restatements of the basic tenet ng else of the Pythagoreans: the one is substance and not a predicate of somethi them(987b22-3), being and the one are nothing else (in need of a substrate) but is a itself one the whether selves substances (1001a4-12, 996a6-8), the problem imply to appears This -13). substance or there is an underlying nature (1053b11 ces in the that finitude and unlimited are thought by Aristotle to be substan when the one (monas) has been constructed immediately the nearest part of the unlimited began to be constrained and limited by the limit (109 1al2-18). Similarly, in the Physics (213b22-27) we are told that “the void enters into the heaven from the infinite breath (pneumatos, cf. MSS), the heaven being supposed to breathe in the actual void.” This idea of a breathing heaven is commented on by Simplicius (In Phys. 651.26): “They say that the void enters the cosmos as if it breathed in a sort of breath from that which lies outside,” and Stobaeus (Ecl. 1, 18, Ic) says that Aristotle in his work on the Pythagoreans writes that the universe from the unlimited draws in time, breath and void (= Arist. frg. 13 Ross). There is here a definite identification of the unlimited with breath and void which seems ancient.° The infinite breath (213b22) is to some extent reminiscent of of Anaximenes’ cosmic air (frg. 2) and possibly also of Anaximander’s fertile should be 1017b24).* Evidently then Aristotle thinks that the Pythagoreans infinite (cf. Phys. 187a20ff, 203b10-15) and is therefore perhaps best conceived of as a chaotic war-fare of Ionian opposites (hot-cold, dry-wet, etc.). Cf. Plato, above mentioned sense of individual things (the “ultimate substratum” first grouped with the materialists (cf. hypokeimenon used of the principle of the (9870.14)).° Polit. 273d: limitless sea of unlikeness and Phil. 24a-e, 25de, Tim. 52df: irregular The Pythagorean interest in definition is seen as something different (note the intransition 987a19-22). The Pythagoreans showed a beginning but superficial solids. Movement belongs to the unlimited because it was thought of as indefiphilosophers (984a22, 29), kai (986016) and kata ton auton [] tropon Met 512.20 terest in definition of essence (987a20-7, cf. 1036b8ff with Alex. in of and frg. 13 (p. 138 in Ross)). Aristotle now deals with substance in the sense e Aristotl have to then seem We 3). essence, the object of definition (cf. 1017b21- nite (Phys. 201b19ff), but it is passive motion in the sense that it is disorderly, without a goal/limit. However, the unlimited is probably evil (EN 1106b28, cf. 1072b30) in contrast to Anaximander’s infinite and Anaximenes’ infinite air that are divine: instead of Ionian material monism we have here cosmological dualas an indiism leading to/based on anthropological dualism (evil body/supernatural soul). subvidual (tode ti), second (now) (b) as an essence (ti esti). (a) implies the essential between ion distinct a s concern (b) stance/attribute distinction, whereas Cf. Orphic dualism and its cosmogonic Night (cf. KRS no 30 and Burkert 37- playing around with two sides of his doctrine of substance: first (a) 39). It appears that the cosmos is generated in time (989b31, 990a20-1, 1091a13, ces of the and accidental predication. Cf. (a) primary and (b) secondary substan 1072b31). real) prinwith secondary substances he is definitely attributing abstract (incorpo form/matter dualism in the Aristotelian sense but with something far more primitive: the primeval unit (with magnitude 1080b21) breathes and imposes dealt Categories. In so far as Aristotle is now suggesting that the Pythagoreans ciples to them. He does not in so many words say that he is still speaking of the The following conclusion can now be drawn: we are not dealing with a

order on the unlimited (1091a15-18). On the other hand, Pythagorean opposites are of another, wider and more abstract order than Ionian opposites (cf. Phys. to 188b30ff) and they are notably value-laden. Hence, when Aristotle seems think, uneasily, that we are still within the confines of materiality, he is not quite right (see below), and when he mentions unlimited only as matter (hyle 988227), he may be interpreting the Pythagoreans in Platonic terms (987b20-27). EARLY PYTHAGOREAN PRINCIPLES 309 it is stressed over and again that it is observed likenesses between things and number that lead to identification (pp. 139 Il. 6-9, 29-32, 140 Il. 12-15), and this applies equally to reciprocal justice and to the heavenly bodies having their distances in a certain ratio. On the other hand, the first way number may be causes (1092b9-13) obviously refers to calculation with pebbles and representing numbers geometrically (cf. 1036b8-13 for the converse method?). It is explicitly Matter is the product of the imposition of peras on apeiron, which is at the and uniquely connected with a named person, Eurytus, a pupil of Philolaus, and may therefore be relatively late. It does not appear to be assumed in the identifications discussed so far.'” (986a16-17). This statement has caused unnecessary exegetical alarm,’ but the 30-33; 1083b14-17), though he oddly, in view of 986a19 and On Pyth. frg. 13 (Ross p. 140 ll. 7-11), confesses ignorance as to how the first unit is constructed V. The Nature of Number same time the production of numbers (1091a12ff), since things are numbers. Aristotle tells us that the Pythagoreans believed that number is the principle both as matter for things and as their modifications (pathe) and states (hexeis) sense must be elucidated by the preceding page and appears to be simply that the Is number then extended? Aristotle claims repeatedly that Pythagorean numbers are not truly monadic (i.e. unextended) but have magnitude (1080b15ff, (1091a15-17) and even traces that ignorance to the Pythagoreans themselves world is or consists of number (986a3, cf. 990a21-22, 1083b11-12 (but note a (1080b20-22). However, Aristotle also complains that their monads in combination cannot either produce bodies or possess weight (Cael. 300a15ff, cf. Met. matter but also as modifications or states. Thus the Pythagoreans noticed that the modificiations and ratios of the musical scales were expressible in numbers 990a8-14), and we are informed by Stobaeus (Ecl. I, 308) that Ecphantus (4th cent. if historical) was the first to conceive of bodily numbers. This leaves the possibility that Pythagorean numbers may have been conceived of as extended hesitation in lines 18-19), 1090a22, 32). True, number is not only a principle as (985b31-32, cf. 1090a24-5). Aristotle states the issue in slightly other terms, when he says that numbers are the causes of the substance of other things (987b24-5, cf. 990220). But he complains that it is not made clear in which way numbers are the causes of substances and of being, whether as boundaries or as ratios of numbers (1092b8- 15). The first alternative is illustrated by points being limits of magnitudes and by a reference to Eurytus who did like those who attribute numbers to shapes like triangle and square (i.e. three and four points/units respectively as the minimum for containing the mentioned figures). The second alternative involves (with magnitude) but without corporeality: matter and geometrical structure coincide, in spite of Aristotle’s protests (1001b10, Sens. 445b11-15). Cf. Melissos’ one which is extended (frg. 3) but without body (frg. 9), Plato’s elementary geometrical configurations in space (Timaeus 53cff), Descartes’ matter = extension only (Princ. Phil. II 4, Letter to More 1649 Phil. Letters, ed. Kenny, 237-45),'' and in modern times, perhaps, Einstein’s and Heisenberg’s notion of fields in space. Aristotle cannot accept the notion of an extended but incorporeal selfsubsistent substance. This accounts for his unsuccessful attempts to fit the Pyanalysis of the proportions in which things are blended or made (e.g. bone or thagoreans into his own metaphysical framework. form: the ratio is the substance, whereas number becomes the matter (hyle). Looking back at Met. A we note that not only extended bodies are composed Conclusion flesh). Aristotle complains that in that case number is not substance nor cause of of numbers but also “abstract” entities like justice, soul and reason (985b30-1), opinion, opportunity, injustice, separation and mixture (990a22-5) and marriage (1078b21, before the time of Democritus and Socrates). The identification in the case of “abstract” entities like justice is with a certain modification (toiondi pathos) of number (985b29), just as several places in the world are associated with the modifications of number (990a26-7), or modifications of numbers are discovered in harmony, in the world, etc. (1090a24-5, cf. 21-22). The modifications and ratios of harmonies are seen in numbers (985b31-32) and numbers and harmonies® are related to modifications and parts of the world (986a3-6). It seems then that the dictum that “things are numbers” may be a handy way of expressing an identification of things and their properties with proportions (modifications of numbers). This appears to be the latter alternative mentioned at 1092b8-15, which is probably the older of the two.’ In On Pyth. frg. 13 (Ross) Is it then a philosophy of form we have been considering or is it rather one of matter? The answer is that the question is anachronistic. Put in Aristotelian terms it is both too modern (later in time) and too primitive. Recent developments in physics appear to have transcended the Aristotelian dichotomy and physicists are now concerned with a mathematical conception of reality that might have raised the interest or even enthusiasm of the ancient Pythagoreans. NOTES * Page references to Aristotle lacking the title of the work are to the Metaphysics. 1. Cf. e.g. Ross, Aristotle Metaphysics I 147, 152, 156, Guthrie HGPh I, 4 and 467 and Greek

Philosophers (UP 1967) ch. II, Kirk and Raven, Presocratic Philosophers 216, Hussey, The Presocratics 76, 154, Armstrong, Intr. Anc. Phil. (UP 1965) p. 8. 2. Theophrastus (De Sensu 1ff) takes Parmenides (frg. 16) as assuming the identity of sensation and thought. 3. Philip (50-1): “not constituted of some other thing like fire” is correct, but unclear what is the reference of “they” in “the substance of the things of which they are predicated.’’ Most other translators are rather free here. Thus Ross translates: “not attributes of certain other things.” The chosen translation preserves the archaic ambiguity inherent in the statement that the principles are other than ordinary elements (difference in number/level). 4, Heidel’s worry (361) that Pythagoreans may not have meant what they said, i.e. the infinite not after all a substance seems incomprehensible (cf. Anaximander’s infinite). Contra Guthrie I 241. 5. Ross I 156 claims that Alexander (47.5) cannot be right in suggesting that the Pythagoreans recognized two material causes for the dubious reason that the preceding summary does not refer to any other thinkers recognizing two material causes. However, in context it appears natural to make just that inference. It is at any rate difficult to see a reference to a formal cause at Met. 987a13-14 as Ross wants. 6. Alcmaeon identified air and void (see Beare 93f), whereas Empedocles (frg. 100) and Anaxagoras (Phys. 213a24-6) distinguished them. 7. See Ross I 147 who wants to import Aristotelian forms here. 8. Note that harmony appears in Aristotle’s account both as sensible object and as a numerical ratio. 9. Empedocles (frg. 96) defines bone as a ratio of three elements and the definition of an “abstract” like justice as a ratio stems from a time before Democritus and Socrates (1078b21-3). The parallel treatment of abstracts and concrete entities points to the early 5th cent. 10. Guthrie (I 256ff) seems rather optimistic in his belief that this aspect is early. 11. I see no need to take the void separating numbers (Phys. 213b24ff) as creating literal voids. It is rather the even separating the uneven (Phys. 203a10-13). 12. A. Einstein: “Felder sind physikalische Zustände des Raumes” (Das Raum-Ather- und FeldProblem der Physik in Forum Philosophicum I, 1930, p. 143). Cf. W. Heisenberg, Physik und Philosophie, 1959, p. 91. BIBLIOGRAPHY Armstrong, A.H.: An Introduction to Ancient Philosophy, London UP 1965 | Barnes, J.: The Presocratic Philosophers, London 1982 Burkert, W.: Lore and science in Ancient Pythagoreanism, Harvard 1972 Diels, H./Kranz, W.: Die Fragmente der Vorsokratiker I-III, Zürich 1966” Furley, D. and Allen, R. (edd): Studies in Presocratic Philosophy I-II, London 1970/5 Guthrie, W.K.C.: (1) A History of Greek Philosophy I-II, Cambridge 1962-5 (2) The Greek Philosophers, From Thales to Aristotle, London 1967 Heath, T.L.: A History of Greek Mathematics vol 1, Oxford 1921 Heidel, W.A.: (1) ITépaç and ” Aneıpov in the Pythagorean Philosophy, AGPh 1901 (2) ‘The Pythagoreans and Greek Mathematics’, repr. in Furley and Allen 1 350-81 EARLY PYTHAGOREAN PRINCIPLES 311 Hicks, R.D.: Aristotle De Anima, Cambridge 1907 Hussey, E.: The Presocratics, London 1972 Kahn, Ch.: (1) Anaximander and the Origins of Greek Cosmology, N.Y. 1960 (2) ‘Pythagorean Philosophy before Plato’, in Mourelatos Kirk, G.S./Raven, J.E./Schofield, M.: The Presocratic Philosophers, 1983° Mourelatos, A.P.D. (ed): The Presocratics, N.Y. 1974 | Philip, J.A.: Pythagoras and Early Pythagoreanism, Toronto 1966 Ross, W.D.: (1) The Works of Aristotle, vol VIII Metaphysica, Oxford 1928? (2) The Works of Aristotle vol XII Select Fragments, Oxford 1952 (3) Aristotelis Fragmenta Selecta, Oxford 1955 (4) Aristotle Metaphysics I-II, Oxford 1958 (5) Aristotle Physics, Oxford 1960 (6) Aristotelis Physica, Oxford 1966 Vlastos, G.: Review of J.E. Raven’s ‘Pythagoreans and Eleatics,’ repr. in Furley and Allen II, 166-176 Zeller, E.: Die Philosophie der Griechen I. Teil, Lpz. 1923° ASS. PROF. DR. ERIK NIS OSTENFELD DEPARTMENT OF CLASSICS UNIVERSITY OF AARHUS DENMARK