by A. A. Long
Philolaus (c. 470-385 B.C.) was the first Pythagorean to write a book (D.L. VIII.84-85) and, after years in limbo because of questions about authenticity, the fragments of that book have now emerged as the crucial primary texts for early Pythagoreanism. Some fragments fit the pattern of the pseudepigrapha, assigning Platonic and Aristotelian ideas to Philolaus, and are thus spurious. However, a core of fragments (DK 44 B1-7, 13, 17) use precisely the concepts that Aristotle assigns to fifth-century Pythagoreanism and are therefore genuine and indications that Philolaus was Aristotle’s primary source.24 They reveal Philolaus as an important thinker in the tradition of early Greek natural philosophy.25
Philolaus began his book with a concise statement of his central thesis:
Nature in the world-order was fitted together both out of things which
are unlimited and out of things which are limiting, both the world-order
as a whole and all the things in it. (B1)
The concepts here (nature = physis, world-order = kosmos) have an important place in earlier Greek thought. Moreover, although the Pythagoreans have often been regarded as sui generis and, beginning with Aristotle, primarily connected to Plato, Philolaus’ basic principles, limiters and unlimited, are a response to the earlier Greek tradition of natural philosophy. For Anaximander the world arose out of the unlimited (apeiron), Anaximenes called his basic stuff, air, unlimited (DK 13 A1 and 6) and, in the generation before Philolaus, Anaxagoras began his book by asserting that in the beginning all things were “unlimited both in multitude and in smallness” (DK 59 B1). It is opposites like hot and cold, dry and wet that come from Anaximander’s unlimited and that are labeled unlimited by Anaxagoras along with materials such as air and aither. These unlimited “stuffs” (i.e., both opposites and materials) dominated early Greek philosophy of nature. However, limit too had found its champion in Parmenides, who stressed that “what is” was held fast within limits and likened it to a sphere (DK 28 B8. 26, 42).
Philolaus draws on both these traditions, but is especially emphatic in his rejection of the dominant trend that made all principles unlimited, asserting instead that they fall into one of three classes:
It is necessary that the things that are be all either limiting, or unlimited, or both limiting and unlimited but not in every case unlimited alone. (DK 44 B2)
He goes on to argue that the world-order manifestly has elements that are limits, for example shapes and structures, and that the concept of order necessarily involves the limitation of the unlimited. Philolaus’ introduction of limiters as basic constituents of reality alongside unlimited stuffs leads him to redefine the essential nature of these stuffs. What makes them a unified class is not their qualitative features, such as hot and cold, but the fact that in themselves they are not determined by any quantity. They mark out a continuum of possible quantities that is then structured by limiters. The continuum of pitch is structured by limiting notes that define a scale; continua like water or earth become lakes or rocks when limited by shapes. Here we have a bold first step toward the matter-form distinction, although Philolaus gives no hint that these two types of element exist in any different way from one another and seems to treat them both as physical components of the cosmos.
In B6 Philolaus makes another crucial point about basic principles:
… the being of things which is eternal and nature in itself admit of divine and not human knowledge, except that it was impossible for any of the things that are and are known by us to have come to be, if the being of the things from which the world-order came together, both the limiting things and the unlimited things, did not preexist.
He is arguing that we cannot specify any particular set of unlimiteds (e.g., earth, air, fire, and water) or any particular set of limiters as eternal being, but that we can be sure that some set of limiters and some set of unlimiteds preexisted, since otherwise the world we know could not have come to be. In B2 and B6 Philolaus is accepting an axiom of early Greek thought sharpened by Parmenides, in not allowing anything to come to be from what is not. If the world has both limiting and unlimited features in it, these cannot have arisen just from what is limiting or just from what is unlimited. Philolaus’ point is not that earlier Greek philosophers had not seen the world as an ordered place but rather that they had failed to make limiters principles in their own right and mistakenly tried to generate an ordered-world out of basic principles that were, in their own nature, unlimited.
Philolaus refers to the limiters and unlimiteds as archai, “starting points.” Different sets of archai appear in other fragments, and it appears that the method followed in B6 with regard to the cosmos as a whole was used in each of the wide variety of subjects that Philolaus discussed. He begins by identifying a minimum set of starting points (archai) without which it is impossible to explain the phenomena. In the case of diseases, he specifies bile, blood, and phlegm as the archai (A27); in the psychic structure of human beings, the brain, the heart, the navel, and the genitals (B13).26 When it comes to the sciences, geometry is the starting point from which the others develop (A7a). Much of this method remains obscure, but Philolaus is struggling towards a generally applicable methodology that resembles the axiomatization of mathematical sciences.
In B6 he argues that yet a third principle is required to explain the world. Since limiters and unlimiteds are unlike, they must be held together by some type of bond determining the specific way in which they combine to form the ordered world we see. Philolaus calls this bond “fitting together” (harmonia), and it involves the last central concept in his system, number. He uses the diatonic scale as a prime example of his system of principles. An unlimited (the continuum of sound) is combined with limiters (points on that continuum). However, this combination is governed by a fitting together according to whole number ratios 1:2, 2:3, 3:4 that define the central musical concords of the octave, fifth, and fourth respectively, so that the result is no chance set of notes but the diatonic scale.
Philolaus also has important things to say about epistemological questions. Fragment B6 belongs in the early Greek tradition of scepticism about human knowledge (cf. Xenophanes DK 21 B34). However, it is original in its almost Kantian thesis, that, since knowledge of “nature in itself” is not available to mortals, the best that they can do is to posit as principles what is necessary to explain the world as we know it, that is limiters, unlimiteds, and harmonia. Moreover, the function of number in Philolaus’ system is to solve problems concerning knowledge of our world, perhaps in response to Parmenides, as is shown by B4:
And indeed all the things that are known have number. For it is not possible that anything whatsoever be understood or known without this.
Number is taken to be the prototype of what is knowable. Nothing is more determinate and certain than a numerical relationship such as 2 + 2 = 4. Philolaus thinks that the cosmos is held together by such numerical relationships and that Parmenides was right to object that the unlimited by itself is not a sufficient basis for human knowledge. In B3 Philolaus argues that, “There will not be anything that is going to know at all, if everything is unlimited.” This argument, that knowing requires an act of limiting, has particular force against Anaxagoras, who both posited basic principles that are all unlimited and also asserted the existence of a cosmic knower, nous. Philolaus may also be responding to Parmenides, arguing that even the unlimited may be knowable in so far as it is determined by number or involved in numerical relationships and that a plural world structured by such numerical connections is also knowable.
Both the strengths and weaknesses of Aristotle’s account of Philolaus and early Pythagoreanism are now evident. Aristotle’s famous assertion that for the Pythagoreans things “were numbers” makes sense as his interpretation of Philolaus. Since what is knowable for Philolaus is numerical, and for Aristotle what is knowable about things is their essence, it was an easy step for Aristotle to say that for the Pythagoreans numbers were the essence of things. At the same time, Aristotle
has seriously distorted the situation, by criticizing the Pythagoreans for constructing physical things out of numbers. Philolaus does not think that things are constructed out of numbers, but out of limiters and unlimiteds (B1), principles mentioned in Aristotle but which appear largely unmotivated there. But Aristotle is right not to project this system of principles back onto Pythagoras. While the contrast between limiters and unlimiteds is not impossible in Pythagoras’ time, these principles as well as the strong epistemological strain make better sense after Parmenides’ reflections on the conditions for knowledge and his insistence that “what is” is limited.
Philolaus’ cosmogony has been similarly distorted under the influence of Aristotle. The common view is that the first thing created was a monad or point. However, the fragments of Philolaus reveal that his cosmogony began with the central fire, the “hearth” of the cosmos and archetypal example of a combination of an unlimited (fire) with a limiter (centre). “The first thing fitted together, the one in the centre of the sphere, is called the hearth” (B7). Next, the central fire draws in the unlimiteds breath, time, and void (Aristotle, fr. 201 Rose). Philolaus draws an explicit parallel between the birth of the cosmos and the birth of a human embryo which, although hot in its own nature (like the central fire), breathes in cooling breath upon birth (A27). The biological analogy is not an archaic feature that goes back to Pythagoras himself as some have maintained but is paralleled in the atomists’ cosmology, where a crucial step was the formation of a “membrane” around the embryo universe (D.L. IX.31).27
Philolaus’ astronomical system has long been famous as the first to move the earth from the centre of the cosmos and make it a planet. Still, the earth does not orbit the sun but rather the central fire, along with the sun, moon, five planets, fixed stars, and a counterearth. Copernicus saw Philolaus as an important predecessor, but scholars have taken this astronomical system to show that he was not a natural philosopher but a number mystic.28 Certain a priori principles of order do play an important role in Philolaus’ system: the counter-earth is introduced to fill out the perfect number ten, and fire is put in the centre because the most valued element belongs in the most valued place. But such considerations do not make the system mere fantasy. A priori principles play a prominent role in most Greek astronomical schemes. A rational astronomy should include a combination of a priori principles and a posteriori information that produces a system open to challenge by appeal to the phenomena. Philolaus’ system does indeed confront a series of such challenges: problems of how to explain night and day and difficulties with parallax resulting from the motion of the earth are addressed (Aristotle, fr. 204 Rose, De caelo II. 13 293b25 ff.). Even the explanation as to why we never see the counter-earth or central fire, that is, that our side is always turned away from the centre of the cosmos, recognizes the importance of the phenomena. Moreover, the Philolaic system was the first to include the five planets known to the ancient Greeks in correct order. Philolaus may have speculated about inhabitants of the moon (A20) but so did such staid rationalists as Anaxagoras (DK 59 A77). In fact, the testimonia for both Anaxagoras’ and the atomists’ astronomical systems show that the Philolaic system is comparatively more sophisticated.29 Philolaus’ natural philosophy may have some origins in Pythagoras’ emphasis on significant numbers, but it is primarily his own response to problems raised by figures like Anaxagoras and Parmenides. Philolaus was a Pythagorean because he lived the sort of life Pythagoras prescribed, not because of his views on natural philosophy. In the modern world, we may say that someone is a Catholic without therefore being at all clear what he believes on a whole range of philosophical issues. A Pythagorean could become a philosopher of the early Greek sort (a physikos), a mathematician, a physician, or even a leading general, but none of these pursuits were demanded of him as a Pythagorean. Philolaus was a natural philosopher who also happened to be a Pythagorean.
Neither Lysis, known mainly as the teacher of the Theban general Epaminondas, nor Eurytus the pupil of Philolaus, wrote anything. Eurytus illustrated the identification of man or horse with a specific number by making pebble drawings of them (Theophrastus, Metaph. 11). Archytas, the last great name in early Pythagoreanism, was a contemporary of Plato and therefore not properly an early Greek philosopher. Nonetheless, because of his sophisticated three-dimensional solution to the problem of the doubling of the cube and his work on musical theory, he fits the popular conception of the Pythagorean as a master mathematician better than anyone else in the early tradition.
Perhaps the most enduring legacy of the Pythagorean tradition was its influence on Plato. It is possible that when Plato went to Italy for the first time in the early 380s B.C. he met an aged Philolaus. Philolaus is mentioned in the Phaedo (61d), perhaps in recognition of Plato’s debt to Pythagoreanism for his views on the soul.30 Moreover, a Platonic adaptation of Philolaus’ metaphysical system of limiters and unlimiteds is at the core of the Philebus.31 Although Archytas is never mentioned by name in the dialogues, the Platonic letters show that Plato had extensive contact with him and owed to him his final rescue from Dionysius II of Syracuse in 361. Plato in fact quotes from one of the three genuine fragments of Archytas (DK 47 B1) in Republic VII 530d8 where he refers to music and astronomy as “sister sciences.”32 Indeed, the mathematical curriculum of the Republic could owe its inspiration to Archytas (B1), and Archytas himself, who was elected seven consecutive times in Tarentum and never suffered a defeat in battle, may be a model of the philosopher king. The specific functions of mathematics in Plato’s philosophy (e.g., turning the soul toward the world of Forms) are largely his own creation, and the Timaeus is a Platonic and not a Pythagorean work. However, the conviction that mathematics could help in addressing important philosophical problems, which began with Philolaus and was shared by Archytas, and Pythagoras’ own vision of the mythic cosmos in which the migrating soul is subject to judgement for its deeds, give Platonism an undeniable Pythagorean content from the middle dialogues onward.33
NOTES
1 For Pythagoras as pioneer mathematician, see A.N. Whitehead, Science and the Modern World (New York, 1925)41. For Pythagoras the shaman, see Dodds [94] 143–45.
2 Burkert [201] 129ff., 218–20.
3 See P. Brown, The Making of Late Antiquity (Cambridge, Mass., 1978) 54–80.
4 O’Meara [224].
5 J. Dillon The Middle Platonists (London, 1977).
6 Burkert [201] 53–83.
7 Guthrie’s great account of Pythagoreanism (in Guthrie [15]) spends 180 pages elucidating this unified Pythagoreanism and just 15 pages on individual Pythagoreans.
8 Guthrie [15] 206 ff.; Kahn [218] is more careful.
9 Guthrie [15] 181.
10 Burkert [201] 28–83.
11 Other passages (e.g., Metaph. VII. 11 1036b8) have been misinterpreted as evidence for a Pythagorean derivation of the sensible world from more ultimate mathematical principles by the derivation sequence of one = point, two = line, three = surface, and four = solid, with physical bodies then arising from the geometrical solids. However, this belongs to the early Academy (Burkert [201] 67). Thus twenty pages ([15] 256–76) of Guthrie’s account of Pythagoreanism are undercut. This sequence turns up in the Pythagorean memoirs (D.L. VIII.24–33) excerpted by Alexander Polyhistor (first century B.C.). Their value as a source is very doubtful (Burkert [201] 53 and Festugière [210]), but Guthrie uses them heavily. See Kahn [218].
12 Burkert [201]; Thesleff [202] and [199].
13 See Burkert [201] 126, 133; West [136] 62; Kahn [217] 166. Metempsychosis may have come into Orphism through Pythagoras (Burkert [201] 126, 133). Late sources assign it to Pherecydes (West [136] 25).
14 Burkert [201] 133ff.
15 See Laks in this volume, pp. 251–2; Burkert [201] 134, n. 78; Claus [486] 4–5, 111–21; Huffman [198] 330–31. Other early texts also stress Pythagoras’ expertise on the soul (Herodotus IV.95, Empedocles DK 31 B129, Ion in D.L. I.120).
16 The historian Timaeus repo
rted that these lines refer to Pythagoras. Diogenes Laertius (VIII.55) says that some thought they refer to Parmenides. The “generations” and “wise deeds” fit Pythagoras better.
17 Minar [221].
18 Burkert [201] 162ff.
19 Some make Pythagoras more of a natural philosopher than I have done. My interpretation is based on a strict reading of the early evidence and seems to be Aristotle’s interpretation. Guthrie [15] 166–67 relies on Republic VII as the sole early evidence of “the scientific side.” But these are fourth-century Pythagoreans (Huffman [216]). Heraclitus’ reference to Pythagoras’ practice of historia is too general to be conclusive, and the Heraclitean concept of harmonia, if it alludes to Pythagoras, can just as easily refer to the Pythagoras I have described as to a Pythagoras who was an Ionian-style cosmologist.