Why one should be passionate about numbers

Draft submitted in 2005 for publication in a volume to be edited by Apostolos Pierris

Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers—not just its physics but perhaps morality too. Was this just fanciful speculation? Is it muddled as a theory of causality, as Aristotle suggests? I shall try to rehabilitate the passion for numbers by linking it with the notions of harmony and proportion in other thinkers who have a higher credibility factor in the philosophical stakes, and by showing that the desire to reduce quality to quantity, and to discover an exact science that can explain human life and meaning, is a serious philosophical passion that doesn't easily go away. And, after all, why should it?

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    Why one should be passionate about numbers
    Catherine Osborne, July 2005 Abstract: Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers—not just its physics but perhaps morality too. Was this just fanciful speculation? Is it muddled as a theory of causality, as Aristotle suggests? I shall try to rehabilitate the passion for numbers by linking it with the notions of harmony and proportion in other thinkers who have a higher credibility factor in the philosophical stakes, and by showing that the desire to reduce quality to quantity, and to discover an exact science that can explain human life and meaning, is a serious philosophical passion that doesn't easily go away. And, after all, why should it?
    
    Why one should be passionate about numbers
    Granted, we don't know very much about Pythagoreanism in the early days when Pythagoras was around. But it seems reasonable, and not hugely controversial, to suggest that a passion for numbers was at least an embryonic part of the early heritage, something that belongs in the Pythagorean core, and not just to the crust of lichen and fungus that grew round the name of Pythagoras in later generations.1
    
    1
    
    According to Burkert's authoritative analysis this is somewhat controversial, since he
    
    excludes most of the stuff about numbers (mainly in Aristotle) from having anything to do with the real Pythagoras. Aristotle, according to Burkert, is careful to tell us about the "so-called" Pythagoreans, namely fifth century Pythagoreans, not Pythagoras himself. By contrast, the authentic material that we have for Pythagoras himself is almost exclusively about mysticism and wonderworking. My claim is not so much that the elaborate theories of number are authentic to early Pythagoreanism, but that an embryonic interest in harmony, and in the mystical aspects of number, such as the oath by the tetraktys, can safely be traced back to the early period. For the more austere division, see Walter Burkert Lore and Science in Ancient Pythagoreanism, trans. E.L. Minar Jr (Cambridge,
    
    2 But what kind of a passion for numbers was it? Was it the kind of enthusiasm that philosophers can safely hail as part of their authentic heritage, and attribute to the real Pythagoras with pride? Or is it an embarrassment, to be hushed up and omitted from the histories of real philosophy? Were some of the Pythagoreans guilty of an unfortunate misjudgement—perhaps even some of the greatest thinkers, perhaps Pythagoras himself, perhaps many of his followers and Neopythagoreans right into late antiquity? Did they fail to see the difference between respectable mathematics and mystical mumbo-jumbo? Did they get carried away with an inappropriate desire to put numbers where numbers should not be? Jonathan Barnes gives a dramatic rendering of the dilemma for the historian of philosophy by inviting us first to approve and then to condemn. First he invites us to approve by sketching the story as a story in which Pythagoras is a great mathematical hero and astronomer: These pious offerings portray an impressive figure: Pythagoras, discoverer and eponym of a celebrated theorem, was a brilliant mathematician; by applying his mathematical knowledge, he made progress in astronomy and harmonics, those sister sirens who together compose the music of the spheres; and finally, seeing mathematics and number at the bottom of the master sciences, he concocted an elaborate physical and metaphysical system and propounded a formal, arithmological cosmogony. Pythagoras was a Greek Newton; and if his intellectual bonnet hummed at times with an embarrassing swarm of mystico-religious bees, we might reflect that Sir Isaac Newton devoted the best years of his life to the interpretation of the number symbolism of the book of Revelations. If Greek science began in Miletus, it grew up in Italy under the tutelage of Pythagoras; and it was brought to maturity by Pythagoras' school, whose members, bound in fellowship by custom and ritual, secured the posthumous influence of their master's voice. Jonathan Barnes The Presocratic Philosophers 100-101 Then he withdraws that story, because it isn't true:
    
    Mass.: Harvard University Press, 1972). Burkert's analysis replaces a previous tradition, including Francis MacDonald Cornford 'Mysticism and science in the Pythagorean Tradition' in Alexander P.D. Mourelatos ed. The Presocratics (Garden City: Anchor Books, 1974) 135-60 originally published in 1922-3 (who saw early Pythagoreans as having a "mystical system" that put them with the Milesians and under criticism from Parmenides, and later ones as having a pluralist system of number atomism which was the object of Zeno's hostility). Against Burkert's minimalist conclusions see the more optimistic assessment in C. J. de Vogel Pythagoras and Early Pythagoreanism: an interpretation of neglected evidence on the philosopher Pythagoras (Assen: Van Gorcum, 1966), close in time to Burkert's original publication date and not directly addressed to his finished publication.
    
    3 What are we to make of this pleasing picture of a Newtonian Pythagoras? It is, alas, mere fantasy: the shears of scholarship soon strip Pythagoras of his philosophical fleece. Jonathan Barnes The Presocratic Philosophers 101 And thirdly he invites us instead to condemn the whole Pythagorean passion for numbers, not just Pythagoras himself but the later tradition in particular: What philosophical use did the Pythagoreans make of mathematics? The cynical will speak dismissively of number mysticism, arithmology, and other puerilities. And it is undeniable that a great quantity of Pythagorean 'number philosophy' is a 'number symbolism' of the most jejune and inane kind. … 'Touching on' arithmetic, the Pythagoreans were impressed by certain properties of the number 10; alas, their impression degenerated into a sort of mysticism: amazement, the nurse of philosophy, soon has her milk soured and turns into silly reverence and superstition. Those with a taste for intellectual folly will have their appetite sated if they go through the Theologoumena Arithmeticae. That Pythagorean work is a late compilation; the earliest examples of such symbolism are found in the acousmata and probably date from the time of Pythagoras himself: from first to last the Pythagoreans engaged in arithmology. Jonathan Barnes The Presocratic Philosophers 381 Barnes offers us two options: we could admire Pythagoras if he was, as suggested, a Newtonian, whose mathematical discoveries were put to fine use in developing a mathematical astronomy and answers to physics that sought confirmation in mathematics. Or we could condemn him (and his followers) if the philosophical use to which they put their number work was mere mysticism and number symbolism. Since the first option seems to be ruled out by the lack of sound historical evidence for the fanciful portrait of the Newtonian Pythagoras, we are left with the second. And so we condemn. My task today is to persuade you that, notwithstanding Barnes's fine rhetoric on the matter, we should nevertheless admire, and not condemn, the Pythagorean enthusiasm for numbers.2
    
    1 Pythagoras and the early Presocratic tradition.
    One not particularly original route to defending the Pythagoreans would be to reject Barnes's claim that the Newtonian Pythagoras is mere fantasy. One could try to show that Pythagoras should be
    
    2
    
    Of course the association with mathematics is only one branch of the many traditions
    
    about Pythagoras as wise man. See Carl Huffman 'The Pythagorean tradition' in A A Long ed. The Cambridge Companion to Early Greek Philosophy (Cambridge: Cambridge University Press, 1999) 66-87: 66-7 for a judicious statement of the general problem, and the need to rescue Pythagoreanism from the problems created by an exaggerated reputation and problematic source materials.
    
    4 reliably credited with certain significant mathematical discoveries, that his work on harmonics was of a serious experimental nature, and that it is not improbable that he applied these studies to astronomy in a spirit of scientific inquiry, so as to show that his work on numbers was not mere speculation or mysticism. That is not the task I have set myself today. For not only is it not a particularly new project, but in any case it is not the case I want to make, for reasons that I shall go on to explain. A second rather more interesting project does strike me as worthwhile however. In this first part of my paper I shall not so much seek to restore Pythagoras's credibility as a practitioner of the exact sciences as to ask why Pythagoras has been downgraded in the contemporary assessments, dismissed as a mystic and wonder-worker, while other Presocratics with rather similar points to make have been exalted as pioneers of embryonic scientific and philosophical thought. Is there really so much difference? It's probably not worth starting with Thales (although clearly one could), but let's begin with Anaximander. First of all we should note the presence of the term 'apeiron' as a technical term in Anaximander's cosmology. When we meet this term in Pythagorean documents, we see a contrast between limit and unlimited, a classification of numbers that spills out into reality more widely.3 It is part of what makes number the basis of the whole of reality, and it gets taken up into the more peculiar bits of Platonic theory, in particular the one and the indefinite dyad.4 Here in our modern accounts of Pythagoreanism, the unlimited or indefinite it is conceived as funny stuff. But in Anaximander it is conceived very differently. Rather than seeing here a primarily mathematical notion being used to create a metaphysical basis for reality, we read Anaximander as speaking of an unlimited material: physical stuff. We read him as a materialist, and for that reason apeiron is defused. It doesn't smack of funny stuff, as it does in the hands of the Pythagoreans. It looks instead like a primitive kind of prime matter, something that Aristotle could look back to, and in which he could trace the origins of the material cause.
    
    3
    
    The terminology is widespread, particularly in material from Philolaus. See Philolaus Aristotle Metaphysics 987b18-988a1. Details discussed in J.N. Findlay Plato; The written and
    
    fragments 1, 2, 3, 6 etc, and Aristotle Metaphysics A chapter 5.
    4
    
    unwritten doctrines (London: Routledge and Kegan Paul, 1974) chapter 2.
    
    5 Secondly we should notice that Anaximander has a universe structured with concentric circles to explain the positions of the heavenly bodies. We do not hear the music of the spheres in Anaximander's universe, but it is surely improbable that they do not utter sounds, for the circles that carry the stars have "flute-like" pipes with breathing holes through which the fire bursts forth when they are not blocked up.5 It seems certain that the pressurised fiery vapour escaping from these ejkpnoaiv must make sounds or notes that reflect the size and diameter of the pipe, rather like the sound of huge pan pipes played across the dark and misty heavens. How are their notes related? Harmonically or out of tune? Who can say? But we do know—roughly speaking—that Anaximander posited some numbers which claim to be the sizes or distances of the circles of the sun and the moon and the stars.6 The numbers seem to form a pattern, a sequence of a geometrical kind, probably 9, 18, 27, with the earth too having the numerical proportion 3:1 between its height and its diameter.7 In fact, it seems that Anaximander was applying a sort of geometrical thinking to his speculations about the shape and the movements of the heavens.8
    
    5 6 7
    
    Hippolytus Ref 1.6 Hippolytus Ref 1.6 The evidence for these numbers in the doxography, and the reconstruction of the
    
    mutilated texts, are discussed by Denis O'Brien 'Anaximander's measurements' Classical Quarterly 17 (1967) 423-32 and Gerard Naddaf 'Anaximander's measurements revisited' in Anthony Preus ed. Before Plato (New York: SUNY Press, 2001) 5-23. See also Charles Kahn Anaximander and the origins of Greek Cosmology (New York: Columbia University Press, 1960) pages 61-3, G S Kirk, J E Raven, and M Schofield The Presocratic Philosophers, Second edition ed. (Cambridge: Cambridge University Press, 1983) pages 133-7, Dirk L Couprie 'The Discovery of Space: Anaximander's Astronomy' in Dirk L Couprie, Robert Hahn, and Gerard Naddaf ed. Anaximander in Context (New York: SUNY Press, 2003) 165-254.
    8
    
    This is widely agreed. Kahn suggests that the inspiration for Anaximander's numbers was
    
    mathematical rather than mystical (Kahn Anaximander and the origins of Greek Cosmology 96-7). A possible link with the geometrical calculations used in architecture has been explored in detail by Robert Hahn Anaximander and the Architects (New York: SUNY Press, 2001) and Robert Hahn 'Proportions and numbers in Anaximander and Early Greek Thought' in Dirk L Couprie, Robert Hahn, and Gerard Naddaf Anaximander in Context (New York: SUNY Press, 2003) 73-163. Much effort has gone into trying to show that, although the work was speculative rather than experimental, Anaximander was not just engaged in arithmology. See also W.A. Heidel 'The Pythagoreans and Greek Mathematics' in David Furley and R.E. Allen ed. Studies in Presocratic Philosophy Vol. 1 (London:
    
    6 When Pythagoras, or some Pythagoreans, posit harmonic proportions, based on ratios of numbers, as the explanatory principles in determining the structures of the heavens, and do this on the basis of discovering that physical sounds can be analysed mathematically as related to the size and shape of the physical object that produces them, their speculations are frequently dismissed as idle speculation.9 When Anaximander engages in some similar, though less well grounded, fantasy about the relative sizes of the heavenly rings, he is hailed as a pioneer and an impressive forerunner of mathematical astronomy.10 Of course, Anaximander lived a little earlier than Pythagoras, in the early sixth century rather than the second half of it. But really there is not a lot of difference in the dates,
    
    Routledge and Kegan Paul, 1940/1970) 350-81: 379 . The alternative line that this is a rationalisation of some mythic or poetic materials is proposed by Martin L. West Early Greek Philosophy and the Orient (Oxford: Clarendon Press, 1971) 94, Dirk L Couprie 'Anaximander's discovery of space' in Anthony Preus ed. Before Plato (New York: SUNY Press, 2001) 23-48: 40-1.
    9
    
    It is hard to find references that explicitly present this attitude as strongly as Barnes does in
    
    the passage cited above. Yet I think that the judgement is evident in the general approach to the study of the Presocratic philosophers and in the extent to which Pythagorean number theory is marginalised in the main-stream collections of work on Presocratic philosophy. Compare the evidence presented by Kingsley for a tradition in intellectual history that tries to cleanse the early Pythagoreans of the mystical, by implying that it was a feature of a decadent, late, pseudoPythagoreanism, not the true philosophical period of Pythagoras and his early followers (Peter Kingsley Ancient Philosophy, Mystery and Magic: Empedocles and Pythagorean Tradition (Oxford: Oxford University Press, 1995) 317-20). That is not quite the pattern I am seeking to highlight here, but the two are closely related. See also Geoffrey E R Lloyd Early Greek Science Thales to Aristotle (London: Chatto and Windus, 1970) 26-7 ("Secondly many of the resemblances that the Pythagoreans claimed to find between things and numbers were quite fantastic and arbitrary…. Obviously while the search for numerical ratios proved fruitful in such fields as the analysis of musical harmonies, and mathematics itself, it also and more often led to mumbo-jumbo and crude number mysticism"); and David Furley The Greek Cosmologists, vol. 1 (Cambridge: Cambridge University Press, 1987) 58 who dismisses some parts of Pythagorean astronomy as fantasy, though he does this in the service of a more positive assessment of their particular emphasis on form and structure.
    10
    
    "The importance of this theory is that it is the first attempt at what we may term a
    
    mechanical model of the heavenly bodies in Greek astronomy" (Lloyd Early Greek Science @17); "His theory of equilibrium was a brilliant leap into the realms of the mathematical a priori" Kirk, Raven, and Schofield The Presocratic Philosophers 134. Recent fashion has been rather more low key in its estimate of Anaximander (e.g. Keimpe Algra 'The beginnings of cosmology' in A A Long ed. The Cambridge Companion to Early Greek Philosophy (Cambridge: Cambridge University Press, 1999) 45-65: 55).
    
    7 and we might suppose that by appealing to harmonic structures, Pythagoras was attempting to put more plausibility and accuracy into the speculative arithmetic, on the basis of his discoveries that related audible harmony to numerical proportions and the sizes of physical objects,11 in place of whatever obscure patterns lay behind Anaximander's guesswork. So it seems that when Anaximander indulges in pure numerical fantasy, plucking multiples of 3 and 9 apparently out of the air, he is hailed as the pioneer who saw for the first time that the plausibility of one's cosmological theory can be enhanced by showing that it makes mathematical sense. When Pythagoras does a more sophisticated version of the same thing, selecting harmonic and geometrical sequences in preference to arbitrary patterns, he is accused of superstitious and fanciful numerological speculation. And yet the reason for positing harmonic ratios in nature is that harmonic ratios are found in nature, and are perceptible by us because we are naturally attuned so that we find such ratios beautiful, when they occur. Certainly, to suggest that the heavens manifest a harmonic structure which we find beautiful is not to engage in empirical science of quite the sort we are used to. But if we are in the business of speculative astronomy, rather than empirical astronomy, then Pythagoras (or whatever Pythagorean invented this idea) has at least as good a grounding for his approach as Anaximander seems to have. And if empirical support is a virtue, at least Pythagoras can point to his work on harmony, which evidently does have some empirical support.12 Why, then, do we cite Pythagorean harmony theory as a type of worthless superstition, but cite Milesian science as an impressive forerunner of modern mathematical techniques in empirical science? Could it be that the idea that something is beautiful, or that there should be music in it, seems not to be a good reason for supposing it to be true? But if that is so, we need to examine our preconceptions. For it seems, first, that we have come with a built in prejudice in favour of the idea that nature is random, disordered or arbitrary, rather than systematic, displaying patterns and orders at more than one level. Why should it not be more likely that harmonic patterns figure in the structure of the heavens?
    
    11
    
    See Xenocrates fragment 9 Heinze, apud Porphyry Commentary on Ptolemy's Harmonics 30 Two key texts are Xenocrates fragment 9 Heinze, apud Porphyry Commentary on Ptolemy's
    
    1-6 Düring.
    12
    
    Harmonics 30 1-6 Düring; Aristoxenus fr 77 Müller, in a scholiast on Plato Phaedo 108d, Greene p.15.
    
    8 Secondly, it seems plausible to suppose, as I have suggested, that Anaximander too thought that the heavenly bodies uttered a flute-like whistle. Perhaps he, too, was moved by that thought in composing his theory about the sizes and shapes of the hoops that circle the earth. So that even if we do, sadly, start from a post-enlightenment prejudice in favour of seeing the world as meaningless and lacking in beauty, still there seems to be some inconsistency in our preference for the speculations of Anaximander over those of Pythagoras. Is that just because we don't happen to have any texts on the music of the spheres —or rather wheels—in Anaximander? Moving on from Anaximander to Heraclitus, let us ask a different question, this time about logos. It has become customary in writing about Heraclitus to leave the word 'logos' untranslated even when writing for a Greekless readership.13 Alternatively translators look for a standard formulaic or non-committal translation (such as "account")14 in order to avoid giving any specific meaning to the term in any particular occurrence.15 These high-minded practices have an unforeseen consequence, it seems to me. First they make the term logos stand out as something like a key concept, inviting us to think that Heraclitus has a theory of "the Logos". This is reinforced by the habit of adding the definite article, which is there in some but not all of Heraclitus's references to logos,16 but would not always be there, typically, in English even where it is normal in Greek (for instance if oJ lovgo" were translated by a term such as 'discourse', 'language', 'reason', 'proportion,' 'rationality' or 'logic'). The result is that the Logos becomes almost imperceptibly hypostasised, until we find it occupying a place
    
    13
    
    Robin Waterfield in the commentary in 'The First Philosophers' in Oxford World's Classics,
    
    ed. Robin Waterfield (Oxford: Oxford University Press, 2000) ; Kirk, Raven, and Schofield The Presocratic Philosophers ; Richard McKirahan Philosophy before Socrates (Indianapolis: Hackett, 1994);
    14
    
    "Account" Barnes in 'Early Greek Philosophy' in Penguin Classics, ed. Jonathan Barnes
    
    (Harmondsworth: Penguin, 1987) ; Robinson in Heraclitus Fragments edited by T.M. Robinson, Phoenix Presocratics, Toronto: University of Toronto Press, 1987; Charles Kahn The Art and Thought of Heraclitus (Cambridge: Cambridge University Press, 1979); "principle" Waterfield in 'The First Philosophers'
    15
    
    The motives for both practices are, of course, admirable in their way, in so far as the
    
    translator tries to avoid imposing an interpretation by rendering the term one way rather than another, or concealing the same term under unrecognisable variant translations. I am not suggesting that there is a better solution, but rather that translation is inherently unsatisfactory.
    16
    
    Logos occurs with the definite article in frr. 1, 2, 31B; 50; without in frr. 39, 45, 72, 87, 108,
    
    115.
    
    9 that looks plausibly god-like, whereupon we are tempted to identify it with the 'god' of fragment 67. What might well have looked like an immanent pattern in the behaviour of the world takes on a metaphysical role as a divine entity that explains or dictates the reciprocal patterns in the world. This seems to happen at least in part because our translation practices, and exegetical practices, irresistibly privilege the term logos and exalt it to become a term of art. Because it looks as though the Logos is a kind of god in Heraclitus's system, we fail to notice the resemblance between Heraclitus's interest in proportion and ratios and the same topics in Pythagoreanism. So the failure to translate logos, the adoption of a special systematic pseudo-translation, and the addition of the definite article and capital letter as though the logos were an hypostasis or divinity, mask the links between the notion of logos and the notion of "harmony", which is also seen as a key concept in Heraclitus's thought, as it is in Pythagorean thought. The Pythagorean resonances in Heraclitus would, of course, be much more apparent if Heraclitus's logos were read as 'ratio' or 'proportion', and the numerical significance of harmony were allowed to emerge in the context of both opposition (in Heraclitus fragment 8), of ratio (in fragments 49, 79, 82-3), and of measure, limit and the unlimited (in fragments 120, 94, 30,31 and 45). Granted, Heraclitus uses a range of terms for limit and boundary, including ou\ro" (120), tevrmata (120), mevtron (94), as well as the peivrata of 45, so that even in the Greek we are not alerted very strongly to the link between these obsessions with measure and limits, whereas in Pythagorean sources there is a kind of technical terminology— the a[peira kai; peraivnonta of Philolaus and the pevra" kai; a[peiron of the first pair in Aristotle's table of Pythagorean opposites—which draws attention to the theme more prominently than the Heraclitean vocabulary does.17 Still it remains undeniable that Heraclitus uses the idea of logos or proportion to bring order to the measured processes of the world, and that he draws connections between patterns of opposition and the idea of harmony (hidden or otherwise). So while the Pythagoreans' attempts to put numbers on the patterns and proportions that they saw in the cosmos, and to link those numbers to geometrical and harmonic proportions, are
    
    17
    
    The impression that the Pythagoreans have a systematic technical terminology may be
    
    exaggerated because of the prominence of Philolaus as a prime source for Presocratic Pythagoreanism.
    
    10 easily dismissed by Barnes (and not just Barnes)18 as unreasoning numerical fantasies, Heraclitus's mysterious logos is given a definite article, a capital letter, and hailed as an attempt to bring reason and order into a world of opposition and strife.19 What is the difference? Is it that Heraclitus does not give us the numbers but only hints at the existence of measures and ratios (the sea returns "to the same measure as was there before it became earth", fragment 31)? But when the Pythagoreans tell us that there are numbers (that the number of the heavenly bodies is ten, for example) we find that their choice of numbers is motivated by the desire for perfection, not the desire to save the phenomena? Or is it that (by tradition) we translate out, or interpret out, the references to harmony and ratio in Heraclitus: we don't see them as number mysticism because instead we interpret Heraclitus's interest in logos as a kind of monotheism—something in the tradition of Xenophanes in which a Zeus-like divinity is rationalised and demythologised to become an immanent physical regulatory principle in the world, a guiding principle that sees to it that we don't need to appeal to weird metaphysical structures or mystical number patterns. We try to see in Heraclitus a step on the route from Milesian materialism to post-enlightenment physics, and we let him have his logos in the guise of personified reason—exalted, but not transcendent, much as Cartesian theism accounts for the existence and orderly functioning of a mathematically regular cosmos. Sextus Empiricus provides a lengthy analysis of the role of reason as a criterion of knowledge in the Presocratic philosophers, an analysis heavily coloured by the interests and concerns of Hellenistic epistemology. Having dealt with Anaxagoras ("the most physical" of the Presocratics)20 he introduces the passage on the Pythagoreans thus:
    
    18
    
    I have used Barnes because he represents an extreme end of a certain kind of anglo-saxon
    
    tradition. Not all historians of Presocratic philosophy display such antagonism towards the Pythagorean school, so that my generalisations should be taken to have distinguished exceptions. I am happy for the reader to identify with me or with my opponents as he/she feels most at home.
    19
    
    Barnes is careful to warn us against taking the logos as a technical term and the key to
    
    Heraclitus's secrets (Jonathan Barnes The Presocratic Philosophers, 2nd ed. (London: Routledge and Kegan Paul, 1982) 59) but he goes on to allow that there may be a metaphysical logos doctrine, and that Heraclitus is to be placed with the Milesian rational tradition and sheltered from the pejorative term "mystic" (which is doubtless reserved for those who fall into the Pythagorean mire), Barnes The Presocratic Philosophers 80-1.
    20
    
    Sextus Empiricus Adv Math 7.90
    
    11
    
    w{ste oJ me;n ∆Anaxagovra" koinw'" to;n lovgon e[fh krithvrion ei\nai: oiJ de; Puqagorikoi; to;n lovgon mevn fasin, ouj koinw'" dev, to;n de; ajpo; tw'n maqhmavtwn periginovmenon, kaqaper e[lege kai; oJ Filovlao", qewrhtikovn te o[nta th'" tw'n o{lwn fuvsew" e[cein tina; suggevneian pro;" tauvthn, ejpeivper uJpo; tou' oJmoivou to; o{moion katalambavnesqai pevfuken... h\n de; ajrch; th'" tw'n o{lwn uJpostavsew" ajriqmov": dio; kai; oJ krith;" tw'n pavntwn lovgo" oujk ajmevtoco" w]n th'" touvtou dunavmew" kaloi'to a]n ajriqmov".21 Sextus Empiricus Adv Math 7.92 (= DK 44A29)22 Sextus goes on to provide six or seven further paragraphs of evidence in support of the claim that mathematical reason is the criterion for the Pythagoreans before proceeding to apply the same kind of treatment to Xenophanes, Parmenides, Empedocles, Democritus and Heraclitus.23 It is no surprise that Heraclitus too is said to make the logos a criterion of truth.24 In Heraclitus's case it is not cashed out as mathematical reason, naturally enough, since that is distinctively Pythagorean. Instead Sextus delivers the fragments that we standardly use as evidence for the Logos doctrine in Heraclitus.25 So we accept from this passage an account of Heraclitus and his notion of the logos—this survives almost unscathed into our interpretation of great and crucial moves in Presocratic thinking, but we ignore the bits about logos in Pythagoreanism, and its basis in number and the idea that like is known by like so that if the universe is numerically ordered, then our understanding of it will be similarly ordered as a mathematical kind of science.26
    
    21
    
    So that Anaxagoras said that the logos generally was the criterion. The Pythagoreans also
    
    say that it is the logos—but this time not the logos generally but the logos that is acquired from studies (mathematics?)—just as Philolaus also said—and that given that it contemplates the nature of the universe, it has a certain affinity with that nature, if like is by nature grasped by like… But number was the principle of the structure of the universe; hence the logos that is the judge of all things is not devoid of this power, and would therefore be called number.
    22
    
    Text as in Carl Huffman Philolaus of Croton (Cambridge: Cambridge University Press, 1993) Adv math 7 92-140. Adv Math 7.126. That is fragments 1 and 2. In defence of this practice one might appeal to the fact that we do have genuine fragments
    
    199.
    23 24 25 26
    
    of Heraclitus in which he uses the term logos, whereas there is no textual support for attributing that term in that sense to Philolaus. But we should notice that Sextus is talking in terms that are alien to the Presocratic discourse throughout, and this applies as much to his search for a criterion of truth in Heraclitus as it does in Philolaus. On the anachronism of the terms of the enquiry see Huffman Philolaus of Croton 199-201.
    
    12 That bit—the appeal to a specifically mathematical type of calculation—is not exactly what we find in Heraclitus, although there are hints of a kind of thinking that invokes proportions and ratios in Heraclitus too.27 But it is not obvious that the addition of mathematics as a criterion of sound understanding of the world is a development that we should dismiss as mystical mumbo-jumbo, or despise as a fairy story, by comparison with the rather vaguer and more general notion of logos in Heraclitus. On the contrary, we should probably agree, nowadays, that cosmology requires not just a generic brand of rational enquiry but more specifically a mathematically trained investigator, whose criterion for whether he has reached something worthy to count as knowledge will be whether the mathematics works. So why should we think of Heraclitus as a model of Presocratic philosophy at its best, while dismissing Pythagorean theory as an embarrassing disfigurement of an otherwise pure stream of increasingly rational investigation? Let me offer a proposal for what lies behind this widespread preference for the Heraclitean over the Pythagorean harmony theory. It is this. It seems to me that Barnes (as well as others who share his judgements) is following a tradition that prefers signs of materialism and reductionism over any kind of metaphysical or teleological picture. It is a tradition that sees the pure materialist reductionism of the atomists as the culmination and high point of the Presocratic achievement, and it assesses the contribution of earlier thinkers by how closely they approximate to that ideal—an ideal that is seen as a kind of no-nonsense physics, even if it has little ambition to provide genuine empirical support for its speculations.28 It is true that I have suggested that Heraclitus's logos gets hypostasised as "The Logos" with a capital letter, and in the process takes on a quasi-god-like role as the governor of cosmic processes. That might suggest that our admiration for Heraclitus is not because we see him as eliminating metaphysical and religious entities. But, as I suggested above, we tend to conceive of that move as
    
    27 28
    
    E.g. fragments 30, 31, 79, 82, 83. Barnes himself is careful to warn us against over-enthusiastically assimilating ancient
    
    atomism to modern science, and he points out many ways in which the atomism of Leucippus and Democritus raise problems that they cannot answer (Barnes The Presocratic Philosophers 343-4, 76-7). But these warnings leave untouched the general sense that the atomists' theory is virtuous just in so far as it approximates to the ideal of modern science — so that ancient atomism is not quite admirable because it fails to live up quite fully to that ideal in all respects.
    
    13 somewhat reductionist, like Xenophanes's theological endeavours. On that reading of Heraclitus, God, or the Logos, just is the world, when all's said and done: God is day, night, winter, summer, war, peace, hunger satiety…29 God is the processes that once seemed mysterious; but really (according to this version) they are not so much mysterious as regular, not unpredictable but reasonable. Reason, not religion, is the way to get control of the physical events. That is what we take Heraclitus to be saying, in his rather obscure and difficult way. So God turns out to be no more than our picture of how the world works to a predictable pattern. Harmony theory is then read not as a metaphysical thesis but as a materialist one. Perhaps this is correct as an interpretation of Heraclitus? In support one might notice the way in which Heraclitus appeals to the notion of measure in connection with specific physical processes that manifest this kind of proportionality—the notion that we find this logos in the quantity of sea that you get back when earth has been turned back into water (fr. 31) and that there are observable limits to the passage of the sun across the sky.30 It seems that no matter how little evidence Heraclitus gives for observing such regularities in nature, he does mean that the regularities in question are part of nature.31 Similarly, to return to Anaximander's picture of the cosmos, those heavenly pipes seem to be substantial and very concrete material chariot wheels revolving in the sky, invisible only because they can't be seen for the mist. Anaximander too is giving us material explanations with numbers on them, not numerical explanations with no matter to do the work. Pythagorean mathematics, by contrast, tries to make numbers do the work. If you're looking for an account of material and efficient causes in the cosmos, it's odd to point to numbers as such (as
    
    29 30
    
    Heraclitus fragment 67. This is one traditional interpretation of the claim in fragment 94 that "the sun will not
    
    overstep its measures". The Derveni papyrus (which appears to combine what we used to know as fragments 3 and 94) opens the possibility that the measures are the sun's size (a foot across, fr 3) rather than its tropics. See Gabor Betegh The Derveni Papyrus: Cosmology, Theology and Interpretation (Cambridge: Cambridge University Press, 2004) 10-1. This alternative is also compatible with the traditional reductionist interpretation that I am sketching.
    31
    
    Here perhaps we assume too readily that the image of the Erinues in fr.3 ("otherwise the
    
    Furies, ministers of Justice, will find it out") is just a picturesque metaphor to convey what we take to be a natural constraint on the behaviour of the sun. Again the Derveni papyrus, with its obsession with the daimonic, tells us that Heraclitus was not always read so rationalistically in antiquity (see previous note).
    
    14 opposed to applying numbers to quantities of other things, quantities of some material stuffs or physical forces that could do the work). So if you conceive the Presocratic project as a project to suggest and improve explanatory factors that are to be invoked in the interests of a reductionist thesis about how the world works, numbers as such seem to be the wrong kind of thing.32 This is a complaint that is closely related to an objection made by Aristotle in Metaphysics N: oiJ de; Puqagovreioi dia; to; oJra'n polla; tw'n ajriqmw'n pavqh uJpavrconta ãejnà toi'" aijsqhtoi'" swvmasin, ei\nai me;n ajriqmou;" ejpoivhsan ta; o[nta, ouj cwristou;" dev, ajll∆ ejx ajriqmw'n ta; o[nta. dia; tiv dev… o{ti ta; pavqh ta; tw'n ajriqmw'n ejn aJrmoniva/ uJpavrcei kai; ejn tw/' oujranw'/ kai; ejn polloi'" a[lloi" … oij me;n ou\n Puqagovreioi kata; me;n to; toiou'ton oujqeni; e[nocoi eijsin, kata; mevntoi to; poiei'n ejx ajriqmw'n ta; fusika; swvmata, ejk mh; ejcovntwn bavro" mhde; koufovthta e[conta koufovthta kai; bavro", ejoivkasi peri; a[llou oujranou' levgein kai; swmavtwn ajll∆ ouj tw'n aijsqhtwvn. 1090a20-25, 30-35. 33 Aristotle exonerates the Pythagoreans from any charges to the effect that they treat numbers as separate from sensible things or as intermediate between sensible things and forms.34 But this is only because they think that sensible things are numbers, which is, if anything, an even more peculiar idea, albeit one that avoids the problem of duplication of Platonic entities. Aristotle suggests that the Pythagoreans adopted this theory because they perceived that there were pavqh of numbers, such as harmonies, ratios and the like, which can be seen as numerical characteristics of things, and that these show up all over the place in astronomy and the other aspects of nature. Seeing those mathematical phenomena made the Pythagoreans go for the idea that things actually were numbers. Although the tendency to exclude Pythagorean speculations from the serious history of Presocratic philosophy comes from a tradition that is heavily indebted to Aristotle, and to Aristotle's
    
    32 33
    
    See Furley The Greek Cosmologists 52-3 for a brief discussion of this thought. But the Pythagoreans made things be numbers (because they saw many numerical effects
    
    existing in perceptible bodies) but not separate numbers, but things being constituted of numbers. But why so? Because the numerical effects exist in harmony in the heavens and in many other things. … On the one hand the Pythagoreans seem not to be liable to any charge on this matter (sc. separating the mathematicals). But in respect of making physical bodies out of numbers, making things that have lightness and heaviness out of things that have no heaviness or lightness, they seem to be talking about a different heaven and different bodies, but not the perceptible ones.
    34
    
    The Platonists and Speusippus are under attack for a variety of modifications of the idea of
    
    numbers that are separate from aistheta.
    
    15 reconstruction of early cosmology, it does not seem to me that the modern objections to Pythagorean numerology reflect quite the same motivation as Aristotle's bemused comments in Metaphysics N. Aristotle suggests that numbers aren't the right kind of thing to be the constitutive substance of things that have mass. Numbers don't have heaviness and lightness, so you can't make things out of them. But Aristotle (rightly I think) does not dismiss the Pythagoreans as stupid or muddled. Rather, he just sees that they can't be talking about things that have weight and so on at all. "They seem to be talking about a different heaven and different bodies, but not the perceptible ones."35 Aristotle's observations about whether people 'separate' numbers is pertinent here. In the case of Plato, he says, the theory allows that there are things that have physical mass, on the one hand, and then, on the other hand, there are ajriqmoiv that are cwristoiv – a separate realm of things that are not in the same category as the physical things which manifest the numerical effects. It is a two world view. But with the Pythagoreans you do not have a two world view. They neither separate numbers, nor make them intermediates.36 Instead you have just one of the two worlds, namely the incorporeal one. That is why Aristotle says that they seem not to be liable to any charge of separating numbers from the perceptible things.37 That is because there is just one set of things, namely the numbers.38 So we have just the one world, but it is composed of incorporeals. The result is that (as Aristotle observes) they seem to be talking about a different world and different bodies, 1090a34. If things are made of numbers, then they are a very funny kind of "things". This is not particularly an objection. It avoids canvassing the sort of objection that one might expect from a scientist, that it is naïve and stupid to try to create bodies out of numbers, or that this
    
    35 36
    
    1090a34-5. Cf. Metaphysics A, 987b27: kai; e[ti oJ me;n tou;" ajriqmou;" para; ta; aijsqhtav, oiJ d∆ 1090a30 Aristotle's comment at 1090a30 seems to conflict with the passage in Metaphysics A
    
    ajriqmou;" ei\naiv fasin aujta; ta; pravgmata, kai; ta; maqhmatika; metaxu; touvtwn ouj tiqevasin.
    37 38
    
    (987b11) where he tells us that the Pythagoreans have a notion comparable to Plato's notion of mevqexi". It seems that at 987b11 Aristotle is assimilating Plato and Pythagoreanism — a tendency that was to have a long subsequent history— by contrast with the careful attempt to draw distinctions between different kinds of Platonism about numbers in the passages in Metaphysics N.
    
    16 is the wrong kind of matter to do the job. Instead, Aristotle observes that the question must have been different. The realities to be explained must have been incorporeals, a different world from the perceptible things that Plato was talking about. Nevertheless, we can see behind Aristotle's discussion here an assumption that the Pythagorean project (including the appeal to numbers) was the same project as the Ionian scientists' project. Aristotle thinks that the question was how to explain the world in terms of its constitutive matter. Once the constitutive matter has been specified as non-separated numbers, Aristotle concludes that the Pythagoreans were evidently explaining a rather incorporeal world, with all its entities composed of numbers; but he does not drop the idea that the Pythagoreans are to be assessed for their competence at reducing reality to explanatory components that are immanent and not transcendent.39
    
    2 From incomprehension to admiration
    I have suggested that there are striking similarities between the Pythagorean concerns with number, ratio and harmony, and some of the material that has been regarded as pioneering and profound in Anaximander and Heraclitus. In asking why these moves should be considered important and profound in the latter cases, I have suggested that they meet with approval among modern scholars because the modern scholars are assessing the Presocratic thinkers for their progress in a sequence of developments in the direction of reductive materialist physics. Because they see the numerical patterns in the universe as immanent pavqh of things, and do not presuppose a set of theoretical entities that are 'numbers' with properties of their own, the numerical fantasies of these early Ionian thinkers are seen as an acceptable—or even progressive — part of that generally materialist project.
    
    39
    
    The problem is surely only that Aristotle takes the reduction to be materialist in its
    
    outlook. Without that assumption the project to reduce ontology to a system of numbers is not selfevidently flawed. W.V. Quine entertains precisely this project and investigates what it lacks, if anything, as a serious candidate in a number of works (see W. V. Quine 'Ontological reduction and the world of numbers' in Quine The Ways of Paradox and other essays (New York: Random House, 1966) 199-207, W. V. Quine 'Ontological Relativity' in Quine Ontological Relativity and other essays (New York: Columbia University Press, 1969) 26-68: 58-68, W. V. Quine 'Propositional Objects' in Quine Ontological Relativity and other essays (New York: Columbia University Press, 1969) 137-60: 14752). I am grateful to Nick Denyer for pointing me to these references.
    
    17 It seems to me that this evaluation of the early Greek philosophers displays an agenda that is built into our heritage of modern Presocratic scholarship. The agenda is very evident in Barnes, but that is only because he is particularly blatant about expressing his prejudices in outspoken terms. In practice he is following an existing tradition. One would say that the tradition was Aristotelian in origin— the rejection of fancy metaphysical entities, the down-to-earth preference for specifying that the world of particulars is what is real, the analysis of the Presocratics as engaged in diagnosing the material cause, all these seem to be Aristotle's prejudices— except that I think that it is a modern version of Aristotelianism that has acquired a great deal of baggage from the Enlightenment, from logical positivism and from a more recent scientism that equates truth with what can be proved by empirical methods. Aristotle is, of course, opposed to some of the things he finds in Platonism, such as the separation of Forms, but his opposition is not for quite the same reasons as the reasons that modern Presocratic scholarship would offer for why it doubts that Pythagorean numerology was a valuable contribution to Western Philosophy's overall development. In this second part of the paper I shall move beyond my initial thought, that Pythagorean speculations are no worse than the comparable bits of Heraclitus or Anaximander, if one is looking for empirically verifiable reasons in favour of a particular theory. My second thought is more ambitious. I want to propose that there is something in the Pythagorean enthusiasm for numbers that is far more significant philosophically, and has had far greater ramifications in the story of Western philosophy than anything Anaximander or Heraclitus ever did, despite their superior credibility in the contemporary western system of values. That is, I am suggesting that we should not apologise for the Pythagoreans' tendency to idolise numbers, or certain particular numbers, nor should we try to discard those bits and look for some cleaner bits of respectable doctrine instead. Rather we should celebrate them. A number of thinkers from Plutarch to Leibniz have anticipated my point. We should start however with a well known passage attributed to the fifth century Pythagorean Philolaus, which is quoted by Stobaeus.40 The point that is relevant to my topic is where Philolaus says that without number it is not possible for anything to be thought or known:
    
    40
    
    Philolaus fragments 4-5; Stobaeus 1.21.7b-c.
    
    18
    
    kai; pavnta ga ma;n ta; gignwskovmena ajriqmo;n e[conti: ouj ga;r oJtiw'n ãoi|ovnà te oujde;n ou[te nohqh'men ou[te gnwsqh'men a[neu touvtw.41 Philolaus fr. 4 The idea that "having a number" is a criterion of knowability is not one that is often proposed or given prominence in epistemological discussions, but it is worth comparing it to the point made by Parmenides about the relation between being and knowability: taujto;n d'∆ ejsti noei'n te kai; ou{neken e[sti novhma. ouj ga;r a[neu tou' ejovnto", ejn w|/ pefatismevnon ejstin euJrhvsei" to; noei'n.42 Parmenides B8.34-6 It seems that Philolaus is giving to numbers a role very similar to the role that is served by being and truth in Parmenides. There is no true thinking without being in Parmenides. There is no true thinking without numbers in Philolaus. It is a very severe epistemology, in which nothing counts as knowing unless it is knowledge of numbers. There is only one set of knowable objects, namely mathematicals. So study of mathematics is not just one of the sciences, alongside physics, but mathematics is the only science that relates to knowable objects.43 Secondly we may notice a quotation from Archytas (fragment 1) preserved by Porphyry in his commentary on Ptolemy's Harmonics, and the echo of the same thought that is known from Plato's Republic: parakeivsqw de; kai; nu'n ta; ∆Arcuvta tou' Puqagoreivou, ou| mavlista kai; gnhvsia levgetai ei\nai ta; suggravmata: levgei d∆ ejn tw'i peri; maqhmatikh'" eujqu;" ejnarcovmeno" tou' lovgou tavde: kalw'" moi dokou'nti toi; peri; ta; maqhvmata diagnwvmen, kai; oujde;n a[topon ojrqw'" aujtouv", oi|av ejnti, peri; eJkavstou fronevn:
    
    41
    
    And indeed all the things that are known have number. For without this it is not possible
    
    for anything to be thought or known. Text from Huffman Philolaus of Croton .
    42
    
    It's the same thing —thinking and that whose thought it is. For you won't find thinking without the reality, in which it is an utterance.
    43
    
    Huffman argues (against Martha C Nussbaum 'Eleatic conventionalism and Philolaus on the
    
    conditions of thought' Harvard Studies in Classical Philology 83 (1979) 63-108) that the meaning of "having a number" in fragment 4 has to be more than just being countable or limited, and that the criterion of knowledge here must involve knowing the specific number of a thing, Huffman Philolaus of Croton 173-7. My suggestion that the knowable things are the mathematicals, so that other things are knowable just in virtue of having numbers (which are, then, what we know about them) goes rather beyond what is justified by the text. It presupposes that things that are known are known because, and only because, they have a number to be known.
    
    19
    
    peri; ga;r ta;" tw'n o{lwn fuvsio" kalw'" diagnovnte" e[mellon kai; peri; tw'n kata; mevro", oi|av ejnti, kalw'" ojyei'sqai. periv te dh; ta'" tw'n a[strwn tacuta'to" kai; ejpitola'n kai; dusivwn parevdwkan aJmi'n safh' diavgnwsin kai; peri; gametriva" kai; ajriqmw'n kai; oujc h{kista peri; mwsika'". tau'ta ga;r ta; maqhvmata dokou'nti ei]men ajdelfeav:44 Porphyry In Ptolem. Harm. 1.3, (p. 56 Düring (quoting Archytas fragment1)45 This passage confirms Plato's claim, at Republic 530d, that the Pythagoreans called arithmetic, geometry, harmonics and astronomy 'sister sciences' and suggests that it was Archytas who coined the phrase. But my interest is not in that point, but in Archytas's suggestion that one would expect the experts in these sciences – which are grouped together because they work by theoretical manipulation of abstracted mathematicals, not empirical data from physical bodies – one would expect these experts to be the ones who correctly discern the nature of things, and of the universe as a whole. It is oujqe;n a[topon, says Archytas, that these people think correctly about things. But notice also the idea that they do this kalw'" – they discern the workings of the universe beautifully. And this surely links in to the idea that the workings of the universe are themselves a fine object of attention. That point is made more explicit by Plutarch in a passage in his Quaestiones conviviales: pa'si me;n ou\n toi'" kaloumevnoi" maqhvmasin, w{sper ajstrabevsi kai; leivoi" katovptroi", ejmfaivnetai th'" tw'n nohtw'n ajlhqeiva" i[cnh kai; ei[dwla: mavlista de; gewmetriva kata; to;n Filovlaon ajrch; kai;mhtrovpoli" ou\sa tw'n a[llwn ejpanavgei kai; strevfei th;n diavnoian, oi|on ejkkaqairomevnhn kai; ajpoluomevnhn ajtrevma th'" aijsqhvsew". dio; kai; Plavtwn aujto;" ejmevmyato tou;" peri; Eu[doxon kai; ∆Arcuvtan kai; Mevnaicmon eij" ojrganika;" kai; mhcanika;" kataskeua;" to;n tou' stereou' diplasiasmo;n ajpavgein ejpiceirou'nta", w{sper peirwmevnou" divca lovgou duvo mevsa" ajna; lovgon, h/| pareivkoi, labei'n: ajpovllusqai ga;r ou{tw kai; diafqeivresqai to; gewmetriva" ajgaqo;n au\qi" ejpi; ta; aijsqhta; palindromouvsh"
    
    44
    
    And now let us set alongside the words of Archytas the Pythagorean, to whom the writings are most reliably attributed. He says in the work on mathematics, right at the beginning, the following: "The people who are versed in learned subjects (mathematics?) seem to me to discern beautifully, and there is nothing absurd in their thinking correctly about each of the things just what it is like; for, since they discern beautifully with regard to the nature of the universe as a whole, it is to be expected that they will observe beautifully about the particular things, just what they are like. They have handed down to us clear knowledge concerning the speed of the stars and their risings and settings, and about geometry and numbers, and not least about music. For these subjects seem to be sister-subjects."
    45
    
    Text as in Carl Huffman Archytas of Tarentum: Pythagorean, Philosopher and Mathematician
    
    King (Cambridge: Cambridge University Press, 2005)
    
    20
    
    kai; mh; feromevnh" a[nw mhd∆ ajntilambanomevnh" tw'n ajidivwn kai; ajswmavtwn eijkovnwn, pro;" ai|sper w]n oJ qeo;" ajei; qeov" ejstin (Plat. Phaedr. 249c).46 Plutarch Quaestiones conviviales 8.2.1, 718E (= DK 44A7a) The thought is supposed to go back to Philolaus in some sense,47 and indeed it is faintly reminiscent of the passage from Sextus Empiricus which we noticed above,48 where Philolaus was said to have claimed that one gets an affinity with the harmony of the universe from assimilation to mathematical knowledge. Here too in Plutarch's passage the thought attributed to Philolaus—or built upon Philolaus's foundations by Plutarch— is that handling numbers does something splendid for you.49 And the less empirical the science is the better it is at this task. This thought is supported by reference to a legend according to which Plato is supposed to have raised objections in relation to the method for duplicating the cube attributed (here) to Eudoxus, Archytas and Menaichmus. The details of the mathematics are not immediately relevant for now (except perhaps to note that Archytas should be exempt from the criticism). 50 Plutarch's point is simply this: that we should not be attending to the
    
    46
    
    In all the so called (mathematical?) studies, the traces and images of the truth of intelligible
    
    objects are reflected, as in even and polished mirrors; and most of all geometry, according to Philolaus, being the source and mother-city of the other studies, leads the mind up and converts it, like a mind purified and released effortlessly from perception. Hence Plato himself criticised the followers of Eudoxus and Archytas and Menaichmus, who tried to divert the doubling of the cube to instrumental and mechanical devices, as though they were trying to obtain the two mean proportionals, however practicable, aside from rationality. For this is to destroy and corrupt the good of geometry, when it is dragged back to perceptible things and not carried up and not grasping eternal and bodiless icons instead—"those things closeness to which makes god always be god".
    47
    
    The name Philolaus is obtained by a plausible emendation of a corrupt reading fivlaon in Sextus Empiricus Adv Math 7.92 (= DK 44A29) Huffman takes the material from Philolaus to be very brief, only the reference to geometry
    
    the manuscripts.
    48 49
    
    as the source and mother city, with the reflections on that being from Plutarch's Platonist context (Huffman Philolaus of Croton 193-4). On the other hand it is not clear why Plutarch would be prompted to cite Philolaus at this point if there were not some invitation to this line of thinking in the text to which he is alluding.
    50
    
    There is something wrong with the story, though how exactly it has ended up in this form
    
    in Plutarch is not entirely clear. In fact, it would appear that Archytas's solution to the problem of obtaining the two mean proportionals was more theoretical and did not resort to practical methods as implied here. It makes no sense to suggest that Archytas was one of the offenders against whom Plato would have laid such a charge, therefore. The allusion does not seem to be to any existing text
    
    21 material examples. We should be handling numbers. That's how we get to think of incorporeals… This is the Platonic thought, that we need to get close to God by escaping from perceptible things. Of course, Plutarch is feeding us Platonism served up on a bed of Philolaus. I'm not meaning to pretend that the importance or the significance of the Pythagoreans' devotion to the elegance of numbers was explicitly appreciated by them at the time that they first developed their fascination. On the contrary, I want to suggest that it is with hindsight that we can see that this was one of the most profound and lasting legacies of Presocratic thought—the discovery of incorporeals. It is a point made by Plutarch, and by Plato too of course,51 that learning mathematics and geometry helps us to turn our gaze upwards, or to abstract from material things: that is a point about the usefulness of a particular kind of abstract discipline. But I would also want to add that there is something important about the idea that the most important reality might be the one that obeys the mathematical rules, not the one that falls short and approximates to mathematical accuracy. In other words the idea that maths talks about the pathe that are numbers, but it does not treat them as pathe of things (things that are more real) but it treats them as the most perfectly real things (or at least the more perfectly real, as compared with bodily things). This was an approach with which Aristotle was only partly out of sympathy. It is true that he does not think that the Platonic separation of mathematicals and forms is a good philosophical move. He prefers to think of the numbers as pathe of bodies, and he is pretty sceptical of the weird results when the Pythagoreans imagine that things just are numbers. But it is less alien to Aristotle's way of thinking than it is to the agenda with which modern scholarship has approached the Pythagoreans— those prominent parts of modern scholarship which have been responsible for relegating them to the status of superstitious mystics.52
    
    of Plato, although the issue of the need for two mean proportionals between cubic numbers figures in Plato's Timaeus 31c-32b, and there is a passage in the Republic VII 528a-d, which criticises stereometry for some failings that commentators have tried to link to the dispute mentioned by Plutarch. See Huffman Archytas 344-401.
    51 52
    
    Plato Republic 527b. There are, of course, notable exceptions in modern scholarship on the Presocratics. Most
    
    importantly perhaps, Carl Huffman, who has done much to bring the contributions of neglected Pythagorean thinkers such as Philolaus and Archytas to our attention in their own right, and show that they have serious philosophical and theoretical meat to offer. Others, including most prominently
    
    22 What do I mean? I mean that there is a seamless continuity between Pythagorean awe at the perfection of the number system, Parmenides's awe at the eternity of Being, Platonic awe at the Form of the Good, and Aristotle's awe at the Unmoved Mover. All these are objects of love and admiration, but their power is derived entirely from their beauty and perfection, not from any efficient or material causal efficacy.53 At the end of the day, Aristotle too would locate the best and most perfect causal power in the teleological cause, a cause discovered by abstract reasoning, not by experimental science. The reason why Aristotle found the Pythagoreans puzzling was because he thought that they must be looking for the material cause. He could not fit into his system a weird attempt to explain bodies that have mass by appeal to entities that have no mass; but his bemusement is created by the story that he has to tell about the development of Presocratic philosophy. Presocratic phusikoi are (he thinks) trying to answer the question "Out of what, ejk tivno", did the world come?" (or "out of what, ejk tivno", is the world made?").54 Aristotle glosses the 'stuff', out of which they suppose the world comes, as their arche, and his account of his predecessors tends to be couched as an analysis of their various attempts to cope with the logical complexity of the notion of coming to be out of something, while supposing that the something was to be the material substrate. So when the Pythagoreans say that the arche is number, something very weird seems to be going on. But it is only weird if you don't start by wondering whether they are noticing beauty, structure, form, and indeed teleology, in the universe.55 If that is what they are doing, then the suggestions are not just methodologically sound, but also extremely perceptive.
    
    Peter Kingsley, have sought to show why the religious and mystical side of Pythagorean traditions needs to be taken seriously.
    53
    
    Some sources credit the Pythagoreans (Hippasus in particular) with the discovery of
    
    irrational numbers, or particularly the incommensurability of the side and diagonal in a square, and suggest that this was a challenge to their belief in the mathematical perfection of the universe. (Porphyry Life of Pythagoras 246-7; Clement Stromateis 5.58). But equally one might suppose that the fact that the side and diagonal are commensurable when squared (effectively Pythagoras's theorem) would reveal a hidden rationality, a virtual rationality, in numbers that were apparently irrational when treated as lines, and restore one's faith in the idea of a mathematically coherent universe.
    54 55
    
    Aristotle Metaphysics 983b6-11; cf. Physics 187a12-26. Indeed there is evidence that Aristotle was partially aware of this alternative construal, as
    
    for instance in his comments at Metaphysics N 1092b8, where he admits that it is not clear in what sense numbers are explanatory of being, and suggests (as the second alternative) that it is because
    
    23 Thus in Aristotle the incomprehension is created by an agenda that he has in his investigation and presentation of the pre-Aristotelian history of ideas, but it does not reflect any ideological antipathy to the possibility that there might be other worthy causes to investigate besides the material cause. In modern thought the lack of appreciation for this pioneering work on numbers, which paved the way for all the most influential thinking from Plato to the Renaissance,56 is due to something worse. It is due, I think, to a fundamental prejudice in favour of reductionism, materialism, and mechanistic conceptions of the physical world. It is not that we think, like Aristotle, that the Presocratics were primitive in trying to explain things by reference to matter alone. We actually think that we should congratulate them for that, and we think that progress in their understanding was manifested in their move towards the more and more mechanistic theories of Anaxagoras and Democritus. We are embarrassed by Empedocles, and we try to rescue him by giving him one physical poem in which mechanistic forces explain everything and there are no appeals to immaterial values. and we find ourselves quite unable to stomach the Pythagoreans, when we discover that even abstract maths has come down laden with ethical and religious value. They just shouldn't be talking about numbers with that kind of language, we fear. It's superstitious. It worries us. The reason why we think that is because we have an agenda that is more ideologically blinkered than Aristotle's. Our ideological preference for mechanistic theory prevents us from seeing that one might want to explain what is beautiful and awe-inspiring about the world; and that the explanation should not explain it away in such a way that there is no awe and no beauty to move us after all. To encapsulate the point I want to make, and to remind ourselves that the prejudice against the Pythagoreans is not universal, we should look at a passage from Leibniz. It is rightly said in the paper given to the Princess of Wales, and which Her Royal Highness was kind enough to send to me, that next to vicious passions, the principles of the Materialists contribute much to support impiety. But I do not think that there were grounds to add that the Mathematical Principles of Philosophy are opposed to those of the Materialists. On the contrary, they are the same except that the Materialists, following in the footsteps of Democritus, Epicurus, and Hobbes, restrict themselves to mathematical principles alone, and admit nothing but bodies, whereas the Christian Mathematicians also admit immaterial substances. Thus it is
    
    harmony is (explained as) a ratio of numbers and that this idea extends to explanations of other things.
    56
    
    It would be good to say something about Plato, and particularly about those bits of Plato
    
    that have been taken to be somewhat Pythagorean in inspiration (including parts of the Phaedo), but that will have to wait for a more substantial opportunity to treat it in its own right.
    
    24 not Mathematical Principles (in the ordinary sense of the term) but Metaphysical Principles which ought to be opposed to those of the Materialists. Pythagoras, Plato, and to some extent Aristotle, had some knowledge of these, but I claim to have established them demonstratively, although in my Theodicy this was done in a popular manner. Gottfried Wilhelm Leibniz: Correspondence with Clarke: Leibniz's second paper.57 There is a distinction to be made between the materialist way of doing numbers, which uses numbers to give exact accounts of the behaviour of bodies, and the metaphysical move, which admits immaterial substances. What the Pythagoreans give us, according to Leibniz, is the metaphysical, which is, I would claim, the origin of real philosophy. It wasn't Anaximander who started philosophy, nor Democritus, nor even perhaps Heraclitus. It was Pythagoras, and he did it when he told us to take our oaths by the tetraktys, and that justice is the number 4 and kairos is the number 7.58 Don't get me wrong. I am not trying to say that metaphysical theories are right, or that a metaphysical theory is better (as a theory) than a materialist one. I am just saying that we would not have had a history of western philosophy if there hadn't been Pythagoreans and Platonists: that the metaphysical turn is what is distinctive and sets the debate about the nature of reality going. Parmenides does it, and the Pythagoreans do it, but the materialists don't. And that is one reason why one might want to say that philosophy started in southern Italy, not on the coast of Turkey, and that Parmenides was, after all, an honorary Pythagorean.59
    
    Bibliography
    
    Algra, Keimpe 'The beginnings of cosmology' in The Cambridge Companion to Early Greek Philosophy, edited by A A Long, Cambridge: Cambridge University Press, 1999, 45-65 Barnes, Jonathan The Presocratic Philosophers. 2nd ed. London: Routledge and Kegan Paul, 1982 Betegh, Gabor The Derveni Papyrus: Cosmology, Theology and Interpretation. Cambridge: Cambridge University Press, 2004
    
    57
    
    Gottfried Wilhelm Leibniz Die Philosophischen Schriften von Gottfried Wilhelm Leibniz edited
    
    by C. I. Gerhardt, Berlin: Weidmann, 1890 volume 7 page 355. I am grateful to Lloyd Strickland who kindly supplied this new translation for me.
    58 59
    
    Alexander In Metaph. 38.10). This paper has benefited greatly from discussion with various audiences, including
    
    participants at the Samos colloquium in 2005, the audience at a meeting of the B club in Cambridge, and members of the pure mathematics seminar at UEA Norwich in 2006.
    
    25 Burkert, Walter Lore and Science in Ancient Pythagoreanism Translated by E.L. Minar Jr. Cambridge, Mass.: Harvard University Press, 1972 Cornford, Francis MacDonald 'Mysticism and science in the Pythagorean Tradition' in The Presocratics, edited by Alexander P.D. Mourelatos, Garden City: Anchor Books, 1974, 135-60 Couprie, Dirk L 'Anaximander's discovery of space' in Before Plato, edited by Anthony Preus, New York: SUNY Press, 2001, 23-48 ——— 'The Discovery of Space: Anaximander's Astronomy' in Anaximander in Context, edited by Dirk L Couprie, Robert Hahn and Gerard Naddaf, New York: SUNY Press, 2003, 165-254 Early Greek Philosophy translated by Jonathan Barnes, Penguin Classics, Harmondsworth: Penguin, 1987 Findlay, J.N. Plato; The written and unwritten doctrines. London: Routledge and Kegan Paul, 1974 The First Philosophers translated by Robin Waterfield, Oxford World's Classics, Oxford: Oxford University Press, 2000 Furley, David The Greek Cosmologists. Vol. 1. Cambridge: Cambridge University Press, 1987 Hahn, Robert Anaximander and the Architects. New York: SUNY Press, 2001 ——— 'Proportions and numbers in Anaximander and Early Greek Thought' in Dirk L Couprie, Robert Hahn and Gerard Naddaf Anaximander in Context, New York: SUNY Press, 2003, 73163 Heidel, W.A. 'The Pythagoreans and Greek Mathematics' in Studies in Presocratic Philosophy, edited by David Furley and R.E. Allen, Vol. 1, London: Routledge and Kegan Paul, 1940/1970, 350-81 Heraclitus Fragments edited by T.M. Robinson, Phoenix Presocratics, Toronto: University of Toronto Press, 1987 Huffman, Carl Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King. Cambridge: Cambridge University Press, 2005 ——— Philolaus of Croton. Cambridge: Cambridge University Press, 1993 ——— 'The Pythagorean tradition' in The Cambridge Companion to Early Greek Philosophy, edited by A A Long, Cambridge: Cambridge University Press, 1999, 66-87 Kahn, Charles Anaximander and the origins of Greek Cosmology. New York: Columbia University Press, 1960 ——— The Art and Thought of Heraclitus. Cambridge: Cambridge University Press, 1979 Kingsley, Peter Ancient Philosophy, Mystery and Magic: Empedocles and Pythagorean Tradition. Oxford: Oxford University Press, 1995 Kirk, G S, J E Raven, and M Schofield The Presocratic Philosophers. Second edition ed. Cambridge: Cambridge University Press, 1983 Leibniz, Gottfried Wilhelm Die Philosophischen Schriften von Gottfried Wilhelm Leibniz edited by C. I. Gerhardt, Berlin: Weidmann, 1890 Lloyd, Geoffrey E R Early Greek Science Thales to Aristotle. London: Chatto and Windus, 1970 McKirahan, Richard Philosophy before Socrates. Indianapolis: Hackett, 1994 Naddaf, Gerard 'Anaximander's measurements revisited' in Before Plato, edited by Anthony Preus, New York: SUNY Press, 2001, 5-23 Nussbaum, Martha C 'Eleatic conventionalism and Philolaus on the conditions of thought' Harvard Studies in Classical Philology 83 (1979): 63-108 O'Brien, Denis 'Anaximander's measurements' Classical Quarterly 17 (1967): 423-32 Quine, W. V. 'Ontological reduction and the world of numbers' in Quine The Ways of Paradox and other essays, New York: Random House, 1966, 199-207 ——— 'Ontological Relativity' in Quine Ontological Relativity and other essays, New York: Columbia University Press, 1969, 26-68 ——— 'Propositional Objects' in Quine Ontological Relativity and other essays, New York: Columbia University Press, 1969, 137-60 Vogel, C. J. de Pythagoras and Early Pythagoreanism: an interpretation of neglected evidence on the philosopher Pythagoras. Assen: Van Gorcum, 1966 West, Martin L. Early Greek Philosophy and the Orient. Oxford: Clarendon Press, 1971

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Heinz Duthel, PCU  27th March, 2009 Ms. Osborne, just amazing and surprisingly how deep you look into the matter of your own investigations. Congratulations. I wish you have been my teacher.