The pluralism of Greek mathematics G E R Lloyd
8 The pluralism of Greek ‘mathematics’ G . E . R. Ll oy d Greek mathēmatikē, as has often been pointed out, is far from being an exact equivalent to our term ‘mathematics’. The noun mathēma comes from the verb manthanein that has the entirely general meaning of ‘to learn’. A mathēma can then be any branch of learning, or anything learnt, as when in Herodotus (1 207) Croesus refers to the mathēmata – what he has learnt – from his own bitter experiences. So the mathēmatikos is, strictly speaking, the person who is fond of learning in general, as indeed it is used in Plato’s Timaeus at 88c where the point at issue is the need to strike a balance between the cultivation of the intellect and that of the body, the principle that later became encapsulated in the dictum ‘mens sana in corpore sano’. Yet Plato also recognizes certain special branches of the mathēmata, as when in the Laws at 817e the Athenian Stranger speaks of those that are appropriate for free citizens as those that relate to numbers, to the measurement of lengths, breadths and depths, and to the study of the stars, in other words, very roughly, arithmetic, geometry and astronomy. In Hellenistic Greek mathēmatikos is used more often of the student of the heavens in particular (whether what we should call the astronomer or the astrologer) than of the mathematician in general in our sense. Whether we should think of either what we call mathematics or what we call philosophy as well-defined disciplines before Plato is doubtful. I have previously discussed the problems so far as philosophy is concerned.1 Those whom modern scholars conventionally group together as ‘the Presocratic philosophers’ are a highly heterogeneous set of individuals, most of whom would not have recognized most of the others as engaged in the same inquiry as themselves. Their interests spanned in some, but not all, cases what we call natural philosophy (the inquiry into nature), cosmology, ontology, epistemology, philosophy of language and ethics, but the ways in which those interests were distributed among the different individuals concerned varied considerably. It is true that we have one good fifth-century bce example of a thinker most of whose work (to judge from the very limited information we have 294 1 Lloyd 2006b. The pluralism of Greek ‘mathematics’ about that) related to, or used, one or other branch of mathematics, namely Hippocrates of Chios. He was responsible not just for important particular geometrical studies, on the quadrature of lunules, but also, maybe, for a first attempt at systematizing geometrical knowledge, though whether he can be credited with a book entitled (like Euclid’s) Elements is more doubtful. Furthermore in his other investigations, such as his account of comets, reported by Aristotle in the Meteorology, he used geometrical arguments to explain the comet’s tail as a reflection. Yet most of those to whom both ancient and modern histories of preEuclidean Greek mathematics devote most attention were far from just ‘mathematicians’ in either the Greek or the English sense. Philolaus, Archytas, Democritus and Eudoxus all made notable contributions to one or other branches of mathēmatikē, but all also had developed interests in one or more of the studies we should call epistemology, physics, cosmology and ethics. A similar diversity of interests is also present in what we are told of the work of such more shadowy figures as Thales or Pythagoras. The evidence for Thales’ geometrical theorems is doubtful, but Aristotle (who underlines the limitations of his own knowledge about Thales) treats him as interested in what he, Aristotle, termed the material cause of things, as well as in soul or life. Pythagoras’ own involvement in geometry and in harmonics has again been contested,2 and the more reliably attested of his interests relate to the organization of entities in opposite pairs, and, again, to soul. These remarks have a bearing on the controversy on the question of whether deductive argument, in Greece, originated in ‘philosophy’ and was then exported to ‘mathematics’,3 or whether within mathematics it was an original development internal to that discipline.4 Clearly when neither ‘philosophy’ nor ‘mathematics’ were well-defined disciplines, it is hard to resolve that issue in the terms in which it was originally posed, although, to be sure, the question remains as to whether the Eleatic use of reductio arguments did or did not influence the deployment of arguments of a similar type by such figures as Eudoxus. If we consider the evidence for the investigation of what Knorr, in other studies,5 called the three ‘traditional’ mathematical problems, of squaring the circle, the duplication of the cube and the trisection of an angle, those who figure in our sources exhibit very varied profiles. Among the ten or so individuals who are said to have tackled the problem of squaring the circle 2 3 4 5 Burkert 1972. Szabó 1978. Knorr 1981. Knorr 1986. 295 296 geoffrey Ll oyd it is clear that ideas about what counts as a good, or even a proper, method of doing so differed.6 At Physics 185a16–17 Aristotle distinguishes between fallacious quadratures that are the business of the geometer to refute, and those where that is not the case. In the former category comes a quadrature ‘by way of segments’ which the commentators interpret as lunules and forthwith associate with the most famous investigator of lunules, whom I have already mentioned, namely Hippocrates of Chios. Yet even though there is another text in Aristotle that accuses Hippocrates of some mistake in quadratures (On Sophistical Refutations 171b14–16), it may be doubted whether Hippocrates committed any fallacy in this area.7 In the detailed account that Simplicius gives us of his successful quadrature of four specific types of lunules, the reasoning is throughout impeccable. Quite what fallacy Aristotle detected then remains somewhat of a mystery. But two other attempts are also referred to by Aristotle and dismissed either as ‘sophistic’ or as not the job of the geometer to disprove. Bryson is named at On Sophistical Refutations 171b16–18 as having produced an argument that falls in the former category: according to the commentators, it appealed to a principle about what could be counted as equals that was quite general, and thus far it would fit Aristotle’s criticism that the reasoning was not proper to the subject-matter. Antiphon’s quadrature by contrast is said not to be for the geometer to refute (Physics 185a16–17) on the grounds that it breached the geometrical principle of infinite divisibility. It appears that Antiphon proceeded by inscribing increasingly many-sided regular polygons in a circle until – so he claimed – the polygon coincided with the circle (which had then been squared). The particular interest of this procedure lies in its obvious similarity to the so-called but misnamed method of exhaustion introduced by Eudoxus in the fourth century. This too uses inscribed polygons and claims that the difference between the polygon and the circle can be made as small as one likes. It precisely does not exhaust the circle. If Antiphon did indeed claim that after a finite number of steps the polygon coincided with the circle, then that indeed breached the continuum assumption. But of course later mathematicians were to claim that the circle could nevertheless be treated as identical with the infinitely-sided inscribed rectilinear figure. Other solutions were proposed by other figures, by a certain Hippias for instance and by Dinostratus. Whether the Hippias in question is the famous sophist of that name has been doubted, precisely on the grounds that the 6 7 Mueller 1982 gives a measured account. Lloyd 2006a reviews the question. The pluralism of Greek ‘mathematics’ device attributed to him, the so-called quadratrix, is too sophisticated for the fifth century. Although much remains obscure about the precise claims made in different attempts at quadrature, it is abundantly clear first that different investigators adopted different assumptions about the legitimacy of different methods, and second that those investigators were a heterogeneous group. Some were not otherwise engaged in mathematical studies at all, at least to judge from the evidence available to us. An allusion in Aristophanes (Birds 1001–5) suggests that the topic of squaring the circle had by the end of the fifth century become a matter of general interest, or at least the possible subject of anti-intellectual jokes in comedy. Among those I have mentioned in relation to quadratures several are generally labelled ‘sophists’, this too a notoriously indeterminate category and one that evidently cannot be seen as an alternative to ‘mathematician’. As is well known Plato does not always use the term pejoratively, even though he certainly has severe criticisms to offer, both intellectual and moral, of several of the principal figures he calls ‘sophists’. Yet Plato himself provides plenty of evidence of the range of interests, both mathematical and non-mathematical, of some of those he names as such. As regards the Hippias he calls a sophist, those interests included astronomy, geometry, arithmetic, but also, for instance, linguistics: however, whether the music he also taught related to the mathematical analysis of harmonics or was a matter of the more general aesthetic evaluation of different modes is unclear. Again, the fragments that are extant from Antiphon’s treatise Truth deal with questions in cosmology, meteorology, geology and biology.8 Protagoras, who is said by Plato to have been the first to have taught for a fee, famously claimed, according to Aristotle Metaphysics 998a2–4, that the tangent does not touch the circle at a point, a meta-mathematical objection that he raised against the geometers. Thus far I have suggested some of the variety within what the Greeks themselves thought of as encompassed by mathēmatikē together with some of the heterogeneity of those who were described as engaged in ‘mathematical’ inquiries. But in view of some persistent stereotypes of Greek mathematics it is important to underline the further fundamental disagreements (1) about the classification of the mathematical sciences and the hierarchy within them, (2) about the question of their usefulness, and 8 The identification of the author of this treatise with the Antiphon whose quadrature is criticized by Aristotle is less disputed than the question of whether the sophist is identical with the author called Antiphon whose Tetralogies are extant. 297 298 geoffrey Ll oyd especially (3) on what counts as proper, valid, arguments and methods. Let me deal briefly with the first two questions before exemplifying the third a little more fully. (1) Already in the late fifth and early fourth centuries bce a divergence of opinion is reported as between Philolaus and Archytas. According to Plutarch (Table Talk 8 2 1, 718e) Philolaus insisted that geometry is the primary mathematical study (its ‘metropolis’). But Archytas privileged arithmetic under the rubric of logistikē (reckoning, calculation, Fr. 4). The point is not trivial, since how precisely geometry and arithmetic could be considered to form a unity was problematic. According to the normal Greek conception, ‘number’ is defined as an integer greater than 1. In this view, arithmetic dealt with discrete entities. But geometry treated of an infinitely divisible continuum. Nevertheless both were regularly included as branches of ‘mathematics’, sister branches, indeed, as Archytas called them (Fr. 1). The question of the status of other studies was more contested. For Aristotle, who had, as we shall see, a distinctive philosophy of mathematics, such disciplines as optics, harmonics and astronomy were ‘the more physical of the mathēmata’ (Physics 194a7–8). The issue of ‘mechanics’ was particularly controversial. According to the view of Hero, as reported by Pappus (Collection Book 8 1–2), mechanics had two parts, the theoretical which consisted of geometry, arithmetic, astronomy and physics, and the practical that dealt with such matters as the construction of pulleys, war machines and the like. However, a somewhat different view was propounded by Proclus (Commentary on Euclid’s Elements 41.3 – 42.8) when he included what we should call statics, as well as pneumatics, under ‘mechanics’. (2) That takes me to my next topic, the issue of the usefulness of mathematics, howsoever construed. Already in the classical period there was a clear division between those who sought to argue that mathematics should be studied for its practical utility, and those who saw it rather as an intellectual, theoretical discipline. In Xenophon’s Memorabilia 4 7 2–5 Socrates is made to insist that geometry is useful for land measurement, astronomy for calendar regulation and navigation, and so on, and he there dismissed the more theoretical or abstract aspects of those subjects. Similarly Isocrates too distinguished the practical and the theoretical sides of mathematical studies and in certain circumstances favoured the former (11 22–3, 12 26–8, 15 261–5). Yet Plato of course took precisely the opposite view. It is not for practical, mundane, reasons that mathematics is worth studying, but rather as a training for the soul in abstract thought. But even some who emphasized practical utility sometimes defined that very broadly. It is striking that The pluralism of Greek ‘mathematics’ in the passage just quoted from Pappus he included both the construction of models of planetary motion and that of the marvellous gadgets of the ‘wonder-workers’ among ‘the most necessary of the mechanical arts from the point of view of the needs of life’. Meanwhile the most ambitious claims for the all-encompassing importance of ‘mathematics’ were made by the neoPythagorean Iamblichus at the turn of the third and fourth centuries ce. He argued in On the Common Mathematical Science (ch. 32: 93.11–94.21) that mathematics was the source of understanding in every mode of knowledge, including in the study of nature and of change. (3) From among the many examples that illustrate how the question of the proper method in mathematics was disputed let me select just five. (3.1) In a famous and influential passage in his Life of Marcellus (ch. 14, cf. Table Talk 8 2 1, 718ef) Plutarch interprets Plato as having banned mechanical methods from geometry on the grounds that these corrupted and destroyed the pure excellence of that subject, and it is true that Plato had protested that to treat mathematical objects as subject to movement was absurd. The first to introduce such degenerate methods, according to Plutarch, were Eudoxus and Archytas. Indeed we know from a report in Eutocius (Commentary on Archimedes Sphere and Cylinder 2, 3 84.12–88.2) that Archytas solved the problem of finding two mean proportionals on which the duplication of a cube depended by means of a complex three-dimensional kinematic construction involving the intersection of three surfaces of revolution, a right cone, a cylinder and a tore. Plutarch even goes on to suggest that Archimedes himself agreed with the Platonic view (as Plutarch represents it) that the work of an ‘engineer’ was ignoble and vulgar. Most scholars are agreed first that that most probably misrepresents Archimedes, and secondly that few practising mathematicians would have shared Plutarch’s expressed opinion as to the illegitimacy of mechanical methods in geometry. (3.2) My second example comes from Archimedes himself and concerns precisely how he endorsed the usefulness of mechanics, as a method of discovery at least. In his Method (2 428.18–430.18) he sets out what he describes as his ‘mechanical’ method which depends first on an assumption of indivisibles and then on imagining geometrical figures as balanced against one another about a fulcrum. The method is then applied to get the area of a segment of a parabola, but while Archimedes accepts the method as a method of discovery, he puts it that the results have thereafter to be demonstrated rigorously using the method of exhaustion standard throughout Greek geometry. At the same time the method is useful ‘even for the proofs of the theorems themselves’ in a way he explains (Method 299 300 geoffrey Ll oyd 428.29–430.1): ‘it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof, than it is to find it without any previous knowledge’. We should note that what is at stake is not just the question of admissible methods, but that of what counts as a proper demonstration. (3.3) For my third example I turn to Hero of Alexandria.9 Although he frequently refers to Archimedes as if he provided a model for demonstration, his own procedures sharply diverge, on occasion, from his. In the Metrica, for instance, he sometimes gives an arithmetized demonstration of geometrical propositions, that is he includes concrete numbers in his exposition. Moreover in the Pneumatica especially he allows exhibiting a result to count as a proof. Thus at 1 16.16–26 and at 26.25–28 he gives what we would call an empirical demonstration of propositions in pneumatics, expressing his own clear preference for such by contrast with the merely plausible reasoning used by the more theoretically inclined investigators. In both respects his procedures breach the rules laid down by Aristotle in the Posterior Analytics, both in that he permits ‘perceptible’ proofs and does not base his arguments on indemonstrable starting points and in that he moves from one genus of mathematics to another. If we think of precedents for his procedures, then they have more in common with the suggestion that Socrates makes to the slave-boy in Plato’s Meno (84a), namely that if he cannot give an account of the solution to the problem of doubling the square, he can point to the relevant line. (3.4) Fourthly there is Ptolemy’s redeployment of the old dichotomy between demonstration and conjecture in two contexts in the opening books of the Syntaxis and of the Tetrabiblos. In the former (Syntaxis 1 1, 1 6.11–7.4) he discusses the difference between mathēmatikē, ‘physics’ and ‘theology’. The last two studies are conjectural, ‘physics’ because of the instability of what it deals with, ‘theology’ because of the obscurity of its subject. Mathēmatikē, by contrast, which here certainly includes the mathematical astronomy that he is about to expound in the Syntaxis, alone of these three is demonstrative, since it is based on the incontrovertible methods of geometry and arithmetic. Whatever we may think about the difficulties that Ptolemy himself registers, in practice, in living up to this ideal when it comes, for instance, to his account of the movements of the planets in latitude, it is clear what his ideal is. Moreover when in the Tetrabiblos (1 1, 3.5–25, 1 2, 8.1–20) he speaks of the other branch of the study of the heavens, that which engages not in the prediction of the movements of the 9 Cf. Tybjerg 2000: ch. 3. The pluralism of Greek ‘mathematics’ heavenly bodies, but in that of events on earth on their basis – astrology, in other words, on our terms – that study is downgraded precisely on the grounds that it cannot deliver demonstration. It is conjectural, though he would claim that it is based on tried and tested assumptions. (3.5) Fifthly and finally there are Pappus’ critical remarks, in the opening chapters (1–23) of Book 3 of his Collection, on certain procedures based on approximations that had been used in tackling the problem of finding two mean proportionals in order to solve the Delian problem, of doubling the cube.10 Although certain stepwise approximations can yield a result that is correct, they fall short, in Pappus’ view, in rigour. Pappus himself distinguishes between planar, solid and linear problems in geometry and insists that each has its own procedures appropriate for the subject matter in question. What we find in all of the cases I have taken is a sensitivity not just to the correctness of results or the truth of conclusions, but to the appropriateness or otherwise of the methods used to obtain them. It is not enough just to know the truth of a theorem: nor is it enough to have some means of justifying the claim to such knowledge. No: what is required is that the method of justification be the correct one for the field of inquiry concerned according to the particular standards of correctness of the author in question. That is the recurrent demand: yet it is clearly not the case that all Greek investigators who would have considered themselves mathēmatikoi agreed on what is appropriate in each type of case or had uniform views on what counts as a demonstration. Similar second-order disputes recur in most other areas of inquiry that the Greeks engaged in, and this too is worth illustrating since it suggests that the phenomenon we have described in mathematics is symptomatic of more general tendencies in Greek thought. Sometimes we find such disagreements within what is broadly the same discipline, sometimes across different disciplines. In medicine the Hippocratic treatise On Ancient Medicine provides examples of both kinds. The author first castigates other doctors who try to base medical practice on what he calls ‘hypotheses’, arbitrary postulates such as ‘the hot’, ‘the cold’, ‘the wet’, ‘the dry’ and anything else they fancy (CMG 1 1, 36.2–21). In this author’s view, that is wrong-headed since medicine is and has long been based on experience. The investigation of what happens under the earth or in the sky may be forced to rely on such postulates, but they are a disaster in medicine, where they have the result of narrowing down the causal principles of diseases. While that drives a wedge between medicine and ‘meteorology’, he goes on in chapter 20 (51.6–18) 10 I may refer to the detailed analysis in Cuomo 2000: ch. 4. 301 302 geoffrey Ll oyd specifically to attack the importation into medicine of methods and ideas that he associates with ‘philosophy’, by which he here means speculative theories about such topics as the constitution of the human body. For good measure he insists that if one were to engage in that study, the proper way of doing so would be to start from medicine. Medicine provides particularly striking examples of second-order debates parallel to those in mathematics: indeed in the Hellenistic period the disagreements among the medical sects were as much about methods and epistemology as they were about medical practice. But other fields too exhibit similar fundamental divisions between competing approaches. In music theory, Barker has explored the analogous disputes first between practitioners on the one hand, and theoretical analysts on the other, and then, among the latter, between those who treated musical sound in geometrical terms, as an infinitely divisible continuum, and those who adopted an analysis based rather on arithmetic.11 Further afield I may simply remark that the methods and aims of historiography are the subject of explicit comment from Herodotus onwards. His views were criticized, implicitly, by his immediate successor Thucydides, who contrasts history as entertainment with his own ambition to provide what he calls a ‘memorial for eternity’ (1 21). But to achieve that end depended, of course, on the critical evaluation of eyewitness accounts, as well as an assumption that certain patterns of behaviour repeat themselves thanks to the constancy of human nature. With the development of both the practice and the teaching of rhetoric – the art of public speaking – goes a new sense of what it takes to persuade an audience of the strength of your case – and of the weakness of your rivals’ position. Both the orators and the statesmen deployed a rich vocabulary of terms, such as apodeiknumi, epideiknumi and cognates, to express the claim that they have proved their point, as to the facts of the matter in question, as to the guilt or innocence of the parties concerned, or as to the benefits that would accrue from the policies they advocated. Yet that very same vocabulary was taken over first by Plato and then by Aristotle to contrast what they claimed to be strict demonstrations on the one hand with the arguments that they now downgraded as merely plausible or persuasive, such as were used in the law courts and political assemblies – and this takes us back to mathematics, since it provides the essential background to the claims that some, but not all, mathematicians made about the strictest mode of demonstration that they could deliver. 11 Barker 1989, 2000. The pluralism of Greek ‘mathematics’ Aristotle was, of course, the first to propose an explicit definition of rigorous demonstration, which must proceed by way of valid deductive argument from premisses that are not just true, but also necessary, primary, immediate, better known than, prior to and explanatory of the conclusions. Furthermore Aristotle draws up a more elaborate taxonomy of arguments than Plato had done, distinguishing demonstrative, dialectical, rhetorical, sophistic and eristic reasoning according first to the aims of the reasoner (which might be the truth, or victory, or reputation) and secondly to the nature of the premisses used (necessary, probable, or indeed contentious). Yet while the ideal that Aristotle sets for philosophy and for mathematics is rigorous, axiomatic–deductive, demonstration, he not only allows that the rhetorician will rely on what he calls rhetorical demonstration, but concedes that in philosophy itself there may be stricter and looser modes, appropriate to different subject matter.12 The goal the philosophers set themselves was certainty – where the conclusions reached were, supposedly, immune to the types of challenges that always occurred in the law courts and assemblies. Yet from some points of view the best area to exemplify this was not philosophy itself (ontology, epistemology or ethics) but, of course, mathematics. However, the attitudes of both Plato and Aristotle themselves towards mathematics were distinctly ambivalent – not that they agreed on the status of that study. For Plato, the inquiries the mathematician engages in are inferior to dialectic itself: they are part of the prior training for the philosopher, but do not belong to philosophy itself. The grounds for this that he puts forward in the Republic are twofold, that the mathematician uses diagrams and that he takes his ‘hypotheses’ for granted, as ‘clear to all’.13 So although mathematics studies intelligible objects and so is superior to any study devoted to perceptible ones, it is inferior to dialectic which is purportedly based ultimately on an ‘unhypothesised starting point’, the idea of the Good. Aristotle, by contrast, clearly accepts that mathematical arguments can meet the requirements of the strictest mode of demonstration, since he privileges mathematical examples to illustrate that mode in the Posterior 12 13 Lloyd 1996: ch. 1. The interpretation of the expression ‘as clear to all’, hōs panti phanerōn, in the Republic 510d1, is disputed. My own view is that Plato is unlikely not to have been aware that many of the hypotheses adopted by the mathematicians were contested (including for example the definitions of straight line and point). When Socrates says that the mathematicians give no account to themselves or anyone else about their starting-points, it would seem that this is their claim, rather than (as it has generally been taken) their warrant. Burnyeat (2000: 37), however, has argued that there is no criticism of mathematics in this text, but simply an observation of an inevitable feature of their methods. 303 304 geoffrey Ll oyd Analytics. But mathematics suffers from a different shortcoming, in his view, which relates to the ontological status of the subject matter it deals with. Unlike Plato, who suggested that mathematics studies separate intelligible objects that are intermediate between the Forms and sensible particulars, Aristotle argued that mathematics is concerned with the mathematical properties of physical objects.14 While physical objects meet the requirements of substance-hood, what mathematics studies belongs rather to the category of quantity than to that of substance. While Plato and Aristotle disagreed about the highest mode of philosophizing, ‘dialectic’ in Plato’s case, ‘first philosophy’ in Aristotle’s, they both considered philosophy to be supreme and mathematics to be subordinate to it. Yet mathematics obviously delivered demonstrations, and exemplified the goal of the certainty and incontrovertibility of arguments, far more effectively than metaphysics, let alone than ethics. Once Euclid’s Elements had shown how virtually the whole of mathematical knowledge could be represented as a single, comprehensive system, derived from a limited number of indemonstrable starting points, that model exerted very considerable influence as an ideal, not just within the mathematical disciplines, but well beyond them.15 Euclid’s own Optics, like many treatises in harmonics, statics and astronomy, proceeded on an axiomatic–deductive basis, even though the actual axioms Euclid invoked in that work are problematic.16 More remarkably Galen sought to turn parts of medicine into an axiomatic–deductive system just as Proclus did for theology in his Elements of Theology.17 The prestige of proof ‘in the geometrical manner’, more geometrico, made it the ideal for many investigations despite the apparent difficulties of implementing it. The chief problem lay not with deductive argument itself, but with its premisses. Aristotle had shown that strict demonstration must proceed 14 15 16 17 Lear 1982. As noted, the question of whether Hippocrates of Chios had a clear notion of ultimate starting-points or axioms in his geometrical studies is disputed. In his quadratures of lunes he takes a starting-point that is itself proved, and so not a primary premiss. Ancient historians of mathematics mention the contributions of Archytas, Eudoxus, Theodorus and Theaetetus leading up to Euclid’s own Elements, but while the commentators on that work identify particular results as having been anticipated by those and other mathematicians, the issue of how systematic their overall presentation of mathematical knowledge was remains problematic. Thus one of Euclid’s definitions in the Optics (def. 3, 2.7–9: cf. Proposition 1, 2.21–4.8) states that those things are seen on which visual rays fall, while those are not seen on which they do not. That seems to suggest that visual rays are not dense, a conception that conflicts with the assumption of the infinite divisibility of the geometrical continuum. See Brownson 1981; Smith 1981; Jones 1994. Lloyd 2006c. The pluralism of Greek ‘mathematics’ from premisses that are themselves indemonstrable – to avoid the twin flaws of circular argument and an infinite regress. If the premisses could be proved, then they should be, and that in turn meant that they could not be considered ultimate, or primary, premisses. The latter had to be self-evident, autopista, or ex heautōn pista. Yet the actual premisses we find used in different investigations are very varied. To start with, the kinds or categories of starting points needed were the subject of considerable terminological instability. Aristotle distinguished three types, definitions, hypotheses and axioms, the latter being subdivided into those specific to a particular study, such as the equality axiom, and general principles that had to be presupposed for intelligible communication, such as the laws of non-contradiction and excluded middle. Euclid’s triad consisted of definitions, common opinions (including the equality axiom) and postulates. Archimedes in turn begins his inquiries into statics and hydrostatics by setting out, for example, the postulates, aitēmata, and the propositions that are to be granted, lambanomena, and elsewhere the primary premisses are just called starting points or principles, archai. As regards the actual principles that figure in different investigations, they were far from confined to what Aristotle or Euclid would have accepted as axioms. In Aristarchus’ exploration of the heliocentric hypothesis, he set out among his premisses that the fixed stars and the sun remain unmoved and that the earth is borne round the sun on a circle, where that circle bears the same proportion to the distance of the fixed stars as the centre of a sphere to its surface. Archimedes, who reports those hypotheses in the Sand-Reckoner 2 218.7–31, remarks that strictly speaking that would place the fixed stars at infinite distance. The assumption involves, then, what we would call an idealization, where the error introduced can be discounted. But in his only extant treatise, On the Sizes and Distances of the Sun and Moon, Aristarchus’ assumptions include a value for the angular diameter of the moon as 2°, a figure that is far more likely, in my view, to have been hypothetical in the sense of adopted purely for the sake of argument, than axiomatic in the sense of accepted as true. Meanwhile outside mathematics, we find Galen, for example, taking the principles that nature does nothing in vain, and that nothing happens without a cause, as indemonstrable starting points for certain deductions in medicine. In Proclus, the physical principles that natural motion is from, to, or around the centre, are similarly treated as indemonstrable truths on which natural philosophy can be based. The disputable character of many of the principles adopted as axiomatic is clear. Euclid’s own parallel postulate was attacked on the grounds that it should be a theorem proved within the system, not a postulate at all, 305 306 geoffrey Ll oyd although attempts to provide a proof all turned out to be circular. Yet the controversial character of many primary premisses in no way deterred investigators from claiming their soundness. The demand for arguments that are unshakeable or immovable, unerring or infallible, inflexible in the sense of not open to persuasion, indisputable, irrefutable or incontrovertible is expressed by different authors with an extraordinary variety of terms. Among the most common are akinēton (immovable), used for example by Plato at Timaeus 51e, ametapeiston or ametapiston (not subject to persuasion), in Aristotle’s Posterior Analytics 72b3 and Ptolemy’s Syntaxis 1 1 6.17–21, anamartēton (unerring), in Plato’s Republic 339c, ametaptōton (unchanging) and ametaptaiston (infallible), the first in Plato’s Timaeus 29b and Aristotle’s Topics 139b33, and the second in Galen, K 17(1) 863.3, and especially the terms anamphisbētēton, incontestable (already in Diogenes of Apollonia Fr. 1 and subsequently in prominent passages in Hero, Metrica 3 142.1, and in Ptolemy, Syntaxis 1 1 6.20 among many others) and anelegkton, irrefutable (Plato, Apology 22a, Timaeus 29b, all the way down to Proclus in his Commentary on Euclid’s Elements 68.10).18 The pluralism of Greek mathematics thus itself has many facets. The actual practices of those who in different disciplines laid claim to the title of mathēmatikos varied appreciably. They range from the astrologer working out planetary positions for a horoscope, to the arithmetical proofs and use of symbolism discussed by Mueller and Netz in their chapters, to the proof of the infinity of primes in Euclid or that of the area of a parabolic segment in Archimedes. There was as much disagreement on the nature of the claims that ‘mathematics’ could make as on their justification. One group asserted the pre-eminence of mathematics on the grounds that it achieved certainty, that its arguments were incontrovertible. Many philosophers and quite a few mathematicians themselves joined together in seeing this as the great pride of mathematics and the source of its prestige. But the disputable nature of the claims to indisputability kept breaking surface, either in general or in relation to particular results. Moreover while there was much deadly serious searching after certainty, there was also much playfulness, the ‘ludic’ quality that Netz has associated with other aspects of the 18 It is striking that the term anamphisbētēton may mean indisputable or undisputed, just as in Thucydides (1 21) the term anexelegkton means beyond refutation (and so also beyond verification). In neither case is there any doubt, in context, as to how the word is to be understood. That is less clear in the case of the chief term for ‘indemonstrable’, anapodeikton, which Galen has been seen as using of what has not been demonstrated (though capable of demonstration) although in Aristotle it applies purely to what is incapable of being demonstrated (see Hankinson 1991). The pluralism of Greek ‘mathematics’ aesthetics that began to be cultivated in the Hellenistic period.19 In the case of mathematics, there were occasions when its practitioners delighted in complexity and puzzlement for their own sakes. From a comparative perspective what are the important lessons to be learnt from the material I have thus cursorily surveyed in this discussion? The points made in my last paragraph provide the basis for an argument that tends to turn a common assumption about Greek mathematics on its head. While one image of mathematics that many ancients as well as quite a few modern commentators promoted has it that mathematics is the realm of the indisputable, it is precisely the disputes about both first-order practices and second-order analysis that mark out the ancient Greek experience in this field. Divergent views were entertained not just about what ‘mathematics’ covered, but on what its proper aims and methods should be. The very fluidity and indeterminacy of the boundaries between different intellectual disciplines may be thought to have contributed to the construction of that image of mathematics as the realm of the incontrovertible – contested as that image was. But we may remark that that idea owed as much to the ruminations of the philosophers – who used it to propose an ideal of a ‘philosophy’ that could equal and indeed surpass mathematics – as it did to the actual practices of the mathematicians themselves. It may once have been assumed that the development of the axiomatic– deductive mode of demonstration was an essential feature of the development of mathematics itself. But as other studies in this volume amply show, there are plenty of ancient traditions of mathematical inquiry that got on perfectly well, grew and flourished, without any idea of the need to define their axiomatic foundations. In Greece itself, as we have seen, it is far from being the case that all those who considered themselves, or were considered by others, to be mathematicians thought that axiomatics was obligatory. This raises, then, two key questions with important implications for comparativist studies. First how can we begin to account for the particular heterogeneity of the Greek mathematical experience and for the way in which the axiomatic–deductive model became dominant in some quarters? Second what were the consequences of the hierarchization we find in some writers on the development and practice of mathematics itself? In relation to the first question, my argument is that there was a crucial input from the side of philosophy, in that it was the philosopher Aristotle who first explicitly defined rigorous demonstration in terms of valid deductive argument from indemonstrable primary premisses – an ideal 19 Netz 2009. 307 308 geoffrey Ll oyd that he promoted in part to create a gap between demonstrative reasoning and the merely plausible arguments of orators and others. Whether or how far Aristotle was influenced by already existing mathematical practice is a question we are in no position to answer definitively. But certainly his was the first explicit definition of such a style of demonstration, and equally clearly soon afterwards Euclid’s Elements exemplified that style in a more comprehensive manner than any previously attempted. From this it would appear that it was the particular combination of cross-disciplinary and interdisciplinary rivalries in Greece that provided an important stimulus to the developments we have been discussing. Elsewhere in other mathematical traditions there was certainly competition between rival practitioners. It is for the comparativist to explore how far the rivalries that undoubtedly existed in those traditions conformed to or departed from the patterns we have found in Greece. Then on the second question I posed of the consequences of the proposal by certain Greeks themselves of a hierarchy in which axiomatic–deductive demonstration provided the ideal, we must be even-handed. On the one hand we can say that with the development of axiomatics there was a gain in explicitness and clarity on the issue of what assumptions needed to be made for conclusions that could claim certainty. On the other there was evidently also a loss, in that the demand for incontrovertibility could detract attention from heuristics, from the business of expanding the subject and obtaining new knowledge. This is particularly evident when Archimedes remarks that conclusions obtained by the use of his Method had thereafter to be proved rigorously using the standard procedures of the method of exhaustion. 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