WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 What to Say to a Geometer Michael J. White W HEN EPICURUS (D.L. 10.6) advised Pythocles to "flee all paideia) '" it seems extremely likely that education in the es­ tablished geometrical science was part of what he had in mind. Indeed, Epicurean aversion to the geometers' pulvis eruditus eventually became something of a rhetorical commonplace. Cicero (Nat.D. 2.47f) comments on this aversion and the consequent mathematical ignorance of Epicureans, attributing to it, among other deficiencies) the aesthetic blindness that renders them in­ capable of recognizing that a sphere is more beautiful than a cone, cylinder, or pyramid. While there were some Epicureans who were-or previously had been-mathematicians,l it seems that ac­ ceptance of the Epicurean world-view generally involved, for these individuals, a conversion from the practice of geometry. Polyaenus, an eminent 'first-generation' Epicurean, who was perhaps the most distinguished of the converted mathematicians, is described by Cicero (Acad. 2.106) as having come to believe that "all geometry is false'" after he had accepted the views of Epicurus. Zeno of Sidon apparently was an exceptional figure who, according to the account of Proclus, seems to have continued his mathematical work as an Epicurean, arguing that (all? many? some?) theorems of geometry do not follow without some additions to the (normally accepted?) set of postulates. 2 Despite a few problematic cases such as that of Zeno, it seems there is good reason to agree with David Sedley that "the wholesale rejection of geometry was still orthodox Epicureanism'" (24). Even 'wholesale rejection' can come in several varieties, however. Whole­ sale rejection can be grounded in ignorance, prejudice, or prepos­ session. And of course (witness Cicero) the Epicurean rejection of 1 See the summary account in D. SEDLEY, "Epicurus and the Mathematicians of Cyzicus," ChronErco/6 (1976 [hereafter 'Sedley']) 2-26. 2 Proclus, In primum Euclidis elementorum commentarium, ed. G. Friedlein (Leipzig 1873) 199. 297 WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 298 WHAT TO SAY TO A GEOMETER geometry was often so interpreted. But wholesale rejection can also be based on undestanding and prudence. A rejection of geometry -as geometry had developed in antiquity-is an intelligent and prudent response (perhaps the only viable response) for a philoso­ pher seriously committed to the doctrine of indivisible quanta of magnitude. For such a philosopher, the best stratagem available in antiquity probably would have been an entirely defensive stone­ wall: to claim that geometrical concepts simply are not applicable to such quanta, and consequently to refuse to pursue geometrical ar­ guments pertaining to them. When one introduces indivisible quanta of magnitude, in the true sense of 'quanta' (i.e., the sense connoting 'positive measure' or bulk), protestations in the form of pointed geometrical questions are almost certain to be voiced. What is the shape of such a quan­ tum? (Assumption: any shape implies a geometrical organization of spatial proper parts, which contradicts the notion of a conceptually or theoretically indivisible spatial magnitude.) What are the dimen­ sions of such a quantum? That is, how far is it from one side of it to another? (Assumption: any positive distance, specified in terms of some real-valued measure, means that we should, theoretically or conceptually, be able to talk about half that distance, a quarter of that distance, etc.-even if, as a matter of fact, there are no separable bodies or particles that small.) Probably the most effective way of dealing with such questions is to deny their applicability to indi­ visible quanta. The questions only arise, it might plausibly be main­ tained, because the questioner mistakenly persists in (tacitly or ex­ plicitly) conceiving the quanta as embedded in a matrix or medium that is infinitely divisible and continuous. To what extent did ancient proponents of the quantum model of magnitude make use of this strategy of rebuttal to geometrical criticism of their doc­ trine? The evidence is scanty. One apparently early and apparently non-Epicurean examrle comes from the pseudo-Aristotelian treatise On Individua Lines (968b15-17), where the classification 'rational/irrational' is withheld from lines constructed from a sup­ posedly minimal two-dimensional figure. An Epicurean example that has been much discussed comes from the Letter to Herodotus 58f. In this passage, Epicurus appears to conclude analogically (from the case of perceptible minima) that minimae partes within an atom are arranged successively or dis­ cretely but are not contiguous and that, by their number, they de- WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 MICHAEL J. WHITE 299 termine the bulk or size of the whole atom. Epicurus is not obliged to answer-and he wisely resists answering-questions that arise because of the tacit assumption of an infinitely divisible and con­ tinuous spatial matrix in which the quanta are embedded: what are the shapes of the minima? how, precisely, can they be successively ordered without touching? are there interstices between them? etc. It seems clear that wholesale rejection of geometry can come in less- and more-sophisticated varieties. But there are other possi­ bilities with respect to the relation between Epicurean physics and geometry. Vlastos has, I believe, adopted a much too restrictive assumption with respect to this matter. He maintains, in effect, that mathematically interested and educated Epicureans had only two options. The first is the development of a finitistic geometry in the rather narrow sense of a postulate set, analogous to that of Euclid but with an intended model of discrete elements. The second option, which arises in the absence of such an axiomatic develop­ ment of finitistic geometry, is the "consign[ment of] the whole of geometry to the devil" 3 or the acceptance of the developing 'Eu­ clidean' geometry with its mathematical assumptions of the con­ tinuity and infinite divisibility of magnitude. Vlastos is surely correct in claiming that there is no evidence for the existence of any ancient postulate set the intended model of which contained discretely ordered (and, at least for bounded con­ structions, a finite number of) Urelemente. It is not surprising that there should be no evidence of ancient axiomatic development of such a finitistic geometry. Vlastos comments, quite reasonably, "I find it very hard to see what the finitist geometry ... would be like" (127). In contemporary mathematics, such finite models typically arise in the algebraic study of 'absolute' geometry, in which funda­ mental concepts that are very abstract from the physical point of view replace the more intuitively 'comfortable' concepts of tra­ ditional elementary geometry.4 I believe that Vlastos is on firm ground in pointing to the a priori implausibility of the ancient axiomatic development of finitistic geometry: and I agree with Sed- 3 G. V LASTOS, '"Minimal Parts in Epicurean Atomism," Isis 56 (1965 [hereafter 'Vlastos'J) 127. 4 Absolute geometry pertains to geometry without anything corresponding to the Euclidean parallel postulate. See e.g. H. Wolff and A. Bauer, "Absolute Geometry," in H. Behnke et al., edd., Fundamentals of Mathematics, tr. S. H. Gould (Cambridge [Mass.]/London 1974) II 129-73. WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 300 WHAT TO SAY TO A GEOMETER ley (26 n.24) that Vlastos "spells out a strong argumentum e silentio against the existence of a special Epicurean geometry." In fact) there seems to be no serious opposition to Vlastos on this point. Although Jiirgen Mau has sometimes been cast in the role of a defender of the idea of a 'special Epicurean mathematics') the claims that he advances in his paper on that topic are actually quite modest: (a) the Epicurean rejection of de; l:btEtpov 'tOil" is the rejection of "the mathematical axiom that any size can be bisected again and again ad infinitum," and, consequently) the Epicurean £AUXto''tOV or minimal quantum is) in some sense) geometrically minimal and indivisible; (b) from the Epicurean standpoint, "reason­ ing without the axiom of division ad infinitum is legitimate because from the reliable testimony of Archimedes [Mau has in mind the M £8080e;] we know that it is possible to find new and true theorems without that axiom."5 With respect to the second option, perhaps some mathematically sophisticated converts to Epicureanism did choose to consign geometry to the devil. Cicero's references to Polyaenus may plausibly be interpreted as implying something of the aversion, on Polyaenus' part) of the reformed smoker to cigarettes. But there were doubtless other reactions on the part of Epicurean mathe­ matical cognoscenti. As Vlastos notes, Demetrius of Laconia wrote a treatise on geometry; and Zeno) mentioned above, seems to have pursued mathematical matters (as an Epicurean) with a diligence sufficient to elicit a book-length response from Posidonius.6 Did any of these mathematical Epicureans work within the tradi­ tion of Euclidean geometry, perhaps contributing to its develop­ ment? Vlastos offers as an instance an Epicurean Zeno of Sidon. 7 I believe, however, that the plausibility of Vlastos' view of Zeno's mathematical endeavors largely rests upon Vlastos' claim (135-47) that Epicurean minimae partes are not geometrically or mathemati­ cally indivisible quanta. Vlastos' interpretation of minima is summarized by a proposition) 5 J. Mau, "Was There a Special Epicurean Mathematics?" in E. N. Lee et at., edd., Exegesis and Argument: Studies in Greek Philosophy Presented to Gregory Vlastos (Assen 1973) 422, 428, 429. 6 Vlastos 127 n.35; Procl. In EueL 199f. 7 G. Vlastos, "Zeno of Sidon as a Critic of Euclid," in L. Wallach, ed., The Classical Tradition: Literary and Historical Studies in Honor of Harry Caplan (Ithaca 1966) 148f. WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 MICHAEL J. WHITE 301 characterized by him as a "law of nature" and a "physical statement about the atoms" (designated L[I]): atoms are so constituted that variations in atomic lengths occur only in integral multiples of the smallest atomic length (138). With resrect to L(I), Vlastos com­ ments (147) that "if this were the law 0 nature it is meant to be, it should be compatible with any (self-consistent) mathematical system." But would it be? In order to avoid inconsistency, the notion of variations in atomic lengths must be interpreted in such a restricted, artificial sense that L(I) becomes virtually vacuous. First, it is important to note that the smallest atomic length, designated q by Vlastos, does not represent any dimension of the parts into which atoms may be physically divided because all atoms are qua atoms physically indivisible. Whether the dimensions or atomic lengths of a particular atom are all multiples of q depends on what we count as 'dimensions' or 'atomic lengths'. For example, if the edges of a cubic atom count as dimensions and are each of atomic length 2q, then the diagonal of the cube cannot count as a dimen­ sion/atomic length, for it will not be commensurable with the lengths of the edges of the cube. But we could just as well count the diagonal as a dimension/atomic length-by an appropriate geometrical construction of the cube-in which case its edges cannot be atomic lengths. Could we rather arbitrarily limit the concept of atomic lengths to the edges of solids? Not unless we want to rule out a variety of regular solids as possible atomic shapes. For a regular pyramid with square base of side 2q and height 2q will have edges with lengths incommensurable with q. Vlastos suggests that we might try to preserve q as minimal atomic length by supposing that "all atoms, no matter how irregular might be their contours, could be (theoretically) broken down in the last analysis into parts of parallelepipedal shape, and the atomic lengths of the whole atom would be the set composed of all the atomic lengths of these component parts" (138 n.86). Presumably, the "atomic lengths of these component parts" refers to the lengths of the sides of the parallelepipeds, which will be mutltiples of q. But the prob­ lem is that, since we are referring to a theoretical 'breaking down' rather than a physical one and since the theory in question is, ac­ cording to Vlastos' assumption, Euclidean geometry, there are any number of ways of geometrically breaking down or carving up such atoms. Why, for example, think of a cube as geometrically constituted of smaller cubes with edges commensurable in length to the edges of the larger cube rather than as geometrically WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 302 WHAT TO SAY TO A GEOMETER constituted of pyramids, which have edges not commensurable in length with the edges of the larger cube? And why think of atomic lengths as sides of parallelepipeds rather than, say, their heights (which may not be commensurable with their sides)? There do not seem to be any reasons, other than arbitrary geometrical stipulation, for ruling out geometrical cdecompositions' of atoms that do not yield cparts' having edges the lengths of which are all multiples of some q. But when we invoke such geometrical stipulations, we certainly appear to be placing some sort of limitation on Euclidean geometry. It might be granted to Vlastos that the Epicurean doctrine that atoms are found only in a limited variety of sizes and shapes is a claw of nature' or Cphysical statement' without mathematical import. But I cannot see how restrictions on how we are to conceptualize, geo­ metrically, the structure of atoms-which are ex hypothesi physi­ cally indivisible--can be anything but geometrical restrictions; and these restrictions must appear entirely arbitrary if we are assuming that Euclidean geometry characterizes the spatial matrix of atomic solids. Other criticisms of Vlastos' interpretation of Epicurean minimal quanta have been given. Furley points out, for example, that since Vlastos' interpretation does not attribute to the Epicureans the doc­ trine of minimal, discrete spatial distances, an infinite number of atomic shapes could easily be produced by slight spatial reorienta­ tion of the finite number of parallelepipedal solids into which each atom is properly (geometrically) analyzed, according to Vlastos. In order to preclude an infinite variety of atomic shapes, then, Vlastos needs another principle (geometrical or physical?) restricting possible Cgeometrical conjunctions' of his canonical parallelepidedal solid atomic parts. 8 Furthermore, Sedley notes (26 n.26) that the "mathematical fragments of Demetrius of Laconia make it appear that the £A.clXtcJ'tOV posed a threat to geometry; which on Vlastos' in­ terpretation it would not do.'" In sum, then, I think that there is reason to reject Vlastos' contentions that L(I), qua Cphysical state­ ment', captures all that the Epicureans intended by talk of cleast' and cpartless' magnitudes and that mathematically sophisticated but Cphysically orthodox' Epicureans need not have had any funda­ mental objections to Euclidean geometry. At the very least, this 8 D. J. Furiey, Two Studies in the Greek Atomists (Princeton 1967) 42f. WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 MICHAEL J. WHITE 303 contention would seem to need to be supported by a variety of stipulations and restrictions, which appear ad hoc and which are not obviously physical, rather than mathematical, in character. It is easy to see, however, that so long as Vlastos' interpretation of the minimum stands, there is some considerable reason to maintain that Zeno's criticisms of Euclidean geometry must have been con­ structive. For example, according to Vlastos' interpretation of the doctrine of the minimum, Zeno would have no obvious mathe­ matical objection to the additional assumption that (according to Prod. In Eucl. 214.21£) he considers necessary in order to con­ struct an equilateral triangle (the first proposition of the first book of Euclid). Zeno argues that in order for the construction to be legitimate, it must be assumed that two lines do not share a com­ mon segment. According to Vlastos' interpretation, the Epicurean doctrine of the minimum could not constitute the basis of a denial of the existence of lines, points, surfaces, in the 'true limit' sense, nor the denial of a mathematical conception of divisibility ad in­ finitum. If, however, Vlastos' interpretation is rejected-if, in other words, the doctrine of spatial minima was understood as having geometrical significance-it becomes very difficult to see Zeno as a basically constructive (although perhaps niggling and not very penetrating) critic of the Euclidean axiomatization of geometry. If Zeno subscribed to a doctrine of indivisible quanta of magnitude, it seems likely that, as Sedley says, "Zeno regarded the additional premise as false," indeed, mathematically false (25). Since a line cannot be unextended in one dimension (because there are no such limit entities according to a geometrical doctrine of indivisible quanta), it is entirely possible-indeed necessary-for two non­ parallel straight lines to have a common segment. To show that a given theory requires, in order to derive its theorems, a postulate that one takes to be false-or that the theory entails a false proposi­ tion-can easily be rresented as a quite destructive criticism of the theory: a reductio 0 the theory, in fact. I very much suspect that such was the nature of Zeno's criticism of Euclidean geometry. But where does this leave Zeno and other Epicurean mathematicians? If we accept Vlastos' exhaustive di­ chotomy of adherence to Euclid or the consignment of geometry in toto to the devil, the answer is obvious. But there are other op­ tions. Sedley mentions a sort of piecemeal, non-axiomatic "applied geometry," the exchange of a mathematical science for "an inexact but serviceable discipline" (26). And I believe that Mau perhaps WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 304 WHAT TO SAY TO A GEOMETER entertains the same hypothesis. Another possibility, not inconsis­ tent with this hypothesis, is the dialectical use of geometry. By cdialectical use' I mean the criticism of Euclidean geometry not necessarily as an end in itself but as a tool in the development of atomistic physical doctrines. Specific evidence for such an under­ taking is, alas, scanty to say the least. But in what follows I develop a speculative cplausible story' of Epicurean criticism of Euclid's parallel postulate as a means toward clarifying the doctrine of the XCXp£"(KAtO't<;. I quite realize that I am skating on very thin ice in­ deed, in terms of actual history of Epicurean thought. But I hope that my story, even if suspect as an historical hypothesis, has some conceptual interest as an example of a way in which the mathe­ matically sophisticated Epicurean could have used geometry to his own purposes. In his discussion of Euclid's parallel postulate (1.5), Proclus (In Eucl. 368.27-369.1) reports an argument that purports to establish the contrary of the parallel postulate, i.e., that "it is impossible that lines produced at angles less than two right angles should meet." The argument is the following (see fig. 1). Take two straight lines AB A B E D c Figure 1 and CD and a straight line segment AC connecting them in such a way that the sum of the interior angles ("on the right") is less than two right angles. Bisect AC at E and measure off a length AF equal to AE on AB and a length CG equal to CE on CD. It is clear that AB and CD do not meet at F and G (i.e., it is clear that F and G are not, in fact, the same point). For if AB and CD did so meet, the sum of two sides of a triangle (i.e., of AF and CG) would be equal to the third side AC, WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 MICHAEL J. WHITE 305 "which is impossible." SO FG must be a line segment having some length. Bisect it at H; and again measure out line segment FJ on AB equal to FH and line segment GK on CD equal to GH. The same kind of argument can be used to show that lines AB and CD do not meet at points J and K. Since this process can be continued infinitely, which results in taking further and further segments on AB and CD, the argument's proponents conclude (according to Proclus 369.1- 20) that the straight lines do not meet anywhere. In his discussion of this argument, Proclus makes a promising beginning (369.21-370.2): Although [the proponents of the argument] speak the truth, they do not say as much as they believe. That it is not possible to define the point of intersection in this straightforward way is true. However, it is not true that the lines do not meet at all. The argument provides a method of generating a sequence of pairs of points on AB and CD, respectively, and shows that, for each of these pairs, the members of the pair cannot coincide. However, as Proclus claims, it does not follow from this construction that the lines do not intersect. There is the case where the two angles (CAB and ACD) are not equal and the point of intersection falls between successive points on one of the lines (see fig. 2). Proclus may have B E Figure 2 had this case in mind in the first part of his more detailed response to the argument (370.2-10). But this part of his response seems WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 306 WHAT TO SAY TO A GEOMETER quite confused, and I shall not go into details. In general, after his promising initial comment quoted above, Proclus' analysis of the argument is disappointing. 9 Another case that I wish to consider more closely is that in which angles CAB and ACD are equal. Here, according to the parallel postu­ late, line segment AC is the base of an isosceles or equilateral triangle having as (equal-length) sides segments of lines AB and CD. In this case the sequence of points A, F,j ••• (on line AB) and the sequence C, G, K ••• (on line CD) converge to the intersection of the lines (i.e., the vertex of the triangle) as a limit of both sequences. For each n , the pair of points which are the n-th members of each sequence are some finite distance E from the point of intersection; but for any distance E, there is a natural number N such that for all n > N, the distance between each of the n-th members and the point of intersection is less than E. When Heath notes a certain similarity between this criticism of the parallel postulate and Zeno's paradox of Achilles and the tortoise, he is evidently thinking of this par­ ticular case. 10 Although the construction on lines AB and CD can be continued ad infinitum, the sequences of points resulting from the continued construction converge to a point finitely distant from A and from c. Proclus does not indicate the provenance of this argument, and, so far as I know, there is no evidence that would come close to deciding the issue. However, an interesting hypothesis, which I shall entertain, is that the argument is an Epicurean one. The one explicit principle employed in the argument that is mentioned by Proclus is Euclid 1.20 to the effect that the sum of any two sides of a triangle is greater than the remaining side. Proclus reports that 9 After the opaque argument at 370.2-10, Proclus states the obvious, that the ar­ gument proves too much, noting that it will be refuted if a line can be drawn from A to G. But if it cannot, the first as well as the fifth postulate of the first book of Euclid will be contradicted. Finally, Proclus comments that ·someone could say" (evidently, someone not assuming the parallel postulate) that straight lines whose interior angles total less than two right angles but are greater than or equal to some angle a • remain nonsecant"; but if the total is less than a, they intersect (370.4-10). If one lets one of the interior angles remain a right angle and thinks of a as the angle of rotation of the other line, less than a right angle, at which the lines become nonsecant, a is the 'angle of parallelism' in Lobachevskian geometry, as noted by T. L. Heath, The Thirteen Books of Euclid's Elements2 I (New York 1956) 207. 10 Heath (supra n.9) 206. WHITE, MICHAEL J., What to Say to a Geometer , Greek, Roman and Byzantine Studies, 30:2 (1989) p.297 MICHAEL J. WHITE 307 Epicureans used this theorem as an example of the inutility of geometry because the proposition is "clear even to an ass and requires no demonstration"-i.e., the proposition belongs among 'tel EJ.L