Ascending to problems: Astronomy and harmonics in Republi VII

A Bh, d ‘ASS ASCENDING TO PROBLEMS: ASTRONOMY AND HARMONICS IN REPUBLIC VII Ian Mueller* In book VII of the Republic (521c ff.) Socrates describes five sciences (mathemata)—arithmetic, geometry, stereometry, astronomy, harmonics—which would play a fundamental role in the education of the rulers of the ideal state which he is constructing. From a modern point of view there is an important and obvious difference between the first three, which are purely deductive, and the last two, in which observation is of paramount importance. Socrates, however, appears to go out of his way to deny this difference. His students will “let be the things in the heavens” (530b7)' in studying astronomy and pay no attention to “heard concords” (531c1-2) in connection with harmonics. Most commentators have found Socrates’ descriptions of astronomy and harmonics puzzling, if not outrageous. I would like to try to minimize the seeming anomaly of these descriptions by invoking certain Greek scientific texts which, I believe, make clearer the kind of astronomy and harmonics Plato has in mind in the Republic. I shall argue that Plato assimilates astronomy to geometry (including stereometry) and harmonics to arithmetic. and that his doing so is not unreasonable in the light of these texts. I shall also attempt to explain what distinctions are needed in order to undermine the assimilations made by Plato. I shall assume that underlying what Socrates says is a conception of scientific knowledge frequently ascribed to Plato. For my purposes there are two fundamental features of this conception; one is logical, the other ontological.? The former is relatively easy to state; for Plato, scientific know*lan Mueller is Professor of Philosophy. University of Chicago.

ledge is deductive in structure. Although Plato does not intend such an extension, he certainly made no attempt to describe the principles governing deduction, he does stress the role of assumptions and hypotheses and the development apprise the readers of the Republic of the fact. In describing the position of stereometry between plane geometry and astronomy, the only relevant contrast Socrates makes is that astronomy studies “the solid in motion” (528a9) or the “move- 104 of a theory from them.’ From a linguistic point of view, the most important of these hypotheses are precise definitions. 105 ment of Solids” (528e1) whereas stereometry studies the solid From a more ontological point of view, the hypotheses are assumptions that there exist objects satisfying precisely in itself (auto kath’ auto 528b1). Socrates does not emphasize the introduction of motion into science, but treats it as a matter of course. Similarly, he does not say explicitly that determined conditions. Plato develops this ontological point of view by arguing that ordinary physical objects do not satisfy the precisely determined conditions specified in scithe objects of astronomy are something other than the heaentific hypotheses. Hence, he concludes, the objects of science exist in an ideal realm outside the physical world. Socrates stresses this point in his discussion of the sciences, since their principal role in his educational scheme is to draw the minds of the students away from sensible becoming (and observation) to intelligible being (521d2-3). According venly bodies. However it is natural to take him to mean this, given his remarks about arithmetic and geometry and his disparagement of astronomic observation. Thus if Socrates’ description of arithmetic and geometry is based on a notion of intermediates satisfying the hypotheses of those sciences, about the accuracy of this interpretation of Platonic theory. then the description of astronomy implies the existence of moving intermediates, even if Plato never explicitly asserted the existence of such objects. On the other hand, if Socrates’ description of arithmetic and geometry presupposes that these sciences are the study of forms, then his description of astronomy implies that it is likewise and that there is a form of motion or of the solid-in-motion or of some such thing. In other words, in whatever sense arithmetic is the study of “units which can only be conceived by thought” (526a6-7), geometry the study of the “eternally existent” (527b7), and stereometry the study of the “solid in itself” (528b1), astronomy would seem to be the study of the “solid in revolution” (528a9), and, to extend the discussion to the last part of Socrates’ mathematical curriculum, harmonics the study of “concordant numbers” (531c3). The inclination of moderns to accept the ascription to Plato of an ontology which includes non-sensible arithmetic I hope to be able to by pass this controversy. For my purand geometric objects but does not include non-sensible poses it is sufficient that Plato assumes every science to be astronomic and harmonic objects seems to derive in part from to Aristotle, Plato held that the objects of the mathematical sciences exist in a realm between the forms and sensible things: ... Besides sensible things and forms [Plato] says there are objects of mathematics (mathematika), which occupy an intermediate position, differing from sensible things in being eternal and unchangeable, from forms in that there are many alike, while the form itself is in each case unique. (Met. A.6. 987b14-18) In antiquity this account of Plato’s view of science was taken for granted. The later neo-Platonists associated it with the view that science is a function of a faculty called dianoia, the objects of which are dianoetika, as opposed to nous, the objects of which include the forms.‘ In the twentieth century there has been a great deal of controversy the study of some kind of real but non-sensible object or objects.’ Most commentators have not hesitated to ascribe to Plato an ontology of non-sensible arithmetic, geometric, and stereometric objects. Many of them have balked at extending this ontology to astronomy and harmonics. If Plato did not OAS+E©eR? the modern versions of related sciences. Modern astronomy and harmonics obviously do study physical things, and modern pure mathematics, the outgrowth of Greek arithmetic and geometry, includes disciplines having no apparent connection with the physical world. It is important to realize that no such sharp dichotomy exists for the ancient versions of the

sciences in Socrates’ curriculum. Some of the astronomic and harmonic texts which I-will be considering shortly are lan Mueller Sph. II. 7. If in a sphere a greatest circle touches some circle of those in the sphere and some other greatest circle oblique not easily distinguished from their arithmetic and geometric to the circles parallel [to the one touched by the first greatest analogues. These analogues themselves are very closely tied circle] but to the physical world. Greek geometry studies the spatial touched by the first, the points of contact being on the first properties of idealized physical objects, Greek arithmetic the properties of finjte sets of these objects, when the objects are conceived of as units, i.e., as single things. Aristotle's conception of mathematics as the study of physical objects thought of in abstraction from certain of their properties from the oblique circle on the same side of the greatest of the parallel circles and parallel circles drawn through the generated points, they will cut off between them unequal arcs of the first greatest circle and the arc nearer to the greatest of of the parallel circles will always be greater than the further away one, metic and geometry than Plato’s postulation of separate which means Plato’s mathematical ontology derives at least as much from his reflection on general philosophical problems as from a detailed study of the character of mathematics.’ Given the fairly loose connection between the Greek arithmetic and geometry of Euclid’s Elements and this ontology, one should nol expect any stronger connection between the astronomy and harmonics of analogous texts and what Socrates says or implies about these sciences in the Republic. The astronomical texts to which I shall be referring are touching parallel circles larger than the one greatest circle, and if equal arcs are cut off consecutively seems much more natural and appropriate for Greek arithmathematical entities. It seems reasonable to assume that 107 Of arcs of the horizon determined by consecutive planes parallel to the equator and passing through beginning points of signs of the zodiac on one side of the equator, an arc closer to the equator is greater than one further away, as is shown by Euclid’s use of the proposition in Ph. 8. The signs of the zodiac rise and set in unequal sections of the horizon, those toward the equator In the biggest sections, those which follow in less, those toward the tropics in the least. A comparison of this proposition from the Phenomena with from the so-called Lesser Astronomy. They are Theodosius’s Spherica III. 7 reveals two major and characteristic differences. Theodosius’s terminology is geometric and his proposi- Spherica, Autolycus’s On a Moving Sphere, and Euclid's tions more general, since he does not take as given the rela- Phenomena. Although the Spherica was compiled at least tive position of the horizon and such lines and points as the two centuries after the death of Plato, there is no reason to ecliptic and the poles. (These features of the Spherica acdoubt that it differs little in content or form from works count for the wordiness of 111.7.) Euclid substitutes astroextant in the late fourth century. For Autolycus and Euclid nomical terms for geometric descriptions and sometimes not only presuppose much of the content of the Spherica, takes for granted actual relative positions.’ they also quote propositions from it verbatim. The treatise A third difference is the presence of motion in the Pheof Autolycus is considered to be one of the earliest extant nomena and its absence from the Spherica. Mathematically examples of Greek deductive mathematics, dating from the this change is hardly noticeable, as can be seen by looking at Autolycus's On a Moving Sphere, which is a more fundamental treatise than the Phenomena in the sense that its late fourth century. The Phenomena is assumed to be approximately contemporary." The Spherica is, in one sense, simply the theory of circles content is presupposed in the Phenomena. Autolycus estabformed on the surface of a sphere by planes passing through lishes certain properties of a sphere which rotates “uniformit. But only excessively elaborate verbal paraphrase keeps ly.” He assumes: the astronomical content out of sight. For example. Theodosius states and proves Points are said to move uniformly when they pass through equal and similar magnitudes in an equal time. If some point moving uniformly on some line passes through two lines [Le., two segments of the line], the time in which the point passes

through one of the lines will have the same ratio to lan Mueller the other time as the line has to the line. (Aut. 195.3-8) With these assumptions, questions about the time of movements are reduced to questions about lines which can be handled in an ordinary geometric way.!° Autolycus's term- 109 tifies the postulation of ideal objects, so does the character of astronomy. Although the treatises under consideration can be applied to relatively few astronomic phenomena, there is no doubt that the techniques involved could be extended, at least in inology is almost as purely geometric as Theodosius's. However, he does use the phrase ‘a great circle dividing (horiprinciple, to cover a much wider range of facts. Nor would dzon) the visible and invisible hemispheres of the sphere’, a terms like ‘horizon’, ‘ecliptic’, ‘star’ for their more elaborate phrase which he shortens to ‘horizon’. He also speaks of geometrical paraphrases. However, one is inclined to think points rising and setting.'' that general -statements like Spherica 111.7 do not become Theodosius clearly strove to eliminate all astronomical terminology from the Spherica. Since Autolycus represents but does not mention the equator, the tropics, the ecliptic or zodiac, it seems likely that he strove to do the same, treating ‘visible’, ‘invisible’, ‘rising’, ‘setting’ as terms with an obvious enough geometric paraphrase not to need definition. Whatever terminology is used, a Platonist would have no trouble in assimilating both Autothere be any problem in substituting ordinary astronomical astronomical statements simply by the substitution of astronomical terms for geometrical phrases. The statements also need to be given a specific content by fixing the positions and movements of various lines and points. What is missing from Socrates’ account of astronomy is any attempt to explain how such a specific content would be justified. Euclid's Phenomena provides an example of how such justifications lycus's On a Moving Sphere and Theodosius’s Spherica to Euclid's geometry. Both use geometrical reasoning to derive theorems from initial assumptions. In this sense they salisfy the Platonic methodological requirement that a science be deductive. In addition, since the deductions presuppose that the objects treated satisfy exact geometric conditions, it would ordinarily proceed. It includes a preamble in which Euclid argues for a certain geometric representation of astronomical phenomena. The beginning of the preamble is would be possible to argue along Platonic lines that the objects must exist outside the physical world. Aristotle provides us with such an argument: simultaneously are always seen rising simultaneously, and typical of the whole: Since the fixed stars are always seen rising from the same place and setting in the same place, and those which rise those which set simultaneously are always seen setting simultaneously, and in their motiuns their distances from one another are always seen to be the same, and since this only But on the other hand astronomy cannot be dealing with peroccurs with things moving in a circle when the eye is equally ceptible magnitudes nor with this heaven. For neither are perceptible lines such lines as the geometer speaks of (for distant from the entire circumference, as is proved in the no perceptible thing is straight or round in the way in which circle and are attached in one body and that the eye is equally he defines ‘straight’ and ‘round’; for a hoop touches a straight edge not at a point, but as Protagoras used to say it did in distant from their circumferences. (Ph. 2.1-10) his refutation of the geometers), nor are the movements and spiral orbits in the heavens like those of which astronomy treats, nor have points the same nature as the stars. Met. B.2. 997b34-998a8) Aristotle here refers to the idealization involved in geometrical astronomy as a reason for saying that it does not deal with the perceptible heaven. He points out, and I would say correctly, that there is no difference in this regard between astronomy and geometry. If the character of geometry jus- Optics, let it be postulated (theteon) that the stars move in a Here, of course, rising and setting are taken for granted, as is a long series of observations. The reference to the Optics is puzzling, since no corresponding proposition occurs in our texts of Euclid’s Optics. Clearly the claim that the observed movements of the stars are only compatible with the hypothesis of a circular motion seen from a point equidistant from the whole circumference involves tacit assumptions about the forms which the movement could possibly take and our own stationariness. Similar points could be made about

SWumeas fan Mueller 111 the entire preamble. However the important point is that jects. It is now a philosophical commonplace that reference Euclid does attempt to justify certain astronomic-geometric to physical objects or other phenomena cannot provide a correlations and explain the geometrical account of astronomical terms by reference to allegedly observable phenomlogical effect such references might have on an individual. ena. Once these correlations have been established, the trea- The explanation for this commonplace is that ordinary obtise proceeds in a strictly deductive way." jects do not satisfy geometric hypotheses because of the justification for geometrical hypotheses, whatever psycho- It seems unlikely to me that Socrates would countenance idealization involved in geometry. In the quotation given such justifications in the astronomy taught to the future rulers of his ideal state. For in book VI of the Republic he involves idealization. My suggestion is that this idealization above, Aristotle points out that geometrical astronomy also says that within mathematics itself the justification of hypomay have been Plato's only ground for ignoring the role of phenomena in the justification of astronomical hypotheses. theses plays no role: The last question I would like to consider in connection ... Students of geometry and reckoning and such subjects first postulate the odd and the even and the figures and three with Platonic astronomy concerns the tenability of Plato's kinds of angles and other things akin to these in each branch assimilation of astronomy to geometry. There are, of course, of science, regard them as known, and, treating them as hymany ways in which a person might try to argue against potheses, do not deign to render any further account of them this assimilation. One might be to invoke falsification. Astroto themselves or others. . . (510c2-7) nomical hypotheses would seem to be falsifiable in a way in If this account were applied to astronomy, then the purported justifications of hypotheses in the preamble to the Phenomena would have no place in Platonic astronomy. Rather the astronomer would simply lay down some hypotheses and derive consequences from them. Socrates may have such derivations in mind when he says that astronomy should be studied “by means of problems. . .as in geometry.” (530b6-7)" Socrates’ answer to the question ‘How are astronomical hypotheses justified?’ would seem, then, to be that hypotheses are not justified within astronomy itself. Such an answer is, of course, appropriate within Socrates’ educational which geometric ones are not. I doubt that such an argument PeRcOTha would work against Plato. For, if I have interpreted him rie to distinguish between the ways in which phenomena fail correctly, he believes that the phenomena do not satisfy either geometric or astronomic hypotheses (although, of course, they do not falsify them either, since the hypotheses don't concern phenomena at all). To distinguish geometry and astronomy in a way satisfactory to Plato, one would have to satisfy the hypotheses of the two sciences. Geometrical hypotheses involve idealization. So do astronomical hypotheses, but these also involve what might be called approximascheme, in which the justification of hypotheses is reserved tion. For example, Euclid treats the earth as an eye, the eye for dialectic. However it would seem that Plato must have as a point. His doing so is justified by the assumption that the stars are extremely far away relative to the earth. Can realized that in ordinary astronomy hypotheses are thought to be justified by reference to the things in the heavens, as they are justified in the Phenomena. How is it possible to explain Socrates’ explicit denial that the phenomena are one distinguish conceptually between the idealization inrelevant to astronomy as he conceives it? I am inclined to think that the answer to this question lies in a Platonic asearth in the Phenomena, which, I suggest, involves both similation of astronomy to geometry. In other words, I believe to this question, but it seems to me that without such a disthat Plato thought the relationship of the hypotheses of tinction Socrates’ account of astronomy is as tenable as his geometrical astronomy to the heavens to be the same as the account of geometry. relation of geometric hypotheses to ordinary physical obvolved in Euclid’s assumption in the Elements that two points determine a unique straight line and his treatment of the idealization and approximation? I do not know the answer When Socrates turns from astronomy to harmonics, he

describes them as “kindred sciences” which study two forms which are composed of parts are said to have the ratio of a number to one another so that notes are also necessarily said to be in the ratio of a number to one another. Of numbers, some are said to be in a multiple ratio, some in an epimorios (expressible in the form n+2/n+1), some in an epimeres (ex- 112 of movement. “We may venture to suppose,” he says, “that as the eyes are framed for astronomy so the ears are framed for the movements of harmony.” (530d6-7) After he has dis-. paraged both the attempt to determine minimal musical intervals by experimentation and the search for numbers in heard concords, he calls for a harmonics which ascends to problems and considers “which numbers are concordant and which not and why in each case.” (530e1-531c4) The subject matter of Platonic harmonics would appear, then, to be numbers or, more exactly, relations between numbers, ratios. The principal distinction between harmonics and arithmetic will be the numerical properties with which each is concerned. Whereas arithmetic will deal with such properties as being composite or relatively prime, harmonics will be concerned with numerical concordance and discordance in general and such particular examples of them as being an octave or a tone. In modern times the idea of concordant numbers has found few defenders; yet in antiquity the notion of harmonics as a branch of arithmetic seems to have flourished. The text which best embodies this notion is the Sectio Canonis, usually ascribed to Euclid. In it Euclid attempts to establish numerical expressions for the concordant intervals using only arithmetic and some further assumptions. Like the Phenomena, the Sectio begins with an explanatory preamble. If there were rest and motionlessness there would be silence; if there were silence and nothing moved, nothing would be heard. ‘Therefore if something is to be heard there must be beforehand a blow and motion. Thus since all notes occur when a blow occurs but it is impossible for a blow to occur unless a movement occurs beforehand, and [since] of motions some are denser, some rarer, and the denser make higher notes, the rarer deeper. necessarily there are some higher notes since they are composed of more compact and more numerous movements, and some deeper since they are composed of rarer and fewer movements. Thus those notes which are higher than fitting attain the fitting when relaxed by a subtraction or reduction of motion, the deeper attain the fitting when tensed by an addition or increase of motion. Therefore notes are said to be composed of parts since by addition and subtraction they attain the fitting. All things 113 pressed in least numbers by m+1-+n+2/n+2)" so that notes are said necessarily to be in such ratios to one another. We also know that of notes some are concordant, some discordant, and the concordant make one blend from two things, the discordant do not. It is therefore reasonable (eikos) that the concordant notes be among the numbers which are called by one name in relation to one another, i.e., the multiples and the epimoria, since they make one blend of voice out of two things. (Sectio 158. 1-160.4) In this preamble Euclid makes a general reference to empirical phenomena in order to “justify” a frequency theory of pitch and thereby the numerical representation of pitch relations.'S This seems to be the only purpose of the preamble. For once the representation is given, the Sectio proceeds in a purely arithmetic way except for a curious blend of musical and arithmetic terminology.'f The main difference between the preambles of the Sectio and the Phenomena is that the preamble of the former does not even attempt to provide any genuine observational basis for the mathematical model adopted in the way that the preamble of the latter does. In particular, the frequency theory is not used and, given Greek experimental capacity, could not have been used to justify the assignment of particular kinds of ratios to consonant intervals. From a modern point of view the justification of this assignment in the preamble is extremely tenuous. In Greek, epimoria and multiples can be expressed by one word, the nth multiple by adding the suffix '-plasios’ to a form of the word for n, the ratio n+1/n by adding the prefix ‘epi’ to a stem meaning nth. (The latter rule is true for n greater than 2; 3/2 is called hemiholios.) On the other hand, ratios of the general form m+n/n are expressed by ‘m+n nths’ and hence not called by one name. Euclid “argues” that concordant intervals should be associated with multiples and epimorios ratios, since notes separated by such intervals combine to produce a unified sound. Although no trace of this particular piece of reasoning is found elsewhere, there is no reason to doubt that it is seriously meant. Others

equally insubstantial are ascribed by ancient authors to the morion is never the square of a ratio, so that the double octave is a multiple, but then by SC5, according to which a ratio with a multiple as square is itself a multiple, the octave is a multiple. Although Euclid does not actually succeed in carrying out his program due to a fallacy'in the argument for SC11,'8 there seems to me no doubt that the body of the Sectio can be adequately characterized as the investigation, by means of problems, of which numbers are concordant and which are not. If the preamble were simply replaced with SCA-SCD, the result would be a Platonic harmonics. An interesting feature of the Sectio is its incompatibility with empirical facts. I have already mentioned that the failure to include the octave-plus-fourth among the concordant intervals is simply a case of ignoring musical facts. Other interesting cases are provided by Euclid's mathematical refutations of the claims of more empirical music theorists. For example, in SC16 Euclid proves that there is no division of the tone into equal intervals. In mathematical terms this proposition is simply the true assertion that the 114 Pythagoreans. For example, Ptolemy says that they chose to assign multiples and epimoria to the concordant intervals “because of the simplicity of the comparison: in epimoria the excess [of the greater term over the less] is a part, in multiples the less is a part of the greater.” (Har. 11.15-17) Ptolemy chastises the Pythagoreans for their arbitrary procedure, particularly on the grounds that it leads to the exclusion from the concordant intervals of the octave-plusfourth. which turns out to have the ratio 8/3. There is no reason to doubt that Ptolemy and others who repeat his criticism are correct from an empiricist point of view. For the Greeks the octave-plus-fourth is a concordant interval so that any theory, like that of the Sectio, which excludes it is simply contradicting the phenomena. The preamble of the Sectio purports to provide a justification for the fundamental hypothesis of harmonics: the concordant notes are among the numbers which are called by one name in relation to one another. This assumption can be expressed as SCA. If two numbers are concordant, the greater is either a multiple or an epimorion of the lesser. The basic program of the Sectio is, first, the derivation from SCA of specific numerical assignments for the concordant intervals and the tone, and, second, the “refutation” of cer- 115 equation ‘9xm+1 =gymt1' has no integral solutions. However, because of the limits of human auditory discrimination, any careful experiment with a stringed instrument would falsify this claim, interpreted musically. In this respect one might compare proposition 16 with the assertion of the incommensurability of the side and the diagonal of a square, which of musical intervals as the multiplication and division of likewise is always disconfirmed by careful measurement. The separation of harmonics from observation seems to have been a tradition in the subject.'? When Aristoxenus ratios respectively. He also takes for granted the following: opposes his own procedures and views to those of people who SCB. The concordant intervals are, in order of increasing size. “reason in another way and turn away from perception as fourth, fifth, octave, octave-plus-fifth, double octave. not being accurate, constructing intelligible causes and tain claims made by more empirical music theorists.'” In making his derivations Euclid treats addition and subtraction ple or an epimorion (SCA and SCB) and invokes two presaying that high and low pitch come to be in certain numerical ratios and relative speed” (Har. 32.21-26), he is undoubtedly referring to the kind of theorizing found in the Sectio Canonis. Even Ptolemy, who insists upon the importance of observation and experiment in music, allows logos to be the ultimate judge, since “sensation finds the approximate and receives from elsewhere what is accurate, logos receives the approximate from elsewhere and finds what is accurate” viously proved arithmetic facts; according to SC3 an epi- (Har. 3.6-8). SCC. The octave is composed of a fourth plus a fifth. SCD. A tone is the difference between a fifth and a fourth. As has already been mentioned, Euclid carries out his program in a strictly arithmetical way. Thus, for example, to prove SC10. The octave interval is a multiple, Euclid argues that the double octave must be either a multi-

117 It seems reasonably clear, then, that Plato would have had appear more or less equal too, but we do not insist that they no trouble in interpreting the Sectio, purged of its preamble, be exactly equal. But when we arrange objects in m rows of n objects each, we expect there to be exactly nXm objects in the array. In other words, although we may concede that as a piece of deductive arithmetic based on SCA-SCD and ordinary arithmetic hypotheses. Presumably he would not have felt it incumbent upon the music theorist to justify either his hypotheses or such terms as ‘concordant’ and ‘octave’. These terms and hypotheses are simply ways of referring to and describing perfectly clear characteristics of numbers. Perhaps the first thing that strikes us as wrong with the Platonic conception, as I have explained it, is that it does not treat the numbers of harmonic theory as numbers of anything. We understand ratios in harmonics as ratios between physical magnitudes: string lengths or frequencies, for example. Plato may have understood the ratios of harmonics in this way, but there is no evidence of his doing so in the Republic. Socrates speaks of harmonics as the study of numbers and not as the study of measurable characteristics.?' A similar conception of harmonics is found in a Pythagorean division of the sciences, which has often been compared with the scientific curriculum of the Republic: The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitude as either stationary or in motion. Arithmetic then studies quantity as such, music the relation between quantities, geometry magnitude at rest, spherics magnitude inherently moving. (Proclus, In Pr. Eucl. 35.21-36.3) there are no perfect physical realizations of geometric truths, there is no reason for us to concede the same thing in the case of arithmetic truths. When Socrates contrasts pure and applied arithmetic in the Republic (525c8-526a7), he stresses the difference between the abstract units of pure numbers, which are “equal to every other without the slightest difference and admitting no division into parts,” and the unequal divisible things which we count in everyday life. The distinction which Socrates makes here is a conceptual one and does not affect the question whether the truths of arithmetic are perfectly realized in the physical world, whether, for example, there are always twenty cows in five rows of four cows each. I am unable to find any evidence that Plato was conscious of the difference between the two contrasts: the geometric one between idealized picture and physical realization and the arithmetic one between abstract object and physical one. Nevertheless I would like to make use of the difference in discussing briefly the question of how we should understand the relation between arithmetic harmonics and the physical world. Clearly the answer to this question depends upon what we take the physical embodiment of sound to be: in other words, what we take the physical interpretation of harmonics to be. If the numerical ratios are understood as If harmonics is simply the study of certain properties of ratios of string lengths, the relation of pure harmonics to the numbers, then the relation between Platonic harmonics and the physical world will probably not be the same as the relation between Platonic astronomy and the physical world. physical world would presumably be like the relation of geometry to it; just as two strings cannot be exactly equal, neither can they be exactly in the ratio of two to one. To argue against the assimilation of harmonics to pure mathematics, one would have to construe harmonics as involving approximation like astronomy. On the other hand, if Plato believed that the ratios of harmonics are properly interpreted as ratios of frequencies.?? For, whereas points on a rotating sphere are a direct representation or idealized picture of the sphere of the fixed stars, the ratio of two to one is not in the same sense a picture of the octave. This difference between Platonic harmonics and astronomy has a parallel in the case of geometry and arithmetic, which might be brought out in the following way. When we measure the angles of a triangle havhe might claim that, like the truths of arithmetic, the truths of harmonics have perfect realizations. For clearly, if strings ing sides which appear to be equal, we expect the angles to a and b are vibrating at frequencies in the ratio of three to

119 NOTES two and b and c are vibrating in the ratio of four to three, then a and c are vibrating in the ratio of two to one. The difficulty arises, however, when we try to translate this truth into a statement about sound. Presumably Plato believed that our perceptions of fourths, fifths, and octaves are This paper is based on research done while the author held a grant from the American Council of Learned Societies. The author would like to express his gratitude to the Council and to critics of an earlier draft of the paper: Malcolm Brown, Myles Burnyeat, and Gregory Vlastos. too imprecise to enable us to determine ratios between fre- 1. In this paper I use, with occasional changes, the following translaquencies, and hence believed that we are never in a position tions: Shorey's Republic of Plato, Ross's Metaphysics of Aristotle, and to confirm or disconfirm the claim that an octave is composed Morrow's Commentary on Book I of Euclid's Elements of Proclus. Other translations are mine. References are made to the standard editions listed of a fourth and a fifth. It is perhaps from thisipoint of view that he belittles the Pythagorean search for numbers in heard concords. If the position I have just been describing is Plato’s, then perhaps the relevant way to distinguish harmonics and arithmetic is.in terms of relative accuracy of perception. Although my senses never enable me to determine the exact ratio of the frequencies of two vibrating strings, they do allow me to determine the number of people in a room. The differences between harmonics and arithmetic would seem to involve a notion of approximation, but a slightly different one from that involved in the difference between astronomy and geometry. Both arithmetic and harmonics can be treated as pure sciences of number, but, whereas the truths of arithmetic can sometimes be determined to hold exactly in the physical world, the laws of harmonics are never determined to be more than approximately true. in the bibliography. . 2. What I take as the two fundamental features correspond to points ll and VI of A. Wedberg's account of Plato's philosophy of arithmetic and geometry, although Wedberg ascribes to Plato a theory of intermediate objects. See A. Wedberg, Plato’s Philosophy of Mathematics (Stockholm, 1955), pp. 61, 62, 65, 67. 3. See especially Rep. VI 510b2 ff. 4. See, e.g.. Proclus, In Pr. Eucl., 10.16-11.25. The view is, of course, a plausible interpretation of the middle books of the Republic. 5. I must admit that some views | ascribe to Plato in this paper are more simply interpreted with intermediates than without them, particularly if forms are construed as abstract concepts rather than perfect exemplars. It seems to me, however, that the same could be said of many of Plato's remarks about arithmetic and geometry. 6. For substantiation of these claims about Greek mathematics and Aristotle's interpretation of it, see my papers "Euclid's Elements and the axiomatic method," British Journal for the Philosophy of Science 20(1969): 289-309, and “Aristotle on geometric objects,” Archiv fiir Geschichte der Philosophie 52 (1970): 156-71. 7. Thus it is necessary to distinguish between Plato's mathematical Platonism and the modern mathematical philosophy called Platonism. The: latter is based on particular features of modern mathematics which are foreign to Greek mathematics. For a description of modern mathematical Platonism, see P. Bernays, “On Platonism in mathematics,” in P. Benacerraf and H. Putnam (eds.), Readings in the Philosophy of Mathematics (Englewood-Cliffs, N.]., 1964), pp. 274-86. 8. See T. Heath, A History of Greek Mathematics (Oxford, 1921), Vol. I, pp. 348-53 or, for more detail, F. Hultsch, “Autolykos und Euklid,” Berichte über die Verhandlungen der Kôniglich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Philologisch-Historische Klasse, 38(1888): 12855. I have chosen to treat the mathematical texts with which I deal in this paper as entirely genuine rather than enter into the thorny questions which have been raised about the authenticity of various parts of them. 9. Euclid does not always do so, however. He adds to the antecedent of proposition 2 the condition that the pole of the horizon be between the summer tropic and the visible pole and imposes a similar condition in

proposition 7. In his treatise On Risings and Settings Autolycus always takes such things for granted. by any assignment of ratios of the following form: tone: m+n+1/n+1; 10. This is a slight exaggeration, since the transformation from spatiotemporal to purely spatial considerations presupposes the description of fifth: (m+n+1)%n+1)/1; double octave: (m Fn+1){n +1)2/1. the path of a point on the surface of a rotating sphere and an extension of Autolycus's assumptions to such points moving on different paths. Autolated by E. L. Minar Jr. (Cambridge, Mass., 1972), pp. 383-886. lycus establishes these presuppositions in propositions 1 and 2 with argufrom a modern point of view. For example, he describes (Har. 26.3-14) an ments which are in fact question-begging. 11. ‘Earth’ and ‘stars’ each occur once in proposition 9. Hultsch (“Autoexperiment with an eight-stringed canon to show (against Aristoxenus) 121 fourth: n+1/1; fifth: m+n+1/1; octave: (m+n+1)(n+1)/1; octave-plus19. See W. Burkert, Lore and Science in Ancient Pythagoreanism, trans20. Ptolemy's notion of an experiment is, however, rather unsatisfactory that the octave is less than six tones; but the experiment presupposes that lykos und Euklid," 144, fn.) ascribes their occurence to “later reworkings the lengths of strings producing notes a tone apart must be in the ratio of of the original text.” 9/8. He also describes (Ibid., 25.5-11) a more neutral experiment in which 12. There is one puzzling exception to this generalization, namely, the a “very musical" person produces seven consecutive pitches, each a tone first proposition of the Phenomena. It says that the earth is the center of above its predecessor, and checks to see whether the last is an octave the cosmos.To prove this assertion Euclid invokes a measuring instrument, above the first. However, Ptolemy concludes that, if the experiment the diopter, and describes “observations” which establish that the zodiac doesn't refute Aristoxenus, the tones must have been inexact. is bisected by the horizon, a claim already invoked in the preamble. Since actual observations with a diopter would not yield the results envisaged 21. Van der Waerden makes this point in “Platon et les sciences exactes des Pythagoriciens,” Bulletin de la Societé Mathématique de Belgique 21 by Euclid and since the preamble includes material for proving proposition 1 in a strictly geometric way, it is difficult to determine why Euclid proclear distinction between sounds and musical intervals on the one hand ceeds as he does. and numbers and their ratios on the other" in Ptolemy's Harmonics. 13. I am inclined to think that the word ‘problem’ here does not have the technical sense of ‘construction’ but the more general sense of ‘something to Plato in An Examination of Plato's Doctrines (London, 1963), vol. II, set out for proof or refutation’. See, e.g., Aristotle, Topics A.4.101b28-37, pp. 182-83. or the scholium, perhaps by Proclus, on the similar use of the word at Th. 180c5. (W. C. Greene (ed.), Scholia Platonica [Haverford, 1938], p. 33) There does not seem to be any particular reason for Plato to focus on constructions in particular in the Republic passage. 14. Theon (Expos. 78.6-22) and Nicomachus (Int. 1.20) both restrict the term epimeres to ratios expressed in least numbers in the form m+1 +n+2/n+2 with n+2 greater than m+1. If Euclid does the same, he has not, of course, accounted for all ratios of a greater to a lesser number in his threefold classification. 15. See B. L. van der Waerden, “Die Harmonielehre der Pythagoreer,” Hermes 78(1943): 192-97. The justification of the numerical representation is, of course, illegitimate. since frequencies can be irrational. 18. The claim that the Sectio proceeds purely arithmetically is true of all but the last four propositions, which involve the determinalion of certain musical “systems” (tunings or scales). However these propositions too are totally independent of the frequency theory of the preamble. An example of the curious terminology is the use of the word diastema to mean both ‘ratio’ and ‘interval’. 17. The doctrines of these theorists are most easily found in the Harmonics of Aristoxenus. 18. The fallacy was first pointed out by P. Tannery, “Inauthenticité de la ‘Division du Canon’ attribuée à Euclide," Mémoires Scientifiques (Toulouse and Paris, 1911 ff.), vol III, p. 215. It amounts to taking SCA as asserting that all multiples are concordant. In fact SCA-SCD are satisfied (1969): 121. I think, however, he exaggerates when he refers to a “very 22. I. M. Crombie ascribes something like a frequency theory of pitch

BIBLIOGRAPHY Aristotle, Metaphysics, trans. W. D. Ross, Oxford 1928. ____-___.__. Opera, ed. 1. Bekker, Berlin, 1831. Aristoxenus, Harmonics, ed. and trans. H. S. Macran, Oxford 1902. Autolycus, De Sphaera Quae Movetur et.De Ortibus et Occasibus, ed. J. Mogenet, Louvain 1950. _ Euclid, Opera Omnia, ed. J. Heiberg and H. Menge, Leipzig 1883-1916. (Phenomena and Sectio Canonis in vol. VIII) Nicomachus, Introductio Arithmeticae, ed. R. Hoche, Leipzig 1886. Plato,‚Opera, ed. J. Burnet, Oxford 1901. D WESER . Republic, trans. P. Shorey, London and New York 1930, 1935. Proclus, A Commentary on the First Book of Euclid’s Elements, trans. G. Morrow, Princeton 1970. e . In Primum Kuclidis Elementorum Librum Commentarii, ed. G. Friedlein, Leipzig 1873. Ptolemy, Harmonica, ed. I. Düring, Göteborg 1930. Theodosius, Sphaerica, ed. J. Heiberg, Berlin 1927. Theon, Expositio Rerum Mathematicarum ad Legendum Platonem Utilium, ed. E. Hiller, Leipzig 1878.