Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries. The studies focused first on the Accepted September 1997 I This article was written during my fellowship at the Centre for Hellenic Studies (Washington, D.C.). An earlier version of it was given as a talk at the Departmentof Classical Studies, Yale University. I am very grateful to Heinrich von Staden for his kind invitation to Yale and to Charles Price for his generous help in preparing the final English version. In quoting Greek authors I have used standardEnglish transla- tions where available. C) Koninklijke Brill NV, Leiden, 1998 Phronesis XLIII13 This content downloaded from 134.34.5.113 on Tue, 7 May 2013 02:14:34 AM All use subject to JSTOR Terms and Conditions 212 LEONID ZHMUD mathematicalpassages of Plato's dialogues and second on the evidence (usually late) about his mathematicaldiscoveries, such as the golden sec- tion, the method of analysis, the method of founding the "Pythagorean triplets,"etc. The generalconclusion of these studieswas that Plato him- self was not an activescientist and that the scientificdiscoveries and hypothe- ses attributedto him in the ancienttradition do not really belong to him.2 In the second half of the 20th centuryit seems there have been no seri- ous attemptsto debate this conclusion,3and the discussionhas been con- cerned with the two other aspects - philosophicaland methodological. In the first case the main question usually focused on the extent to which Platonismstimulated the developmentof the exact sciences in an- tiquity and/orhow much it hinderedthe formationof the appliedsciences and empiricallyoriented research. I recentlystated my positionregarding this question,4the essence of which can be summedup as follows: there is no groundfor believing that in ancientGreece mathematicswas much more influencedby philosophy(including Plato's philosophy)than it has been in the modernperiod. Because of the fundamentalepistemological heterogeneityof science and philosophy,they had to be developed in a differentway from their very beginning,i.e. from the sixth centuryB.C. on, and all the evidence available to us shows that they were actually developed in a differentway. In fact, the relationshipbetween the exact sciences and philosophywas essentially the same in antiquityas it is in the modernperiod: it was mathematicsthat influencedphilosophy and not vice versa. As W. Knorr,one of the leading expertsin ancientmathema- tics, emphasizes: . mathematical studies were autonomous, almoust completely so, while the philosophical debates... frequently drew support and clarification from mathe- matical work.... My view conforms to what one may observe as the usual rela- tion between mathematics and philosophy throughout history and especially recently.5 2 See, for example C. Blass, De Platone mathematico, Bonn 1861; G.J. Allman, Greek Geometry from Thales to Euclid, Dublin 1889 (Repr. New York 1976); M. Simon, Geschichte der Mathematikim Altertum, Berlin 1909 (Repr. Amsterdam 1973), 183ff; T.L. Heath, A History of Greek Mathematics,V. I, Oxford 1921, 284ff. 3 See, however Ch. Mugler, Platon et la recherche mathematiquede son epoque, Strasbourg 1948; cf. a long review of Mugler by H. Cherniss, "Plato as Mathema- tician," Rev.Met. 4 (1951), 395-425 (= Selected Papers, Ed. L. Tarin, Leiden 1977, 222-252). 1 L. Zhmud,"Die Beziehungenzwischen Philosophieund Wissenschaft in der Antike," SudhoffsArchiv 78 (1994), 1-13. 5 W. Knorr, "The Interaction of Mathematics and Philosophy in Antiquity," in: This content downloaded from 134.34.5.113 on Tue, 7 May 2013 02:14:34 AM All use subject to JSTOR Terms and Conditions PLATO AS "ARCHITECTOF SCIENCE" 213 It is revealing that we can discern the obvious influenceof the exact sciences on Plato's convictionthat any firmknowledge of physicalreality is impossible, or on his belief in the mathematicalstructure of the uni- verse, but we can hardlyprove that these ideas in turnhad any immediate influenceon those who carriedout researchin ancientGreece. In the fieldof methodologythe argumentconcerned not so muchPlatonism as the exact sciences in the Platonicschool.6 Many suggestedthat even if Plato did not achieve any success in the exact sciences, he did play a con- siderablerole as an organiserof scientificresearch and as a methodolo- gist, who defined the problemsmathematicians and astronomersstudied and the methods they used.7I quote only one typical opinion: Die traditionelle Platosauffassung,wie sie auch von den beteiligten Mathema- tikem im wesentlichen geteilt wird, besagt: Plato hat natiirlich keine mathema- tische Entdeckungengemacht; die Uberlieferung,die ihm Dodekaederzuschreibt, ist wegzulegen; aber Plato hat der Mathematikdie allgemeinen Direktiven gege- ben, die axiomatische Strukturder Elemente, die Beschrankung auf Konstruk- tionen mit Zirkel und Lineal allein, die analytische Methode sind Platos Werk; die groBen Mathematikerseines Kreises, Theatet und Eudoxus, haben die soge- nannte Euklidische Mathematikunter seinem EinfluBgeschaffen.1 Despite the criticism of this position frequently expressed both by philologists and by historiansof mathematics,9in the last decades it has N. Kretzman (ed.), Infinity and Continuity in Ancient and Medieval Thought, Ithaca 1982, 112. 6 See the review article by M. Isnardi Parente, "Caratteree strutturadell' Acca- demia antica," in: E. Zeller, R. Mondolfo, La filosofia dei Greci nel suo sviluppo storico, II,3, Firenze 1974, 867-877. 7 H. Usener, "Organisation der wissenschaftlichen Arbeit" (1884), in: idem, Vortrdge und Aufsatze, Leipzig 1907, 69-102; U. von Wilamowitz-Moellendorff, Antigonos von Karistos, Berlin 1889, 279ff; I.L. Heiberg, Geschichte der Mathematik und Naturwissenschaftim Altertum,Leipzig 1912, 9f.; P. Shorey, "Platonism and the Unity of Science" (1927), Selected Papers. (Ed. L. Taran) New York 1980, 434ff; F. Solmsen, "Platons EinfluB auf die Bildung der mathematischenMethode," Q&St. Abt. B, 1 (1929), 93-107 (= K. Gaiser (ed.), Das Platonbild, Hildesheim 1969, 125- 139); H. Herter, Platons Akademie, Bonn 1946; G. Hauser, Geometrie der Griechen von Thales bis Euklid, Luzern 1955, 127-138. 0O. Toeplitz, "Mathematikund Antike,"Die Antike 1 (1925), 201 (italics are mine). It is worth pointing out that Toeplitz himself understood the vulnerability of this position. I For example E. Howald, Die platonische Akademie und die moderne universitas litterarum,Bern 1921; E. Frank,"Die Begrundungder mathematischenWissenschaften durch Eudoxos" (1932), in: L. Edelstein (ed.), Wissen, Wollen, Glauben, Zurich 1955, 144f; A. Szab6, "Anfange des Euklidischen Axiomensystem," AHES 1 (1960), 99ff (= 0. Becker (ed.), Zur Geschichte der griechischen Mathematik, Darmstadt 1965, This content downloaded from 134.34.5.113 on Tue, 7 May 2013 02:14:34 AM All use subject to JSTOR Terms and Conditions 214 LEONID ZHMUD been developed in many importantstudies.'0 While differentin approach, these studies share the tendency to presentthe Academy as a kind of a researchinstitution, where the best mathematiciansand astronomersof the time worked underPlato's methodologicalsupervision." In the following sections of my paperI will discuss variousaspects of this issue. Sections II and III deal with two specific scientific problems, the duplicationof the cube and the "saving of the appearances,"where Plato is supposedto play the role of the scientificorganiser and method- ologist. Section IV considers a recently restoredpapyrus text of Philo- demus that directly calls Plato an architectof the mathematicalsciences. The focus of this section is the authorshipof this particularpassage and its similarityto the well-knownCatalogue