The Other Plato: The Tübingen Interpretation of Plato's Inner-Academic Teachings

FOUR Plato’s Synopsis of theMathematical Sciences

FOUR PlatO’s synOPsis OF the MatheMatical sciences Konrad Gaiser This essay was presented at several occasions where there was an opportunity to convey the “Tübingen Plato interpretation” and discuss the question of how it should be represented.1 Hence, here again we are pursuing the attempt to relate Plato’s literary dialogues, which were intended for the public, to the “esoteric” doctrine that was present within the school. Furthermore , we will establish a connection between the students’ testimonies about Plato’s theory of the principles and the relevant allusions in the dialogues.2 Most importantly, I want to draw on a particular and central Platonic motif in order to reevaluate and prove that there is a close connection between his literary statements and the testimonies concerning his oral doctrine. The guiding question shall be that of determining in what sense Plato expects mathematics to provide a crucial aid for the cognition of ideas, especially cognition of the “idea of the good.” This question is addressed in the Republic, yet it also touches upon the core of the “unwritten doctrine,” because the “idea of the good” is the highest cause of all being. According to the accounts given by his students, Plato referred to this idea from a mathematical perspective as “the one.” Most of what Plato says about the imperative of mathematical studies for the philosopher has often enough been rightly interpreted and is generally familiar. However, it seems to me that what he says about a “synopsis” [Zusammenschau] of the mathematical sciences, which is supposed to be 83 84 THE OTHER PLATO especially instructive for dialectic, needs further explanation.3 I would like to try to analyze passages from the texts of the dialogues more precisely and completely than has been done so far in Tübingen and elsewhere. 1. PlatO’s DevalUatiOn OF the PRactical Utility OF MatheMatics When one is asked today what good mathematics serves, what first comes to mind is its eminent importance for natural science and technology. One might also mention that to become aware of the theoretical beauty of mathematical structures causes pleasure. During antiquity such practical interest was also more dominant. All of early mathematics prior to the Greeks— Babylonian and Egyptian—was oriented to purpose and application. Even the Greeks valued the science of numbers, geometry (i.e., “the art of land measurement”), and astronomy because of their importance for trade and transportation, architecture and underground engineering and other technical constructions, and creating calendars; naturally, music for them was primarily not a theoretical science. However, already before the time of Plato mathematics approached the level of a theoretical science in two ways. On the one hand, it played an important role within religious speculative traditions, especially in the Pythagorean mathematical ideas that in part were kept concealed as mysterious wisdom. On the other hand, there were already Greek thinkers prior to Plato who carried out mathematical research and formulated some universal mathematical propositions—presumably Thales, who is credited with having discovered the proposition concerning a right angle inscribed in a semicircle, and this is certainly true of mathematicians such as Hippocrates of Chios and Theodorus of Cyrene. In Plato’s case, both tendencies coincide: on the one hand mathematics was included in a comprehensive interpretation of the world, and on the other hand it was developed to the point of full scientific autonomy.4 What is new in Plato is a conscious reflection on the importance of mathematical sciences within the larger context of an interpretation of being [Seinserklärung] and approaching the question of how to lead one’s life [Lebensgestaltung ]. In particular, one must mention the conception, which was also firmly defended by later thinkers, that the proper value of mathematics does not lie in its practical utility but in its ability to lead thinking on the way to true knowledge. Plutarch tells us that Plato even rejected the use of mechanical tools for solving geometrical-stereometric problems. Archytas, Eudoxus, and other mathematicians had constructed devices in order to solve the problem of doubling the cube (i.e., for the purpose of finding the cube root of a given magnitude). These devices made it possible to represent the curves of a 85 PLATO’S SYNOPSIS OF THE MATHEMATICAL SCIENCES higher degree by performing the required operation of rotation. Thus, Plutarch notes (Vita Marcelli 14, 5–6): However, since Plato grew reluctant and accused them of...