Plato on Geometry and the Geometers

The present paper aspires to explain fully both the supreme importance of Geometry for Plato, and also the nature of the serious ongoing criticism that Plato Plato (428–348 BCE) (and the Academy) directs against the geometers, an explanation that has eluded modern scholars of Plato Plato (428–348 BCE) (since M. Ficino in the fifteenth century to our present day). In order to understand the criticism, it is necessary first to have a true understanding of the nature of Plato’s philosophy. The most crucial concept in understanding Plato’s philosophy, and essentially the only one, is the geometrical concept of periodic anthyphairesis Anthyphairesis of two magnitudes, say line segments, established with the Logos criterion Logos criterion (Sect. 1.2). The Platonic true Being, the intelligible Platonic Idea, is a dyad of opposite parts in the philosophic analogue, Division and Logos-CollectionDivision and Logos-collection , in close imitation of periodic anthyphairesis Anthyphairesis periodic . Plato Plato (428–348 BCE) in effect isolates a method for acquiring full and complete knowledge, as it exists in a small but vital part of Geometry, namely in incommensurability Incommensurability by periodic anthyphairesis Anthyphairesis periodic , and develops a general theory of knowledge, Division and Collection Division and collection , of the Platonic Ideas, in close imitation to the complete acquisition of knowledge provided by the Logos criterion Logos criterion in periodic anthyphairesis Anthyphairesis periodic . The anthyphairetic Anthyphairesis nature of Platonic intelligible Beings was examined in detail by the author in earlier publications and is outlined in the present paper: the One of the second hypothesis of the Parmenides and its close relation to Zeno’s Zeno of Elea (c. 490–c. 430 BCE) arguments and paradoxes Zeno of Elea (c. 490–c. 430 BCE) paradoxes (outlined in Sect. 1.3) and the Division and Collection Division and collection in the Sophistes and Politicus, where the genera and kinds in the Division are considered as hypotheses (outlined in Sect. 1.5). Furthermore, we establish in the present paper that the dialectics of the Politeia coincide with Division and Collection Division and collection -Logos (Sect. 1.4), described also equivalently in analysis fashion as ascent from hypotheses to the “anhupotheton Anhupotheton ”, the hypothesis-free (531d-e, 532a-b,d-e, 534b-d) (Sect. 1.6). Plato (428–348 BCE) Parmenides Plato (428–348 BCE) Politeia Without understanding the anthyphairetic Anthyphairesis nature of Plato’s dialectics it is impossible to understand Plato’s praise of Geometry and criticism of its practice by geometers. Criticism I: the rejection by Plato of the indivisible geometric pointGeometric point in favor of the “indivisible line”, according to Aristotle’s Metaphysics 992a, 19–22, coincides Plato (428–348 BCE) Timaeus with the rejection of the One of the first hypothesis as a true Being, a One similar to the geometric point, Geometric point and the adoption of the One of the second hypothesis as a true Being, a One described by Zeno, Zeno of Elea (c. 490–c. 430 BCE) Plato, and Xenocrates in terms akin to those of the indivisible line, Indivisible line in the Parmenides (Sect. 1.7). Criticism II: the rejection by Plato of the use of hypotheses (namely basic definitions and postulates on lines, circles, angles in geometry, on units and numbers in Arithmetic) as principles, and not as stepping stones towards the true Being, the anhupothetonAnhupotheton . The rejection of the axiomatic method on epistemological grounds, since hypotheses, namely definitions and postulates, are arbitrarily accepted and hence these, with all its consequences, cannot be known; knowledge, by Division and Collection Division and collection , is achieved only when the generation of these hypotheses, the basic geometric (lines, circles, angles) and arithmetical (units, numbers) concepts and their Postulates, takes place within the Platonic true Beings, something possible because of their periodic anthyphairetic Anthyphairesis structure in the Politeia(510–511, 527a-b) (Sects. 1.8, 1.9, 1.11, and 1.12). Praise of Geometry: Plato Plato (428–348 BCE) makes the extraordinary claim that the method of Division and Collection Division and collection must be the method employed for the acquisition of true knowledge for all of Geometry (in place of Euclidean axiomatics). With this method he constructs the numbers, the straight line, and the circle (Sect. 1.10), and the three kinds of angle (Sect. 1.11). Criticism III: the use of geometric diagrams is rejected by Plato, not simply because they are visible/sensible, but because they are sensible representations not provoking to the mind, as they should be, if they were represented as true sensibles, participating anthyphairetically Anthyphairesis in the intelligible by means of the receptacle/diakena, as presented analytically in the Timaeus 48a-58c and in preliminary manner in the Politeia 522e–525a. The geometers fall into this faulty use of geometric diagrams, as a result of their “dianoia” way of constructing their arguments (Criticism II) (Sects. 1.13 and 1.14). Criticism IV: the rejection by Plato [1] of Archytas’ non-anthyphairetic proofs of quadratic and cubic incommensurabilities (based on the arithmetical Book VIII of the Elements and eventually on the arithmetical Proposition VII.27 of the Elements), possibly expressed in the distinction between the eristic and dialectic way of going to infinity in the Philebus 16d-e, replacing Theaetetus’ anthyphairetic proofs of quadratic incommensurabilities, [2] of Eudoxus’ Dedekind-like theory of ratios of magnitudes (Book V of the Elements), expressed clearly in the second part of Scholion In Euclidem X.2, replacing Theaetetus’ anthyphairetic one, and thus moving away from Plato’s philosophy based on periodic anthyphairesis, and [3] of Archytas’ non-anthyphaireic stereometric solution of the problem of the duplication of the cube, in the Politeia 527d-528e (Sect. 1.15). All of Plato’s criticisms of the geometers have a common strain: the practice of geometers is distancing away from periodic anthyphairesis Anthyphairesis periodic and from Platonic true Beings, based on periodic anthyphairesis Anthyphairesis periodic .