Późna nauka Platona : związki ontologii i matematyki

Abstract

The aim of the present work is to show the content of Plato’s unwritten “late teaching” which constitutes the last stage in the evolutionary process of the development of his thought, the stage called “the unwritten (or esoteric) teaching” . It grew quite naturally from the earlier stages of Plato’s philosophy (“the written teaching”). Plato decided to focus his attention on the ontological status of mathematics, with particular emphasis on the necessary conditions for the existence of mathematics, and on the question about the obligatoriness, independent from the subjective statuting, of mathematical theorems. As a result of this analysis, “the theory of ideas”, formulated in the exoteric stage, became transformed into “the theory of ideal numbers”. This is indeed the essence of the transformation that took place in the late period of Platonic thinking, when the theory of ideas was interpreted in a mathematical context, and served the purpose of explaining the ontological status of mathematical objects. This situation resulted, first of all, from the role that mathematics played in the Academy, where - as we happen to know - mathematical studies were an important part of education, and where the most outstanding mathematicians of the time used to work. The above is witnessed by the relation of Aristotle, a member of the Academy, who in his Metaphysics contained some information about the philosophical disputes taking place there. Let it be noticed that Aristotle’s criticism of Plato’s ideas refers in the first place to the dispute on the status of mathematical objects, the special case of which we may find in the M and N books (it seems that it is exactly for this reason that we cannot fully and properly understand Aristotle’s attitude towards Plato’s theory of ideas if we fail to take into account the matters connected with the ontology of mathematics). In the late stage of Plato’s activity, we find a particularly strong confirmation of the links between ontology and mathematics when analysing the documents connected with Plato’s successors in the Old Academy: Speusippus, Xenocrates, Eudoxus, or Philip of Opus. It can be seen very clearly to what extent they continue Plato’s “late thought” in becoming involved in the dispute concerning the status of ontological principles, on the relations between mathematical objects and the concept of mathematical natural sciences. A completion of this picture we find in the documents of the “intermediate tradition” through which the ancient commentators of Plato tell us about the late form of his philosophy. We have here again to do with the problem of analysing the links that exist between ontology, mathematics and what can be called mathematical natural sciences. The said documents have been collected by K. Gaiser under the general title Testimonia Platonica. Bearing in mind the above mentioned facts, it seems justifiable to try to reconstruct Plato’s late teaching, and its interpretation in the context of the theory associated with period o f the dialogues. In this sense the present work is a continuation and expansion of the analyses carried out in the work The Theory of Ideas - The Evolution of Plato ’s Thought (Katowice 1997 [the first edition], and 1999 [the second edition]) where I attempted to justify the thesis that “the theory of ideas” is not a random and heterogeneous collection of statements and philosophical intuitions scattered all over various dialogues, but rather an orderly process of development in which several stages can be discerned. I assumed then that Plato’s thought has an evolutionary nature, and this evolution can be fully exemplified by “the theory of ideas” . This is because Plato started to construct his theory on the basis of Socratic inspirations, completing Socrates’ conception with an ontological dimension. “The theory of ideas” thus constructed in the middle-Academic period, was later subjected to a thorough analysis and reinterpretation, and given a new form that was called “the theory of ideal numbers”. Since the latter demanded its own, ultimate legitimisation, Plato decided to adopt the conception called “the theory of ontological principles” . However, neither “the theory of ideal numbers” nor “the theory of ontological principles” can be observed in Plato’s dialogues. Plato’s disciples and ancient commentators described these theories only later. This is why it became conventional to call this ultimate, late form of Plato’s thought by the name of “the unwritten teaching”. Its essence was already in ancient times an object of interpretative controversies. The modem research on Plato’s thought brought about a great intensification of that controversy. Some scholars try to belittle the significance of “the unwritten teaching” (among other names Cherniss, M. Insardi-Parente, G. Reale, G. Vlastos, J.N. Findlay, or E. Dont), but others emphasise its importance for the understanding of Plato’s entire philosophy (here belong H J. Krämer, G. Kaiser, T.A. Szlezal, G. Reale, G. Halfwassen, J.N. Findlay, or V. Hosle). The representatives of the latter position assumed that, in order to fully understand the philosophical sense of the dialogues, it is necessary to refer to “the unwritten teaching”, so that it was accepted that the said teaching contains the essence of the Platonic thought. In the present work, I suggest a different interpretative stand. I assume, namely, that “the unwritten teaching” (the theory o f ideal numbers and theory of principles) constitute a natural consequence of the development o f Plato’s ontological thought, and should be considered from the point of view o f that evolution, that is in the context of the maturation of his thought. Thus, “the unwritten teaching” appears the crowning of Plato’s ontological conceptions. The point is that already in the period of the dialogues we can see Plato grappling with the difficulties inherent in his “theory of ideas”, which is confirmed by the disputes within the Academy, and debates with representatives of other philosophical positions. Plato seeks new solutions, striving to find the conclusive arguments, which results in new conceptions. Consequently, there arose “the theory of ideal numbers” and “theory of principles” . How can we not recognise that we have to do here with a peculiar case of the evolution of the philosopher’s views? Do we find a philosopher whose thought would not undergo a process of this kind? How can we then claim that, in the light of the testimony of the disciples and ancient commentators, “the unwritten teaching” do not constitute an essential part of Plato’s doctrine? And how can we, on the other hand, maintain that it is only “the unwritten teaching” that express the Platonic thought? Particularly strange appears to be the statement that Plato, already when he was writing his dialogues, possessed a conception proper to “the unwritten teaching”, but decided not to reveal it, so that we should reconstruct it on the basis of an analysis of the dialogues. I think that the above-described positions, original as they may be, are rather radical and thus not necessarily corresponding to the actual state of affairs. The suggestion that emphasises the evolution of Plato’s views, and the development of his thought stemming from his being aware of the limitations connected with particular aspects of his theory, and his desire to legitimise it, may seem then more natural, even though it may be less original. I shall try to justify this thesis in the present work, using as an example an analysis of the links between the Platonic concepts of ontology and mathematics. Thus, the present dissertation finds its place in the debate, conducted by the most outstanding modem commentators, on our understanding of Plato’s philosophy. It would be useless to enumerate here those commentators, their names and conceptions will turn up while discussing various particular problems. I have made an effort to use their research achievements by integrating them with my analyses, or by entering into a discussion with them. I have tried to distance myself only from such interpretations that are based on much later theories (particularly the contemporary ones) whose connection with Plato’s thought seems dubious. I mean particularly such interpretations that stipulate the reading of Plato from the point of view of modem logical, mathematical, and philosophical theories. It is impossible, naturally, to become liberated from the temporal context in which a given text is commented upon. We should, however, strive to limit such conditioning so that a given interpretation is to a greater extent predicated on the inner logic of the original text than on the commentator’s (however inevitable) assumptions. Consequently, the present analyses are based on Plato’s own writings and on those of ancient commentators and doxographers informing us of his conceptions.