From One to one, to many: the Parmenides 142b-144b Florin George Călian Department of Philosophy Central European University A Dissertation Submitted in Partial Fulfilment of the Requirements for the Doctor of Philosophy Supervisor: Professor Gábor Betegh CEU eTD Collection Budapest, Hungary 2016 Declaration This dissertation is the result of my own work and includes nothing, which is the outcome of work done in collaboration except where specifically indicated in the text. It has not been previously submitted, in part or whole, to any university of institution for any degree, diploma, or other qualification. Signed: ____________________________________________________________ Date: ______________________________________________________________ CEU eTD Collection Abstract This dissertation is a contribution to the reassessment of Platonic ontology and mathematical conceptions as found in the beginning of the second hypothesis of the second part of the Parmenides (142b-144b), in what I argue to be Plato’s ontological argument and the argument for the generation of numbers. I support the reading of these two specific arguments as a coherent continuum: how the being of one was conceived by Plato as necessarily different from one, differentiating thus himself from Eleatic philosophy and setting the grounds for multiplicity, which leads Plato into a discussion on the primordia of numbers. There are numbers (the mathematical argument) only because one and being can be separated (the ontological argument). My analysis of the argument for the generation of numbers is substantiated by an evaluation of Aristotle’s testimony in Metaphysics A6, a testimony which, I argue, is built upon this passage of the Parmenides. The dissertation provides an analytical commentary on the stages of the arguments, in an attempt to place the arguments on the map of Plato’s philosophy. I demonstrate that the ontological and mathematical arguments are actual Platonic arguments, and not merely dialectic exercises, which have traceable conclusions in Plato’s philosophy of the so-called late dialogues, CEU eTD Collection especially the Theaetetus, the Sophist, and the Timaeus. Acknowledgements First and foremost I would like to express indebted thanks to my supervisor Professor Gábor Betegh, for his very valuable pieces of advice and critical comments. Professor Betegh has constantly helped me to pass from intuitions to clarifying arguments and it has been a privilege to work under his supervision. I owe the highest debt of gratitude for his thoroughness, irreplaceable expertize and patience to make sense of my Balkan writing. His vivid and enamoured interest in, and study of, everything related to ancient philosophy has been a profound and determinative influence. I want to extent my thanks to the whole of the Department of Philosophy at Central European University which was not only a very good working place from a practical point of view; it is an institution which has shaped my philosophical thinking and moulded my approach to academic research. I mention with gratitude Prof. István Bondár, for the discussions related mostly to subjects in Ancient Philosophy, but also to other topics. Another important encounter was that of Prof. Hanoch Ben-Yami: the debates that I witnessed and have been part of on the miscellanea of philosophy at the Department were very instructive. Writing this dissertation was made possible also with the support I have received from people I have encountered outside the Department. I want to thank to Prof. Brad Inwood who facilitated my access to Robarts library in 2012. The three months period that I spend at Fribourg, through the Doctoral Research Grant from Central European University, helped me a lot in developing important chapters of the dissertation. I want to give special thanks to Prof. Filip Karfik for all his support and philosophical discussion on my research, as well as on other philosophical issues, and also to Prof. Dominic O’Meara for all his kind advices. I have also hugely profited from the time spent in 2013 at Plato Centre, Trinity College, Dublin, where the discussions with Prof. John Dillon, Prof. Vasilis Politis, Dr. Brendon O’Brien, and Dr. Manfred Weltecke have proven inspirational for my research. New Europe College has also generously encouraged this research through the grant it has given me in 2013/2014, especially through the travel grant offered to meet Prof. Leslie Brown and Prof. Michael Inwood at University of Oxford. Their knowledgeable comments have substantially contributed to the delimitation of my understanding of Plato’s Parmenides. And last, but not least, I would like to thank to CEU eTD Collection Professor Edward Kanterian who has been very supportive, in key moments, to settle my academic hesitations. Special thanks must go to my two elective affinities, to my wife Antoaneta, who sensibly put up with my writing blocks and other caprices all these years, and to our daughter, Sophie. This thesis is dedicated to Ciprian Aristotel (1977-2000) CEU eTD Collection CEU eTD Collection Contents No table of contents entries found. CEU eTD Collection List of Figures FIGURE 1 THE STRUCTURE OF THE ARGUMENTS (142B1-144A1) .............................................8 FIGURE 2 THE DIVISION OF ONE AND BEING ..........................................................................17 FIGURE 3 VISUAL REPRESENTATION OF FORMULA 2K+1 .......................................................97 FIGURE 4 TRIANGULAR NUMBERS ........................................................................................100 CEU eTD Collection “If someone were to reduce Plato to a system, he would render a great service to mankind…” Leibniz (Philosophical Papers and Letters, 1989, 659) “There is an argument, suggested by a passage in Plato's Parmenides, to the effect that, if there is such a number as 1, then 1 has being; but 1 is not identical with being, and therefore 1 and being are two, and therefore there is such a number as 2, and 2 together with 1 and being gives a class of three terms, and so on. This argument is fallacious, partly because “being” is not a term having any definite meaning, and still more because, if a definite meaning were invented for it, it would be found that numbers do not have being—they are, in fact, what are called logical fictions’.” Russell (Introduction to Mathematical Philosophy, 1993, 138) “Anybody who sets out to report Plato's opinions can properly be asked to tell us on what principles he interprets the evidence at his disposal” Crombie (An Examination of Plato's Doctrines, 1962, 14) CEU eTD Collection 1 CEU eTD Collection 2 1. The Parmenides Question The Parmenides is a key Platonic dialogue, which raises controversial and contradictory interpretations. The first part of the dialogue contains both the most coherent and detailed depiction of one central version of the theory of forms as it is to be found in the middle dialogues and, at the same time, the most upsetting counter-arguments to the theory. 1 Each counter-argument leaves us with the impression of undermining the edifice of the theory of forms as it is displayed in the Phaedo and the Republic. The first part of the Parmenides does not reach any theoretical philosophical conclusion regarding the ontological status of the forms, except that they are necessary for thought and speech. The second part of the dialogue is introduced by Plato as a necessary dialectical exercise for a better training in philosophy. How to understand these complicated structures and Plato’s critique of his own theory of forms is not certain at all. One might think that referring to previous scholarship would offer some guidance in reading the Parmenides; however, even from within the Academy, through the Neo-Platonists, and up to contemporary commentators, there is no scholarly agreement on a generally accepted modus of interpretation: the Parmenides remains a puzzling dialogue 2 and there is no real consensus in respect to its CEU eTD Collection 1 I am aware that it is only a textual artifice to constantly mention a theory of forms; Plato’s dialogues propose no definitive theory of forms that remains unchanged in the entire corpus. In several dialogues versions of the theory of forms are presented, and in different parts of Plato’s work different facets of the theory are displayed with subsequent emendations, testifying thus to a continuous re-thinking process on the theory of forms. The discussions in the Parmenides of aspects pertaining to the theory of forms are representative for the development of the theory in what has been identified as the middle period. 2 Mary Louise Gill, and Paul Ryan, Plato Parmenides (Hackett Publishing, 1996), 1. 3 purpose3, topic, and techniques4. Similar to ancient readings of the dialogue, with the complex exception of the Neo-Platonists5 - as even in their case the overall interpretation was not fully unanimous – contemporary studies demonstrate that the dialogue’s structure and arguments are far from clear. The second part of the dialogue has been even more puzzling and contentious for scholars. Although it formally makes up a whole with the first part of the dialogue, the articulation of these two components is highly problematic; moreover, at first glance the second part seems difficult to link not only with the first part, but with ideas expressed in Plato’s other dialogues as well. This part of the dialogue remains a piece of curiosity (both in the Platonic corpus and in the history of philosophy), and it has often been left aside as not providing any reliable philosophical implications that might prove relevant for reconstructing Plato’s philosophy6. 3 G. Vlastos thinks that the dialogue is an example of ‘honest perplexities’; see “The Third Man Argument in the Parmenides.” In Studies in Plato’s Metaphysics, ed. R. E. Allen, (London: Routledge & Kegan Paul, 1965), 231–63. 4 G. E. L. Owen considers the dialogue as an example of sophistry. See Owen, G. E. L. “Notes on Ryle’s Plato.” In Logic, Science and Dialectic, (London: Duckworth, 1986), 85–103.