On Plato's Ontology and on Plato's Theaetetus (first Part, the math. Dynameis) E-Book

Peter Georgi

• Herausgeber: BoD - Books on Demand
• Kategorie: Geisteswissenschaft
• Sprache: Englisch

Seitenzahl: 325

Veröffentlichungsjahr: 2024

Beschreibung

The Ontology part of the book is shown first in the title because of its more general, weightier meaning; but it has emerged from the Theaetetus part and is thus found after it. Both parts of the book can be read largely independently of each other. On the Theaetetus part: The dialogue Theaetetus is dedicated to the question: Knowledge - what is it actually? In the dialogue, it is problematized how the concept of something at all, so also that of knowledge, can be determined. The 'famous' dynamis passage plays an essential role in this. To the passage substantial new issues are shown. In addition, there is a new perspective on the attempts in the initial dialogue part to determine what knowledge is. On the Ontology part: Here, starting from the dialogue Phaedo, a model of Plato's ontology is developed with provided means of mathematical logic. The model, in particular his version of concept, enables (to the author's knowledge) a partially new understanding of Plato's so-called theory of ideas.

Leseprobe

Πάντες ἄνθρωποι τοῦ εἰδέναι ὀρέγονται φύσει.

All humans strive for knowledge by nature.

(Aristotle, Metaphysics 980a21)

Il y a assurément un autre monde, mais il est dans celui-ci.

There is certainly another world, but it is in this one.

(Paul Éluard)

Introduction

The present book is a translation of the German book: Zu Platons Ontologie und zu Platons Theaitetos (erster Teil, die math. Dynameis), third edition, published in 2025; in the translation, a correction of a note on Carnap was made.

(Because of my maybe sometimes 'unconventional' English I ask for indulgence.)

Regarding the title of the book and its structure: In the title, "On Plato's Ontology" is mentioned as the more general and important topic before the more specific topic "On Plato's Theaetetus (first part, the math. Dynameis)"; but in the treatment of the topics the order is reversed, according to the genesis of the ontology part of the book: it is – with the intention of clarifying what is meant by a concept – emerged from the Theaetetus part. Both parts of the book can be read largely independently of each other.

I On Plato's Theaetetus: The mathematical Dynameis, the first Part of the Dialogue

The matter of the mathematical dynameis in the opening part of Plato's dialogue Theaetetus is of particular importance (a) with regard to the issue of how a concept (in dialogue especially that of knowledge) is to be determined – which essentially brings Plato's ontology into focus – and no less (b) with regard to the history of mathematics.

The question of how the ("famous" named) mathematical Dynamis passage is to be understood is a much-discussed topos of the Theaetetus interpretation; this question entails the question of how the passage is thematically related to the context (the initial course of the dialogue) or to the central problem of the dialogue, namely to determine what knowledge actually is, in the process the response to this follow-up question has mostly been neglected. The efforts to adequately understand the Dynamis passage or also its function in the initial course of the dialogue have a long history. Perhaps soon, perhaps as soon as Plato himself or his immediate disciples could no longer be consulted, a need for interpretation arose. The first evidence of such a need is an anonymous commentary on Theaetetus from the first or second century A.D. And especially in modern times (since about 1900), the above two questions (the first more, the second less) are the subject of discussion.

A definitive understanding of the Dynamis passage in every respect is arguably hardly achievable; for example, it does not seem to be possible to clarify definitively why just the word dynamis is used to designate certain square sides (or squares, as other interpreters think), even if a possible origin of the term dynamis is shown here.

However, the present work endeavours to achieve an essential gain in understanding of the Dynamis passage – whose subject is primarily a concept determination and only secondarily a summary presentation of Theaetetus' mathematical achievements – and its context; in the process results, in particular, also an essentially new perspective (as far as I know) on the attempts made in the initial course of the dialogue to determine what knowledge actually is (the dialogue is, as known, dedicated essentially to this question).

II On Plato's Ontology

Since Plato's ontology is also essentially addressed in the topic of Part I, a model for this is developed in Part hII. In this model, in addition to the things of the perception world, there are the 'otherworldly', ideal things. These are assigned to property expressions (propositional forms with exactly one free variable), in the process only one ideal object is assigned to such a property expression, and are therefore also called properties. The same property is assigned to certain property expressions, e.g. to those which are equal in meaning/sense (the problem of their determination is only hinted at). In the assignment the property expressions and the properties assigned to them have a certain (called congruence) relationship to each other: an arbitrary object participates (in the Platonic sense) in the property assigned to a property expression if and only if the object fulfils the property expression (in a well-defined sense). Concepts are now special properties (which are assigned to certain equally constructed property expressions): For every ideal object A there is exactly one property B, called concept (of A), so that B is the totality (in a well-defined sense) of all the ideal objects in which the same objects participate as in A. Text passages / formulations suggest that Plato in the course of his Idea-theoretical considerations had these concepts in mind, albeit in a still vague, undeveloped way. In addition, the concept of a property A has an essential Idea-theoretical function: it brings about the participation of objects in property A.

In the modified/extended version of the model (§ 21), one has analogous states of affairs for relation expressions, relations and concepts of relations as for the property expressions, properties and concepts of properties; here, however, it should be noted in particular: an ideal object is assigned either to a property expression or to a relation expression, concepts of relations remain properties.

For the understanding of Part II (among other basic knowledge of logic), familiarity with the recursive definition of the validity of an object-language proposition (based on Tarski's definition), as found in textbooks of (mathematical) logic, is advantageous – but this is not assumed; therefore, the said definition (in § 8.2), modified for the intentions of Part II, is presented in as much detail as necessary for them.

Preliminary Remarks of a technical Nature