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

1. Auflage 2024
ISBN: 978-3-7693-9083-4
Verlag: BoD - Books on Demand
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 148 Seiten

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. A reasoned new view of the passage is 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.

The author studied philosophy, mathematics and ancient Greek and has previously published (in German) on mathematics and philosophy in early Greek antiquity and on Plato's dialogues Phaedo and Timaeus.

Autoren/Hrsg.

Georgi, Peter

Leseproben

Part II On Plato's Ontology
§ 7 Introductory (Thesis)
Main thesis, in brief: With a certain degree of probability Plato had – not everywhere but in certain passages where the ?d?a, the µ??f?, the e?d?? of a concrete property (an a?t?-entity) or the µ??f? of any property is mentioned – approximately in mind, without being able "to capture" it sufficiently "in a logos",109 what is defined here (in § 8.3) as concept.110 To this end, it should be explained how in particular elementary mathematical considerations, as they were certainly also known to Plato, suggest arriving at this definition. Unconnected with a genitive noun (= GN), ?d?a, e?d?? (µ??f? plays no role here, see n.162) has the meaning: property in general (good-, beautiful-, just-being, etc.). Connected with a GN that denotes a property, however – what is essential to note in a Plato reception –, two meanings are possible: (a) in the sense of an appositive genitive, the expression ?d?a/e?d?? + GN (µ??f? plays no role here) only emphasizes that the entity denoted by the GN is a property/a?t?-entity, (b) in the sense of a possessive genitive, ?d?a/µ??f?/e?d?? + GN refers to the concept of the property designated by the GN, whereby concept is defined here as the totality (specified in the def.) of all the properties in which the same objects participate as in the property designated by the GN. If Plato should not have had said definition and the conceptions in its context (e.g. property expression) in mind or only subsignificantly (no doubt he had them not formulated explicitly/technically as in the following in mind), then the presentation in this part arguably nevertheless (though perhaps not in every respect) offers a plausible and instructive model for Plato's ontology; synonymous with this (and more frequently in the literature) is spoken of Plato's theory of Ideas.111 Three notes for the following: • Axioms (postulates, demands, basic assumptions) and other important points are preceded by "(*...)", where "..." stands for e.g. "Cm", "Co", "A1", "An". • A distinction must be made between the beautiful in itself (property), the beautiful itself (concept), the good in itself, the good itself, etc. • A distinction must be made between participating of an object in a property and partaking of an object in a concept, even though both are 'almost the same'. § 8 A Model to Plato's Ontology (Property, Concept)
The following is a formal sketch of Plato's ontology in the form of a model, based on "the first clear outline of his ontology ... in the Phaedo".112 This sketch is presented in a frame theory, which has much of a set theory, although this is not explicitly mentioned (as is usually the case with representations of formal/mathematical logic), and is structured in paragraphs § 8.1 – § 8.3. The frame theory is about things and a relationship (governed by axioms) ? that exists or does not exist between two things. If a thing A is in relationship ? to a thing B, let it be written: A ? B, or: A is an element of B, or in short: A is B. § 8.1 Universe, Objects, u-Properties, u-Relations, Universe structure
(1) Let a thing U be given as an infinite totality of certain things, i.e. the certain things (see p.? on the division of U), and only they are elements of U. U is called universe (in the sense of a universe of discourse). If a thing A is an element of U, then A is called an object (of the universe) or sometimes an entity. A thing T is called a subtotality (of U) if we have for every thing A: if A is an element of T, then A is also an element of U (A ? T ? A ? U). An element of a subtotality is therefore always also an object (of the universe). (2) Let be given, as things of the frame theory, infinitely many subtotalities of U; these are called u-properties; they are well to be distinguished from the objects introduced in § 8.3 as properties, although there is a connection between the two (see p.?, 1212). Two of these u-properties form a division of the universe, the one is the subtotality of the perceptible objects, the other the subtotality of the ideal (otherworldly) objects. The ideal objects are characterized in opposition to the perceptible objects (see Phd. 78c-9d) as: imperceptible to the senses, unchanging, always being.113 They include not only entities such as the beautiful (in itself), the good (in itself), the just (in itself), etc. – for the designation with "in itself" see p.? – but also the ideal mathematical objects of an arithmetic or geometrical nature; these stand in contrast to the mathematical objects of the perception world, which are embodiments (concretizations) of the former.114 Furthermore, as u-properties one has about the subtotality of the beautiful objects, good objects, red objects, of men, lines, triangles, etc., these u-properties are designated in turn with: "beautiful", "good", "red", "man", "line", "triangle", etc. There may be different u-properties with the same elements (if one had in the frame theory as extensionality axiom: u-properties with the same elements are equal, this would not be possible), see to this p.?. (3) Let there be given, as things of the frame theory, infinitely many relationships which exist between objects of the universe U; these are called u-relations; such one is called n-digit (n = 2) if it exists between n objects; the u-relations are well to be distinguished from the objects introduced in § 19 as relations, although there is a connection between the two (see p.?). (A note on relationships: These generally exist between things of frame theory. For example, the fulfilment relationship introduced in § 8.2 exists between objects and expressions. Relationships are well to be distinguished from the objects called in § 19 relations.) Of the u-relations, the 2-digit u-relation of the so-called participation, which is frame-theoritcally also called the participation relationship, should be emphasized; if one has for two objects A, B: A participates in B (A has part in B, A shares in B), let it be written: A e B, for frame-theoretically explicitly: ? e. If in each case between n objects (n = 2) given in a sequence exist simultaneously an n-digit u-relation A and an n-digit u-relation B (A and B apply to the same n-tuples of objects), then A and B (as things of the frame theory) are not necessarily equal (which could not be the case if one had an extensionality axiom for the u-relations in the frame theory). Some in connection with the participation relationship, where A is an object: • Definition: A is object-containing/non-empty :### ?B [B e A], A is empty :### A is not object-containing. • It is required: (*Ne) A is not empty ### A is an ideal object. This provides (arguably in the sense of Plato) a further characterisation of the ideal objects. The non-ideal objects, the objects of the perception world, are therefore empty. • If an object has a part in an object, then A is an object of the perception world or an ideal object; examples of the latter case: the trinity in itself, which participates in the number in itself, an ideal circle of a certain size, which participates in the circle in itself. (4) The universe, together with the given u-properties and u-relations, forms a structure, in the sense of mathematical logic, called the universe structure; it consists, roughly speaking, of a listing of U, the u-properties, and the u-relations (in this order); the u-properties and u-relations are collectively referred to as u-attributes. § 8.2 Syntax and Semantics of a Language on the Universe structure Things of the frame theory function as signs of the language on the universe structure (these signs are noted in bold below): • there are so-called o-signs to U, where to every o-sign A belongs exactly one object A and every object belongs to an o-sign; it is OS := the totality of all o-signs to U • there are so-called p-signs, where to every p-sign belongs exactly one u-property, different u-properties belong to different p-signs, and each u-property belongs to a p-sign; e.g. the u-property beautiful belongs to the p-sign beautiful • there are so-called n-digit (n = 2) r-signs, where to every n-digit r-sign belongs exactly one u-relation, namely an n-digit one, different u-relations belong to different r-signs, and each u-relation belongs to an r-sign; e.g. to the 2-digit r-sign e belongs the...