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

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

Expressions in Part I, which are given with superscript single quotation marks ('...'), actually belong to a language constructed in Part II (§ 8.2) for the presentation of Plato's ontology in a frame theory.

The quotation of Greek text (single words, syntagmata, sentences and sentence periods) is usually not indicated by quotation marks (...) in order to relieve the type-face.

Quotations from German literature are translated into English.

When is talk of numbers, always numbers of the sequence 1, 2, 3, 4, ... are meant.

Line is always to be understood as straight line, except in § 4.7.2 (p.39-41).

ἐπιστήμη, a central word, is not translated uniformly, but depending on the aspect as cognition or knowledge (cognition = result of recognizing).

idea (beginning with a small letter) is to be understood in the ordinary, non-philosophical sense, whereas Idea (beginning with a capital letter) in the philosophical, epistemological sense.

On the subject concept: If, for example, there is talk of the concept of beautiful (in itself), it consistently refers to a certain entity belonging to the beautiful, assigned to it (possessive genitive), and thus does not indicate that the beautiful is to be understood as a concept (appositive genitive).

In the case of nouns such as being-given, being is to be understood as a gerund and not as a participle.

Part I

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

§ 0 Preliminary Remarks on the Subject of Property and Concept

In § 1 – 4 and in § 6 there is (more or less still kept vague) talk of concept, also of property. What exactly is meant by property and by concept is given in Part II (§ 8.3) within the frame of a sketchy model of Plato's ontology after some thematic preparation. Part II could easily be anticipating read up to the determination of concept given there, since only afterwards, especially when dealing with how the equivalence of property expressions, also of relation expressions, leads to the equality of single-digit concepts, is reference made to Part I.

However, a brief introduction to the subject of property and of concept is already given here, so that for a first understanding, when property or concept is mentioned in Part I, Part II does not need to be read in advance:

The basis for the determination of concept is (interpreting Plato) the view that (1) a property expression (e.g. 'X is beautiful') denotes an ideal entity, called a 'property', where different property expressions can denote the same property (e.g. the property expressions 'X is spruce' and 'X is red fir'), and (2) objects (of the perception world, but also ideal ones) participate in the properties – in the sense of Plato's participation relationship. Properties can be simultaneous, i.e. the same objects participate in them (the properties belong to the same objects), but without being the same, e.g. the property 'is equilateral triangle' and the property 'is equiangular triangle'. The totality of properties that are simultaneous to each other is defined as concept. The intuitive, vague sense of totality of simultaneous properties can be specified, based on the participation relation, in such a way that the totality of properties simultaneous to a given property A is to be understood as a certain ideal object, which is called the concept of A (see p.→). Moreover, the concept of a property A has an essential ideal-theoretical function: it brings about the participation of objects in property A (see § 10).

Terminology: In the property expression 'X is beautiful', the corresponding property is named: the property 'is beautiful', the beautiful (in itself). The corresponding concept is called: the concept of the property 'is beautiful', the concept of beautiful (in itself), the concept 'beautiful', the beautiful itself. Analogous to the property expression 'X is dynamis', etc.

§ 1 Overview

The main subject of Part I is the dialogue section 147c7 – 148d7, in particular the part 147c7 – 148b2, hereinafter referred to as Dynamis passage. In it for the first time, the irrationality of the square roots of the non-square numbers (2, 3, 5, 6, 7, 8, 10, 11, etc.) is discussed.¹ The passage is dealt with in § 1 (only briefly) – § 5. The context of the passage is dealt with in § 1 (only briefly), § 2 (only briefly), § 4 and § 6.

In the initial dialog course, the passage has two functions:

(1) On the one hand, it is intended to show that Theaetetus has understood how Socrates would like a concept to be determined, by specifying a mathematical property expression 'A(X)'² under the concept of which an infinite number of entities fall, which therefore cannot be given by an enumeration that remains eo ipso finite.

Theaetetus's first answer, in 146c-d, to the question of what actually/basically ἐπι-στήμη (itself) is – this is the fundamental question of the dialogue, first posed in 145e, posed next, but fallaciously formulated in 146c – consisted (as known) in a list of various ἐπιστῆμαι, which was criticized by Socrates.³ (Theaetetus was led to this enumeration answer by the fallacious form of Socrates' question in 146c, where it says: what does ἐπιστήμη seem to you to be? For the fallacious nature of this form of questioning when for a concept is asked – in 146c is actually asked for the concept 'ἐπιστήμη', for the ἐπιστήμη itself – see p.→.)

The mathematical property expression 'A(X)' is specified as part of the specification of an all-equivalence 'ⱯX [A(X) ↔ B(X)]' where 'B(X)' is a 'labelling' property expression introduced by examples and by an and so on, and 'A(X)' articulates, so to speak, the so of the and so on; the property expression 'A(X)' results 'inductively inferring from the examples' and denotes a common/identical of these examples,⁴ which should turn out to be the intended common/identical of those objects to which 'B(X)', seen from a transcendent perspective, applies (cf. n.21). In this context, let it be said that the explicandum 'B(X)' is explicated to 'A(X)', let the all-equivalence 'ⱯX [A(X) ↔ B(X)]' be denoted as explicating all-equivalence and 'A(X)' as an explicatum of 'B(X)'.⁵

(2) On the other hand, the passage contains, presents mathematical results (theorems) as an original achievement of Theaetetus. At that two of these theorems also have the form of an all-equivalence 'ⱯX [A(X) ↔ B(X)]'.⁶

For Plato, from the assertion or validity of an all-equivalence 'ⱯX [A(X) ↔ B(X)]' or an all-equivalence 'ⱯXY [A(X,Y) ↔ B(X,Y)]', where 'A(X,Y)' and 'B(X,Y)' are relation expressions,⁷ results the assertion or fact, respectively, that the concepts corresponding to the equivalence members are the same, in the process in the case of all-equivalences with relation expressions, to these may also correspond single-digit concepts.⁸

This is already the case in the determination of σοφία as ἐπιστήμη by Socrates, which immediately precedes his first question, what ἐπιστήμη actually is (145e). In this determination, σοφία and ἐπιστήμη – it stands to reason to see it in this way – are regarded as being the same as a consequence of a rudimentary all-equivalence 'ⱯXY [S(X,Y) ↔ E(X,Y)]', which is regarded as valid.⁹ Analogously, Theaetetus makes his second attempt to determine what ἐπιστήμη actually is (his first determination attempt consisted, as known, of a list of various ἐπιστῆμαι), which is now in methodical terms approved by Socrates: he determines: ἐπιστήμη is nothing more than αἴσθησις, whereby he sees this equality as resulting from a rudimentary all-equivalence 'ⱯXY [E(X,Y) ↔ A(X,Y)]' which he considers valid (151e).¹⁰

If σοφία, ἐπιστήμη and αἴσθησις are understood as concepts within the meaning of § 0 or, more precise, of § 8.3, then it can be precisely stated, axiom-based, how the equality of σοφία and ἐπιστήμη and also that of ἐπιστήμη and αἴσθησις follows from the respectively associated all-equivalence (see above) regarded as valid (see p.→ on this).

§ 2 Text and Translation of the Dynamis Passage (147c7 – 148b2)

The text follows the Oxford edition of 1900 (Burnet 1900), with one exception: there is a comma between προσαγο-ρεύσομεν and τὰς δυνάμεις in the present text. In the 1995 edition of Oxford (Duke 1995), ἀποφαίνων is missing. ἀποφαίνων is not found in Codex T (Venetus), but (as far as I know) in all other codices, as well as in a papyrus with an anonymous commentary on Theaetetus from the first or second century A.D. (Diels/ Schubart 1905). Further information on ἀποφαίνων as a textual occurrence in: Knorr 1975, p.70 with n.33 (p.→).

Occasionally, the translation is 'bumpy'. Namely, when emphasis has been placed on the reproduction of the word order of the Greek text, but correct (good) word order in English cannot be adhered to.

Theaetetus follows on from the example of a concept determination given by Socrates in 147c4-6. In response to the question of what clay actually is (147a1-2, c4-5), Socrates argues, one could arguably explain plainly and simply: clay is earth that is (by nature) moistened and can be kneaded (into a permanent form); It should be added: if one tries to grasp what is common to all earths called clay (in a teaching situation) and only to them (see p.→-→, where is returned to the determination of what clay is).

[147c] ...

Θεαίτητος: ῾Ρᾴδιον, ὦ Σώκρατες, νῦν γε οὕτω φαίνεται· ἀτὰρ κινδυνεύεις ἐρωτᾶν οἷον καὶ αὐτοῖς ἡμῖν ἔναγχος [d] εἰσῆσλθε διαλεγομένοις, ἐμοί τε καὶ τῷ σῷ ὁμω-νύμῳ τούτῳ Σωκράτει.

[147c] ...

Theaetetus: The matter now seems to be easy, Socrates (sc. to determine a concept). You well seem to be asking for something as it came into our minds recently in our reflections, me and your namesake, Socrates here.

Σωκράτης: Τὸ ποῖον δή, ὦ Θεαίτητε;

Socrates: What kind of thing (matter) was that, Theaetetus?

ΘΕ: Περὶ δυνάμεών τι ἡμῖν Θεόδωρος ὅδε ἔγραφε, τῆς τε τρίποδος πέρι καὶ πεντέ-ποδος [ἀποφαίνων] ὅτι μήκει οὐ σύμμετροι τῇ ποδιαίᾳ, καὶ οὕτω κατὰ μίαν ἑκάσ-την προαιρούμενος μέχρι τῆς ἑπτακαιδεκάποδος· ἐν δὲ ταύτῃ πως ἐνέσχετο. ἡμῖν οὖν εἰσῆλθέ τι τοιοῦτον, ἐπειδὴ ἄπειροι τὸ πλῆθος αἱ δυνάμεις ἐφαίνοντο, πειρα-θῆναι συλλαβεῖν εἰς [e] ἕν, ὅτῳ πάσας ταύτας προσαγορεύσομεν, τὰς δυνάμεις.

TH: About the dynameis Theodorus demonstrated us something, namely, of the three-footed and five-footed he showed (pointed out), that they are by length not commensurable with the one-footed straight line;¹¹ and so he took each one by one, up to the seventeen-footed; in this he somehow (without any particular reason) stopped.¹² The following now came to our minds, since the dynameis are evidently infinitely many, to try to grasp into one by which we can all 'say' (specify/determine/characterize)¹³ them, the dynameis.

ΣΩ: Ἦ καὶ ηὕρετέ τι τοιοῦτον;

SO: And did you find something like that?

ΘΕ: Ἔμοιγε δοκοῦμεν· σκόπει δὲ καὶ σύ.

TH: We seem to me indeed . But you too should examine !

ΣΩ: Λέγε.

SO: Tell me!

ΘΕ: Τὸν ἀριθμὸν πάντα δίχα διελάβομεν· τὸν μὲν δυνάμενον ἴσον ἰσάκις γίγνεσ-θαι τῷ τετραγώνῳ τὸ σχῆμα ἀπεικάσαντες τετράγωνόν τε καὶ ἰσόπλευρον προσ-είπομεν.

TH: We divided the number totality into two classes: the numbers, which are capable of being formed as equal-times-equal, we compared with the square in shape and called them quadratic or equilateral .

ΣΩ: Καὶ εὖ γε.

SO: Very good!

ΘΕ: Τὸν τοίνυν μεταξὺ τούτου, ὧν καὶ τὰ τρία καὶ [148a] τὰ πέντε καὶ πᾶς ὃς ἀδύνατος ἴσος ἰσάκις γενέσθαι, ἀλλ᾿ ἢ πλείων ἐλαττονάκις ἢ ἐλάττων πλεονάκις γίγνεται, μείζων δὲ καὶ ἐλάττων ἀεὶ πλευρὰ αὐτὸν περιλαμβάνει, τῷ προμήκει αὖ σχήματι ἀπεικάσαντες προμήκη ἀριθμὸν ἐκαλέσαμεν.

TH: Now the numbers between these, to which belong the # three, the # five, and every number which cannot have been formed as equal-times-equal, but which is formed either as fewer-times-more or as more-times-fewer, i.e. a larger and smaller side always comprise it, we compared with the rectangle, again in shape, and called them rectangular numbers.

ΣΩ: Κάλλιστα. ἀλλὰ τί τὸ μετὰ τοῦτο;

SO: Very nice! But what (how) next?

ΘΕ: Ὅσαι μὲν γραμμαὶ τὸν ἰσόπλευρον καὶ ἐπίπεδον ἀριθμὸν τετραγωνίζουσι, μῆκος ὡρισάμεθα, ὅσαι δὲ [b] τὸν ἑτερομήκη, δυνάμεις, ὡς μήκει μὲν οὐ συμμέτ-ρους ἐκείναις, τοῖς δ᾽ἐπιπέδοις ἃ δύνανται. καὶ περὶ τὰ στερεὰ ἄλλο τοιοῦτον.

TH: We determined the straight lines which square the equilateral and plane number (in each case) as length,¹⁴ but which the different-sided as the dynameis, as by length not commensurable with those, but by the surfaces which (sent. object) they (sent. subject, sc. the straight lines of both kinds combined) are able .¹⁵ And concerning the room sizes, there is a corresponding state of affairs.

§ 3 Interpretation at a Glance

In the Dynamis passage, three states of affairs are interwoven:

(1) An explication of the property expression 'X is dynamis' by Theaetetus, in the process the dynameis have been introduced by Theodorus only exemplarily and by an and so on.

(2) Formulations of mathematical results achieved by Theaetetus (as a student of Theodorus).

(3) A proof of the adequacy of the explication of the property expression 'X is dynamis'.

To (1):

The dynameis may be seen, in the frame of the lesson as sketched by Theaetetus, introduced by Theodorus something like this:

Let's look at the straight line that squares the 2, the one that squares the 3, the one that squares the 5, the one that squares the 6, the one that squares the 7, the one