The mathematical passage at Tht. 147d–148b is an important piece of evidence for the reconstruction of pre-Euclidean mathematics, in particular the theory of irrational lines. The literature on it is huge.Footnote 1 The passage comprises a concise description of ‘Theodorus’ lesson’ on lines that are incommensurable with a reference line, and Theaetetus’ attempt at defining some of the involved mathematical objects. I would like to point out some features of this attempt, most of which have gone apparently unnoticed until now, and draw a conclusion on the ability of Theaetetus as a definition-maker. The features are as follows:
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Unless hermeneutic contortions are allowed, the δυνάμεις at 147d3–6 are indisputably plane regions (see the specimina named at 147d4: the repeated preposition περὶ … πέρι is crucial),Footnote 2 those defined at 148a7–b1 are indisputably (straight) lines (they are defined as a subset of the γραμμαί at 148a6);Footnote 3
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hence, either (if the previous statement is conceded) the defined δυνάμεις (lines) carry the name already assigned to a structurally different mathematical object (a plane region), the latter designation being endorsed just before the definition is given,Footnote 4 or (if it is not conceded, that is, if the δυνάμεις are lines throughout) they are designated by the name they already had, which makes one wonder whether there is anything to be defined here, or (again if it is not) they are designated by the genus name of the elements of a larger set, which includes the lines that ‘square’ square numbers (148a6–7)—one wonders even more;Footnote 5
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consequently (if it is conceded), either the relational predicate μήκει οὐ σύμμετρος is first applied to plane regions at 147d4–5 and then to lines at 148b1 (which entails that the extension of the focus-pointer determiner μήκει is not the same in the two cases), or, worse still, in both cases (and indisputably at 148b1–2 δυνάμεις … συμμέτρους … τοῖς ἐπιπέδοις ἃ δύνανται) the two slots of the commensurability relation are occupied by non-homogeneous mathematical objects, namely, a plane region and a line.
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Any number in one of the two classes of numbers upon which the definition of δυνάμεις hinges is defined as προμήκης, but two lines later, within the definition of δυνάμεις, it is referred to as ἑτερομήκης:Footnote 6 consequently, the definition of δυνάμεις is empty because the latter term has no referent;
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any number in the complementary class of numbers is defined as τετράγωνός τε καὶ ἰσόπλευρος, but four lines later, within the definition of μῆκος, it is referred to as ἰσόπλευρος καὶ ἐπίπεδος: consequently, the definition of μῆκος is empty because the latter term has no referent.
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The composite determiner ἰσόπλευρος καὶ ἐπίπεδος contains an adjective, namely, ἐπίπεδος, that is used in the characterization of the lines in the complementary class (τοῖς ἐπιπέδοις ἃ δύνανται at 148b1–2);
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in the composite determiner τετράγωνός τε καὶ ἰσόπλευρος, if τετράγωνος means ‘square’, as is suggested by the expression τῷ τετραγώνῳ τὸ σχῆμα ἀπεικάσαντες at 147e6–7, ἰσόπλευρος is redundant;
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likewise, the three-tier characterization of non-square numbers at 148a1–3 is redundant, for any of the three disjoints would have sufficed.
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The definition of δυνάμεις contains a form of the verb δύναμαι: hence, the definition is either circular or based on the meaning of an operation that is not defined, but whose designation obviously interferes with the term to be defined;
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if the definition is not circular, necessarily the operation also applies to plane regions ‘squared’ by the lines in the complementary class, for any line δύναται a suitable region, as is borne out by τοῖς ἐπιπέδοις ἃ δύνανται at 148b1–2;
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accordingly, as the two horns of the definition are perfectly analogous, δύναμαι and τετραγωνίζω are synonyms, which introduces a further redundancy;
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a line may τετραγωνίζειν a surface only, not a number, contrary to what is stated at 148a6–7 (compare 148b2); a reference to a unit of measurement (like the ποδιαία at 147d5, be it a plane region or a line) is missing;
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a sustained pun involving the verb δύναμαι (also at 147e7 and 148b6) and the adjective ἀδύνατος (148a1) pervades the passage.
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In the definition of the first class of lines, the term μήκη (pl.) should be used, not μῆκος (sing.);
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the key term of the definiens of one of the two classes of lines, namely προμήκης, is an adjective whose stem coincides with the term that designates the other class of lines, namely, μῆκος.
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Contrary to the final claim at 148b2, there is hardly room for an analogous (τοιοῦτον) definition in the case of solids because the partition of numbers at 147e5–148a4 is exclusive and exhaustive, and hence the related classification of the relevant lines referred to a standard length is, too: lines that τετραγωνίζουσι resp. κυβίζουσι a given number would belong to different—depending on the classification used—yet identically named classes; were not the classes identically named, the confusion would be even greater.Footnote 7
These remarks show that Theaetetus’ definition of δυνάμεις is inadequate as to clarity, conciseness, lack of ambiguity and pertinence. Since the goal of the passage is exactly and explicitly to present someone as perfectly fit for definition-making, the well-known hermeneutic commodity of Plato’s stylistic aims cannot fix any of the above-listed quirks. These cannot be fixed by invoking a comparable relaxation in other Platonic passages in which definitions are proposed,Footnote 8 or by invoking a presumed relaxation in a not-yet stabilized mathematical lexicon.Footnote 9 I conclude that Plato introduces Theaetetus as an inept definition-maker,Footnote 10 thereby preparing for the former being repeatedly savaged by Socrates in the much subtler powergame of defining knowledge.
