Abstract
The present paper aspires to explain fully both the supreme importance of Geometry for Plato, and also the nature of the serious ongoing criticism that Plato (and the Academy) directs against the geometers, an explanation that has eluded modern scholars of Plato (since M. Ficino in the fifteenth century to our present day). In order to understand the criticism, it is necessary first to have a true understanding of the nature of Plato’s philosophy. The most crucial concept in understanding Plato’s philosophy, and essentially the only one, is the geometrical concept of periodic anthyphairesis of two magnitudes, say line segments, established with the Logos criterion (Sect. 1.2).
The Platonic true Being, the intelligible Platonic Idea, is a dyad of opposite parts in the philosophic analogue, Division and Logos-Collection , in close imitation of periodic anthyphairesis . Plato in effect isolates a method for acquiring full and complete knowledge, as it exists in a small but vital part of Geometry, namely in incommensurability by periodic anthyphairesis , and develops a general theory of knowledge, Division and Collection , of the Platonic Ideas, in close imitation to the complete acquisition of knowledge provided by the Logos criterion in periodic anthyphairesis .
The anthyphairetic nature of Platonic intelligible Beings was examined in detail by the author in earlier publications and is outlined in the present paper: the One of the second hypothesis of the Parmenides and its close relation to Zeno’s arguments and paradoxes (outlined in Sect. 1.3) and the Division and Collection in the Sophistes and Politicus, where the genera and kinds in the Division are considered as hypotheses (outlined in Sect. 1.5). Furthermore, we establish in the present paper that the dialectics of the Politeia coincide with Division and Collection -Logos (Sect. 1.4), described also equivalently in analysis fashion as ascent from hypotheses to the “anhupotheton ”, the hypothesis-free (531d-e, 532a-b,d-e, 534b-d) (Sect. 1.6).
Without understanding the anthyphairetic nature of Plato’s dialectics it is impossible to understand Plato’s praise of Geometry and criticism of its practice by geometers.
- Criticism I::
-
the rejection by Plato of the indivisible geometric point in favor of the “indivisible line”, according to Aristotle’s Metaphysics 992a, 19–22, coincides with the rejection of the One of the first hypothesis as a true Being, a One similar to the geometric point, and the adoption of the One of the second hypothesis as a true Being, a One described by Zeno, Plato, and Xenocrates in terms akin to those of the indivisible line, in the Parmenides (Sect. 1.7).
- Criticism II::
-
the rejection by Plato of the use of hypotheses (namely basic definitions and postulates on lines, circles, angles in geometry, on units and numbers in Arithmetic) as principles, and not as stepping stones towards the true Being, the anhupotheton . The rejection of the axiomatic method on epistemological grounds, since hypotheses, namely definitions and postulates, are arbitrarily accepted and hence these, with all its consequences, cannot be known; knowledge, by Division and Collection , is achieved only when the generation of these hypotheses, the basic geometric (lines, circles, angles) and arithmetical (units, numbers) concepts and their Postulates, takes place within the Platonic true Beings, something possible because of their periodic anthyphairetic structure in the Politeia(510–511, 527a-b) (Sects. 1.8, 1.9, 1.11, and 1.12).
- Praise of Geometry::
-
Plato makes the extraordinary claim that the method of Division and Collection must be the method employed for the acquisition of true knowledge for all of Geometry (in place of Euclidean axiomatics). With this method he constructs the numbers, the straight line, and the circle (Sect. 1.10), and the three kinds of angle (Sect. 1.11).
- Criticism III::
-
the use of geometric diagrams is rejected by Plato, not simply because they are visible/sensible, but because they are sensible representations not provoking to the mind, as they should be, if they were represented as true sensibles, participating anthyphairetically in the intelligible by means of the receptacle/diakena, as presented analytically in the Timaeus 48a-58c and in preliminary manner in the Politeia 522e–525a. The geometers fall into this faulty use of geometric diagrams, as a result of their “dianoia” way of constructing their arguments (Criticism II) (Sects. 1.13 and 1.14).
- Criticism IV::
-
the rejection by Plato [1] of Archytas’ non-anthyphairetic proofs of quadratic and cubic incommensurabilities (based on the arithmetical Book VIII of the Elements and eventually on the arithmetical Proposition VII.27 of the Elements), possibly expressed in the distinction between the eristic and dialectic way of going to infinity in the Philebus 16d-e, replacing Theaetetus’ anthyphairetic proofs of quadratic incommensurabilities, [2] of Eudoxus’ Dedekind-like theory of ratios of magnitudes (Book V of the Elements), expressed clearly in the second part of Scholion In Euclidem X.2, replacing Theaetetus’ anthyphairetic one, and thus moving away from Plato’s philosophy based on periodic anthyphairesis, and [3] of Archytas’ non-anthyphaireic stereometric solution of the problem of the duplication of the cube, in the Politeia 527d-528e (Sect. 1.15).
All of Plato’s criticisms of the geometers have a common strain: the practice of geometers is distancing away from periodic anthyphairesis and from Platonic true Beings, based on periodic anthyphairesis .
θεὰ σκέδασ΄ ἠέρα, εἴσατο δὲ χθών·Odusseia, Book XIII, line 352[G]oddess [Athena] dispersed the mist, and the land was recognized.
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Notes
- 1.
Joannes Philoponus, In Aristotelis libros de anima Commentaria 15,117,26-27 Πυθαγόρειος δὲ ὁ Πλάτων, οὗ καὶ πρὸ τῆς διατριβῆς ἐπεγέγραπτο ‘ἀγεωμέτρητος μὴ εἰσίτω’. Olympiodoros, Prolegomena (tes Logikes) 9,1 ἐπιγεγράφθαι ἐν τῷ τοῦ Πλάτωνος μουσείῳ ‘ἀγεωμέτρητος μηδεὶς εἰσίτω’.
- 2.
“μάλιστα προστακτέον ὅπως οἱ ἐν τῇ καλλιπόλει σοι μηδενὶ τρόπῳ γεωμετρίας ἀφέξονται.”
- 3.
A “magnitude” is used to render “megethos” in the Elements, meaning line, surface, or volume, and in opposition to ‘number’ (“arithmos”), which is always a natural number starting with 1 (or in fact with 2, as strictly speaking 1 is the unit generating all the numbers).
- 4.
Negrepontis [23].
- 5.
“ΣΩ. ῎Ιθι δή—καλῶς γὰρ ἄρτι ὑφηγήσω—πειρῶ μιμούμενος τὴν περὶ τῶν δυνάμεων ἀπόκρισιν, ὥσπερ ταύτας πολλὰς οὔσας ἑνὶ εἴδει περιέλαβες, οὕτω καὶ τὰς πολλὰς ἐπιστήμας ἑνὶ λόγῳ προσειπεῖν.” Theaetetus 148d4-7.
- 6.
Negrepontis [24]. That Plato was greatly influenced by Zeno is attested by the comparison of Zeno’s Fragments and paradoxes with Plato’s Parmenides.
- 7.
- 8.
In Euclidem 211,18–22.
- 9.
More details can be found in Negrepontis and Lamprinidis [26].
- 10.
Aristotle in Analytics, 72b5-73a5 mentions some unnamed, avoiding to name Plato explicitly, who have such an objection to axiomatisation (“Ενίοις μὲν οὖν διὰ τὸ δεῖν τὰ πρῶτα ἐπίστασθαι οὐ δοκεῖ ἐπιστήμη εἶναι”). Aristotle rejects this objection, because it would lead, by introducing higher and higher axioms, to a regress to infinity, something impossible (“ἀδύνατον γὰρ τὰ ἄπειρα διελθεῖν΄). Cf. also “`εἰ γὰρ εἰσὶν ἀρχαί, οὔτε πάντ΄ ἀποδεικτὰ οὔτ΄ εἰς ἄπειρον οἷόν τε βαδίζειν” 84a32-33; “ἔτι αἱ ἀρχαὶ τῶν ἀποδείξεων ὁρισμοί, ὧν ὅτι οὐκ ἔσονται ἀποδείξεις δέδεικται πρότερον—ἢ ἔσονται αἱ ἀρχαὶ ἀποδεικταὶ καὶ τῶν ἀρχῶν ἀρχαί, καὶ τοῦτ΄ εἰς ἄπειρον βαδιεῖται, ἢ τὰ πρῶτα ὁρισμοὶ ἔσονται ἀναπόδεικτοι.” 90b24-27.
- 11.
- 12.
Most Platonists fail to associate the Politeia dialectics with Division and Collection . Stenzel [31] had rejected the association, while Cornford [8] and Hare [12] do associate Politeia dialectics with Division and Collection; but again do not have an anthyphairetic interpretation of Division and Collection . Furthermore the anthyphairetic interpretation of Logos in the Parmenides, Sophistes-Politicus and Politeia passages has not, to the best of my knowledge, been considered by any modern student of Plato.
- 13.
Cf. “he must not desist till (‘me proaphistasthai prin’, 285b1-2)” in the description of Division and Collection in the Politicus 285a4-b6.
- 14.
In this passage the idea of the Good is, like any other true Being (“hosautos”), knowable by the method of Division and Collection. It is then difficult to reconcile its description as “epekeina ousias”, beyond Being, in the Politeia 509b. Perhaps Plato regarded, apparently only in the Politeia, the Idea of the Good as the supreme true Being, in the senses of an intelligible analogue of the “apokatastatikos arithmos” for the visible universe, according to which there is a periodic restoration of the cosmic cycle (cf. Proclus, Eis Politeian 2, 15–19): when the (anthyphairetic) period of the Idea of the Good is completed, then the (anthyphairetic) period of every true Being will be completed as well, and there will be universal restoration.
- 15.
Cf. the comment in Sect. 1.8.3.2 about the Phaedrus 264b3-c5 passage.
- 16.
The second advantage of the abbreviated, tree-like Division will be discussed in Sect. 1.10.2 in connection with the generation of the intelligible line.
- 17.
We leave this crucial term untranslated for the moment, yet uncommitted as to its meaning, and discuss it below in this section.
- 18.
Proclus, In Parmeniden 622,29-623,28 and 655,16-656,2, clearly identifies “hikanon” and “anhupotheton ”. “hikanos” in the Sophistes 221b2 also fits perfectly (cf. Sect. 1.5.2).
- 19.
Thus the second part of Cherniss [6, p. 418, Footnote 54], claiming that “tas hupotheseis anairousa” in the Politeia 533c8 is not a reference to division in kinds, is not correct; furthermore his claim, that Robinson’s [29, p. 171]) notion that “Proclus seems to have understood division as belonging to the upward path.” is mistaken, is also incorrect. Indeed, Proclus, in both In Euclidem 211, 23-27, and Commentary to Cratulus 173,1-5, identifies Division with “anairesis”, and regards Division in the upward path.
- 20.
This identification is explicitly confirmed by Proclus, In Parmeniden 1033,32–35.
- 21.
Cf. ἀλλὰ γὰρ ἡμῖν ἔτι καὶ τοῦτο σκεπτέον, ἆρ’ ἄτομον ἤδη ἐστὶ πᾶν ἤ τινα ἔχον διαίρεσιν ἀξίαν ἐπωνυμίας. Sophistes 229d5-6.
- 22.
ἔτι αἱ στιγμαὶ ἐκ τίνος ἐνυπάρξουσιν; τούτῳ μὲν οὖν τῷ γένει καὶ διεμάχετο Πλάτων ὡς ὄντι γεωμετρικῷ δόγματι, ἀλλ΄ ἐκάλει ἀρχὴν γραμμῆς—τοῦτο δὲ πολλάκις ἐτίθει—τὰς ἀτόμους γραμμάς.
- 23.
Cf. ἀλλὰ γὰρ ἡμῖν ἔτι καὶ τοῦτο σκεπτέον, ἆρ΄ ἄτομον ἤδη ἐστὶ πᾶν ἤ τινα ἔχον διαίρεσιν ἀξίαν ἐπωνυμίας. Sophistes 229d5-6.
- 24.
One may conjecture that the name “indivisible line” was given, by Platonic circles, so as to make the indivisible line (which according to Plato replaces the point in the foundation of Geometry) appear as closely as possible similar to the partless, indivisible point.
- 25.
Morrow [20, p. 79], changes “aulon”–“immaterial” to its opposite “enulon”–“material”, which involves a serious misunderstanding.
- 26.
Def. I.4. A straight line is a line which lies evenly with the points on itself.
- 27.
Def. I.15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
- 28.
Def. I.10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
Def. I.11. An obtuse angle is an angle greater than a right.
Def. I. 12. An acute angle is an angle less than a right angle.
- 29.
Def. VII.1. An unit is that by virtue of which each of the things that exist is called one.
Def. VII.2. A number is a multitude composed of units.
- 30.
Post. I.1.- To draw a straight line from any point to any point.
Post. I.2. To produce a finite straight line continuously in a straight line.
Post. I.3. To describe a circle with any centre and distance.
Post. I.4. That all right angles are equal to one another.
Post. I.5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
There are no arithmetic Postulates.
(Translation of basic definitions and Postulates following Heath [13]).
- 31.
ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ΄ ἀρχὴν πορευομένη ἀλλ’ ἐπὶ τελευτήν (510b4-6); ὑποθέσεσι δ’ ἀναγκαζομένην ψυχὴν χρῆσθαι 511α4 περὶ τὴν ζήτησιν αὐτοῦ, οὐκ ἐπ’ ἀρχὴν ἰοῦσαν, ὡς οὐ δυναμένην τῶν ὑποθέσεων ἀνωτέρω ἐκβαίνειν (511a3-b2).
- 32.
τὸ ὑπὸ τῶν τεχνῶν καλουμένων, αἷς αἱ ὑποθέσεις ἀρχαὶ καὶ διανοίᾳ μὲν ἀναγκάζονται ἀλλὰ μὴ αἰσθήσεσιν αὐτὰ θεᾶσθαι οἱ θεώμενοι 511c6-8, in the passage 511c3-d5, examined in Sect. 1.8.2.
- 33.
“tetragonizein” (squaring): possible by the hypothesis Proposition I.46.
- 34.
“prostithenai” (adding): possible by the hypotheses Postulate 2, Proposition I.2.
- 35.
“parateinein” (extending, stretching): possible by the hypothesis Postulate 5. The usual translation is “applying”, presumably in the sense of application of areas (Propositions VI.28,29, where the term used is “parabalein”), presumably because Plato in Meno 87a5 is using “parateinein” in exactly this sense. But “parateinein” has the principal meaning of “stretching out along”, “extend”, cf. [17, p. 1327].
- 36.
“προστεθῇ”, “τῇ προσκειμένῃ”, “προσκείσθω” (in Proposition II.10).
- 37.
“tetragonizein” (squaring): “τὸ ἀπὸ τῆς ὅλης”, “τὸ ἀπὸ τῆς προσκειμένης”, “τὰ συναμφότερα τετράγωνα”, “τὰ ἀπὸ τῶν ΑΔ, ΔΒ τετράγωνα”, “ἀπὸ τῶν ΑΓ, ΓΔ τετραγώνων” (in Proposition II.10).
- 38.
“ἐκβαλλόμεναι”, “ἐκβεβλήσθωσαν” (in Proposition II.10).
- 39.
Platonists seem to believe that the intelligible straight line is itself a true Being that can somehow be conceived as perfect straight line, without any imperfections of a sensible straight line; but, as we have found, the intelligible straight line is present in every intelligible true Being, and is the manifestation of its anthyphairetic Division.
- 40.
Platonists seem to believe that the intelligible circle is itself a true Being that can somehow be conceived as perfect circle, without any imperfections of a sensible circle (cf. Annas [3, p. 251]: “mathematicians talk about circles and lines, not about the physical diagrams that illustrate them, nor about the unique Form of Circle and Form of Line”); but, as we have found, the intelligible circle is in every intelligible true being, and is the manifestation of its anthyphairetic periodicity and Collection.
- 41.
Since the Pythagoreans used it in the proof of Proposition I.32 of the Elements, according to Eudemus in Proclus, In Euclidem 379,1-18 (in contrapositive form), and in the proof of Proposition II.10 of the Elements, according to Proclus, eis Politeian 2.24-28 (in direct form).
- 42.
Details are given in Negrepontis [22, Section 7].
- 43.
and in modern terms, analogous to the relation between the convergents of a continued fraction and the continued fraction itself.
- 44.
τὰ γὰρ τέτταρα γένη δι’ ἀλλήλων εἰς ἄλληλα ἐφαίνετο πάντα γένεσιν ἔχειν, -οὐκ ὀρθῶς φανταζόμενα· γίγνεται μὲν γὰρ ἐκ τῶν τριγώνων ὧν προῃρήμεθα γένη τέτταρα, τρία μὲν ἐξ ἑνὸς τοῦ τὰς πλευρὰς ἀνίσους ἔχοντος, τὸ δὲ τέταρτον ἓν μόνον ἐκ τοῦ ἰσοσκελοῦς τριγώνου συναρμοσθέν.οὔκουν δυνατὰ πάντα εἰς ἄλληλα διαλυόμενα ἐκ πολλῶν σμικρῶν ὀλίγα μεγάλα καὶ τοὐναντίον γίγνεσθαι, τὰ δὲ τρία οἷόν τε·(Timaeus 54b6-c5).
- 45.
γῆ μὲν συντυγχάνουσα πυρὶ διαλυθεῖσά τε ὑπὸ τῆς ὀξύτητος αὐτοῦ φέροιτ΄ ἄν, εἴτ΄ ἐν αὐτῷ πυρὶ λυθεῖσα εἴτ’ ἐν ἀέρος εἴτ’ ἐν ὕδατος ὄγκῳ τύχοι, μέχριπερ ἂν αὐτῆς πῃ συντυχόντα τὰ μέρη, πάλιν συναρμοσθέντα αὐτὰ αὑτοῖς, γῆ γένοιτο—οὐ γὰρ εἰς ἄλλο γε εἶδος ἔλθοι ποτ΄ ἄν (Timaeus 56d1-6).
- 46.
In retrospect we realize that the reason why Plato takes the half-square triangle a, and the half isopleuron triangle b, is for the purpose of having two right angled triangles with equal hypotenuses.
- 47.
Negrepontis [23].
- 48.
ἔστιν τι πῦρ αὐτὸ ἐφ΄ ἑαυτοῦ καὶ πάντα περὶ ὧν ἀεὶ λέγομεν οὕτως αὐτὰκαθ΄ αὑτὰ ὄντα ἕκαστα. (Timaeus 51b6-52a7).
- 49.
cf. Proclus, eis Timaion 2,49,29-50,2 on the meaning of “kosmous”:
καὶ ταῦτα τῷ τε Πλάτωνι συμφωνότατά ἐστι, νῦν μὲν λέγοντι τὸν οὐρανὸν ἐκ τῶν τεττάρων εἶναι στοιχείων ἀναλογίᾳ συνδεδεμένων καὶ ὅλον τὸν κόσμον συνεστάναι, μικρὸν δὲ ὕστερον τὰ πέντε σχήματα πλάττοντι καὶ πέντε κόσμους ἀποκαλοῦντι (ταῦτα γὰρ καὶ πέμπτην οὐσίαν τῷ οὐρανῷ δίδωσι καὶ τὴν τῶν στοιχείων εἰσάγει τετρακτύν) καὶ τῇ ἀληθείᾳ συνερχόμενα.
- 50.
The revolution (“periodos”) of the All, since it comprehends the Kinds, tightens (“sfiggei”) them all, seeing that it is circular (“kukloteres”) and tends naturally to come together to itself; and thus it suffers not even a single (“outhemian”) hollow space (“kenoteta”) to be left. Wherefore, fire most of all has permeated (“dielekuthen”) all things, and in a second degree air, as it is by nature second in fineness; and so with the rest; for those that have the largest constituent parts have the largest hollow space left in their construction, and those that have the smallest the least. Thus the tightening of the compression (“pileseos sunodos”) forces together the small bodies into the hollow space (“diakena”) of the large. Therefore, when small bodies are placed beside large, and the smaller disintegrate (“diakrinein”) the larger while the larger unite (“sugkrinei”’) the smaller. (58a4-b7).
- 51.
Μέγα μὴν καὶ ὄψις καὶ σμικρὸν ἑώρα, φαμέν, ἀλλ’ οὐ κεχωρισμένον ἀλλὰ συγκεχυμένον τι. Politeia 524c3-4.
- 52.
ἡ δὴ τῆς πιλήσεως σύνοδος τὰ σμικρὰ εἰς τὰ τῶν μεγάλων διάκενασυνωθεῖ. 58b4-5
“pilesis” 58b4-5 compression LSJ 1404,
“sunothein” 53a6, 58b5, force together, compress LSJ 1730.
- 53.
ἡ τοῦ παντὸς περίοδος, ἐπειδὴ συμπεριέλαβεν τὰ γένη, κυκλοτερὴς οὖσα καὶ πρὸς αὑτὴν πεφυκυῖα βούλεσθαι συνιέναι, σφίγγει πάντα 58a4-7
“sphiggein” 58a7 bind tight, tighten LSJ 1741.
- 54.
κενὴν χώραν οὐδεμίαν ἐᾷ λείπεσθαι, 58a7. Note that our rendering of “outhemia” is not as “and thus it suffers no void place to be left” (as translated by W.R.M. Lamb. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1925), but “and thus it suffers not even one single hollow space to be left”, treating the hollow space of B or of A, as unit. Consider a similar meaning of “outhen” in the Parmenides 144c3.
- 55.
νικηθέντα… ὅμοιον τῷ κρατήσαντι γενόμενον, they are defeated and, instead, assume one form similar to the victorious Kind 57b6-7; τὰ δὲ ἀνομοιούμενα ἑκάστοτε ἑαυτοῖς, ἄλλοις δὲ ὁμοιούμενα, those corpuscles which from time to time become dissimilar to themselves and similar to others 57c3-4.
- 56.
τῶν ἐλαττόνων τὰ μείζονα διακρινόντων, 58b6-7.
- 57.
αὐτοῦ σύνοικον μείνῃ, stay dwelling therewith as a united family 57b7.
- 58.
τῶν δὲ μειζόνων ἐκεῖνα [elattona] συγκρινόντων 58b7.
- 59.
Symposium 210e2-211b5.
ὃς γὰρ ἂν μέχρι ἐνταῦθα πρὸς τὰ ἐρωτικὰ παιδαγωγηθῇ, θεώμενος ἐφεξῆς τε καὶ ὀρθῶς τὰ καλά, πρὸς τέλος ἤδη ἰὼν τῶν ἐρωτικῶν ἐξαίφνης κατόψεταί τι θαυμαστὸν τὴν φύσιν καλόν, τοῦτο ἐκεῖνο, ὦ Σώκρατες, οὗ δὴ ἕνεκεν καὶ οἱ ἔμπροσθεν πάντες πόνοι ἦσαν, [1] πρῶτον μὲν ἀεὶ ὂν καὶ οὔτε γιγνόμενον οὔτε ἀπολλύμενον, [2] οὔτε αὐξανόμενον οὔτε φθίνον, [3] ἔπειτα οὐτῇ μὲν καλόν, τῇ δ’ αἰσχρόν, [4] οὐδὲ τοτὲ μέν, τοτὲ δὲ οὔ, [5] οὐδὲ πρὸς μὲν τὸ καλόν, πρὸς δὲ τὸ αἰσχρόν, [6] οὐδ΄ ἔνθα μὲν καλόν, ἔνθα δὲ αἰσχρόν, [7] ὡς τισὶ μὲν ὂν καλόν, τισὶ δὲ αἰσχρόν· [8] οὐδ΄ αὖ φαντασθήσεται αὐτῷ τὸ καλὸν οἷον πρόσωπόν τι οὐδὲ χεῖρες οὐδὲ ἄλλο οὐδὲν ὧν σῶμα μετέχει, [9] οὐδέ τις λόγος οὐδέ τις ἐπιστήμη, [10] οὐδέ που ὂν ἐν ἑτέρῳ τινι, οἷον ἐν ζώῳ ἢ ἐν γῇ ἢ ἐν οὐρανῷ ἢ ἔν τῳ ἄλλῳ, ἀλλ’ αὐτὸ καθ’ αὑτὸ μεθ’ αὑτοῦ μονοειδὲς ἀεὶ ὄν, τὰ δὲ ἄλλα πάντα καλὰ ἐκείνου μετέχοντα τρόπον τινὰ τοιοῦτον, οἷον [1] γιγνομένων τε τῶν ἄλλων καὶ ἀπολλυμένων μηδὲν ἐκεῖνο [11] μήτε τι πλέον μήτε ἔλαττον γίγνεσθαι [12] μηδὲ πάσχειν μηδέν.
- 60.
Politeia 479a5-b8.
Τούτων γὰρ δή, ὦ ἄριστε, φήσομεν, [1] τῶν πολλῶν καλῶν μῶν τι ἔστιν ὃ οὐκ αἰσχρὸν φανήσεται; [2] καὶ τῶν δικαίων, ὃ οὐκ ἄδικον; [3] καὶ τῶν ὁσίων, ὃ οὐκ ἀνόσιον; Οὔκ, ἀλλ’ ἀνάγκη, ἔφη, [1] καὶ καλά πως αὐτὰ καὶ αἰσχρὰ φανῆναι, καὶ ὅσα ἄλλα ἐρωτᾷς. Τί δὲ [4] τὰ πολλὰ διπλάσια; ἧττόν τι ἡμίσεα ἢ διπλάσια φαίνεται; Οὐδέν. [5] Καὶ μεγάλα δὴ καὶ σμικρὰ [6] καὶ κοῦφα καὶ βαρέα μή τι μᾶλλον ἃ ἂν φήσωμεν, ταῦτα προσρηθήσεται ἢ τἀναντία;Οὔκ, ἀλλ’ ἀεί, ἔφη, ἕκαστον ἀμφοτέρων ἕξεται.
- 61.
Platonists in general failed to realize the significance of provoking perceptions in connection with the geometers reliance on sensible images, e.g. Annas [3] writes in p. 278: “Mathematics has two defects compared with dialectic. It relies on visible diagrams, and it does not question its assumptions. Plato never indicates that, or how, these two defects are connected”.
- 62.
μάλιστα δὲ γεωμετρία κατὰ τὸν Φιλόλαον ἀρχὴ καὶ μητρόπολις οὖσα τῶν ἄλλων ἐπανάγει καὶ στρέφει τὴν διάνοιαν, οἷον ἐκκαθαιρομένην καὶ ἀπολυομένην ἀτρέμα τῆς αἰσθήσεως. διὸ καὶ Πλάτων αὐτὸς ἐμέμψατο τοὺς περὶ Εὔδοξονκαὶ ᾿Αρχύτανκαὶ Μέναιχμον εἰς ὀργανικὰς καὶ μηχανικὰς κατασκευὰς τὸν τοῦ στερεοῦ διπλασιασμὸν ἀπάγειν ἐπιχειροῦντας, ὥσπερ πειρωμένους δίχα λόγου δύο μέσας ἀνὰ λόγον, ᾗ παρείκοι, λαβεῖν. ἀπόλλυσθαι γὰρ οὕτω καὶ διαφθείρεσθαι τὸ γεωμετρίας ἀγαθὸν αὖθις ἐπὶ τὰ αἰσθητὰ παλινδρομούσης καὶ μὴ φερομένης ἄνω μηδ΄ ἀντιλαμβανομένης τῶν ἀιδίων καὶ ἀσωμάτων εἰκόνων, πρὸς αἷσπερ ὢν ὁ θεὸς ἀεὶ θεός ἐστιν (718E4-F4).
- 63.
It is not realized by students of the Meno, that in the Meno 82a7-86c3 the recollection (ἀνάμνησις) achieved is the recollection, i.e. repetition, of the logos of periodicity of the duplication of the square, and not simply the Pythagorean theorem for the orthogonal isosceles triangle, resulting in the complete knowledge (ἐπιστήμη) of the duplication ratio.
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Negrepontis, S. (2019). Plato on Geometry and the Geometers. In: Dani, S.G., Papadopoulos, A. (eds) Geometry in History. Springer, Cham. https://doi.org/10.1007/978-3-030-13609-3_1
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