Theaetetus : Knowledge as Continued Learning

Theaetetus: Knowledge as Continued Learning MALCOLM S. BROWN PLATO'S PHILOSOPHICALand mathematical interests come together in the dialogue Theaetetus around Plato's companion at the Academy for whom the dialogue is named. Theaetetus was not only an important member of the circle of mathematicians around Plato, and as such an important part of the mathematical influence on Plato, but he was also in his own right a distinguished contributor to mathematical theory. We do no need to rely exclusively on Plato's testimony to determine the scope and nature of Theaetetus' contributions to mathematics, although his testimony is admittedly important. Another stream of testimony makes these things plain: the evidences from the early historians of mathematics. The chief source in antiquity was the important work (now lost) by Eudemus of Rhodes, History o] Geometry, on which the later commentators Pappus and Proclus drew. These latter commentators were undoubtedly in possession of Eudemus' work. x In turn, Eudemus, who was a pupil of Aristotle's, undoubtedly possessed the works of the pre-Euclidean mathematicians, especially that of Theaetetus, which he discusses . Thus, when one traces back the two lines of evidence--from Plato and his commentators and from the history of mathematics--he finds that they converge on this celebrated companion of Plato, who influenced both the course of mathematics and that of philosophy. 2 The particular character of his mathematical work, which relates closely to the "classical" mathematics of the Pythagoreans, only makes Theaetetus more important in relation to Plato's thought. For what Theaetetus did, to put it roughly, was to subvert the classical mathematics by generalizing it. To borrow from Hegelian jargon, one might say that Theaetetus' work at once "cancelled, preserved, and raised to a higher level" the work of his Pythagorean predecessors. In section n. when I am dealing with Plato's texts, this point will be made more definite. 1 Thomas L. Heath points out that the later commentator Simplicius made remarks on the style of Eudemus and claimed to be quoting him word for word (x,,x&)d~tv) so that even at a date later than Proclus, Eudemus' text was accessible. Eutocius speaks of "examining " the history of Eudemus. (Euclid's Elements, ed. Thomas L. Heath [3 vols., 2nd eel.; New York: Dover, 1956l, I, 35.) Eva Sachs, in her definitive work De Theaeteto Mathematico Atheniensi (Berlin, 1914), finds in Theaetetus Plato's paradigm philosopher: "llle [Theaetetus] re vera philosophus fuit perfectus" (p. 69). She thinks of him in connection with those "admittedly rare but not impossible " Platonic philosophers who are the only guarantee that the intellectual life can survive in society. [3593 360 HISTORY OF PHILOSOPHY Within Plato's own thought, and especially in Meno, Phaedo and Republic, there are close connections ,to the content--mathematical and philosophical---of Theaetetus. F. M. Corrfford drew up a catalogue of similarities 3 between the introductory conversation of Meno and that of Theaetetus which takes in the following four items: (1) Meno, like Theaetetus, begins by erroneously offering a list of answers instead of a single one in response to Socrates' first question; like Theaetetus, he must be corrected. (2) Socrates asks his respondent to form his answer upon the example of a mathematical definition. (3) Socrates echoes in Theaetetus the complaint Meno had entered about Socrates: the complaint about his powers of reducing others to a perplexity (aporia) like his own. (4) In Meno the discussion then turns to the process of recollection, as in Theaetetus it turns to the process of maieutics. One could add to Cornford's list the point Socrates makes about the priority of the "what" to the "what kind" (the ti to the poion). At Theaetetus 196D Socrates echoes the point he made repeatedly in Meno (71B, 86D, 100B). These two dialogues are drawn still closer together by the common topic of incommensurability, which appears explicitly in Theaetetus 147-148 and is implied in Meno 82-85. Where Meno had dealt by implication (and Republic 546 is to deal explicitly) with V~, Theaetetus carries the problem forward to the cases of V3, V~, etc.' Between Theaetetus and Phaedo there is a more fundamental topical connection in their discussions of Equality/~nequality...