Journal of the History of Philosophy

Plato's instructions entail that the line of Republic VI is divided so that the middle two segments are of equal length. Yet I argue that Plato's elaboration of the significance of this analogy shows he believes that these segments are of unequal length because the domains they represent are not of equally clear mental states, nor perhaps of objects of equal reality. I label this inconsistency between Plato's instructions and his explanation the "overdetermination problem." The overdetermination problem has been a perennial concern, and a substantial amount of work has been produced which attempts to deal with it. I offer a classification of approaches to the overdetermination problem as a way of documenting the problem's significance, and show why these approaches are all inadequate as solutions. My novel resolution of the overdetermination problem rests upon a demonstration that the contradiction is intentional. The later recapitulation of the ratio at 534a reveals that Plato was himself aware that the middle two segments are equal. I argue that this contradiction is a sophisticated device designed to lead the reader of the Republic through the four epistemic stages represented by the line itself. Most significantly, recognition of this mathematical contradiction acts as a goad, spurring independent philosophical reflection just in the way that Plato advocates in the Republic more generally.