Philosophy of Mathematics and Philosophy of History

mathematical and one philosophical. In the first, Socrates leads the slave boy to correct, refine, and extend his intuition of a pair of geometrical diagrams by examining his presuppositions in the light of certain background knowledge (about arithmetic), as they solve a problem together. In the second, Socrates leads Meno to correct, refine, and extend his intuition about virtue as well as about method by examining his presuppositions in the light of certain background knowledge, as they engage in philosophical dialectic together. In both cases, an intelligible unity is apprehended, but imperfectly, and analysis leads to increased understanding via a search for the conditions of intelligibility. The form of dramatic dialogue expresses something essential to the process of analysis, for the dialogue may be read both as a set of arguments (in all of which reductio ad absurdum plays a pivotal role) which uncover the logical presuppositions of various claims, but it may also be read as a narrative, a process in time and history. A narrative, as Aristotle tells us in the Poetics, has a beginning, middle, and end. The beginning introduces us to characters who act in a certain situation, one or a few of which have a special claim on our empathy and interest; the middle introduces one or more surprising contingencies or reversals, and the skill of the storyteller lies in maintaining the continuity of the story not in spite of but by means of those discontinuities, deepening our understanding of the characters and their actions; and the end draws us along throughout the tale: how will events ultimately turn out for these characters? By contrast, an argument has the structure of a series of premises that support, in more or less materially and formally successful ways, a conclusion. The priority of premisses with respect to conclusion is logical, not temporal or historical; an argument has no beginning or end, no before and after. The logician takes the cogency of an argument to hold atemporally: if such and such is the case, then such and such must follow, universally and necessarily. Pragmatic or rhetorical considerations may superimpose an historical dimension on the argument, but that is not the concern of the logician. EMILY R. GROSHOLZ *