Mark Faller

This dissertation defends the thesis that Plato's employs methods of philosophical analysis that are akin to and based upon mathematical analysis. It identifies and describes three kinds of ancient mathematical analysis, rectilinear, dioristic, and poristic. It then shows that there are corresponding philosophical modes of analysis in portions of the Meno and Phaedo. Recognizing Plato's method in these dialogues provides insight into the doctrines Plato advances there and the arguments that support them. It also makes it possible to address three scholarly controversies. First, there is a dispute between F. Cornford and R. Robinson on whether mathematical analysis is a deductive or non-deductive procedure. The dissertation shows that some types are deductive and some are not. Second, there is an issue raised by K. Dorter whether the philosophical method of hypothesis proceeds ultimately to an unconditioned first principle, the good. The dissertation shows how philosophical analysis that is modeled on mathematical poristic could ultimately arrive at an unconditioned principle. Third, there is the question raised by Charles Kahn and others whether the divided line of the Republic delineates methods, each of which is applicable to a different type of being. The dissertation shows that, though some methods are most properly applied to particular types of being, Plato applies all the different types of philosophical analysis to all the types of being

The Compossible, the Contingent and the Conditionally Necessary: Leibniz on the Spectrum of Necessity for Possible Worlds The meaning of the concepts ‘necessity’ and ‘possibility’ have become vitiated and equivocal as the dual ideologies of relativism and empirical skepticism have undermined their metaphysical moorings. Empirical skepticism from its foundations with Heraclitus and Gorgias up to its heyday with Hume have cast a ‘duality’ spell on our vision of the world. They preach an ‘evening’ knowledge that idealizes the past and the dead as the model for the living. Knowledge of the world is like, and therefore reducible to the stable and inert facts that have already passed. Anything that is not of this nature is ethereal or merely of the mind. Under this spell, necessity is a mythical narrative we tell ourselves, but not fully applicable to the world. The only type of necessity that is relevant to reality is one based on habit and empirical probabilities. Relativism on the other hand, attempts to transfix our vision toward the ‘morning’ knowledge of an ever unformed future, where all things are equally possible. Berkeley’s ‘threeness’ perspective rejects any robust distinctiveness between truth and fiction and instead holds that the justified beliefs of any community or individual are equally ‘true’. Within this Protagorean vision, necessity is in the eye of the beholder, with rhetoric the only tool of negotiating whether one view is more useful than another. We need to return to a time before the great transfixing webs of Hume and Berkeley to appreciate the possibility of a more balanced idea of truth. It is in the subtle and complex metaphysical nuances of Leibniz that we may begin to purge our souls of these dual contaminants of modernity. I will make the case that Leibniz had a metaphysics of possibility and necessity that attempted to clarify and develop the usages of the two concepts of possibility and necessity. I will further show that this system attempted to bridge the reciprocal weaknesses of both relativism and skepticism in order to nurture an organically realist view of the nature of truth and its corresponding reality. Working out the details of his full system is complicated by two factors. He never lays out the full details of this system in any extended work. And he often overstates the conditions of one or another of these categories in response to [controversial] debates he is involved with at the time. I will try to show that we can develop a unified reading of his position by comparing a number of his works on freedom, determinism and causality into a seamless and consistent view on the nature of possibility and necessity. One of the first misconstruels we have to overcome is that regarding the status of mathematical reasoning. Leibniz is frequently heralded by the logicist school for holding that all mathematical truths just are logical truths. In fact, the reality is a little more complicated. Leibniz believed in the power of logical expression to maximally clarify conceptual relationships. To this end, he believed that any necessary relationship could be eventually analyzed into one of logical necessity. But it is in the details of how this process proceeds that the fuller implications of what he means by the many senses of necessity and possibility emerge.