The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

Gonzalo Rodriguez-Pereyra presents an original study of the place and role of the Identity of Indiscernibles in Leibniz's philosophy. The Principle of the Identity of Indiscernibles rules out numerically distinct but perfectly similar things; Leibniz derived it from more basic principles and used it to establish important philosophical theses. Rodriguez-Pereyra aims to establish what Leibniz meant by the Principle of Identity of Indiscernibles, what his arguments for and from it were, and to assess those arguments and Leibniz's claims about the (...) Principle of Identity of Indiscernibles. He argues that Leibniz had a very strong version of the principle, according to which no possibilia are intrinsically perfectly similar, where this excludes things that differ in magnitude alone. The book discusses Leibniz's arguments for the Identity of Indiscernibles in the Meditation on the Principle of the Individual, the Discourse on Metaphysics, Notationes Generales, Primary Truths, the letter to Casati of 1689, the correspondence with Clarke, as well as the use of the Identity of Indiscernibles in Leibniz's arguments against the Cartesian conception of the material world, atoms, absolute space and time, the Lockean conception of the mind as a tabula rasa, and freedom of indifference. Rodriguez-Pereyra argues that the Identity of Indiscernibles was a central but inessential principle of Leibniz's philosophy. (shrink)

It can be shown by means of a paradox that, given the Principle of Sufficient Reason, there is no conjunction of all contingent truths. The question is, or ought to be, how to interpret that result: _Quid sibi velit?_ A celebrated argument against PSR due to Peter van Inwagen and Jonathan Bennett in effect interprets the result to mean that PSR entails that there are no contingent truths. But reflection on parallels in philosophy of mathematics shows it can equally be (...) interpreted either as a proof that there are "too many" contingent truths to combine in a single conjunction or as a proof that the concept _contingent truth_ is indefinitely extensible and there is no such thing as "all contingent truths." Either interpretation would reconcile PSR with contingent truth, but the natural rationales of those interpretations are at odds. This essay argues that the second interpretation is a more satisfactory explanation of why, if PSR is true, there should be no conjunction of all contingent truths. This sheds new light on the nature of the explanatory demand embedded in PSR and uncovers a number of surprising implications for the commitments of rationalism. (shrink)