This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

Crispin Wright tried to refute classical 'Cartesian' skepticism contending that its core argument is extendible to a reductio ad absurdum (_Mind<D>, 100, 87-116, 1991). We show both that Wright is mistaken and that his mistakes are philosophically illuminating. Wright's 'best version' of skepticism turns on a concept of warranted belief. By his definition, many of our well-founded beliefs about the external world and mathematics would not be warranted. Wright's position worsens if we take 'warranted belief' to be implicitly defined by (...) the general principles governing it. Those principles are inconsistent, as shown by a variant of Godel's argument. Thus the inconsistency Wright found has nothing to do with the special premises of Cartesian skepticism, but is embedded in his own conceptual apparatus. Lastly, we show how a Cartesian skeptic could avoid Wright's critique by reconstructing a skeptical argument that does not use the claims Wright ultimately finds objectionable. (shrink)

_Sweet Reason: A Field Guide to Modern Logic, 2nd Edition_ offers an innovative, friendly, and effective introduction to logic. It integrates formal first order, modal, and non-classical logic with natural language reasoning, analytical writing, critical thinking, set theory, and the philosophy of logic and mathematics. An innovative introduction to the field of logic designed to entertain as it informs Integrates formal first order, modal, and non-classical logic with natural language reasoning, analytical writing, critical thinking, set theory, and the philosophy of (...) logic and mathematics Addresses contemporary applications of logic in fields such as computer science and linguistics A web-site linked to the text features numerous supplemental exercises and examples, enlightening puzzles and cartoons, and insightful essays. (shrink)

The nature of mathematical knowledge can be understood only by locating the knowing mathematician in an epistemic community. This claim is defended by extending Kripke's version of the Private Language Argument to include informal rules and using Godelian results to argue that such rules rules necessary in mathematics. A committed formalist might evade Kripke's original argument by positing internal mechanisms that determine rule -governed behavior. However, in the presence of informal rules, the formalist position collapses into the extreme skepticism that (...) the Private Language Argument works against. The existence of a community of rule followers provides the only viable alternative to such skepticism. (shrink)

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way (...) of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. (shrink)