Marianna Antonutti Marfori

This chapter first argues that while there are solid objections to be raised to Quine’s view, certain widespread arguments against result from overly crude and uncharitable interpretations of Quine. It then turns to the question of what kind of evidence it would take for a Quinean naturalist to change their mind about certain theses, such as the size of the set theoretic universe. It argues that Quineans might be moved to embrace further set-theoretic ontology in the light of the mathematical utility of large cardinals, and potentially even the ‘multiverse’ position on set theory.

Anti-revisionism, according to which any revision to a practice should come autonomously from within the practice itself, is a key component of all naturalistic approaches to science and mathematics, yet it is rarely discussed in detail. In this paper, I focus on naturalistic anti-revisionism about mathematics. After clarifying the reasons why naturalists should be anti-revisionists, I examine two different ways of spelling out a criterion for mathematical anti-revisionism. The first criterion maintains that we should not revise mathematical practice on external (i.e. extra-mathematical) grounds. On the basis of case studies, I argue that this criterion is not supported by a sufficiently sharp distinction between revisions prescribed on internal vs. external grounds. The second criterion maintains that we should not countenance philosophical accounts of mathematics which involve abandoning accepted theorems. I argue that while in line with naturalistic tenets, this criterion does not fully capture what is important to the mathematical naturalist. This leads to the formulation of a third criterion, which adds to the second criterion the clause that we should not abandon established mathematical methodology.

This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the Quine-Putnam indispensability argument. It has been argued that Maddy’s mathematical naturalism makes inconsistent assumptions on the role of mathematics in scientific explanations to the effect that it cannot distinguish mathematics from pseudo-science. I shall clarify Maddy’s arguments and show that the objection can be overcome. I shall then reformulate a novel version of the objection and consider a possible answer, and I shall conclude that mathematical naturalism does not ultimately provide a viable strategy for accommodating an anti-revisionary stance on mathematics within a Quinean naturalist framework.

The History of Philosophical and Formal Logic introduces ideas and thinkers central to the development of philosophical and formal logic. From its Aristotelian origins to the present-day arguments, logic is broken down into four main time periods: Antiquity and the Middle Ages The early modern period High modern period Early 20th century Each new time frame begins with an introductory overview highlighting themes and points of importance. Chapters discuss the significance and reception of influential works and look at historical arguments in the context of contemporary debates. To support independent study, comprehensive lists of primary and secondary reading are included at the end of chapters, along with exercises and discussion questions. By clearly presenting and explaining the changes to logic across the history of philosophy, The History of Philosophical and Formal Logic constructs an easy-to-follow narrative. This is an ideal starting point for students looking to understand the historical development of logic.