: In the Cambridge Conferences Lectures of 1898 Peirce defines a continuum as a "collection of so vast a multitude" that its elements "become welded into one another." He links the transinfinity (the "vast multitude") of a continuum to the confusion of its elements by a line of mathematical reasoning closely related to Cantor's Theorem. I trace the mathematical and philosophical roots of this conception of continuity, and examine its unresolved tensions, which arise mainly from difficulties in Peirce's theory of (...) collections. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the (...) infinitelysmall. (shrink)

In the last decade of his life C.S. Peirce began to formulate a purely geometrical theory of continuity to supersede the collection-theoretic theory he began to elaborate around the middle of the 1890s. I argue that Peirce never succeeded in fully formulating the later theory, and that while that there are powerful motivations to adopt that theory within Peirce’s system, it has little to recommend it from an external perspective.

This article makes three main claims: (1) that the philosophy of Ludwig Wittgenstein, properly understood, has no normative or political implications whatsoever; (2) that scholars with otherwise dramatically conflicting interpretations of Wittgenstein should nonetheless all agree with this conclusion; and (3) that understanding the (non-) implications of Wittgenstein’s philosophy helps to answer the two motivating questions of the literature on value pluralism — whether values are (or can be) plural (yes), and whether value pluralism leads to, requires, or reveals some (...) particular normative or political response (no). (shrink)

Social expectations guide people's evaluations of others’ behaviors, but the origins of these expectations remain unclear. It is traditionally thought that people's expectations depend on their past observations of others’ behavior, and people harshly judge atypical behavior. Here, we considered that social expectations are also influenced by a drive for reciprocity, and people evaluate others’ actions by reflecting on their own decisions. To compare these views, we performed four studies. Study 1 used an Ultimatum Game task where participants alternated Responder (...) and Proposer roles. Modeling participants’ expectations suggested they evaluated the fairness of received offers via comparisons to their own offers. Study 2 replicated these findings and showed that observing selfish behavior (lowball offers) only promoted acceptance of selfishness if observers started acting selfishly themselves. Study 3 generalized the findings, demonstrating that they also arise in the Public Goods Game, emerge cross‐culturally, and apply to antisocial punishment whereby selfish players punish generosity. Finally, Study 4 introduced the Trust Game and showed that participants trusted players who reciprocated their behavior, even if it was selfish, as much as they trusted generous players. Overall, this research shows that social expectations and evaluations are rooted in drives for reciprocity. This carries theoretical implications, speaking to a parallel in the mechanisms driving both decision‐making and social evaluations, along with practical importance for understanding and promoting cooperation. (shrink)

The Archimedean Axiom is often held to be an intuitively obvious truth about the geometry of physical space. After a general discussion of the varieties of geometrical intuition that have been proposed, I single out one variety which we can plausibly be held to have and then argue that it does not underwrite the intuitive obviousness of the Archimedean Axiom. Generalizing that result, I conclude that the Axiom is not intuitively obvious.

Despite the recent upsurge of interest in comparative political theory, there has been virtually no serious examination of Buddhism by political philosophers in the past five decades. In part, this is because Buddhism is not typically seen as a school of political thought. However, as Matthew Moore argues, Buddhism simultaneously parallels and challenges many core assumptions and arguments in contemporary Western political theory. In brief, Western thinkers not only have a great deal to learn about Buddhism, they have a great (...) deal to learn from it. To both incite and facilitate the process of Western theorists engaging with this neglected tradition, this book provides a detailed, critical reading of the key primary Buddhist texts, from the earliest recorded teachings of the Buddha through the present day. It also discusses the relevant secondary literature on Buddhism and political theory, as well as the literatures on particular issues addressed in the argument. Moore argues that Buddhist political thought rests on three core premises--that there is no self, that politics is of very limited importance in human life, and that normative beliefs and judgments represent practical advice about how to live a certain way, rather than being obligatory commands about how all persons must act. He compares Buddhist political theory to what he sees as Western analogues--Nietzsche's similar but crucially different theory of the self, Western theories of limited citizenship from Epicurus to John Howard Yoder, and to the Western tradition of immanence theories in ethics. This will be the first comprehensive treatment of Buddhism as political theory. (shrink)

Mathematical naturalism forbids philosophical interventions in mathematical practice. This principle, strictly construed, places severe constraints on legitimate philosophizing about mathematics; it is also arguably incompatible with mathematical realism. One argument for the latter conclusion charges the realist with inability to take a truly naturalistic view of the Gödel Program in set theory. This argument founders on the disagreement among mathematicians about that program's prospects for success. It also turns out that when disagreements run this deep it is counterproductive to take (...) too narrow a view of how philosophers of mathematics may legitimately proceed. (shrink)

Can a scientific naturalist be a mathematical realist? I review some arguments, derived largely from the writings of Penelope Maddy, for a negative answer. The rejoinder from the realist side is that the irrealist cannot explain, as well as the realist can, why a naturalist should grant the mathematician the degree of methodological autonomy that the irrealist's own arguments require. Thus a naturalist, as such, has at least as much reason to embrace mathematical realism as to embrace irrealism.

The philosophy of mathematics plays a vital role in the mature philosophy of Charles S. Peirce. Peirce received rigorous mathematical training from his father and his philosophy carries on in decidedly mathematical and symbolic veins. For Peirce, math was a philosophical tool and many of his most productive ideas rest firmly on the foundation of mathematical principles. This volume collects Peirce’s most important writings on the subject, many appearing in print for the first time. Peirce’s determination to understand matter, the (...) cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic. (shrink)