Russell Marcus

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe to exist. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument present an intriguing attempt to provide reasons to favor the indispensabilist’s criterion against some contemporary eleatics. I show that Colyvan’s argument is not decisive against the eleatic and then sketch a way to capture some of the important intuitions behind both views.El debate sobre si deberíamos creer en la existencia de los objetos matemáticos conduce rápidamente a la cuestión de cómo determinar lo que deberíamos creer que existe. Los indispensabilistas declaran que deberíamos creer en la existencia de los objetos matemáticos por sus funciones ineliminables en la teoría científica. Los eleáticos argumentan que sólo existen los objetos que tienen propiedades causales. La defensa reciente de Mark Colyvan del argumento de la indispensabilidad de Quine representa un interesante intento de proporcionar razones a favor del criterio indispensabilista, en contra de algunos eleáticos contemporáneos. Mostraré que el argumento de Colyvan en contra de los eleáticos no es decisivo y esbozaré a continuación una manera de capturar algunas de las importantes intuiciones que se encuentran tras ambos puntos de vista.

In this chapter, we explore how to manage student errors in non-traditional philosophy classrooms—where student learning and autonomy are prioritized. We examine the benefits and drawbacks of two models for managing student errors: (1) the philosophical expertise model, whereby instructors are responsible for maximizing the number of true or reasonable claims made in a classroom, including correcting student errors; and (2) the student empowerment model, whereby instructors should be wary of any kind of correction, since more importance is accorded to student autonomy.

Rigorous yet intuitive and accessible, Introduction to Formal Logic provides a focused, "nuts-and-bolts" introduction to formal deductive logic that covers syntax, semantics, translation, and natural deduction for propositional and predicate logics. For instructors who want to go beyond a basic introduction to explore the connection between formal logic techniques and philosophy, Oxford also publishes Introduction to Formal Logic with Philosophical Applications, an extended version of this text that incorporates two chapters of stand-alone essays on logic and its application in philosophy and beyond.

Philosophy students often struggle to master the complex skills needed to succeed in their work, especially in writing thesis-driven essays. Research over the past forty years on instructional scaffolding, both generally and as applied in philosophy, has helped teachers to refine both instruction and assignment design to improve students’ performance on complex philosophical tasks. This essay reviews the fundamentals of scaffolding in order to motivate and support some innovative in-class exercises and writing assignments that can help students develop even finer-grained skills. These skills are useful both intrinsically and for their transfer to longer-form essays, to other philosophical work, and to the general academic and intellectual development of our students.

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not decisive against the eleatic and sketch a way to capture the important intuitions behind both views.

The new explanatory or enhanced indispensability argument alleges that our mathematical beliefs are justified by their indispensable appearances in scientific explanations. This argument differs from the standard indispensability argument which focuses on the uses of mathematics in scientific theories. I argue that the new argument depends for its plausibility on an equivocation between two senses of explanation. On one sense the new argument is an oblique restatement of the standard argument. On the other sense, it is vulnerable to an instrumentalist response. Either way, the explanatory indispensability argument is no improvement on the standard one

The indispensability argument is sometimes seen as weakened by its reliance on a controversial premise of confirmation holism. Recently, some philosophers working on the indispensability argument have developed versions of the argument which, they claim, do not rely on holism. Some of these writers even claim to have strengthened the argument by eliminating the controversial premise. I argue that the apparent removal of holism leaves a lacuna in the argument. Without the holistic premise, or some other premise which facilitates the transfer of evidence to mathematical portions of scientific theories, the argument is implausible.

Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science.

Brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject.

The access problem for mathematics arises from the supposition that the referents of mathematical terms inhabit a realm separate from us. Quine’s approach in the philosophy of mathematics dissolves the access problem, though his solution sometimes goes unrecognized, even by those who rely on his framework. This paper highlights both Quine’s position and its neglect. I argue that Michael Resnik’s structuralist, for example, has no access problem for the so-called mathematical objects he posits, despite recent criticism, since he relies on an indispensability argument. Still, Resnik’s structuralist does not provide an account of our access to traditional mathematical objects, and this may be seen as a problem.

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not decisive against the eleatic and sketch a way to capture the important intuitions behind both views.