Online research in philosophy

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

Quine’s (1969a) arguments for naturalizing epistemology have sparked dissent from the ranks of traditional epistemologists. For example, Kim (1988) complains that by making epistemology a “chapter of psychology,” the naturalist robs epistemology of its normative force and thereby its status of genuine epistemology. I agree with the spirit of Kim’s objection, but it is not likely that naturalized epistemology must abandon normativity in an explicit and obvious way, or that normative alternatives of the kind Kim envisions are available (Bayer 2007). Quine himself insists his epistemology is normative (1986a; 1992), and generic proposals for naturalizing normativity abound. But even if Quine nominally retains a major element of traditional epistemology, it may still be asked whether this element is enough, given the overall nature of Quine’s project.

One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set-theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is that these latter issues are of direct relevance to philosophical arguments connected to the applicability of mathematics. In particular, the possibility of there being distinct alternative foundations for mathematics blocks the standard argument from the indispensable role of mathematics in science to the existence of specific mathematical objects.

Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics. Here mathematicians pursue mathematical theories that are closely connected to the use of mathematics in the sciences and engineering. This area of mathematics seems to proceed using different methods and standards when compared to much of mathematics. I argue that applied mathematics can contribute to the philosophy of mathematics and our understanding of mathematics as a whole.

This paper argues that Philip Kitcher's epistemology of mathematics, codified in his Naturalistic Constructivism, is not naturalistic on Kitcher's own conception of naturalism. Kitcher's conception of naturalism is committed to (i) explaining the correctness of belief-regulating norms and (ii) a realist notion of truth. Naturalistic Constructivism is unable to simultaneously meet both of these commitments.

The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, and an extension to a social constructivist position (which might well have been unacceptable to Lakatos).

Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the only available argument for it. I argue that this argument, the so-called Failure Argument, itself fails. My overall conclusion is that although there is no self-refutation argument against reinterpretation naturalism, there are as yet no good reasons to accept it. Naturalism in mathematics The consistency of mathematical naturalism The failure argument Objections to the failure argument Philosophy as the default.

In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.

This paper contains answers to the following Five questions, posed by the editors are answered: (1) Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics? (2) What example(s) from your work (or the work of others) illustrates the use of mathematics for philosophy? (3) What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science? (4) What do you consider the most neglected topics and/or contributions in late 20th century philosophy of mathematics? (5) What are the most important open problems in the philosophy of mathematics and what are the prospects for progress?