Grounding and the indispensability argument

There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or non-causal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a liberalized version of Field’s response—and I discuss its significance. I argue that if it succeeds, it provides a new refutation of the indispensability argument; and that, even if it fails, its failure may bolster some of the fictionalist responses to the indispensability argument already under discussion. In addition, I use grounding to reply to a recent challenge to these responses.     

Methodological naturalism and the truth seeking objection

Methodological naturalism, the exclusion of the supernatural from the natural sciences, has drawn critique from both proponents of Intelligent Design and some philosophical naturalists who argue that the methods of science can also be used to evaluate supernatural claims. One principal objection to methodological naturalism has been what I call the truth seeking objection. In this article I develop an understanding of methodological naturalism capable of answering the truth seeking objection. I further also argue that methodological naturalism as a convention of science can be best defended by abandoning scientism. In this way methodological naturalism can be reconnected to the original theistic context in which it was first developed.     

Default reasonableness and the mathoids

In this paper I will argue that (principled) attempts to ground a priori knowledge in default reasonable beliefs cannot capture certain common intuitions about what is required for a priori knowledge. I will describe hypothetical creatures who derive complex mathematical truths like Fermat’s last theorem via short and intuitively unconvincing arguments. Many philosophers with foundationalist inclinations will feel that these creatures must lack knowledge because they are unable to justify their mathematical assumptions in terms of the kind of basic facts which can be known without further argument. Yet, I will argue that nothing in the current literature lets us draw a principled distinction between what these creatures are doing and paradigmatic cases of good a priori reasoning (assuming that the latter are to be grounded in default reasonable beliefs). I will consider, in turn, appeals to reliability, coherence, conceptual truth and indispensability and argue that none of these can do the job.     

Dispensability in the Indispensability Argument

One of the most influential arguments for realism about mathematical objects is the indispensability argument. Simply put, this is the argument that insofar as we are committed to the existence of the physical objects existentially quantified over in our best scientific theories, we are also committed to the mathematical objects existentially quantified over in these theories. Following the Quine–Putnam formulation of the indispensability argument, some proponents of the indispensability argument have made the mistake of taking confirmational holism to be an essential premise of the argument. In this paper, I consider the reasons philosophers have taken confirmational holism to be essential to the argument and argue that, contrary to the traditional view, confirmational holism is dispensable.     

Quine's Weak and Strong Indispensability Argument

Quine's views on indispensability arguments in mathematics are scrutinised. A weak indispensability argument is distinguished from a strong indispensability thesis. The weak argument is the combination of the criterion of ontological commitment, holism and a mild naturalism. It is used to refute nominalism. Quine's strong indispensability thesis claims that one should consider all and only the mathematical entities that are really indispensable. Quine has little support for this thesis. This is even clearer if one takes into account Maddy's critique of Quine's strong indispensability thesis. Maddy's critique does not refute Quine's weak indispensability argument. We are left with a weak and almost unassailable indispensability argument. This revised version was published online in August 2006 with corrections to the Cover Date.     

Normative Naturalism and Epistemic Relativism

In previous work, I defended Larry Laudan against the criticism that the axiological component of his normative naturalism lacks a naturalistic justification. I argued that this criticism depends on an equivocation over the term ‘naturalism’ and that it begs the question against what we are entitled to include in our concept of nature. In this paper, I generalize that argument and explore its implications for Laudan and other proponents of epistemic naturalism. Here, I argue that a commitment to naturalism in the methods and aims of science inevitably entails a kind of epistemic relativism. However, I argue that this should not be interpreted as a reductio of naturalism, since the admission of contextually based standards and aims of science does not result in quietism when it comes to important questions concerning scientific rationality.     

Can Interventionists Be Neo-Russellians? Interventionism,the Open Systems Argument,and the Arrow of Entropy

Several proponents of the interventionist theory of causation have recently argued for a neo-Russellian account of causation. The article discusses two strategies for interventionists to be neo-Russellians. First, I argue that the open systems argument—the main argument for a neo-Russellian account advocated by interventionists—fails. Second, I explore and discuss an alternative for interventionists who wish to be neo-Russellians: the statistical–mechanical account. Although the latter account is an attractive alternative, it is argued that interventionists are not able to adopt it straightforwardly. Hence, to be neo-Russellians remains a challenge for interventionists.     

Can the new indispensability argument be saved from Euclidean rescues?

The traditional formulation of the indispensability argument for the existence of mathematical entities (IA) has been criticised due to its reliance on confirmational holism. Recently a formulation of IA that works without appeal to confirmational holism has been defended. This recent formulation is meant to be superior to the traditional formulation in virtue of it not being subject to the kind of criticism that pertains to confirmational holism. I shall argue that a proponent of the version of IA that works without appeal to confirmational holism will struggle to answer a challenge readily answered by proponents of a version of IA that does appeal to confirmational holism. This challenge is to explain why mathematics applied in falsified scientific theories is not considered to be falsified along with the rest of the theory. In cases where mathematics seemingly ought to be falsified it is saved from falsification, by a so called ??Euclidean rescue??. I consider a range of possible answers to this challenge and conclude that each answer fails.     

Analyticity and Incorrigibility

The traditional point of view on analyticity implies that truth in virtue only of meaning entails a priori acceptability and vice versa. The argument for this claim is based on the idea that meaning as it concerns truth and meaning as it concerns competence are one and the same thing. In this paper I argue that the extensions of these notions do not coincide. I hold that truth in virtue of meaning — truth for semantic reasons—doesn't imply a priori acceptability, and that a priori reflection based only on knowledge of meaning—in the sense of competence—doesn't necessitate true conclusions.The main consequence of this view concerns conceptual analysis, as it presupposes we have a privileged—incorrigible in the face of empirical evidence—access to non-trivial truths about the world on the basis of mere a priori reflection founded on meaning. If, as I argue, such access is not incorrigible the project of conceptual analysis loses its special epistemological status.     

A New Kantian Solution to the Third Antinomy of Pure Reason and to the Free Will Problem

The goal of this paper is to articulate a new solution to Kant's third antinomy of pure reason, one that establishes the possibility of incompatibilist freedom—the freedom presupposed by our traditional conceptions of moral responsibility, moral worth, and justice—without relying on the doctrine of transcendental idealism (TI). A discussion of Henry Allison's “two‐aspect” interpretation of Kant's TI allows me both to criticize one of the best defenses of TI today and to advance my own TI‐free solution to the third antinomy by appeal to a thesis of epistemic modesty based on Paul Guyer's realist interpretation of Kant's theory of experience. According to this interpretation, the a priori forms of our sensibility and understanding are not forms that the mind imposes on a material whose real properties are unknowable to us but are instead forms that limit or filter the kinds of things we can experience and know. In particular, being causally determined is a real feature of things as they are in themselves, but the necessity and universality of our deterministic claims are relative, restricted to the objects of possible experience. Consequently, though a causally determined event cannot be free, the necessity and universality of determinism does not entail that free events (choices) cannot exist but that they cannot constitute objects of possible experience. After arguing that freedom is possible, I outline an argument for the reality of freedom, based on the requirements of morality. Finally, I argue that my view, though opposed to metaphysical naturalism, is consistent with scientific realism and methodological naturalism.     

Phenomenal Contrast Arguments for Cognitive Phenomenology

According to proponents of irreducible cognitive phenomenology some cognitive states put one in phenomenal states for which no wholly sensory states suffice. One of the main approaches to defending the view that there is irreducible cognitive phenomenology is to give a phenomenal contrast argument. In this paper I distinguish three kinds of phenomenal contrast argument: what I call pure—represented by Strawson's Jack/Jacques argument—hypothetical—represented by Kriegel's Zoe argument—and glossed—first developed here. I argue that pure and hypothetical phenomenal contrast arguments face significant difficulties, but that there is a sound glossed phenomenal contrast argument for irreducible cognitive phenomenology.     

Resisting Explanation

Although explanation is widely regarded as an important concept in the study of rational inquiry, it remains largely unexplored outside the philosophy of science. This, I believe, is not due to oversight as much as to institutional resistance. In analytic philosophy it is basic that epistemic rationality is a function of justification and that justification is a function of argument. Explanation, however, is not argument nor is belief justification its function. I argue here that the task of incorporating explanation into the theory of rational inquiry poses a serious challenge to our basic concept of epistemic rationality as well as the a priori method of inquiry that still lies at the heart of analytic philosophy. Specifically, it pushes us toward a much stronger form of naturalism than is generally thought necessary, one in which argument and explanation are recognized as distinct and equally fundamental cognitive processes whose dynamic relationship is one of the central issues in the theory of rationality.     

Science as Structured Imagination

This paper offers an analysis of scientific creativity based on theoretical models and experimental results of the cognitive sciences. Its core idea is that scientific creativity — like other forms of creativity — is structured and constrained by prior ontological expectations. Analogies provide scientists with a powerful epistemic tool to overcome these constraints. While current research on analogies in scientific understanding focuses on near analogies, where target and source domain are close, we argue that distant analogies — where target and source domain differ widely — are especially useful in periods of intense conceptual change. To argue this point, we discuss three case studies from the history of science: early physiologists like Harvey, early evolutionary biologists like Darwin, and recent theorists on the evolution of the human mind like Mithen.     

Can We Infer Naturalism from Scepticism?

Hume, like a number of more recent writers, claims that epistemological scepticism gives us reason to think that our beliefs are non-epistemically determined. Because some body of propositions that we believe are all unjustifiable, the argument goes, our beliefs in those propositions must be determined by non-truth-conducive considerations. I argue initially that scepticism does not by itself entail Humean naturalism. I then then develop an argument from scepticism to naturalism which has considerable promise. This more complex argument is built around two considerations: (i) if a subject accepts a local sceptical argument against one of his beliefs and still does not give it up, then we have very good, if not conclusive, reason to think that his belief is non-epistemically determined; (ii) it seems initially plausible that global scepticism can have no affect on the beliefs it targets, even if we were to accept it. Unfortunately, however, even this argument ultimately fails to establish any connection between scepticism and Humean naturalism.     

Evidential Holism and Indispensability Arguments

The indispensability argument is a method for showing that abstract mathematical objects exist (call this mathematical Platonism). Various versions of this argument have been proposed (§1). Lately, commentators seem to have agreed that a holistic indispensability argument (§2) will not work, and that an explanatory indispensability argument is the best candidate. In this paper I argue that the dominant reasons for rejecting the holistic indispensability argument are mistaken. This is largely due to an overestimation of the consequences that follow from evidential holism. Nevertheless, the holistic indispensability argument should be rejected, but for a different reason (§3)—in order that an indispensability argument relying on holism can work, it must invoke an unmotivated version of evidential holism. Such an argument will be unsound. Correcting the argument with a proper construal of evidential holism means that it can no longer deliver mathematical Platonism as a conclusion: such an argument for Platonism will be invalid. I then show how the reasons for rejecting the holistic indispensability argument importantly constrain what kind of account of explanation will be permissible in explanatory versions (§4).     

The holistic presumptions of the indispensability argument

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.     

The Kuhnian mode of HPS

In this article I argue that a methodological challenge to an integrated history and philosophy of science approach put forth by Ronald Giere almost forty years ago can be met by what I call the Kuhnian mode of History and Philosophy of Science (HPS). Although in the Kuhnian mode of HPS norms about science are motivated by historical facts about scientific practice, the justifiers of the constructed norms are not historical facts. The Kuhnian mode of HPS therefore evades the naturalistic fallacy which Giere’s challenge is a version of. Against the backdrop of a discussion of Laudan’s normative naturalism I argue that the Kuhnian mode of HPS is a superior form of naturalism: it establishes contact to the practice of science without making itself dependent on its contingencies.     

Mathematics and conceptual analysis

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light.     

Methodological naturalism and its misconceptions

Methodological naturalism has been defended on both intrinsic and pragmatic grounds. Both of these defenses agree that methodological naturalism is a principle of science according to which the scientist ought to eschew talk of causally efficacious disembodied minds. I argue that this is the wrong interpretation of methodological naturalism. Methodological naturalism does not constrain the theories that scientists may conjecture, but how those theories may be justified. On this view, methodological naturalism is a principle of science according to which supernatural methods of justification, such as faith, are eschewed.