In ‘The varieties of indispensability arguments’ Marco Panza and Andrea Sereni argue that, for any clear notion of indispensability, either there is no conclusive argument for the thesis that mathematics is indispensable to science, or the notion of indispensability at hand does not support mathematical realism. In this paper, I shall not object to this main thesis directly. I shall instead try to assess in a naturalistic spirit a family of objections the authors make along the way (...) to the use of indispensability premises in indispensability arguments. (shrink)

I will contrast two conceptions of the nature of mathematical objects: the conception of mathematical objects as preconceived objects, and heavy duty platonism. I will argue that friends of the indispensability argument are committed to some metaphysical theses and that one promising way to motivate such theses is to adopt heavy duty platonism. On the other hand, combining the indispensability argument with the conception of mathematical objects as preconceived objects yields an unstable position. The conclusion is that the (...) metaphysical commitments of the indispensability argument should be carefully scrutinized. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract (...) mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

It is claimed that the indispensability argument for the existence of mathematical entities (IA) works in a way that allows a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. This is supposed to be possible by virtue of the appeal to confirmational holism that enters into the formulation of IA. Holism about confirmation is supposed to be motivated in analogy with holism about falsification. I present an account of how holism (...) about falsification is supposed to be motivated. I argue that the argument for holism about falsification is in tension with how we think about confirmation and with two principles suggested by Quine for construing a plausible variety of holism. Finally, I show that one of Quine's principles does not allow a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. (shrink)

Alan Baker’s enhanced indispensability argument supports mathematical platonism through the explanatory role of mathematics in science. Busch and Morrison defend nominalism by denying that scientific realists use inference to the best explanation to directly establish ontological claims. In response to Busch and Morrison, I argue that nominalists can rebut the EIA while still accepting Baker’s form of IBE. Nominalists can plausibly require that defenders of the EIA establish the indispensability of a particular mathematical entity. Next, I argue that (...) IBE cannot establish that any particular mathematical entity is indispensable. Mathematical entities do not compete with each other in the way physical unobservables do. This lack of competition enables alternative formulations of scientific explanations that use different, but compatible, mathematical entities. The compatibility of these explanations prevents IBE from establishing platonism. (shrink)

In this paper, I argue that sufficientarian principles are indispensable in the set of principles that have bearing on issues in distributive ethics. I provide two arguments in favor of this claim. First, I argue that sufficientarianism is the only framework that allows us to appropriately analyze what sort of obligations we have toward individuals who are badly off due to their own faults and choices. Second, I argue that sufficientarianism is the only theory that provides an adequate framework for (...) thinking about the desperately badly off. (shrink)

When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival (...) of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ – but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation. (shrink)

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that (...) these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science. (shrink)

For quite some time the indispensability arguments of Quine and Putnam were considered a formidable obstacle to anyone who would reject the existence of mathematical objects. Various attempts to respond to the indispensability arguments were developed, most notably by Chihara and Field. Field tried to defend mathematical fictionalism, according to which the existential assertions of mathematics are false, by showing that the mathematics used in applications is in fact dispensable. Chihara suggested, on the other hand, that mathema­tics makes (...) true existential assertions, but that these can be interpreted so as to remove the commitment to abstract objects. More recently, there have been various attempts to show that the indispensability arguments contain assumptions that are conceptually misguided in ways having little to do with mathematical content. All of this work is of considerable interest, and the result has been a gathering consensus that the indispensability arguments, as put forth by Quine and Putnam, do not provide convincing reason to accept mathematical realism. The focus here will be on the ways of responding to the indispensability arguments, and in particular on the obstacles to fictionalism that remain after the versions of Quine and Putnam are undercut. (shrink)

Much of the current thought concerning mathematical ontology in volves in some way the Quine/Putnam indispensability argument. The indispensability approach needs to be more thoroughly specified, however, before substantive progress can be made in assessing it. To this end I examine in some detail the ways in which pure mathematics is applied to physical theory; such considerations give rise to three specific issues with which the indispensability approach must come to grips.

The question as to whether there are mathematical explanations of physical phenomena has recently received a great deal of attention in the literature. The answer is potentially relevant for the ontology of mathematics; if affirmative, it would support a new version of the indispensability argument for mathematical realism. In this article, I first review critically a few examples of such explanations and advance a general analysis of the desiderata to be satisfied by them. Second, in an attempt to strengthen (...) the realist position, I propose a new type of example, drawing on probabilistic considerations. 1 Introduction2 Mathematical Explanations2.1 ‘Simplicity’3 An Average Story: The Banana Game3.1 Some clarifications3.2 Hopes and troubles for the nominalist3.3 New hopes?3.4 New troubles4 Conclusion. (shrink)

Book Information The Indispensability of Mathematics. By Mark Colyvan. Oxford University Press. New York. 2001. Pp. 172. Hardback, £30.00.

Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The (...) Element's first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively. (shrink)

The article examines a central methodological tenet of Grace Jantzen's Becoming Divine. In this book she turns her back on what she calls Anglo-American philosophy of religion in favour of what she calls a continental approach. I argue that for her, belief is as indispensable in religion and in the philosophy of religion as it is for the Anglo-American philosophy of religion which she rejects. Further, the only argument that she offers for her position is a genetic argument for the (...) origins of religious belief. Consistently with her position, she does not consider any empirical evidence relevant to this causal claim. However, the logic of such genetic claims is that for every empirically grounded genetic argument for A there is a corresponding genetic argument for not-A. So if such an argument invalidates A, it also invalidates not-A. (shrink)

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

What is successful Christian leadership? How should leadership be developed within a Christian context? This article encourages Christian leaders to seek to identify with Jesus’ mission and paradigm in developing leaders by examining the Scriptural passage in Mark 3:13-19. Jesus’ example in leadership development was based on succession of leadership primarily accomplished through personally shaping his disciples in close, mentoring relationships. This article, in particularly examines Jesus’ practice of having his disciples near him in order that they might best accomplish (...) the task he had purposed for them. Currently, this pattern of leadership development has been given diverse definitions from servant-based leadership to transformational leadership, but to Jesus, developing leaders was best accomplished through simple mentoring. Jesus’ desired goals for his disciples were realized through an intentional nearness to the lives of the twelve. For Christian leadership to be healthy, its success depends on close relationships being developed between the mentor and the mentee. The indispensable mark of Christian leadership is the combined effort of action and agenda while purposing to influence others. (shrink)

Si hoy es tan necesario hablar sobre la capacidad humana de asegurar la convivencia entre nosotros, eso ocurre porque vivimos un período de permanente deterioro de esta convivencia. Nos podemos quejar de eso o podemos estar indiferentes a eso, de todas maneras tenemos que reflexionar este hecho. En este sentido, el presente trabajo hace una reflexión sobre la ética de la convivencia y su importancia en nuestra sociedad, en una era donde lo indispensable es inútil debido a la racionalidad que (...) impone el modelo de la globalización neoliberal. (shrink)

In this note, I discuss David Enoch's influential deliberative indispensability argument for metanormative realism, and contend that the argument fails. In doing so, I uncover an important disanalogy between explanatory indispensability arguments and deliberative indispensability arguments, one that explains how we could accept the former without accepting the latter.

Developing the ideas presented in Jacquette (2011), the paper presents an indispensability argument aimed at justification of induction. First, Hume’s problem of induction is introduced via slightly different reconstructions. Second, several traditional attempts to solve Hume’s problem are presented. Finally, Jacquette’s(2011) proposal to justify induction by an indispensability argument is developed. I conclude with presenting a kind of indispensability argument for induction.

The literature on the indispensability argument for mathematical realism often refers to the ‘indispensable explanatory role’ of mathematics. I argue that we should examine the notion of explanatory indispensability from the point of view of specific conceptions of scientific explanation. The reason is that explanatory indispensability in and of itself turns out to be insufficient for justifying the ontological conclusions at stake. To show this I introduce a distinction between different kinds of explanatory roles—some ‘thick’ and ontologically (...) committing, others ‘thin’ and ontologically peripheral—and examine this distinction in relation to some notable ‘ontic’ accounts of explanation. I also discuss the issue in the broader context of other ‘explanationist’ realist arguments. (shrink)

Robust metanormative realists think that there are irreducibly normative, metaphysically heavy normative facts. One might wonder how we could be epistemically justified in believing that such facts exist. In this paper, I offer an answer to this question: one’s belief in the existence of robustly real normative facts is epistemically justified because so believing is indispensable to being a successful inquirer for creatures like us. The argument builds on Enoch's (2007, 2011) deliberative indispensability argument for Robust Realism but avoids (...) relying on an overly pragmatic account of the sources of basic epistemic justification. Instead, I suggest that the sources of basic epistemic justification are those belief-forming methods which are indispensable for zetetically indispensable projects, that is, projects which are constitutive of being a successful inquirer for embodied, agential creatures like ourselves. (shrink)

Indispensability arguments for realism about mathematical entities have come under serious attack in recent years. To my mind the most profound attack has come from Penelope Maddy, who argues that scientific/mathematical practice doesn't support the key premise of the indispensability argument, that is, that we ought to have ontological commitment to those entities that are indispensable to our best scientific theories. In this paper I defend the Quine/Putnam indispensability argument against Maddy's objections.

The recognition of striking regularities in the physical world plays a major role in the justification of hypotheses and the development of new theories both in the natural sciences and in philosophy. However, while scientists consider only strictly natural hypotheses as explanations for such regularities, philosophers also explore meta-natural hypotheses. One example is mathematical realism, which proposes the existence of abstract mathematical entities as an explanation for the applicability of mathematics in the sciences. Another example is theism, which offers the (...) existence of a supernatural being as an explanation for the design-like appearance of the physical cosmos. Although all meta-natural hypotheses defy empirical testing, there is a strong intuition that some of them are more warranted than others. The goal of this paper is to sharpen this intuition into a clear criterion for the admissibility of meta-natural explanations for empirical facts. Drawing on recent debates about the indispensability of mathematics and teleological arguments for the existence of God, I argue that a meta-natural explanation is admissible just in case the explanation refers to an entity that, though not itself causally efficacious, guarantees the instantiation of a causally efficacious entity that is an actual cause of the regularity. (shrink)

According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand (...) of metaphysics, 'Aristotelian (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage. (shrink)

This paper argues that an intersubjective interpretation of probability is indispensable in understanding the rationality of science from the point of view of confirmation. It shows how intersubjective probabilities are superior to their subjective counterparts in a number of key respects, in a peculiar class of circumstances relevant to scientific practice. It also shows how group probabilities can be superior to individual probabilities even if rationality constraints other than coherence, e.g. as advocated by Objective Bayesians, are required of degrees of (...) belief. Key considerations are the improved error correction that can occur, and the increased background knowledge that becomes available, through interaction and group decision-making. (shrink)

Many of us care about the existence of ethical facts because they appear crucial to making sense of our practical lives. On one tempting line of thought, this idea can also play a central role in justifying our belief in those facts. David Enoch has developed this thought into a formidable new proposal in moral epistemology: that the deliberative indispensability of ethical facts gives us epistemic justification for believing in such facts. This chapter argues that Enoch’s proposal fails because (...) it conflicts with a central fact about epistemic justification: that the norms of epistemic justification have the content that they do in part because of some positive connection between those norms and the truth of the beliefs that these norms govern. Alternatives to Enoch’s attempt to defend the idea that deliberative indispensability confers epistemic justification fail for parallel reasons. The chapter concludes that deliberative indispensability does not provide epistemic justification. (shrink)

The Enhanced Indispensability Argument (Baker [ 2009 ]) exemplifies the new wave of the indispensability argument for mathematical Platonism. The new wave capitalizes on mathematics' role in scientific explanations. I will criticize some analyses of mathematics' explanatory function. In turn, I will emphasize the representational role of mathematics, and argue that the debate would significantly benefit from acknowledging this alternative viewpoint to mathematics' contribution to scientific explanations and knowledge.

There is a way of talking that would appear to involve ascriptions of purpose, goal directed activity, and intentional states to groups. Cases are familiar enough: classmates intend to vacation in Switzerland, the department is searching for a metaphysician, the Democrats want to minimize losses in the upcoming elections, and the US intends to improve relations with such and such country. But is this talk to be understood just in terms of the attitudes and actions of the individuals involved? Is (...) the talk, to take an overly simple proposal as an example, a mere summary of familiar individual attitudes of the group members? Or is the ascription of attitudes and actions to groups to be taken more literally, as suggesting that the group for example believes that P, or intends to A, over and above what the members individually think and do? In short, are there groups with minds of their own? Philip Pettit has deployed the “discursive dilemma” to defend the thesis that there are such group minds. In what follows, I explore the relationship between the group allegedly with a mind of its own and the individuals it comprises, and I consider just how this relationship must be understood in order to give Pettit’s argument for group minds its best chance for success. As I understand it, the discursive dilemma has to be used in conjunction with what might be called an indispensability argument for group minds. It is useful to distinguish two forms of this argument. The explanatory version of the indispensability argument is, very schematically, as follows: there is a compelling explanatory theory T concerning the social, certain indispensable elements of T entail the group mind thesis, so the group mind thesis is true. Several questions immediately arise: What sort of theory is T? In what sense is it indispensable? Are there other forms of indispensability? I don’t have definitive answers to these questions. But how we settle them will have implications for the interaction and support the discursive dilemma provides the indispensability argument. In particular, using the discursive dilemma to defend what I characterize below as a practical version of the indispensability argument commits us to the rationality of individual participants in a way that the explanatory version of the indispensability argument does not. My point in the first part of the paper is that if Pettit wants to avoid the weaknesses of the explanatory indispensability argument and pursue the practical version, then he owes us a story about the rationality of individual participation in groups. Pettit also owes us a story about the agency an individual exercises as part of a group. If it takes the actions of individuals to execute the intentions of the group, how are we to understand those actions in order for the group to count as having a mind of its own? How must group intentions figure in the practical or deliberative perspective of individuals who execute those intentions? I will argue that the proponent of the group mind thesis must proceed with some care here, because some natural ways of answering these questions will undermine the thesis. But in the end, I think that these questions are interesting independently of whether Pettit is right to think that groups do have minds of their own. That’s because investigating Pettit’s arguments might lead to new ideas about how the rationality and agency of individuals can be exercised, and suggests new ways of understanding how individuals can act together, irrespective of whether the groups they compose ever have minds of their own. (shrink)

In this paper I examine Quine''s indispensability argument, with particular emphasis on what is meant by ''indispensable''. I show that confirmation theory plays a crucial role in answering this question and that once indispensability is understood in this light, Quine''s argument is seen to be a serious stumbling block for any scientific realist wishing to maintain an anti-realist position with regard to mathematical entities.

David Enoch has recently proposed that the deliberative indispensability of irreducibly normative facts suffices to support their inclusion in our ontology, even if they are not necessary for the explanation of any observable phenomena. He challenges dissenters to point to a relevant asymmetry between explanation and deliberation that shows why explanatory indispensability, but not deliberative indispensability, is a legitimate guide to ontology. In this paper, I aim to do just that. Given that an entity figures in the (...) actual explanation of some phenomenon, it is not an open question whether that entity exists. Thus, if you can manage to find actual explanations, you can find answers to ontological questions. In contrast, even if some entity is indispensable to deliberation, it still might not exist. For example, even if some form of libertarian free will is deliberatively indispensable, we still might not have libertarian free will. So even if you manage to discover the indispensable commitments of deliberation, there is still more work to do to get to the bottom of things. That additional work is explanatory work, and so deliberative indispensability is not an independent guide to what there is. (shrink)

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form (...) of realism about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view. (shrink)

The indispensability argument comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of (...) IA can be obtained, also through different specifications of the notion of indispensability. We then distinguish between schematic and genuine IA, and argue that no genuine sound IA is available or easily forthcoming. We then submit that this holds also in the particularly relevant case in which indispensability is conceived as explanatory indispensability. (shrink)

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments. (...) -/- There are two main indispensability arguments in the literature, though one has received nearly all of the attention. They correspond to two ways in which we use mathematics in science and in everyday life. We use mathematical language to help us describe non-mathematical reality; and we use mathematical reasoning to help us perform inferences concerning non-mathematical reality using only a feasible amount of cognitive power. The former use is the starting point of the Quine-Putnam indispensability argument ([Quine, 1980a], [Quine, 1980b], [Quine, 1981a], [Quine, 1981b], [Putnam, 1979a], [Putnam, 1979b]); the latter provides the basis for Ketland’s more recent argument ([Ketland, 2005]). I begin by considering the Quine-Putnam argument and introduce instrumental nominalism to defuse it. Then I show that Ketland’s argument can be defused in a similar way. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract (...) mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

Confirmational holism is central to a traditional formulation of the indispensability argument for mathematical realism (IA). I argue that recent strategies for defending scientific realism are incompatible with confirmational holism. Thus a traditional formulation of IA is incompatible with recent strategies for defending scientific realism. As a consequence a traditional formulation of IA will only have limited appeal.

Gilbert Harman has argued that it does not make sense to ascribe character traits to people. The notion of morally virtuous character becomes particularly suspect. How plausible this is depends on how broad character traits would have to be. Views of character as entirely invariant behavioural tendencies offer a soft target. This paper explores a view that is a less easy target: character traits as specific to kinds of situation, and as involving probabilities or real possibilities. Such ascriptions are not (...) undermined by Harman's arguments, and it remains plausible that the agent's character often is indispensable in explanation of behaviour. Character is indispensable also as processes of control that impose reliability where it really matters. (shrink)

Realists persuaded by indispensability arguments af- firm the existence of numbers, genes, and quarks. Van Fraassen's empiricism remains agnostic with respect to all three. The point of agreement is that the posits of mathematics and the posits of biology and physics stand orfall together. The mathematical Platonist can take heart from this consensus; even if the existence of num- bers is still problematic, it seems no more problematic than the existence of genes or quarks. If the two positions just (...) described were the only ones possible, there could be no objection to this melding of numbers with genes and quarks. However, the position I call contrastive empiricism (Sober 1990a) stands opposed to both realism and to Van Fraassen's em- piricism. As it turns out, contrastive empiricism entails that coalesc- ing mathematics with empirical science is highly problematic. I believe that there is an important kernel of truth in abductive ar- guments for genes and quarks. But no counterpart argument exists for the case of numbers. Of course, the existence of this third way would be uninteresting, if contrastive empiricism were wholly implausible. However, I will argue that contrastive empiricism captures what makes sense in standard versions of realism and empiricism, while avoiding the excesses of each. This third way is a middle way; it cannot be dismissed out of hand. (shrink)

A movement asking to take race out of medicine is growing in the US. While we agree with the necessity to get rid of flawed assumptions about biological race that pervade automatic race correction in medical algorithms, we urge caution about insisting on a blanket eliminativism about race in medicine. If we look at racism as a fundamental cause, in the sense that this notion has been introduced in epidemiological studies by Bruce Link and Jo Phelan, we must conclude that (...) race is indispensable to consider, investigate, and denounce the health effects of multilevel racism, and cannot be eliminated by addressing more specific risk factors in socially responsible epidemiology and clinical medicine. This does not mean that realism about human races is vindicated. While maintaining that there are no human races, we show how it is that a non-referring concept can nonetheless turn out indispensable for explaining real phenomena. (shrink)

Far from being an unnecessary appendage to Whitehead’s system, temporal atomism is, in my judgment, the basis for pansubjectivity and other fundamental ideas such as becoming, concrescence, and subjectivity.

The subject with which Mark Colyvan's book deals is timely indeed. While discussions of mathematical ontology have been a mainstay in philosophy of mathematics for the last century (at least), for the last thirty years or so this discussion has begun with (and often not left) the Quine/Putnam indispensability argument. Though the argument is widely cited, to my knowledge this is the first book-length project exclusively dedicated to articulating and defending the Quine/Putnam indispensability argument for mathematical platonism. In (...) the first section below, I shall run through a summary of the book (following its chapter layout) with a few critical remarks. In the second section I shall raise a few critical issues organized topically. (shrink)

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. (shrink)

Recently, nominalists have made a case against the Quine–Putnam indispensability argument for mathematical Platonism by taking issue with Quine’s criterion of ontological commitment. In this paper I propose and defend an indispensability argument founded on an alternative criterion of ontological commitment: that advocated by David Armstrong. By defending such an argument I place the burden back onto the nominalist to defend her favourite criterion of ontological commitment and, furthermore, show that criterion cannot be used to formulate a plausible (...) form of the indispensability argument. (shrink)

In this article I consider what it would take to combine a certain kind of mathematical Platonism with serious presentism. I argue that a Platonist moved to accept the existence of mathematical objects on the basis of an indispensability argument faces a significant challenge if she wishes to accept presentism. This is because, on the one hand, the indispensability argument can be reformulated as a new argument for the existence of past entities and, on the other hand, if (...) one accepts the indispensability argument for mathematical objects then it is hard to resist the analogous argument for the existence of the past. (shrink)

This book offers a compelling account of the evolution of sensibility, weaving together Darwinian and biosemiotic theory. It works along non-anthropomorphic aesthetics of the appreciation and creation of beauty in nature as an end in itself which has practical benefit.

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. (...) Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

In this paper, I propose a solution to a challenge formulated by Judith Jarvis Thomson: We have to explain why the moral asymmetry between doing and allowing harm is a deep feature of our moral thinking. In a nutshell, my solution is this: It could not be otherwise. Accepting the asymmetry is indispensable for the construction and maintenance of stable moral communities. -/- My argument centrally involves mental resource management. Moral communities depend on their members’ commitment to moral norms. And, (...) I argue, community members can only be committed to these norms in the required ways without overextending their mental resources if morality contains the doing/allowing asymmetry. -/- This has two noteworthy implications. Firstly, we can and should stop discussing the asymmetry in normative ethics and, instead, regard it as “moral axiom” we cannot but accept. Secondly, even consequentialists have to accept the asymmetry but, by doing so, their view loses part of its intuitive appeal. Lastly, I turn to an objection. One might think that my indispensability argument confuses issues in moral psychology with issues in normative ethics. I reject the objection on broadly pragmatist grounds. (shrink)