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)

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)

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)

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

Indispensability arguments for mathematical realism are commonly traced back to Quine. We identify two different Quinean strands in the interpretation of IA, what we label the ‘logical point of view’ and the ‘theory-contribution’ point of view. Focusing on each of the latter, we offer two minimal versions of IA. These both dispense with a number of theoretical assumptions commonly thought to be relevant to IA. We then show that the attribution of both minimal arguments to Quine is controversial, and (...) stress the extent to which this is so in both cases, in order to attain a better appreciation of the Quinean heritage of IA. (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)

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)

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)

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)

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)

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)

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set (...) of sentences in the nominalist language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the concrete world that may not be expressible in nominalistically acceptable language. While I grant that this may make quantification over abstracta indispensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantify over abstracta whilst denying their existence and I explain how this apparently contradictory practice (a practice I call 'weaseling') is in fact coherent, unproblematic and rational. Finally, I examine the view that platonistic theories are simpler or more attractive than their nominalistic reformulations, and thus that abstract ought to be accepted into our ontology for the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderstand the kind of simplicity and attractiveness we seek. (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)

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)

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.

Regarding the determination of vulnerability, the bioethics community has univocally jettisoned “labelled groups”, groups whose membership confers a context-invariant “vulnerable” status to their members. While the usual reasons against the sole use of labelled groups to determine the vulnerability of individuals are sound, labelled groups as exemplars of vulnerability can play indispensable roles in bioethical reasoning. In this article, I argue against the wholesale jettisoning of labelled groups by showing how they can be useful.

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. (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)

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)

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.

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)

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.

Recent debates about mathematical ontology are guided by the view that Platonism's prospects depend on mathematics' explanatory role in science. If mathematics plays an explanatory role, and in the right kind of way, this carries ontological commitment to mathematical objects. Conversely, the assumption goes, if mathematics merely plays a representational role then our world-oriented uses of mathematics fail to commit us to mathematical objects. I argue that it is a mistake to think that mathematical representation is necessarily ontologically innocent and (...) that there is an argument from mathematics' representational capacity to Platonism. Given that it is common ground between the Platonist and nominalist that mathematics plays a representational role in science, this representationalist argument is to be preferred over the explanatory, or enhanced, indispensability argument. (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)

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

In Grundgesetze, Vol. II, §91, Frege argues that ‘it is applicability alone which elevates arithmetic from a game to the rank of a science’. Many view this as an in nuce statement of the indispensability argument later championed by Quine. Garavaso has questioned this attribution. I argue that even though Frege's applicability argument is not a version of ia, it facilitates acceptance of suitable formulations of ia. The prospects for making the empiricist ia compatible with a rationalist Fregean framework (...) appear thus much less dim than expected. Nonetheless, those arguing for such compatibility eventually face an hardly surmountable dilemma. (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 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)

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)

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)

Mathematical concepts are explications, in Carnap's sense, of vague or otherwise non-clear concepts; mathematical theories have an empirical and a deductive component. From this perspective, I argue that the empirical component of a mathematical theory may be tested together with the fruitfulness of its explications. Using these ideas, I furthermore give an argument for mathematical realism, based on the indispensability argument combined with a weakened version of confirmational holism.

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)

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)

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)

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)

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.

Contemporary philosophers are attracted by the Indispensability Claim, according to which emotions are indispensable in acquiring knowledge of some important values. The truth of this claim is often thought to depend on that of Emotional Dogmatism, the view that emotions justify evaluative judgements because they (seem to) make us aware of the relevant values. The aim of this paper is to show that the Indispensability Claim does not stand or fall with Emotional Dogmatism and that there is actually (...) an attractive alternative way of developing it. We first argue that the Indispensability Claim requires a deflationary account of the cognitive bases of emotions, which we call the Simple View and defend against a variety of criticisms. Armed with the Simple View, we show that the Indispensability Claim is made true by three roles of the emotions: they help us access the relevant evidence, provide a needed hook-up between evidence and judgement, and contribute to evaluative understanding. (shrink)

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)

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)

The expression ‘indispensability argument’ denotes a family of arguments for mathematical realism supported among others by Quine and Putnam. More and more often, Gottlob Frege is credited with being the first to state this argument in section 91 of the _Grundgesetze der Arithmetik_. Frege's alleged indispensability argument is the subject of this essay. On the basis of three significant differences between Mark Colyvan's indispensability arguments and Frege's applicability argument, I deny that Frege presents an indispensability argument (...) in that very often quoted section of the _Grundegesetze_. (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.

In his influential 1960 paper ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Eugene P. Wigner raises the question of why something that was developed without concern for empirical facts—mathematics—should turn out to be so powerful in explaining facts about the natural world. Recent philosophy of science has developed ‘Wigner’s puzzle’ in two different directions: First, in relation to the supposed indispensability of mathematical facts to particular scientific explanations and, secondly, in connection with the idea that aesthetic criteria (...) track theoretical desiderata such as empirical success. An important aspect of Wigner’s article has, however, been overlooked in these debates: his worries about the underdetermination of physical theories by mathematical frameworks. The present paper argues that, by restoring this aspect of Wigner’s argument to its proper place, Wigner’s puzzle may become an instructive case study for the teaching of core issues in the philosophy of science and its history. (shrink)

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)

In recent years, the so-calledindispensability argument has been given a lotof attention by philosophers of mathematics.This argument for the existence of mathematicalobjects makes use of the fact, neglected inclassical schools of philosophy of mathematics,that mathematics is part of our best scientifictheories, and therefore should receive similarsupport to these theories. However, thisobservation raises the question about the exactnature of the alleged connection betweenexperience and mathematics (for example: is itpossible to falsify empirically anymathematical theorems?). In my paper I wouldlike to address this (...) question by consideringthe explicit assumptions of different versionsof the indispensability argument. My primaryclaim is that there are at least three distinctversions of the indispensability argument (andit can be even suggested that a fourth,separate version should be formulated). I willmainly concentrate my discussion on thisvariant of the argument, which suggests thepossibility of empirical confirmation ofmathematical theories. A large portion of mypaper will focus on the recent discussion ofthis topic, starting from the paper by E.Sober, who in my opinion put reasonablerequirements on what is to be counted as anempirical confirmation of a mathematicaltheory. I will develop his model into threeseparate scenarios of possible empiricalconfirmation of mathematics. Using an exampleof Hilbert space in quantum mechanicaldescription I will show that the most promisingscenario of empirical verification ofmathematical theories has neverthelessuntenable consequences. It will be hypothesizedthat the source of this untenability lies in aspecific role which mathematical theories playin empirical science, and that what is subjectto empirical verification is not themathematics used, but the representabilityassumptions. Further I will undertake theproblem of how to reconcile the allegedempirical verification of mathematics withscientific practice. I will refer to thepolemics between P. Maddy and M. Resnik,pointing out certain ambiguities of theirarguments whose source is partly the failure todistinguish carefully between different sensesof the indispensability argument. For thatreason typical arguments used in the discussionare not decisive, yet if we take into accountsome metalogical properties of appliedmathematics, then the thesis that mathematicshas strong links with experience seems to behighly improbable. (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)