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)
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)
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)
In this paper 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)
Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. (...) Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument. (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)
Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indispensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which reference is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I argue that these two arguments are in conflict with each other. Whereas (...) the indispensability argument supports realism about mathematics, the indeterminacy of reference argument, when applied to mathematics, provides a powerful strategy in support of mathematical anti-realism. I conclude the paper by indicating why the indeterminacy of reference phenomenon should be preferred over the considerations regarding indispensability. In the end, even the Quinean shouldn’t be a realist (platonist) about mathematics. (shrink)
Penelope Maddy and Elliott Sober recently attacked the confirmational indispensability argument for mathematical realism. We cannot count on science to provide evidence for the truth of mathematics, they say, because either scientific testing fails to confirm mathematics (Sober) or too much mathematics occurs in false scientific theories (Maddy). I present a pragmatic indispensability argument immune to these objections, and show that this argument supports mathematical realism independently of scientific realism. Mathematical realism, it turns out, may be even more (...) firmly established than scientific realism. (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.
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. (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)
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)
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.
For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking (...) my cue from Mark Colyvan''s (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational'', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way. (shrink)
Quine and Putnam argued for mathematical realism on the basis of the indispensability of mathematics to science. They claimed that the mathematics that is used in physical theories is confirmed along with those theories and that scientific realism entails mathematical realism. I argue here that current theories of confirmation suggest that mathematics does not receive empirical support simply in virtue of being a part of well confirmed scientific theories and that the reasons for adopting a realist view of scientific (...) theories do not support realism about mathematical entities, despite the use of mathematics in formulating scientific theory. (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.
Mark Colyvan uses applications of mathematics to argue that mathematical entities exist. I claim that his argument is invalid based on the assumption that a certain way of thinking about applications, called `the mapping account,' is correct. My main contention is that successful applications depend only on there being appropriate structural relations between physical situations and the mathematical domain. As a variety of non-realist interpretations of mathematics deliver these structural relations, indispensability arguments are invalid.
When the indispensability argument for mathematical entities (IA) is spelled out, it would appear confirmational holism is needed for the argument to work. It has been argued that confirmational holism is a dispensable premise in the argument if a construal of naturalism, according to which it is denied that we can take different epistemic attitudes towards different parts of our scientific theories, is adopted. I argue that the suggested variety of naturalism will only appeal to a limited number of (...) philosophers. I then suggest that if we allow for some degree of separation between different component parts of theories, IA can be formulated as an argument aimed at more than a limited number of philosophers, but in implementing this strategy the notion of indispensability needs spelling out. The best way of spelling out indispensability is in terms of theory contribution, but doing so requires adopting inference to the best explanation (IBE). IBE is however sufficient for establishing the conclusion that IA is supposed to establish. Thus, IA is a redundant argument. (shrink)
The Quine-Putnam indispensability argument urges us to place mathematical entities on the same ontological footing as (other) theoretical entities of empirical science. Recently this argument has attracted much criticism, and in this paper I address one criticism due to Elliott Sober. Sober argues that mathematical theories cannot share the empirical support accrued by our best scientific theories, since mathematical propositions are not being tested in the same way as the clearly empirical propositions of science. In this paper I defend (...) the Quine-Putnam argument against Sober's objections. (shrink)
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). (shrink)
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. (shrink)
In the early- to mid- 1870s, William James started to argue that if one needs to believe something, then one ought to believe it, even if there is no evidence in its favor. It is not easy to unwind the various things that James said about what he called the will to believe, but one thing is clear. He was initially tempted to put forward a very strong point and despite the refinements he was eventually to make, his is the (...) most contentious version of pragmatist indispensability arguments. Most importantly, it set the stage for how pragmatism was to evolve. In some remarks made in an 1875 review in the Nation2 and in the penultimate draft of “The Will to Believe,” James argues that, given the dearth of .. (shrink)
In this paper I do two things: (1) I support the claim that there is still some confusion about just what the Quine-Putnam indispensability argument is and the way it employs Quinean meta-ontology and (2) I try to dispel some of this confusion by presenting the argument in a way which reveals its important meta-ontological features, and include these features explicitly as premises. As a means to these ends, I compare Peter van Inwagen’s argument for the existence of properties (...) with Putnam’s presentation of the indispensability argument. Van Inwagen’s argument is a classic exercise in Quinean meta-ontology and yet he claims – despite his argument’s conspicuous similarities to the Quine-Putnam argument – that his own has a substantially different form. I argue, however, that there is no such difference between these two arguments even at a very high level of specificity; I show that there is a detailed generic indispensability argument that captures the single form of both. The arguments are identical in every way except for the kind of objects they argue for – an irrelevant difference for my purposes. Furthermore, Putnam’s and van Inwagen’s presentations make an assumption that is often mistakenly taken to be an important feature of the Quine-Putnam argument. Yet this assumption is only the implicit backdrop against which the argument is typically presented. This last point is brought into sharper relief by the fact that van Inwagen’s list of the four nominalistic responses to his argument is too short. His list is missing an important – and historically popular – fifth option. (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)
Some philosophers understand epistemological skepticism as merely presenting a paradox to be solved, a paradox given rise to by some apparently forceful arguments. I argue that such a view needs to be justified, and that the best way to do so is to show that we cannot help seeing skepticism as obviously false. The obviousness (to us) of the falsity of skepticism is, I suggest, explained by the fact that we cannot live without knowledge-beliefs (a knowledge-belief about the world is (...) a belief that a person or a group of people know that p, where p is an empirical proposition about the world). I then go on to argue for the indispensability of knowledge-beliefs. The first line of argument appeals to the practical aspects of our employment of the concept of knowledge, and the second line of argument draws on some Davidsonian ideas concerning understanding and massive agreement. (shrink)
Instances of explanatory reduction are often advocated on metaphysical grounds; given that the only real things in the world are subatomic particles and their interaction, we have to try to explain everything in terms of the laws of physics. In this paper, we show that explanatory reduction cannot be defended on metaphysical grounds. Nevertheless, indispensability arguments for reductive explanations can be developed, taking into account actual scientific practice and the role of epistemic interests. Reductive explanations might be indispensable to (...) address some epistemic interest answering a specific explanation-seeking question in the most accurate, adequate and efficient way. Just like explanatory pluralists often advocate the indispensability of higher levels of explanation pointing at the pragmatic value of the explanatory information obtained on these higher levels, we argue that explanatory reduction—traditionally understood as the contender of pluralism—can be defended in a similar way. The pragmatic value reductionist, lower level explanations might have in the biomedical sciences and the social sciences is illustrated by some case studies. (shrink)
An emphasis on explanatory contribution is central to a recent formulation of the indispensability argument (IA) for mathematical realism. Because scientific realism is argued for by means of inference to the best explanation (IBE), it has been further argued that being a scientific realist entails a commitment to IA and thus to mathematical realism. It has, however, gone largely unnoticed that the way that IBE is argued to be truth conducive involves citing successful applications of IBE and tracing this (...) success over time. This in turn involves identifying those constituents of scientific theories that are responsible for their predictive success and showing that these constituents are retained across theory change in science. I argue that even if mathematics can be shown to feature in best explanations, the role of mathematics in scientific theories does not satisfy the condition that mathematics is always retained across theory change. According to a scientific realist, this condition needs to be met for making ontological claims on the basis of explanatory contribution. Thus scientific realists are not committed to mathematical realism on the basis of this recent formulation of IA. (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)
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)
Indispensability arguments purport to show that empirical data provide evidence for the existence of mathematical entities. In this paper we argue that indispensability arguments fail to show that empirical data bears on the mathematical. In order to show this we attempt to clarify what it is to be mathematical , and separate the question whether abstracta in general exist from whether distinctively mathematical abstracta exist. We introduce the notion of an ‘agnostified’ empirical theory as a heuristic device for (...) clarifying the role of mathematical terminology in empirical theories. The special case of spaces is then briefly considered. (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.
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.
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)
Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especially that of Friedman et al. and Feferman et al. Some recent results (of Pour-El and Richards (...) and of the author) are then presented bearing on the first question on the need for non-constructive analysis, especially for quantum physics. Despite the impressive work of Bishop et al. in constructive analysis, Hilbert's objection to intuitionism still carries significant force, and may be decisive depending in part on one's conception of "physics". (shrink)
Both the defenders and the challengers of the indispensability argument seem to ignore the obvious fact that it is meant to be an analogical inference. In this note, I shall draw attention to this fact so as to avoid unnecessary confusions in any future discussion of the indispensability argument. For this purpose, I shall criticize Maddy’s version of the indispensability argument. After having noted that Quinean holism does not have to be one of the necessary premises, I (...) shall suggest alternative formulations of the indispensability argument as an analogical inference. Also, some further reflections on how to evaluate Maddy’s objections to the indispensability argument will be in due order. (shrink)
We defend Joseph Melia's thesis that the role of mathematics in scientific theory is to 'index' quantities, and that even if mathematics is indispensable to scientific explanations of concrete phenomena, it does not explain any of those phenomena. This thesis is defended against objections by Mark Colyvan and Alan Baker.
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)
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)
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)
By carefully examining one of the most famous thought experiments in the history of science—that by which Galileo is said to have refuted the Aristotelian theory that heavier bodies fall faster than lighter ones—I attempt to show that thought experiments play a distinctive role in scientific inquiry. Reasoning about particular entities within the context of an imaginary scenario can lead to rationally justified concluusions that—given the same initial information—would not be rationally justifiable on the basis of a straightforward argument.
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)
The ramifications are explored of taking physical theories to commit their advocates only to ‘physically real’ entities, where ‘physically real’ means ‘causally efficacious’ (e.g., actual particles moving through space, such as dust motes), the ‘physically significant’ (e.g., centers of mass), and the merely mathematical—despite the fact that, in ordinary physical theory, all three sorts of posits are quantified over. It's argued that when such theories are regimented, existential quantification, even when interpreted ‘objectually’ (that is, in terms of satisfaction via variables, (...) rather than by substitution-instances) need not imply any ontological commitments. (shrink)
Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical (...) objects. The most common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. (shrink)
Philosophy of mathematics for the last half-century has been dominated in one way or another by Quine’s indispensability argument. The argument alleges that our best scientific theory quantifies over, and thus commits us to, mathematical objects. In this paper, I present new considerations which undermine the most serious challenge to Quine’s argument, Hartry Field’s reformulation of Newtonian Gravitational Theory.
The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra-mathematical explanation (the explanation of physical facts by mathematical facts). In this paper, I identify a new case of extra-mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra-mathematical explanation in science.
In this paper I focus on the central role faith plays in the thought of Polanyi and Voegelin. I begin by indicating how both find the modern conception of scientific knowing seriously wanting. What Polanyi terms "objectivism" and Voegelin calls "scientism" is the modern tendency to reduce knowledge to only that which can be scientifically demonstrated. This errant view of knowledge does not occur in a vacuum, though, and both men draw a connection between this and the political pathologies of (...) the twentieth century. I then show the complementary ways in which these two thinkers believe recovery is possible: an epistemological solution encompassed in Polanyi's personal knowledge and an ontological reorientation that is the core of Voegelin's insistence that we must recover an awareness of human participation in transcendent reality. (shrink)
I offer a theory of propositional attitudeascriptions that reconciles a number of independently plausiblesemantic principles. At the heart of the theory lies the claim thatpsychological verbs (such as ``to believe'' and ``to doubt'') vary incontent indexically. After defending this claim and explaining how itrenders the aforementioned principles mutually compatible, I arguethat my account is superior to currently popular hidden indexicaltheories of attitude ascription. To conclude I indicate a number oframifications that the proposed theory has for issues in epistemology,philosophy of mind, (...) and formal semantics. (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.
In this paper I argue that narratives concerning Periclean Athens have mistakenly imposed modern conceptions of enlightenment onto the Greek world,and have therefore been blinded to crucial aspects of Socrates’s practice of moral reason giving. In contrast to the Kantian conception of enlightenment, which puts forth an image of the ideally enlightened person as an autonomous reasoner, one who refuses to be guided by another and who has the courage to throw off the chains of tradition and “think for oneself,” (...) I argue that Socrates provides us with a much different picture of the enlightened individual. Socrates’s practice of moral reasoning does not take the form of autonomous rationality that is antithetical to tradition, but rather his practice recognizes the rightful authority of tradition and custom in moral reason giving. Thus, rather than characterizing Socrates through a Kantian enlightenment reading, this paper argues that we should study Socrates through the lens of the sensus communis, a humanist concept articulated well by Giambattista Vico and Hans Georg-Gadamer. (shrink)
According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the concrete world (...) is just as it in fact is, then T’ bear on this claim. It concludes that, while analogies with theories that attempt to eliminate unobservable concrete entities provide good reason to regard theories of the former sort as explanatorily bad, this reason does not apply to theories of the latter sort. (shrink)
Aidan Lyon has recently argued that some mathematical explanations of empirical facts can be understood as program explanations. I present three objections to his argument.
I call attention to Berkeley’s treatment of a Newtonian indispensability argument against his own main position. I argue that the presence of this argument marks a significant moment in the history of philosophy and science: Newton’s achievements could serve as a separate and authoritative source of justification within philosophy. This marks the presence of a new kind of naturalism. A long the way, I argue against the claim tha t there is no explicit opposition or distinction between “philosophy” and (...) “science” until the nineteenth century. Finally, I argue for the conceptual unity between Berkeley’s immaterialism and instrumentalism. I argue that Berkeley’s commitment to immaterialism requires his reinterpretation of science and, thus, the adoption of instrumentalism. (shrink)
The Critical Imagination is a study of metaphor, imaginativeness, and criticism of the arts. Since the eighteenth century, many philosophers have argued that appreciating art is rewarding because it involves responding imaginatively to a work. Literary works can be interpreted in many ways; architecture can be seen as stately, meditative, or forbidding; and sensitive descriptions of art are often colourful metaphors: music can 'shimmer', prose can be 'perfumed', and a painter's colouring can be 'effervescent'. Engaging with art, like creating it, (...) seems to offer great scope for imagination. Hume, Kant, Oscar Wilde, Roger Scruton, and others have defended variations on this attractive idea. In this book, James Grant critically examines it. The first half explains the role imaginativeness plays in criticism. To do this, Grant answers three questions that are of interest in their own right. First, what are the aims of criticism? Is the point of criticizing a work to evaluate it, to explain it, to modify our response to it, or something else? Second, what is it to appreciate art? Third, what is imaginativeness? He gives new answers to all three questions, and uses them to explain the role of imaginativeness in criticism. The book's second half focuses on metaphor. Why are some metaphors so effective? How do we understand metaphors? Are some thoughts expressible only in metaphor? Grant's answers to these questions go against much current thinking in the philosophy of language. He uses these answers to explain why imaginative metaphors are so common in art criticism. The result is a rigorous and original theory of metaphor, criticism, imaginativeness, and their interrelations. (shrink)
It is a popular view amongst some philosophers, most notably those with Quinean views about ontological commitment, that scientific theories are first-orderizable; that we can regiment all such theories in an extensional first-order language. I argue that this view is false, and that any acceptable account of science needs to take some modal notion as primitive.
Mereological Nihilism is the thesis that no material object has proper parts; every material object is a simple. Recent developments in plural semantics have made it possible to develop and motivate this position. In particular, some have argued that the tools of plural reference and quantification enable us to systematically paraphrase true statements apparently about composites into statements that only concern simples. Are composites really surplus to philosophical requirements? Given the resources of plural semantics, what must the world be like (...) if composites are to be theoretically indispensable? I will describe and defend the possibility of scenario in which mention of composites cannot be paraphrased. We will therefore come to appreciate one way in which the world would have to be in order for composites to be required and for Nihilism to fail. (shrink)
I explicate and defend the claim that, fundamentally speaking, there are no numbers, sets, properties or relations. The clarification consists in some remarks on the relevant sense of ‘fundamentally speaking’ and the contrasting sense of ‘superficially speaking’. The defence consists in an attempt to rebut two arguments for the existence of such entities. The first is a version of the indispensability argument, which purports to show that certain mathematical entities are required for good scientific explanations. The second is a (...) speculative reconstruction of Armstrong's version of the One Over Many argument, which purports to show that properties and relations are required for good philosophical explanations, e.g. of what it is for one thing to be a duplicate of another. (shrink)
In this essay, I defend a view I call “Robust Realism” about normativity. According to this view, there are irreducibly, perfectly objective, normative truths, that when successful in our normative inquiries we discover rather than create or construct. My argument in support of Robust Realism is modeled after arguments from explanatory indispensability common in the philosophy of science and the philosophy of mathematics. I argue that irreducibly normative truths, though not explanatorily indispensable, are nevertheless deliberatively indispensable, and that this (...) kind of indispensability is just as respectable as the more familiar explanatory kind. Deliberative indispensability, I argue, justifies belief in normative facts, just like the explanatory indispensability of, say, theoretical entities like electrons justifies belief in electrons. In the introduction I characterize the view I will be arguing for and sketch the main argument of this essay. In chapter 1 I draw the analogy between explanatory and deliberative indispensability, and argue that there is no non-question-begging reason to take the former but not the latter seriously. Here I also present the master-argument of the thesis, and clarify the argumentative work that needs to be done by each of the following chapters. In chapter 2 I address the worries of the antirealist who is willing to reject arguments from explanatory indispensability as well. In other words, in this chapter I try to justify the move from indispensability (of whatever kind) to belief. In chapter 3 I develop an account of deliberation that supports the premises about deliberation needed for my master-argument to go through. In chapter 4 I reject some alternative views, showing that none of them can allow for sincere deliberation. In this chapter, in other words, I support the indispensability premise: I argue that it really is impossible to deliberate sincerely without believing in irreducibly normative truths.. (shrink)
In Word and Object, Quine acknowledged the "practical indispensability" in daily life of the intentional idioms of belief and desire but disparaged such talk as an "essentially dramatic idiom" rather than something from which real science could be made in any straightforward way.Endnote 1 Many who agree on little else have agreed with Quine about this, and have gone on to suggest one or another indirect way for science to accommodate folk psychology: Sellars, Davidson, Putnam, Rorty, Stich, the Churchlands, (...) Schiffer and myself, to name a few. This fainthearted consensus is all wrong, according to Fodor, whose new book is a vigorous--even frantic--defense of what he calls Intentional Realism: beliefs and desires are real, causally involved, determinately contentful states. "We have no reason to doubt," Fodor says, "that it is possible to have a scientific psychology that vindicates commonsense belief/desire explanation." (p.16). (shrink)
Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...) potential future directions of research for each side in the debate over the existence of abstract mathematical objects. Introduction: Mathematical Explanation Indispensability and Explanation Is the Mathematics Indispensable to the Explanation? 3.1 Object-level arbitrariness 3.2 Concept-level arbitrariness 3.3 Theory-level arbitrariness Is the Explanandum ‘Purely Physical’? Is the Mathematics Explanatory in Its Own Right? Does Inference to the Best Explanation Apply to Mathematics? 6.1 Leng's first argument 6.2 Leng's second argument 6.3 Leng's third argument Conclusions CiteULike Connotea Del.icio.us What's this? (shrink)
A central doctrine of Kant's Critique of Pure Reason holds that the content of human experience is rooted in an affection of sensibility by unknowable things in themselves. This famous and puzzling affection doctrine raises two seemingly intractable old problems, which can be termed the Indispensability and the Consistency Problems. By what right does Kant present affection by supersensible entities as an indispensable requirement of experience? And how could any argument for such indispensability avoid violating the Critique's doctrine (...) of noumenal ignorance? This essay develops a new solution to both problems, setting out from the continuity between Kant's early and mature views on sensibility and mind-world relations. Kant's early writings subscribe to an interactionist cosmology opposed to both Leibniz's preestablished harmony and Malebranche's occasionalism. The modern debate on mind-world relations shaping Kant's early cosmology points us to a widely recognized motivation for interactionism, turning on a constraint on agency within certain noninteractionist cosmologies. In particular, Kant's early conversion to a libertarian theory of freedom, together with his rejection of occasionalism, provides the basis for a compelling argument for the indispensability of world-mind affection relations. Extended to the transcendental idealist framework, the same argument reveals noumenal affection as an indispensable presupposition of some knowledge claims consistently upheld by Kant. This leads in turn to a satisfying solution to the Consistency Problem, showing that the doctrine of noumenal affection is not merely consistent with, but is partly motivated by, Kant's commitment to noumenal ignorance. CiteULike Connotea Del.icio.us Digg Reddit Technorati What's this? (shrink)
In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good (...) argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)
The term queer has recently come into wide use to designate distinctive emphases in the politics and the intellectual study of sexuality. This article explores the unfortunate irony that most work falling under the rubric of queer theory has been undertaken largely at some remove from the discipline of sociology, despite the pioneering role that an earlier generation of sociologists played in formulating influential conceptions of the social construction of sexuality. The article suggests important continuities between the earlier sociological theories (...) and recent queer theory, but also analyzes the new challenges that queer theorists have posed by insisting on the indispensability of questions of sexual "marginality" to the larger understanding of social and cultural organization. The article concludes by suggesting how sociologists might engage with such a project. (shrink)
Many philosophers claim that metaphor is indispensable for various purposes. What I shall call the ‘Indispensability Thesis’ is the view that we use at least some metaphors to think, to express, to communicate, or to discover what cannot be thought, expressed, communicated, or discovered without metaphor. I argue in this paper that support for the Indispensability Thesis is based on several confusions. I criticize arguments presented by Stephen Yablo, Berys Gaut, Richard Boyd, and Elisabeth Camp for the (...) class='Hi'>Indispensability Thesis, and distinguish it from several plausible claims with which it is easily confused. Although I do not show that the thesis is false, I provide seven grounds for suspicion of our sense (if we have it) that some metaphors are indispensable for the purposes claimed by advocates of the Indispensability Thesis. CiteULike Connotea Del.icio.us What's this? (shrink)
I start by reconsidering two familiar arguments against modal realism. The argument from epistemology relates to the issue whether we can infer the existence of concrete objects by a priori means. The argument from pragmatics purports to refute the analogy between the indispensability of possible worlds and the indispensability of unobserved entities in physical science and of numbers in mathematics. Then I present two novel objections. One focusses on the obscurity of the notion of isolation required by modal (...) realism. The other stresses the arbitrary nature of the rules governing the behaviour of Lewisean universes. All four objections attack the reductive analysis of modality that is supposed to be the chief merit of modal realism. (shrink)
Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.
Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)
The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine’s “Indispensability Argument”, which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to (...) enlarge the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics . I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field’s nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field’s fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer’s argument for his thesis that fictionalism is “the best version of anti-realistic anti-platonism”. (shrink)
Deflationsism about truth is a pot-pourri, variously claiming that truth is redundant, or is constituted by the totality of 'T-sentences', or is a purely logical device (required solely for disquotational purposes or for re-expressing finitarily infinite conjunctions and/or disjunctions). In 1980, Hartry Field proposed what might be called a 'deflationary theory of mathematics', in which it is alleged that all uses of mathematics within science are dispensable. Field's criterion for the dispensability of mathematics turns on a property of theories, called (...) conservativeness. I present some technical results, some of which may be found in Tarski (1936), concerning the logical properties of truth theories; in particular, concerning the conservativeness of adding a truth theory for an object level language to any theory expressed in it. It transpires that various deflationary truth theories behave somewhat differently from the standard Tarskian truth theory. These results suggest that Tarskian theories of truth are not redundant or dispensable. Finally, I hint at an analogy between the behaviour of mathematical theories and of standard (Tarskian) theories of truth with respect to their indispensability to, as Quine would put, our 'scientific world-view'. (shrink)
The thought that there is a way to reconcile empiricism with a realist stance towards scientific theories, avoiding instrumentalism and without fearing that this will lead straight to metaphysics, seems very promising. This paper aims to articulate this thought. It consists of two parts. The first (sections 2 and 3) will articulate how empiricism can go for scientific realism without metaphysical anxiety. It will draw on the work of Moritz Schlick, Hans Reichenbach and Herbert Feigl to develop an indispensability (...) argument for the adoption of the realist framework. This argument, unlike current realist arguments, has a pragmatic ring to it: there is no ultimate argument for the adoption of the realist framework. The guiding thought here is that fundamental ontic questions are not dealt with in the same way in which questions about the reality of ordinary entities (be they stones or electrons) are dealt with—the ontic framework must already be in place before questions about the reality of specific entities are raised. The second part (sections 4 and 5) will articulate reasons for avoiding instrumentalism. Most space is given in offering reasons to refrain from adopting P. Kyle Stanford’s (2006) neo-instrumentalism—a very sophisticated version of instrumentalism that seems to work within the realist framework and promises empiricists a way to avoid scientific realism. Scientific realism is alive and well because of Ti(a)na: there is (almost) no alternative. However, in section 6, it will be argued that there is room for rapprochement between contextualist instrumentalism and scientific realism. The paper is accompanied by an appendix in which Reichenbach’s argument for scientific realism is presented and discussed. (shrink)
Following the pattern set by the early German Romantics, Kierkegaard conveys many of his insights through literature rather than academic prose. What makes him a valuable member of this tradition is the theory he develops to support it, his so-called “theory of indirect communication.” The most exciting aspect of this theory concerns the alleged importance of indirect communication: Kierkegaard claims that there are some projects only it can accomplish. This paper provides a critical account of two arguments Kierkegaard offers in (...) defense of this claim. The first argument is that he needs to use indirect communication in order to discourage people from losing themselves in the “crowd”. The second argument is that he needs to use it in order to help people out of a “monstrous illusion”. It is shown that while both arguments justify Kierkegaard’s decision to use indirect communication, neither one supports the original claim about its indispensability. (shrink)
The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in (...) light of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above.. (shrink)
Identity, we're told, is the binary relation that every object bears to itself, and to itself only. But how can a relation be binary if it never relates two objects? This puzzled Russell and led Wittgenstein to declare that identity is not a relation between objects. The now standard view is that Wittgenstein's position is untenable, and that worries regarding the relational status of identity are the result of confusion. I argue that the rejection of identity as a binary relation (...) is perfectly tenable. To this end, I outline and defend a logical framework that is not committed to an objectual identity relation but is nevertheless expressively equivalent to first-order logic with identity. After it has thus been shown that there is no indispensability argument for objectual identity, I argue that we have good reasons for doubting the existence of such a relation, and rebut a number of attempts at discrediting these reasons. (shrink)
