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.

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas (...) is genuinely mathematical, according to Baker :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (shrink)

The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I (...) will call ‘optimal representations’, which are representations that capture all that is relevant to explain a physical phenomenon at a specified level of description. In the end, because the role of mathematics in MEPPs is ultimately representational, they cannot be used to support mathematical Platonism. I finish the paper by addressing the claim, advanced by many EIA defendants, that quantification over mathematical objects results in explanations that have more theoretical virtues, especially that they are more general and modally stronger than alternative explanations. I will show that the EIA cannot be successfully defended by appealing to these notions. (shrink)

Platonists in mathematics endeavour to prove the truthfulness of the proposal about the existence of mathematical objects. However, there have not been many explicit proofs of this proposal. One of the explicit ones is doubtlessly Baker’s Enhanced Indispensability Argument, formulated as a sort of modal syllogism. We aim at showing that the purpose of its creation – the defence of Platonist viewpoint – was not accomplished. Namely, the second premise of the Argument was imprecisely formulated, which (...) gave space for various interpretations of the EIA. Moreover, it is not easy to perceive which of the more precise formulations of the above-mentioned premise would be acceptable. For all these reasons, it is disputable whether the EIA can be used to defend Platonist outlook. At the beginning of this century, Baker has shown that the so-called Quine-Putnam Indispensability Argument can not provide “full” platonism - a guarantee of the existence of all mathematical objects. It turns out, however, that the EIA has a similar disadvantage. (shrink)

The contemporary Platonists in the philosophy of mathematics argue that mathematical objects exist. One of the arguments by which they support this standpoint is the so-called Enhanced Indispensability Argument (EIA). This paper aims at pointing out the difficulties inherent to the EIA. The first is contained in the vague formulation of the Argument, which is the reason why not even an approximate scope of the set objects whose existence is stated by the Argument can be (...) established. The second problem is reflected in the vagueness of the very term indispensability, which is essential to the Argument. The paper will remind of a recent definition of the concept of indispensability of a mathematical object, reveal its deficiency and propose an improvement of this definition. Following this, we will deal with one of the consequences of the arbitrary employment of the concept of indispensability of a mathematical theory. We will propose a definition of this concept as well, in accordance with the common intuition about it. Eventually, on the basis of these two definitions, the paper will describe the relation between these two concepts, in the attempt to clarify the conceptual apparatus of the EIA. (shrink)

This discussion note points to some verbal imprecisions in the formulation of the Enhanced Indispensability Argument. The examination of the plausibility of alternative interpretations reveals that the argument’s minor premise should be understood as a particular, not a universal, statement. Interpretations of the major premise and the conclusion oscillate between de re and de dicto readings. The attempt to find an appropriate interpretation for the EIA leads to undesirable results. If assumed to be valid and sound, (...) the argument warrants the rationality of the belief in an unusual variant of Platonism. On the other hand, if taken as it stands, the argument is either invalid or is unsound or does not support the mathematical Platonism. Thus, the EIA in its present form cannot serve as a useful device for the Platonist. (shrink)

In this paper I shall adopt a possible reading of the notions of ‘explanatory indispensability’ and ‘genuine mathematical explanation in science’ on which the Enhanced Indispensability Argument proposed by Alan Baker is based. Furthermore, I shall propose two examples of mathematical explanation in science and I shall show that, whether the EIA-partisans accept the reading I suggest, they are easily caught in a dilemma. To escape this dilemma they need to adopt some account of explanation and (...) offer a plausible answer to the following ‘question of evidence’: What is a genuine mathematical explanation in empirical science and on what basis do we consider it as such? Finally, I shall suggest how a possible answer to the question of evidence might be given through a specific account of mathematical explanation in science. Nevertheless, the price of adopting this standpoint is that the genuineness of mathematical explanations of scientific facts turns out to be dependent on pragmatic constraints and therefore cannot be plugged in EIA and used to establish existential claims about mathematical objects. (shrink)

The new explanatory or enhanced indispensability argument alleges that our mathematical beliefs are justified by their indispensable appearances in scientific explanations. This argument differs from the standard indispensability argument which focuses on the uses of mathematics in scientific theories. I argue that the new argument depends for its plausibility on an equivocation between two senses of explanation. On one sense the new argument is an oblique restatement of the standard argument. On (...) the other sense, it is vulnerable to an instrumentalist response. Either way, the explanatory indispensability argument is no improvement on the standard one. (shrink)

The author of “Evidence, Explanation, Enhanced Indispensability” advances a criticism to the Enhanced Indispensability Argument and the use of Inference to the Best Explanation in order to draw ontological conclusions from mathematical explanations in science. His argument relies on the availability of equivalent though competing explanations, and a pluralist stance on explanation. I discuss whether pluralism emerges as a stable position, and focus here on two main points: whether cases of equivalent explanations have been (...) actually offered, and which ontological consequences should follow from these. (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)

This paper discusses Baker’s Enhanced Indispensability Argument for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use (...) still does not support realism about mathematical objects. (shrink)

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)

Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

In the debate over whether mathematical facts, properties, or entities explain physical events (in what philosophers call “extra-mathematical” explanations), Aidan Lyon’s (2012) affirmative answer stands out for its employment of the program explanation (PE) methodology of Frank Jackson and Philip Pettit (1990). Juha Saatsi (2012; 2016) objects, however, that Lyon’s examples from the indispensabilist literature are (i) unsuitable for PE, (ii) nominalizable into non-mathematical terms, and (iii) mysterious about the explanatory relation alleged to obtain between the PE’s mathematical explanantia and (...) physical explananda. In this paper, I propose a counterexample to Saatsi’s objections. My counterexample is Frank Jackson’s (1998a) program explanation for color experience, which I argue needs recasting as an extra-mathematical PE due to its implicit reliance on reflectance, a property that suffers conceptual regress unless redefined with Fourier harmonics. Pace Saatsi, I argue that this recast example is an authoritative PE, non-nominalizable, and minimally esoteric. Important for the indispensability debate at large, moreover, is that my counterexample reifies Fourier harmonics without the Enhanced Indispensability Argument (an argument to which Lyon applies PE as a premise). Indispensabilists have long overlooked the conditionalization of a limited mathematical realism on property realism, and my counterexample to Saatsi exploits this conditionalization. (shrink)

Linkage arguments, which defend a controversial right by showing that it is indispensable or highly useful to an uncontroversial right, are sometimes used to defend the right to health care. This article evaluates such arguments when used to defend RHC. Three common errors in using linkage arguments are neglecting levels of implementation, expanding the scope of the supported right beyond its uncontroversial domain, and giving too much credit to the supporting right for outcomes in its area. A familiar linkage (...) class='Hi'>argument for RHC focuses on its contributions to the right to life. Among the problems with this argument are that it requires a positive conception of the right to life that is not uncontroversial and that it only justifies the subset of RHC that seeks to prevent loss of life. A linkage argument for RHC with better prospects claims that a well-realized right to health care enhances the realization of a number of uncontroversial rights. (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)

Enhanced indispensability arguments claim that Scientific Realists are committed to the existence of mathematical entities due to their reliance on Inference to the best explanation. Our central question concerns this purported parity of reasoning: do people who defend the EIA make an appropriate use of the resources of Scientific Realism to achieve platonism? We argue that just because a variety of different inferential strategies can be employed by Scientific Realists does not mean that ontological conclusions concerning which things (...) we should be Scientific Realists about are arrived at by any inferential route which eschews causes, and nor is there any direct pressure for Scientific Realists to change their inferential methods. We suggest that in order to maintain inferential parity with Scientific Realism, proponents of EIA need to give details about how and in what way the presence of mathematical entities directly contribute to explanations. (shrink)

Enhanced indispensability arguments seek to establish realism about mathematics based on the explanatory role that mathematics plays in science. Idealizations pose a problem for such arguments. Idealizations, in a similar way to mathematics, boost the explanatory credentials of our best scientific theories. And yet, idealizations are not the sorts of things that are supposed to attract a realist attitude. I argue that the explanatory symmetry between idealizations and mathematics can potentially be broken as follows: although idealizations contribute to (...) the explanatory power of our best theories, they do not carry the explanatory load. It is at least open however that mathematics is load-carrying. To give this idea substance, I offer an analysis of what it is to carry the explanatory load in terms of difference-making and counterfactuals. (shrink)

Baker (Mind 114:223–238, 2005; Brit J Philos Sci 60:611–633, 2009) has recently defended what he calls the “enhanced” version of the indispensability argument for mathematical Platonism. In this paper I demonstrate that the nominalist can respond to Baker’s argument. First, I outline Baker’s argument in more detail before providing a nominalistically acceptable paraphrase of prime-number talk. Second, I argue that, for the nominalist, mathematical language is used to express physical facts about the world. In endorsing (...) this line I follow moves made by Saatsi (Brit J Philos Sci 62(1):143–154, 2011). But, unlike Saatsi, I go on to argue that the nominalist requires a paraphrase of prime-number talk, for otherwise we lack an account of what that ‘physical fact’ is in the case of mathematics that seemingly makes reference to prime numbers. (shrink)

The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant (...) mathematical results. The nominalist’s challenge is thus to undermine the platonistic force of such explanations by way of alternative nominalistic ones. The author’s contention is that such nominalistic explanations should provide a paraphrase of the proofs of the mathematical results being applied. There are reasons to doubt that proofs, construed here as formal derivations, actually contribute to the platonistc force to be undermined and, by parity, that nominalized proofs should bear responsability for the corresponding undermining. A discussion of two examples and of associated arguments by Lange, Pincock, Steiner and Tallant, point to a a wealth of worries concerning the construal of this explanatory role. Among those figure the distinction between the weak and strong role of proofs, the distinction between causal or “ordinary” explanations and genuine mathematical ones, and the unifying role of optimization explanations. More generally, the very idea that the explanatory advantages yielded by applied mathematical claims may be construed as gradual or progressive and the associated notion that the feasibility of their nominalistic paraphrases decreases as the generality and force of these claim increases, deserves a closer attention. (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)

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...) indispensability argument and Baker’s enhanced indispensability argument. (shrink)

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

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)

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)

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)

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)

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)

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

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

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)

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.

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)

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)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise (...) content to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (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.

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)

This book includes detailed critical analysis of a wide variety of versions of the indispensability argument, as well as a novel approach to traditional views about mathematics.

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 indispensability argument is a method for showing that abstract mathematical objects exist. Various versions of this argument have been proposed. Lately, commentators seem to have agreed that a holistic indispensability argument 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 —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. (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)

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)

Enhancement, the improvement of mental capacities with psychoactive substances and technologies has stimulated one of the largest debates in contemporary bioethics. Surprisingly few participants in this debate take note of the tendentious legal status of psychoactive pharmaceuticals as the primary means of enhancement. -/- Enhancement technologies and substances have measurable effects on specific measurable cognitive functions. A major issue of contention in the debate is how to evaluate these effects, i. e. which theory of value to use. It is contested (...) whether the pleasures and achievements resulting from the use of enhancement can be counted as such or can be ignored or devaluate by calling them fraudulent or inauthentic. -/- The alleged and real benefits of enhancement are not expected to be evenly distributed, nor is it always clear whether enhancement is a zero-sum game or has benefits even for the non-users. This results in a complex structure of risks and benefits for individuals and groups, which needs to be broken down in detail. (shrink)

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)

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.

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

This article elaborates the epistemic indispensability argument, which fully embraces the epistemic contribution of mathematics to science, but rejects the contention that such a contribution is a reason for granting reality to mathematicalia. Section 1 introduces the distinction between ontological and epistemic readings of the indispensability argument. Section 2 outlines some of the main flaws of the first premise of the ontological reading. Section 3 advances the epistemic indispensability argument in view of both applied (...) and pure mathematics. And Sect. 4 makes a case for the epistemic approach, which firstly calls into question the appeal to inference to the best explanation in the defense of the indispensability claim; secondly, distinguishes between mathematical and physical posits; and thirdly, argues that even though some may think that inference to the best explanation works in the postulation of physical posits, no similar considerations are available for postulating mathematicalia. (shrink)

An emphasis on explanatory contribution is central to a recent formulation of the indispensability argument for mathematical realism. Because scientific realism is argued for by means of inference to the best explanation, 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)