The classical (set-theoretic) concept of structure has become essential for every contemporary account of a scientific theory, but also for the metatheoretical accounts dealing with the adequacy of such theories and their methods. In the latter category of accounts, and in particular, the structural metamodels designed for the applicability of mathematics have struggled over the last decade to justify the use of mathematical models in sciences beyond their 'indispensability' in terms of either method or concepts/entities. In this paper, I argue (...) that these metamodels employ structures of different natures and epistemologies, and this diversity does pose a serious problem to the intended justification. (shrink)

This paper concerns an epistemological objection against mathematical platonism, due to Hartry Field.The argument poses an explanatory challenge – the challenge to explain the reliability of our mathematical beliefs – which the platonist, it’s argued, cannot meet. Is the objection compelling? Philosophers disagree, but they also disagree on (and are sometimes very unclear about) how the objection should be understood. Here I distinguish some options, and highlight some gaps that need to be filled in on the potentially most compelling version (...) of the argument. (shrink)

This book is in honour of Professor Anna-Sofia Maurin on her 50th birthday. It consists of eighteen essays on metaphysical issues written by Swedish and international scholars.

This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. It belongs (...) to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists. (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)

I revisit my 1993 paper on Putnam and mathematical realism focusing on the indispensability argument and how it has fared over the years. This argument starts from the claim that mathematics is an indispensable part of science and draws the conclusion, from holistic considerations about confirmation, that the ontology of science includes abstract objects as well as the physical entities science deals with.

Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in science? Can we still consider that (...) the above question can have an affirmative answer? My main aim here is to give an account that takes mathematics, in some of the cases discussed in the literature, as contributing to our understanding of physical phenomena despite not being explanatory. (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)

The conditions involved in the applicability of mathematics in science are the subject of ongoing debates. One of the best‐received approaches is the inferential account, which involves structural mappings and pragmatic considerations in a three‐step model. According to the inferential account, these pragmatic considerations happen in the immersion and interpretation stages, but not during derivation (symbol‐pushing in a mathematical formalism). In this work, I draw inspiration from the later Wittgenstein and make the case that the applicability of mathematics also rests (...) on pragmatic considerations at the mathematical derivation level. I make the further case that pragmatic considerations at the mathematical derivation level may substitute pragmatic considerations in the other two stages, thus showing how they are holistically interrelated. I illustrate these two points with the solution of a simple geometrical problem. (shrink)

The aim of this work is to apply information theoretic ideas to the notion of fundamentality. I will argue that if one adopts pancomputationalism as a metaphysics for the universe, then there are higher-level structures which are just as fundamental for computation as anything from microphysics.

In “Mathematical Truth,” Paul Benacerraf raises an epistemic challenge for mathematical platonists. In this paper, I examine the assumptions that motivate Benacerraf’s original challenge, and use them to construct a new causal challenge for the epistemology of mathematics. This new challenge, which I call ‘Gödel’s Gap’, appeals to intuitive insights into mathematical knowledge. Though it is a causal challenge, it does not rely on any obviously objectionable constraints on knowledge. As a result, it is more compelling than the original challenge. (...) It is also more general; the challenge applies equally to platonistic and non-platonistic accounts of mathematical truth. And it can be generalized beyond the case of mathematical knowledge to pose a challenge for, e.g., moral knowledge. (shrink)

The indispensability argument is sometimes seen as weakened by its reliance on a controversial premise of confirmation holism. Recently, some philosophers working on the indispensability argument have developed versions of the argument which, they claim, do not rely on holism. Some of these writers even claim to have strengthened the argument by eliminating the controversial premise. I argue that the apparent removal of holism leaves a lacuna in the argument. Without the holistic premise, or some other premise which facilitates the (...) transfer of evidence to mathematical portions of scientific theories, the argument is implausible. (shrink)

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not (...) decisive against the eleatic and sketch a way to capture the important intuitions behind both views. (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)

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)

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)

Scientific realism holds that the terms in our scientific theories refer and that we should believe in their existence. This presupposes a certain understanding of quantification, namely that it is ontologically committing, which I challenge in this paper. I argue that the ontological loading of the quantifiers is smuggled in through restricting the domains of quantification, without which it is clear to see that quantifiers are ontologically neutral. Once we remove domain restrictions, domains of quantification can include non-existent things, as (...) they do in scientific theorizing. Scientific realism would therefore require redefining without presupposing a view of ontologically committing quantification. (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)

Indispensability arguments are among the strongest arguments in support of mathematical realism. Given the controversial nature of their conclusions, it is not surprising that critics have supplied a number of rejoinders to these arguments. In this paper, I focus on one such rejoinder, Melia’s ‘Weasel Response’. The weasel is someone who accepts that scientific theories imply that there are mathematical objects, but then proceeds to ‘take back’ this commitment. While weaseling seems improper, accounts supplied in the literature have failed to (...) explain why. Drawing on examples of weaseling in more mundane contexts, I develop an account of the presumption against weaseling as grounded in a misalignment between two types of commitments. This is good news to the weasel’s opponents. It reinforces that they were right to question the legitimacy of weaseling. This account is also beneficial to the weasel. Uncovering the source of the presumption against weaseling also serves to draw out the challenge that the weasel must meet to override this presumption—what is required to be an ‘honest weasel’. (shrink)

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)

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)

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. (...) In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception. (shrink)

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...) the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content that does not imply the existence of mathematical objects. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by (...) mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more about the physical world than we would otherwise be able to. Mathematical objects do not need to exist for mathematics to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences that reveals they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which reveals that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show that the emerging hermeneutic fictionalist position is preferable to any alternative interpretation of mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects. (shrink)

Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these (...) formal constructions and discusses them as an argument for the alternative semantic and propositional-structure accounts of the applicability of mathematics. (shrink)

There is growing debate about what is the correct methodology for research in the ontology of artworks. In the first part of this essay, I introduce my view: I argue that semantic descriptivism is a semantic approach that has an impact on meta-ontological views and can be linked with a hermeneutic fictionalist proposal on the meta-ontology of artworks such as works of music. In the second part, I offer a synthetic presentation of the four main positive meta-ontological views that have (...) been defended in philosophical literature about artworks and of some criticisms that can be lodged against them: Amie Thomasson’s global descriptivism, Andrew Kania’s local descriptivism, Julian Dodd’s folk-theoretic modesty, and David Davies’ rational accountability view. In the conclusion, I show the advantages of my view. (shrink)

Philosophers have always taken an interest not only in what is actually the case, but in what is necessarily the case and what could possibly be the case. These are questions of modality. Epistemologists of modality enquire into how we can know what is necessary and what is possible. This dissertation concerns the meta-epistemology of modality. It engages with the rules that govern construction and evaluation of theories in the epistemology of modality, by using modal empiricism – a form of (...) modal epistemology – as a running example. In particular, I investigate the assumption that it is important to be able to meet the integration challenge. Meeting the integration challenge is a source of serious difficulty for many approaches, but modal empiricism is supposed to do well in this respect. But I argue that once we have a better grasp of what the integration challenge is, it is not obvious that it presents no problem for modal empiricism. Moreover, even if modal empiricism could be said to be in a relatively good position with respect to integration, it comes at the cost of a forced choice between far-reaching partial modal scepticism and non-uniformism about the epistemology of modality. Non-uniformism is the view that more than one modal epistemology will be correct. While non-uniformism might not in itself be unpalatable, it must be defined and defended in a way which squares with the modal empiricist’s other commitment. I explore two ways of doing so, both involving a revised idea of the integration challenge and its role for the epistemology of modality. One involves a bifurcation of the integration challenge, and the other a restriction of the integration challenge’s relevance. Both ways are interesting, but neither is, as it turns out, a walk in the park. (shrink)

In this dissertation, I consider from a philosophical perspective three related questions concerning the contribution of mathematics to scientific representation. In answering these questions, I propose and defend Carnapian frameworks for examination into the nature and role of mathematics in science. The first research question concerns the varied ways in which mathematics contributes to scientific representation. In response, I consider in Chapter 2 two recent philosophical proposals claiming to account for the explanatory role of mathematics in science, by Philip Kitcher, (...) and Otavio Bueno and Mark Colyvan. My novel and detailed critique of these accounts shows that they are too limited to encompass the diverse roles of mathematics in science in historical and contemporary scenarios. The conclusion is that any such philosophical account should aim to faithfully capture the structure of our theories and their use in applied contexts. This insight prompts the second question guiding this dissertation that I consider in Chapter 3, regarding a viable philosophical account of the role of mathematics in scientific theories. I respond by proposing a modified form of the reconstructive frameworks for philosophical analysis developed by Rudolf Carnap for theoretical entities. I propose three amendments to Carnap’s account: i) a semantic view for the representation of theories, ii) a careful consideration of instances of the use of theory in representing target systems, and iii) consideration of the practical complexity of relating theory to experimental data. The final research question for this dissertation asks what, if anything, we can legitimately conclude about the nature of theoretical entities invoked by a theory in light of its success in representing phenomena. In the backdrop of the Carnapian frameworks proposed in Chapter 3, I argue that contemporary ontological debates in the philosophy of science are largely premised on an acceptance of Willard Quine’s epistemological outlook on the world and a dismissal of Carnap’s approach, which can be used to offer a satisfactory deflationary resolution. This is in the service of my contention that a Carnapian attitude to central issues in the philosophy of science is decidedly preferable to the route championed by Quine. (shrink)