Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and (...) propose that they prima facie favor a realist account of numbers. (shrink)

Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a (...) tricle is true or false. A tricle is an object that changes its shape from a triangle to a circle, and then back to a triangle with every second. (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.

Indispensablists contend that accepting scientific realism while rejecting mathematical realism involves a double standard. I refute this contention by developing an enhanced version of scientific realism, which I call interactive realism. It holds that interactively successful theories are typically approximately true, and that the interactive unobservable entities posited by them are likely to exist. It is immune to the pessimistic induction while mathematical realism is susceptible to it.

The dominant approach to analyzing the meaning of natural language sentences that express mathematical knowl- edge relies on a referential, formal semantics. Below, I discuss an argument against this approach and in favour of an internalist, conceptual, intensional alternative. The proposed shift in analytic method offers several benefits, including a novel perspective on what is required to track mathematical content, and hence on the Benacerraf dilemma. The new perspective also promises to facilitate discussion between philosophers of mathematics and (...) cognitive scientists working on topics of common interest. (shrink)

What can we infer from numerical cognition about mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show more similarities. (...) This makes moral beliefs more susceptible to a contingency challenge from evolution compared to mathematical beliefs, and indicates that mathematical beliefs might be less vulnerable to evolutionary debunking arguments. I will then examine to what extent INC can be used to flesh out a positive case for mathematical realism. Finally, I will review two forms of mathematical realism that are promising in the light of the evolutionary evidence about numerical cognition, ante rem structuralism and Millean empiricism. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. (...) In particular, I argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what (...) counts as a mathematical object, and how we can have knowledge about an unchanging object. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that (...) the basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (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)

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over (...) mathematical realism and fictionalism. (shrink)

A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...) treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic. (shrink)

The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which (...) instead explains the attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

Two families of mathematical methods lie at the heart of investigating the hierarchical structure of genetic variation in Homo sapiens: /diversity partitioning/, which assesses genetic variation within and among pre-determined groups, and /clustering analysis/, which simultaneously produces clusters and assigns individuals to these “unsupervised” cluster classifications. While mathematically consistent, these two methodologies are understood by many to ground diametrically opposed claims about the reality of human races. Moreover, modeling results are sensitive to assumptions such as preexisting theoretical commitments to (...) certain linguistic, anthropological, and geographic human groups. Thus, models can be perniciously reified. That is, they can be conflated and confused with the world. This fact belies standard realist and antirealist interpretations of “race,” and supports a pluralist conventionalist interpretation. (shrink)

Scientific knowledge is not merely a matter of reconciling theories and laws with data and observations. Science presupposes a number of metatheoretic shaping principles in order to judge good methods and theories from bad. Some of these principles are metaphysical and some are methodological. While many shaping principles have endured since the scientific revolution, others have changed in response to conceptual pressures both from within science and without. Many of them have theistic roots. For example, the notion that nature conforms (...) to mathematical laws flows directly from the early modern presupposition that there is a divine Lawgiver. This interplay between theism and shaping principles is often unappreciated in discussions about the relation between science and religion. Today, of course, naturalists reject the influence of theism and prefer to do science on their terms. But as Robert Koons and Alvin Plantinga have argued, this is more difficult than is typically assumed. In particular, they argue, metaphysical naturalism is in conflict with several metatheoretic shaping principles, especially explanatory virtues such as simplicity and with scientific realism more broadly. These arguments will be discussed as well as possible responses. In the end, theism is able to provide justification for the philosophical foundations of science that naturalism cannot. (shrink)

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an (...) area, D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

Turning away from entities and focusing instead exclusively on ‘structural’ aspects of scientific theories has been advocated as a cogent response to objections levelled at realist conceptions of the aim and success of science. Physical theories whose (predictive) past successes are genuine would, in particular, share with their successors structural traits that would ultimately latch on to ‘structural’ features of the natural world. Motives for subscribing to Structural Realism are reviewed and discussed. It is argued that structural retention claims (...) lose their force if one gives up merely historical readings of the transition from Galilean-relativistic classical mechanics to the ‘special’ theory of relativity, heeding instead basic requirements that lead to their common derivation. Further cause for scepticism is found upon realising that the basic mathematical framework of quantum theory essentially reflects its predictive purpose, without any necessary input, be it of a ‘structural’ kind, from the physical world. (shrink)

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange argues that in (...) legitimate DMEs, but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (shrink)

In this thesis the author focuses on the metaphysical implications of the realist interpretation of special relativity. The realist interpretation is found wanting in coherence as it utilizes metaphysical concepts (as causation) that are left unjustified if the theory is taken at face value. The author points at a possible re-interpretation of special relativity that allows for a coherent metaphysical basis while containing the mathematical structure of the theory.

This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory -- in particular, Savage's theory of subjective expected utility and personal probability. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealised assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational value (...) of sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage's theory and explore some technical properties in relation to other axioms of the system. (shrink)

This paper aims to reconstruct a possible answer to the classical Newman’s objection which has been used countless times to argue against structural realism. The reconstruction starts from the new strand of structural realism – informational structural realism – authored by Luciano Floridi. Newman’s objection had previously stated that all propositions which comprise the mathematical structures are merely trivial truths and can be instantiated by multiple models. This paper examines whether informational structural realism can overcome (...) this objection by analysing the previous attempts to answer this objection, attempts which either try to save the Ramseyfication of mathematical propositions or give up on it. The informational structural realism way is to attempt a third way, the neo-Kantian way inspired by the work of Ernst Cassirer, but also to change the formalism from a mathematical to an informational one. This paper shows how this original combination of neo-Kantianism, informational formalism and the method of levels of abstraction provide the tools to overcome Newman’s objection. (shrink)

Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but (...) a number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim, John Bigelow, and John Bigelow and Robert Pargetter have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of (...) second-order logical consequence, Ω-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets. (shrink)

Hartry Field has argued that mathematical realism is epistemologically problematic, because the realist is unable to explain the supposed reliability of our mathematical beliefs. In some of his discussions of this point, Field backs up his argument by saying that our purely mathematical beliefs do not ‘counterfactually depend on the facts’. I argue that counterfactual dependence is irrelevant in this context; it does nothing to bolster Field's argument.

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

“Structuralism, Fictionalism, and the Applicability of Mathematics in Science”. This article has two objectives. The first one is to review some of the most important questions in the contemporary philosophy of mathematics: What is the nature of mathematical objects? How do we acquire knowledge about these objects? Should mathematical statements be interpreted differently than ordinary ones? And, finally, how can we explain the applicability of mathematics in science? The debate that guides these reflections is the one between (...) class='Hi'>mathematical realism and anti-realism. On the other hand, the second objective is to discuss the arguments that use the applicability of mathematics in science to justify mathematical realism, and show that none of them reaches its aim. To this end, we will distinguish three aspects of the problem of the applicability of mathematics: the utility of mathematics in science, the unexpected utility of some mathematical theories, and the apparent indispensability of mathematics in our best scientific theories - in particular, in our best scientific explanations. Finally, I argue that none of these aspects constitutes a reason to adopt mathematical realism. (shrink)

In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...) to” them? In this article, I argue that there appears to be no satisfactory answer to this question. The problem is quite general, applying to all arguments with the structure of Genealogical Debunking Arguments aimed at realism about a domain meeting two conditions. The Benacerraf-Field Challenge for mathematical realism affords an important special case. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. (...) There simply is no intelligible problem that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: the Access Problem is eliminated for contentious domains, and realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.

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)

This article, written in Bengali ('Gonit Dorshon' means `philosophy of mathematics' ), briefly reviews a few of the major points of view toward mathematics and the world of mathematical entities, and interprets the philosophy of mathematics as an interaction between these. The existence of these different points of view is indicative that mathematics, in spite of being of universal validity, can nevertheless accommodate alternatives. In particular, I review the alternative viewpoints of Platonism and Intuitionism and present the case that (...) in spite of their great differences, they are not mutually exclusive - that both can be accommodated within the infinite edifice of mathematics. This, in turn, is argued to be consistent with the viewpoint of Category Theory that holds the promise of an entirely new interpretation of the world of mathematics and the relation of that world to the world of our concepts and ideas: mathematics is a human enterprise and mathematical logic is a reflection of how our ideas and concepts are formed and combined with one another. I venture that this, perhaps, is the view of mathematics that Ludwig Wittgenstein would espouse. (shrink)

A study of the making of George Peacock's highly influential, yet disturbingly split, 1830 account of algebra as an entanglement of two separate undertakings: arithmetical and symbolical or formal.

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous (...) approach offers such hope for moral realism. (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 ex- planatory generality is concerned. (shrink)

I examine explanations’ realist commitments in relation to dynamical systems theory. First I rebut an ‘explanatory indispensability argument’ for mathematical realism from the explanatory power of phase spaces (Lyon and Colyvan 2007). Then I critically consider a possible way of strengthening the indispensability argument by reference to attractors in dynamical systems theory. The take-home message is that understanding of the modal character of explanations (in dynamical systems theory) can undermine platonist arguments from explanatory indispensability.

State of the art paper on the topic realism/anti-realism. The first part of the paper elucidates the notions of existence and independence of the metaphysical characterization of the realism/anti-realism dispute. The second part of the paper presents a critical taxonomy of the most important positions and doctrines in the contemporary literature on the domains of science and mathematics: scientific realism, scientific anti-realism, constructive empiricism, structural realism, mathematical Platonism, mathematical indispensability, mathematical (...) empiricism, intuitionism, mathematical fictionalism and second philosophy. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the (...) problem can be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the deeper (...) topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence. (shrink)

In this paper, I defend traditional Platonic mathematical realism from its contemporary detractors, arguing that numbers, understood as abstract, non-physical objects of rational intuition, are indispensable for the act of counting.

Ladyman and Ross argue that quantum objects are not individuals and use this idea to ground their metaphysical view, ontic structural realism, according to which relational structures are primary to things. LR acknowledge that there is a version of quantum theory, namely the Bohm theory, according to which particles do have denite trajectories at all times. However, LR interpret the research by Brown et al. as implying that "raw stuff" or haecceities are needed for the individuality of particles of (...) BT, and LR dismiss this as idle metaphysics. In this paper we note that Brown et al.'s research does not imply that haecceities are needed. Thus BT remains as a genuine option for those who seek to understand quantum particles as individuals. However, we go on to discuss some problems with BT which led Bohm and Hiley to modify it. This modified version underlines that, due to features such as context-dependence and non-locality, Bohmian particles have a very limited autonomy in situations where quantum effects are non-negligible. So while BT restores the possibility of quantum individuals, it also underlines the primacy of the whole over the autonomy of the parts. The later sections of the paper also examine the Bohm theory in the general mathematical context of symplectic geometry. This provides yet another way of understanding the subtle, holistic and dynamic nature of Bohmian individuals. We finally briefly consider Bohm's other main line of research, the "implicate order", which is in some ways similar to LR's structural realism. (shrink)

Non-skeptical robust realists about normativity, mathematics, or any other domain of non- causal truths are committed to a correlation between their beliefs and non- causal, mind-independent facts. Hartry Field and others have argued that if realists cannot explain this striking correlation, that is a strong reason to reject their theory. Some consider this argument, known as the Benacerraf–Field argument, as the strongest challenge to robust realism about mathematics, normativity, and even logic. In this article I offer two closely related (...) accounts for the type of explanation needed in order to address Field's challenge. I then argue that both accounts imply that the striking correlation to which robust realists are committed is explainable, thereby discharging Field's challenge. Finally, I respond to some objections and end with a few unresolved worries. (shrink)

In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. -/- Consequently, this account extends Hartry Field’s program outlined in Science Without (...) Numbers (1980), responds to David Malament’s long-standing impossibility conjecture (1982), and establishes an important first step towards a genuinely intrinsic and nominalistic account of quantum mechanics. I will also compare the present account to Mark Balaguer’s (1996) nominalization of quantum mechanics and discuss how it might bear on the debate about “wave function realism.” In closing, I will suggest some possible ways to extend this account to accommodate spinorial degrees of freedom and a variable number of particles (e.g. for particle creation and annihilation). -/- Along the way, I axiomatize the quantum phase structure as what I shall call a “periodic difference structure” and prove a representation theorem as well as a uniqueness theorem. These formal results could prove fruitful for further investigation into the metaphysics of phase and theoretical structure. (shrink)

In this paper I review three different positions on the wave function, namely: nomological realism, dispositionalism, and configuration space realism by regarding as essential their capacity to account for the world of our experience. I conclude that the first two positions are committed to regard the wave function as an abstract entity. The third position will be shown to be a merely speculative attempt to derive a primitive ontology from a reified mathematical space. Without entering any discussion (...) about nominalism, I conclude that an elimination of abstract entities from one’s ontology commits one to instrumentalism about the wave function, a position that therefore is not as unmotivated as it has seemed to be to many philosophers. (shrink)

This paper focuses on a particular kind of non-naturalism: moral non-naturalism. Our primary aim is to argue that the moral non-naturalist places herself in an invidious position if she simply accepts that the non-natural moral facts that she posits are not explanatory. This has, hitherto, been the route that moral non-naturalists have taken. They have attempted to make their position more palatable by pointing out that there is reason to be suspicious of the explanatory criterion of ontological commitment. That is (...) because other perfectly respectable views fall foul of that criterion, most notably: mathematical realism. Since we don’t want to rule out mathematical realism, we should jettison the explanatory criterion of ontological commitment. Against this manoeuvre, we argue that many contemporary mathematical realists accept the explanatory criterion and provide an account of how mathematical objects are indeed indispensable to our best explanations. Thus, the moral non-naturalist will be left in an awkward dialectical position if she accepts that non-natural moral properties play no such explanatory role. (shrink)

The viability of metaphysics as a field of knowledge has been challenged time and again. But in spite of the continuing tendency to dismiss metaphysics, there has been considerable progress in this field in the 20th- and 21st- centuries. One of the newest − though, in a sense, also oldest − frontiers of metaphysics is the grounding project. In this paper I raise a methodological challenge to the new grounding project and propose a constructive solution. Both the challenge and its (...) solution apply to metaphysics in general, but grounding theory puts the challenge in an especially sharp focus. The solution consists of a new methodology, holistic grounding or holistic metaphysics. This methodology is modeled after a recent epistemic methodology, foundational holism, that enables us to pursue the foundational project of epistemology without being hampered by the problems associated with foundationalism. (shrink)

How does Newton approach the challenge of mechanizing gravity and, more broadly, natural philosophy? By adopting the simple machine tradition’s mathematical approach to a system’s co-varying parameters of change, he retains natural philosophy’s traditional goal while specifying it in a novel way as the search for impressed forces. He accordingly understands the physical world as a divinely created machine possessing intrinsically mathematical features, and mathematical methods as capable of identifying its real features. The gravitational force’s physical cause (...) remains an outstanding problem, however, as evidenced by Newton’s onetime reference to active principles as the “genuine principles of the mechanical philosophy”. (shrink)

One way in which to address the intriguing relations between science and reality is to work via the models (mathematical structures) of formal scientific theories which are interpretations under which these theories turn out to be true. The so-called 'statement approach' to scientific theories -- characteristic for instance of Nagel, Carnap, and Hempel --depicts theories in terms of 'symbolic languages' and some set of 'correspondence rules' or 'definition principles'. The defenders of the oppositionist non-statement approach advocate an analysis where (...) the language in which the theory is formulated plays a much smaller role. They hold that foundational problems in the various sciences can in general be better addressed by focusing on the models these sciences employ than by reformulating the products of these sciences in some appropriate language. My model-theoretic realist account of science lies decidedly within the non-statement context, although I retain the notion of a theory as a deductively closed set of sentences (expressed in some appropriate language), in this paper I shall focus -- against the background of a model-theoretic account of science -- on the approach to the reality-science dichotomy offered by Nancy Cartwright and briefly comment on a few aspects of Roy Bhaskar's transcendental realism. I shall, in conclusion, show how a model-theoretic approach such as mine can combine the best of these two approaches. (shrink)

In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding (...) as inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis. (shrink)