This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.

This paper aims to provide a mathematically tractable background against which to model both modal cognitivism and modal expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics, comprise a materially adequate fragment of the language of thought. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are categorically dual. I examine five methods for modeling the dynamics of conceptual engineering for intensions and (...) hyperintensions. I develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. I examine then the virtues unique to the modal expressivist approach here proffered in the setting of the foundations of mathematics, by contrast to competing approaches based upon both the inferentialist approach to concept-individuation and the codification of speech acts via intensional semantics. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (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)

According to a popular ‘explanationist’ argument for moral or mathematical realism the best explanation of some phenomena are moral or mathematical, and this implies the relevant form of realism. One popular way to resist the premiss of such arguments is to hold that any supposed explanation provided by moral or mathematical properties is in fact provided only by the non-moral or non-mathematical grounds of those properties. Many realists have responded to this objection by urging that the explanations provided by the (...) higher-level, moral and mathematical, properties are informative in a way in which their supposed replacements are not, and that this is because moral and mathematical properties are multiply realizable. This chapter responds to this manoeuvre by proposing a way in which moral and mathematical explanations might preserve the distinctive import that multiple realizability brings, without committing those who accept them to the existence of moral and mathematical properties. (shrink)

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. (...) The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse. (shrink)

I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional truthmaker semantics provide a compelling solution to the access problem. I countenance an abstraction principle for epistemic hyperintensions based on Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory. I apply, further, modal rationalism in modal epistemology to solve the access problem. Epistemic possibility and hyperintensionality, i.e. conceivability, can be a guide to metaphysical possibility and hyperintensionality, when (i) epistemic worlds or epistemic hyperintensional states are interpreted as being (...) centered metaphysical worlds or hyperintensional states, i.e. indexed to an agent, when (ii) the epistemic (hyper-)intensions and metaphysical (hyper-)intensions for a sentence coincide, i.e. the hyperintension has the same value irrespective of whether the worlds in the argument of the functions are considered as epistemic or metaphysical, and when (iii) sentences are said to consist in super-rigid expressions, i.e. rigid expressions in all epistemic worlds or states and in all metaphysical worlds or states. I argue that (i) and (ii) obtain in the case of the access problem. (shrink)

This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, (...) and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (shrink)

This paper aims to contribute to the analysis of the nature of mathematical modality and hyperintensionality, and to the applications of the latter to absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority (...) and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance a topic-sensitive epistemic two-dimensional truthmaker semantics, if hyperintensional approaches are to be preferred to possible worlds semantics. I examine the relation between two-dimensional hyperintensional states and epistemic set theory, providing two-dimensional hyperintensional formalizations of the modal logic of ZFC, large cardinal axioms, $\Omega$-logic, and the Epistemic Church-Turing Thesis. (shrink)

It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for (...) mathematical realism. It is widely thought not to be. In this paper, I argue that the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism. Along the way, I substantially clarify the Evolutionary Challenge, discuss its relation to more familiar epistemological challenges, and broach the problem of moral disagreement. The paper should be of interest to ethicists because it places pressure on anyone who rejects moral realism on the basis of the Evolutionary Challenge to reject mathematical realism as well. And the paper should be of interest to philosophers of mathematics because it presents a new epistemological challenge for mathematical realism that bears, I argue, no simple relation to Paul Benacerraf's familiar challenge. (shrink)

This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.

Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the problem of explaining the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the problem of explaining their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, (...) and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is intelligible. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism. (shrink)

Nazovi epistemološke osnove na kojima bismo racionalno trebali odrediti naše ontološke (ili aletiološke) obveze u vezi s entitetom njegov arbitar postojanja (ili arbitar istine). Uobičajeno je mišljenje da arbitri postojanja i istine mogu biti dani putem naših praksi. Ovaj rad tvrdi da takva gledišta imaju nekoliko implikacija: (1) veza između arbitara i naših metafizičkih obveza sastoji se u neophodnosti, (2) realistička gledišta o vrsti entiteta trebala bi se podudarati s vrstama praksi koje pružaju arbitre za tu vrstu entiteta s obzirom (...) na njihove metafizičke ovisnosti, (3) ako realisti smatraju da vrsta prakse pruža temelje na kojima se potvrđuje postojanje vrste entiteta, trebali bi se referirati na iste temelje koje koriste kada pokušavaju pružiti epistemologiju relevantnog područja. (shrink)

This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...) contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

In this paper, I develop a theory on which each of a thing’s abundant properties is immanent in that thing. On the version of the theory I will propose, universals are abundant, each instantiated universal is immanent, and each uninstantiated universal is such that it could have been instantiated, in which case it would have been immanent. After setting out the theory, I will defend it from David Lewis’s argument that such a combination of immanence and abundance is absurd. I (...) will then advocate the theory on the grounds that it accomplishes all of Lewis’s “new work” while providing a gain in parsimony and a new account of fine-grained content. I will close with a discussion of how the theory also affords a new reply to two objections to uninstantiated universals: Armstrong’s charge that they are inconsistent with naturalism, and a Benacerraf-Field-style objection about epistemic access. (shrink)

The fact that debunkers can turn to the argument from disagreement for help is ofcourse not a surprise. After all, both types of challenge basically pursue the same,skeptical conclusion. What I have tried to show, however, is that they are related in amore intimate way.

It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) we can’t (...) know ourselves to have any such freedom. (shrink)

Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude by arguing that versions of OSR (...) that do not rely on mathematical frameworks for representational purposes need not be vulnerable to Benacerraf’s second problem. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

There are several important arguments in metaethics that rely on explanatory considerations. Gilbert Harman has presented a challenge to the existence of moral facts that depends on the claim that the best explanation of our moral beliefs does not involve moral facts. The Reliability Challenge against moral realism depends on the claim that moral realism is incompatible with there being a satisfying explanation of our reliability about moral truths. The purpose of this chapter is to examine these and related arguments. (...) In particular, this chapter will discuss four kinds of arguments – Harman’s Challenge, evolutionary debunking arguments, irrelevant influence arguments, and the Reliability Challenge – understood as arguments against moral realism. The main goals of this chapter are (i) to articulate the strongest version of these arguments; (ii) to present and assess the central epistemological principles underlying these arguments; and (iii) to determine what a realist would have to do to adequately respond to these arguments. (shrink)

We think of logic as objective. We also think that we are reliable about logic. These views jointly generate a puzzle: How is it that we are reliable about logic? How is it that our logical beliefs match an objective domain of logical fact? This is an instance of a more general challenge to explain our reliability about a priori domains. In this paper, I argue that the nature of this challenge has not been properly understood. I explicate the challenge (...) both in general and for the particular case of logic. I also argue that two seemingly attractive responses – appealing to a faculty of rational insight or to the nature of concept possession – are incapable of answering the challenge. (shrink)

What are intuitions? According to doxastic views, they are doxastic attitudes or dispositions, such as judgments or inclinations to make judgments. According to perceptualist views, they are—like perceptual experiences—pre-doxastic experiences that—unlike perceptual experiences—represent abstract matters as being a certain way. In this paper I argue against doxasticism and in favor of perceptualism. I describe two features that militate against doxasticist views of perception itself: perception is belief-independent and perception is presentational. Then I argue that intuitions also have both features. The (...) upshot is that intuitions are importantly similar to perceptual experiences, and so should not be identified with doxastic attitudes or dispositions. I consider a popular argument from the introspective absence of sui generis intuition experiences in favor of doxasticism. I develop a conception of intuition experiences that helps to defuse this argument. (shrink)

In this chapter, I make the case for platonic realism, the thesis that there are properties that lack spatial locations. After criticizing the one-over-many argument for realism and Lewis's argument for realism, I endorse a modal argument that derives the existence of platonic properties from considerations involving necessary truth. I then defend this argument from various objections. Finally, I argue that epistemic considerations and considerations of parsimony favor a weak form of platonic realism on which there are platonic properties, but (...) each property could have had an instance, and would have been located in its instances if it had any. (shrink)

In a recent paper Weinberg (2007) claims that there is an essential mark of trustworthiness which typical sources of evidence as perception or memory have, but philosophical intuitions lack, namely that we are able to detect and correct errors produced by these “hopeful” sources. In my paper I will argue that being a hopeful source isn't necessary for providing us with evidence. I then will show that, given some plausible background assumptions, intuitions at least come close to being hopeful, if (...) they are reliable. If this is true, Weinberg's new challenge comes down to the claim that philosophical intuitions are not reliable since they are significantly unstable. In the second part of my paper I will argue that and why the experimentally established instability of folk intuitions about philosophical cases does not show that philosopher's expert intuitions about these cases are instable. (shrink)

In this paper, I explore a question about deductive reasoning: why am I in a position to immediately infer some deductive consequences of what I know, but not others? I show why the question cannot be answered in the most natural ways of answering it, in particular in Descartes’s way of answering it. I then go on to introduce a new approach to answering the question, an approach inspired by Hume’s view of inductive reasoning.

Intuition is sometimes derided as an abstruse or esoteric phenomenon akin to crystal-ball gazing. Such derision appears to be fuelled primarily by the suggestion, evidently endorsed by traditional rationalists such as Plato and Descartes, that intuition is a kind of direct, immediate apprehension akin to perception. This paper suggests that although the perceptual analogy has often been dismissed as encouraging a theoretically useless metaphor, a quasi-perceptualist view of intuition may enable rationalists to begin to meet the challenge of supplying a (...) theoretically satisfying treatment of their favoured epistemic source. It is argued, first, that intuitions and perceptual experiences are at a certain level of abstraction the same type of mental state, presentations, which are distinct from beliefs, hunches, inclinations, attractions, and seemings. The notion of a presentation is given a positive explication, which identifies its characteristic features, accounts for several of its substantive psychological roles, and systematically locates it in a threefold division among types of contentful states. Subsequently, it is argued that presentations, intuitive no less than sensory, are by their nature poised to play a distinctive epistemic role. Specifically, in the case of intuition, we encounter an intellectual state that is so structured as to provide justification without requiring justification in turn—something which may, thus, be thought of as ‘given’. (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 discuss Benacerraf's epistemological challenge for realism about areas like mathematics, metalogic, and modality, and describe the pluralist response to it. I explain why normative pluralism is peculiarly unsatisfactory, and use this explanation to formulate a radicalization of Moore's Open Question Argument. According to the argument, the facts -- even the normative facts -- fail to settle the practical questions at the center of our normative lives. One lesson is that the concepts of realism and objectivity, which are widely identified, (...) are actually in tension. (shrink)

In this paper I explore irrealist alternatives to orthodox realism about grounding, and claim that at least some of these alternatives represent fertile areas for future discussion.

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...) mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

This chapter examines the status of inference to the best explanation in naturalistic metaphysics. The methodology of inference to the best explanation in metaphysics is studied from the perspective of contemporary views on scientific explanation and explanatory inferences in the history and philosophy of science. This reveals serious shortcomings in prevalent attempts to vindicate metaphysical "explanationism" by reference to similarities between science and naturalistic metaphysics. This critique is brought out by considering a common gambit of methodological unity: (1) Both metaphysics (...) and science employ inference to the best explanation. (2) One has no reason to think that if explanationism is truth-conducive in science, it is not so in metaphysics. (3) One has a positive reason to think that if explanationism is truth-conducive in science, it is also so in metaphysics. (shrink)

I discuss the structure of genealogical debunking arguments. I argue that they undermine our mathematical beliefs if they undermine our moral beliefs. The contrary appearance stems from a confusion of arithmetic truths with (first-order) logical truths, or from a confusion of reliability with justification. I conclude with a discussion of the cogency of debunking arguments, in light of the above. Their cogency depends on whether information can undermine all of our beliefs of a kind, F, without giving us direct reason (...) to doubt that our F-beliefs are modally secure. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

The idea that there are some facts that call for explanation serves as an unexamined premise in influential arguments for the inexistence of moral or mathematical facts and for the existence of a god and of other universes. This book is the first to offer a comprehensive and critical treatment of this idea. It argues that calling for explanation is a sometimes-misleading figure of speech rather than a fundamental property of facts.

At the core of the recent debate over moral debunking arguments is a disagreement between explanationist and modalist approaches. Explanationists think that the lack of an explanatory connection between our moral beliefs and the moral truths, given a non-naturalist realist conception of morality, is a reason to reject non-naturalism. Modalists disagree. They say that, given non-naturalism, our beliefs have the appropriate modal features with respect to truth -- in particular they are safe and sensitive -- so there is no problem. (...) -/- There is something of a stand-off here. I argue, though, that by looking at the role explanatory and modal factors have to play in theory choice more generally, and, in particular, by considering the practice of theory choice in science, we can see that the explanationist is right. The lack of an explanatory connection between our moral beliefs and the moral truths is a reason to reject non-naturalist realism about morality. (shrink)

This paper aims to provide two abductive considerations adducing in favor of the thesis of Necessitism in modal ontology. I demonstrate how instances of the Barcan formula can be witnessed, when the modal operators are interpreted 'naturally' -- i.e., as including geometric possibilities -- and the quantifiers in the formula range over a domain of natural, or concrete, entities and their contingently non-concrete analogues. I argue that, because there are considerations within physics and metaphysical inquiry which corroborate modal relationalist claims (...) concerning the possible geometric structures of spacetime, and dispositional properties are actual possible entities, the condition of being grounded in the concrete is consistent with the Barcan formula; and thus -- in the geometric setting -- merits adoption by the Necessitist. (shrink)

La teleosemántica es una teoría teleológica de las representaciones mentales, propugnada por David Papineau, que tiene como propósito ofrecer una explicación naturalista y evolucionista de dichas representaciones, en particular de las creencias. Mi objetivo será analizar las creencias y su relación con la verdad en la obra de Papineau. Según Papineau, los contenidos y las representaciones mentales, específicamente las creencias, cumplen funciones derivadas de la evolución biológica de la especie, y correlativamente, la finalidad de las creencias y los contenidos proviene (...) de las funciones biológicas que cada una de ellas tiene. De acuerdo a Papineau, para que las creencias cumplan sus funciones biológicas deben ser verdaderas, es decir, corresponder con la realidad. Defenderé la idea de que, a diferencia de lo que plantea la teleosemántica, la verdad y la falsedad, o más específicamente, los enunciados verdaderos y falsos son doxásticos, es decir, son enunciados que expresan creencias. Las creencias no tienen que ser verdaderas, sino más bien útiles, de manera que el que sean verdaderas o no es un producto secundario de la creencia. (shrink)

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed (...) by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type. (shrink)

BEN ALI (Souad) – STOFFEL (Jean-François), Présentation (pp. 5-6). STOFFEL (Jean-François), Introduction (pp. 7-13). BEN ALI (Souad), Aux origines de l’épistémologie historique: un retour à Pierre Duhem est-il justifiable ? (pp. 15-57). BORDONI (Stefano), De Cournot à Duhem: la naissance d’une tradition critique (pp. 59-93). FORTINO (Mirella), L’épistémologie de Duhem est-elle une ennemie de la rationalité ouverte ? (pp. 95-122). LEITE (Fábio Rodrigo), Quelques notes sur le prétendu réalisme struc­turel attribué à Pierre Duhem (pp. 123-164). ROUMENGOUS (Lucas), La continuité de (...) la physique à la métaphysique: un argument en faveur du néo-thomisme de Pierre Duhem ? (pp. 165-198). SEIDENGART (Jean), Duhem et les limites de son phénoménisme: la théorie physique peut-elle se contenter de sauver les phénomènes ? (pp. 199-219). STOFFEL (Jean-Fran­çois), Pierre Duhem et la revendication d’une tradition phéno­mé­naliste: à propos de son «Essai sur la notion de théorie physique de Platon à Galilée» (pp. 221-268). STOFFEL (Jean-François), L’«Histoire de la physique» de Pierre Duhem: contexte d’une publication singulière et historique de l’usage du terme «révolution» (pp. 271-300). BORDONI (Stefano), L’«Histoire de la physique» de Pierre Duhem: une histoire synthétique et tran­chante (pp. 301-310). DUHEM (Pierre), Histoire de la physique / manuscrit édité par Souad BEN ALI et Jean-François STOFFEL (pp. 311-406). Présentation des auteurs (pp. 407-409). Table des matières (pp. 411-412). (shrink)

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...) to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones. -/- . (shrink)

This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand (...) the significance of the distinguished non‐logical individual and relation terms of even inconsistent theories. (3) is a metaphilosophical form of mathematical pluralism and hasn't been discussed in the literature. In what follows, I show how the analysis of theoretical mathematics in object theory exhibits all three forms of mathematical pluralism. (shrink)

I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine's philosophy between naturalism and realism in mathematics. The argument starts from a consequence of naturalism on the nature of human cognitive subjects, physicalism about cognitive subjects, and concludes that this implies a version of nominalism, which I will carefully characterize. The indispensability of classical mathematics for the sciences and semantic/confirmation holism does not affect the argument. The disquotational (...) theory of reference and truth is discussed but rejected. This argument differs from the Benacerrafian arguments against realism, because it does not rely on any specific assumption about the nature of knowledge or reference. It differs from the popular objections to the indispensability argument for realism as well, because it can admit both indispensability and holism. This argument motivates a new, radically naturalistic and nominalistic approach to philosophy of mathematics. (shrink)