Suppose we are asked to draw up a list of things we take to exist. Certain items seem unproblematic choices, while others (such as God) are likely to spark controversy. The book sets the grand theological theme aside and asks a less dramatic question: should mathematical objects (numbers, sets, functions, etc.) be on this list? In philosophical jargon this is the ‘ontological’ question for mathematics; it asks whether we ought to include mathematicalia in our ontology. The goal of this work (...) is to answer this question in the affirmative, by drawing on considerations on the the applicability of mathematics to natural science. (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.

Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory one.

According to standard (quantum) statistical mechanics, the phenomenon of a phase transition, as described in classical thermodynamics, cannot be derived unless one assumes that the system under study is infinite. This is naturally puzzling since real systems are composed of a finite number of particles; consequently, a well‐known reaction to this problem was to urge that the thermodynamic definition of phase transitions (in terms of singularities) should not be “taken seriously.” This article takes singularities seriously and analyzes their role by (...) using the well‐known distinction between data and phenomena , in an attempt to better understand the origin of the puzzle. *Received April 2009; revised July 2009. †To contact the author, please write to: University of Cambridge, Department of History and Philosophy of Science, Free School Lane, Cambridge CB2 3RH, United Kingdom; e‐mail: sib24@cam.ac.uk. (shrink)

The articulation of an overarching account of scientific explanation has long been a central preoccupation for the philosophers of science. Although a while ago the literature was dominated by two approaches—a causal account and a unificationist account—today the consensus seems to be that the causal account has won. In this paper, I challenge this consensus and attempt to revive unificationism. More specifically, I aim to accomplish three goals. First, I add new criticisms to the standard anti-unificationist arguments, in order to (...) motivate the need for a revision of the doctrine. Second, and most importantly, I sketch such a revised version. Then I argue that, contrary to widespread belief, the causal account and this revised unificationist account of explanation are compatible. Moreover, I also maintain that the unificationist account has priority, since a most satisfactory theory of explanation can be obtained by incorporating the causal account, as a sub-component of the unificationist account. The driving force behind this reevaluation of the received view in the philosophy of explanation is a reconsideration of the role of scientific understanding. (shrink)

The question as to whether there are mathematical explanations of physical phenomena has recently received a great deal of attention in the literature. The answer is potentially relevant for the ontology of mathematics; if affirmative, it would support a new version of the indispensability argument for mathematical realism. In this article, I first review critically a few examples of such explanations and advance a general analysis of the desiderata to be satisfied by them. Second, in an attempt to strengthen the (...) realist position, I propose a new type of example, drawing on probabilistic considerations. 1 Introduction2 Mathematical Explanations2.1 ‘Simplicity’3 An Average Story: The Banana Game3.1 Some clarifications3.2 Hopes and troubles for the nominalist3.3 New hopes?3.4 New troubles4 Conclusion. (shrink)

In his early writings, Kant regarded the autonomy of the will as the supreme principle of morality, as well as the sole principle of all moral laws and of the duties conforming to them. Nevertheless, this impressively sounding principle gradually disappeared from the later Kant’s texts, and there is not much in the literature to explain why. Pauline Kleingeld’s purpose, in the two articles I consider here, is to address this lacuna and to show that there are good philosophical reasons (...) for this principle’s gradual disappearance from the Kantian framework. One of my aims, in this paper, is to formulate and argue for several reservations about Kleingeld’s argument; however, even if these reservations turn out to be apposite, my claim is not that Kant did not actually abandon the Principle of Autonomy; instead, I argue that, even if Kleingeld were right and Kant had abandoned the Principle of Autonomy, he would still not have needed to. (shrink)

In his Remarks on the Foundations of Mathematics, Wittgenstein claims, puzzlingly, that ‘the proof creates a new concept’ (RFM III-41). This paper aims to contribute to clarifying this idea, and to showing how it marks a major break with the traditional conception of proof. Moreover, since the most natural way to understand his claim is open to criticism, a secondary goal of what follows is to offer an interpretation of it that neutralizes the objection. The discussion proceeds by analysing a (...) well-known geometrical proof. (shrink)

The Principle of Indifference is a central element of the ‘classical’ conception of probability, but, for all its strong intuitive appeal, it is widely believed that it faces a devastating objection: the so-called (by Poincare´) ‘Bertrand paradoxes’ (in essence, cases in which the same probability question receives different answers). The puzzle has fascinated many since its discovery, and a series of clever solutions (followed promptly by equally clever rebuttals) have been proposed. However, despite the long-standing interest in this problem, an (...) important assumption, necessary for its generation, has been overlooked. My aim in this paper is to identify this assumption. Since what it claims turns out to be prima facie problematic, I will urge that the burden of proof now shifts to the objectors to PI; they have to provide reasons why this assumption holds. (shrink)

I present a reconstruction of Eugene Wigner’s argument for the claim that mathematics is ‘unreasonable effective’, together with six objections to its soundness. I show that these objections are weaker than usually thought, and I sketch a new objection.

My main aim is to sketch a certain reading (‘genealogical’) of later Wittgenstein’s views on logical necessity. Along the way, I engage with the inferentialism currently debated in the literature on the epistemology of deductive logic.

Name der Zeitschrift: Kant-Studien Jahrgang: 107 Heft: 1 Seiten: 2-33.

Constitutivism aims to justify substantial normative standards as constitutive of practical reason. In this way, it can defend the constructivist commitment to avoiding realism and anti-realism in normative disciplines. This metaphysical debate is the perspective from which the nature of the constitutivist justification is usually discussed. In this paper, I focus on a related, but distinct, debate. My concern will not be whether the substantial normative claims asserted by the constructivist have some elements, which are not constructed, but real, given (...) independently from us; instead, my concern will be more narrowly epistemic – whether those claims can be derived from premises, which are normatively less substantial than the normative conclusions themselves. I focus on Korsgaard’s transcendental articulation of the constitutivist argument. I conclude that more work would need to be done, in order for this argument to function as intended. (shrink)

What is the role of bridge laws in inter-theoretic relations? An assumption shared by many views about these relations is that bridge laws enable reductions. In this article, I acknowledge the naturalness of this assumption, but I question it by presenting a context within thermal physics (involving phase transitions) in which the bridge laws, puzzlingly, seem to contribute to blocking the reduction.

The paper argues that scientific progress is best characterized as an increase in scientists' understanding of the world. It also connects this idea with the claim that scientific understanding and explanation are captured in terms of unification.

In this paper I reconstruct and critically examine the reasoning leading to the famous prediction of the ‘omega minus’ particle by M. Gell-Mann and Y. Ne’eman (in 1962) on the basis of a symmetry classification scheme. While the peculiarity of this prediction has occasionally been noticed in the literature, a detailed treatment of the methodological problems it poses has not been offered yet. By spelling out the characteristics of this type of prediction, I aim to underscore the challenges raised by (...) this episode to standard scientific methodology, especially to the traditional deductive-nomological account of prediction. (shrink)

The paper argues that the phenomenon of first-order phase transitions (e.g., freezing) has features that make it a candidate to be classified as 'emergent'. However, it cannot be described either as 'weakly emergent' or 'strongly emergent'; hence it escapes categorization in terms employed in the current literature on the metaphysics of science.

This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical (...) query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. _Naturalizing Logico-Mathematical Knowledge _contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies. (shrink)

For several decades, theorists interested in Kant’s discussion of welfare have puzzled over Kant’s position on the issue of the redistribution of goods in society. They have done this both in order to clarify his position and as a source of inspiration for current conceptual problems faced by contemporary political philosophers who attempt to reconcile the ideal of equal freedom with the asymmetric interference necessary for redistribution and social provision. In this paper, I start with Kant’s brief discussion of welfare (...) in Rechtslehre and I identify four claims that Kant clearly asserts as characteristic for his view. I then outline five main interpretative directions in the literature, I evaluate and rank them. The most accurate view of Kant’s justification of welfare, which I call the “genuinely Kantian” position is, however, unable to explain the nature of the duty of welfare that it asserts. By going back to Kant’s text, I suggest one solution. This solution, together with some further questions, can be seen as initiating a new interpretative direction in the literature. (shrink)

Steiner defines naturalism in opposition to anthropocentrism, the doctrine that the human mind holds a privileged place in the universe. He assumes the anthropocentric nature of mathematics and argues that physicists' employment of mathematically guided strategies in the discovery of quantum mechanics challenges scientists' naturalism. In this paper I show that Steiner's assumption about the anthropocentric character of mathematics is questionable. I draw attention to mathematicians' rejection of what Maddy calls ‘definabilism’, a methodological maxim governing the development of mathematics. I (...) contend that because definabilism is anthropocentric, its rejection casts doubts on Steiner's assumption. (shrink)

In his masterful essay, “Armchair Knowledge: Some Kantian Reflections”, A.W. Moore claims that Kant lapses into contradiction as a result of invoking transcendental idealism as a solution to the puzzle of what Moore calls “armchair knowledge”. Moore talks about “the incoherence of transcendental idealism” and, through a discussion which includes the question of whether different subjects possess different categories, offers an account of armchair knowledge without transcendental idealism. He suggests we should abandon the Kantian a priori intuitions and, with them, (...) also Kant’s synthetic a priori judgements. In this paper, I examine some of the problems identified by Moore in Kant’s account. I do not aim to show that there is no internal inconsistency in Kant’s thought or that Moore (perhaps continuing in this way Kant’s project of uncovering the deceptions of transcendental judgements) might not be right to point to potential problems in Kant’s texts; my claim is rather that there are resources in the Kantian corpus to explain away the main contradictions and conflicts Moore identifies in Kant. (shrink)

In this volume, Sorin Sabou explores the dependency of happiness on external goods in Aristotle's Nicomachean Ethics. Sabou defends the following thesis: the dependency of happiness on external goods, in EN, is interpreted in the light of its political self-sufficiency, and in the light of our political humanity; this dependency is of three kinds: (1) enhancing-instrumental, (2) constitutive, and (3) subsistent.

The paper discuses whether the mathematical singularities characterizing first-order phase transitions are 'fictions'.

In this paper I criticize one of the most convincing recent attempts to resist the underdetermination thesis, Laudan’s argument from indirect confirmation. Laudan highlights and rejects a tacit assumption of the underdetermination theorist, namely that theories can be confirmed only by empirical evidence that follows from them. He shows that once we accept that theories can also be confirmed indirectly, by evidence not entailed by them, the skeptical conclusion does not follow. I agree that Laudan is right to reject this (...) assumption, but I argue that his explanation of how the rejection of this assumption blocks the skeptical conclusion is flawed. I conclude that the argument from indirect confirmation is not effective against the underdetermination thesis. (shrink)

Some of the post-socialist countries of Europe experienced after the fall of communism what some called a religious revival. Anthropologists and sociologists alike were sure that they discovered serious evidence against the case of secularization theory. What unfortunately most of them failed to observe was the particular shape and form of this religious growth and the structural changes of the religious mentalities occurred in the process of transition from a closed, ideologically monopolized society, to a pluralistic one. After more than (...) half of a century of atheistic ideologization of the public sphere, Romania remains one of the most religious countries of both Eastern and Western Europe. The thesis of this article is that this fact is due to the lack of modernization of the Romanian social system both before and during the post-socialist period. (shrink)

Factivism is the view that understanding why a natural phenomenon takes place must rest exclusively on (approximate) truths. One of the arguments for nonfactivism—the opposite view, that falsehoods can play principal roles in producing understanding—relies on our inclination to say that past, false, now superseded but still important scientific theories (such as Newtonian mechanics) do provide understanding. In this paper, my aim is to articulate what I take to be an interesting point that has yet to be discussed: the natural (...) way in which nonfactivism fits within the unificationist account of scientific explanation. I contend that unificationism gives non-factivists a better framework to uphold their position. After I show why this is so, toward the end of the paper I will express doubts with regard to the viability of de Regt’s (2015) kind of non-factivism, based on the idea that understanding should be captured in terms of (scientific) skill. (shrink)

A survey of the main themes and arguments concerning symmetry and invariance in physics and philosophy of physics.

The paper discusses the emergence of Frege's puzzle and the introduction of the celebrated distinction between sense and reference in the context of Frege's logicist project. The main aim of the paper is to show that not logicism per se is mainly responsible for this introduction, but Frege's constant struggle against formalism. Thus, the paper enlarges the historical context, and provides a reconstruction of Frege's philosophical development from this broader perspective.

Immanuel Kant's work continues to be a main focus of attention in almost all areas of philosophy. The significance of Kant's work for the so-called continental philosophy cannot be exaggerated, although work in this area is relatively scant. The book includes eight chapters, a substantial introduction and a postscript, all newly written by an international cast of well-known authors. Each chapter focuses on particular aspects of a fundamental problem in Kant's and post-Kantian philosophy, the problem of the relation between the (...) world and transcendence. Chapters fall thematically into three parts: sensibility, nature and religion. Each part starts with a more interpretative chapter focusing on Kant's relevant work, and continues with comparative chapters which stage dialogues between Kant and post-Kantian philosophers, including Martin Heidegger, Hannah Arendt, Jean-François Lyotard, Luce Irigaray and Jacques Derrida. A special feature of this volume is the engagement of each chapter with the work of the late British philosopher Gary Banham. The Postscript offers a subtle and erudite analysis of his intellectual trajectory, philosophy and mode of working. The volume is dedicated to his memory. (shrink)

In his recent book,The Evolution of Modern Metaphysics, Adrian W. Moore takes Kant to play a crucial role in the evolution of modern philosophy; yet, for him, Kant’s metaphysics is ultimately and profoundly unsatisfactory. In this article, I examine several of Moore’s objections and provide replies. My claim is that Moore’s reading points to fundamental issues, yet these are not issues of Kant’s transcendental idealism, but of the traditional idealism his view has often been taken to represent.

The Dignity of the Children according to the New Testament teaching should be a landmark for modern era. New Testament raised the dignity of the unborn child, from a simple mass of cells to a child who has the right to be born and live. The child born has the value of a person that to enjoy family and society not only for economic potential but for the soul value as they bear the image of God. Children of different ages (...) have the right to be baptized, have access to the grace of the Sacraments and become saints. In fact the true dignity of man is his power to become a saint. And the child is the image of the saint, heir of the Heaven. (shrink)

I identify and characterize a type of noncausal explanation in physics. I first introduce a distinction, between the physical properties of a system, and the representational properties of the mathematical expressions of the system’s physical properties. Then I introduce a novel kind of property, which I shall call a dual property. This is a special kind of representational property, one for which there is an interpretation as a physical property. It is these dual properties that, I claim, are amenable to (...) noncausal (mathematical, in fact) explanations. I discuss a typical example of such a dual property, and an example of an explanation as to why that dual property holds (the explanation of the quantization of the linear momentum). (shrink)

The traditionalist interpretation of Kant's idealism reads his Critical philosophy as a version of traditional idealism, à la Berkeley. By contrast, a revolutionary account of Kant will assert the threefold distinction between states of mind, external objects of the world and things in themselves, and will reject the attempt to reduce external objects to states of mind. In this paper, I argue that, while Caird's interpretation is clearly not traditionalist, nor is it obviously revolutionary: he is critical of Kant's threefold (...) distinction, although he fully acknowledges Kant's attempt to uphold this distinction. While arguing for this claim, the paper will examine several classical objections to Kant's Critical philosophy, including the view that things in themselves should be rejected and that Kant's system lacks the unity necessary to account for the variety of distinctions he introduces, including the distinction between phenomena and noumena. (shrink)

The paper rebuts a currently popular criticism against a certain take on the referential role of discontinuities and singularities in the physics of first-order phase transitions. It also elaborates on a proposal I made previously on how to understand this role within the framework provided by the distinction between data and phenomena.

In this paper I have two objectives. First, I attempt to call attention to the incoherence of the widely accepted anti-essentialist interpretation of Wittgenstein’s family resemblance point. Second, I claim that the family resemblance idea is not meant to reject essentialism, but to render this doctrine irrelevant, by dissipating its philosophical force. I argue that the role of the family resemblance point in later Wittgenstein’s views can be better understood in light of the provocative aim of his philosophical method, as (...) stated (for instance) in PI 133: “[t]he philosophical problems” - associated with essentialism in this case, "should completely disappear". (shrink)

: Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting to (...) clarify where the appeal of this Pythagorean view comes from and what are the arguments favoring its acceptance or rejection. Along the way, I sketch the historical context in which this heuristic interpretation gained credibility (the quantum crisis in physics in the 1920s), as well as the more general implications of this thesis for physicists' metaphysical outlook. (shrink)

This paper explores the connections between K. Wynn's well-known experiments in cognitive psychology and later Wittgenstein's views on the philosophy of mathematics.

This book challenges the standard view of the relationship between Kant's and Sartre's practical philosophies, making a case for regarding Kant as one of Sartre's most significant predecessors. By using an original comparative methodology, the book identifies several fundamental theses of Sartre's practical philosophy despite the common reading of Sartre as a philosopher without a practical philosophy. Furthermore, the book shows that Sartre's practical philosophy proves to be closer to Kant than dominant contemporary Kantian theories are. Starting from the similarities (...) between Kant and Sartre, the book uncovers the project of a critical ethics which is philosophically more compelling than dominant contemporary Kantian theories. (shrink)

In spite of its dominance, there are well-known problems with Rawls’s method of reflective equilibrium (MRE), as a method of justification in meta-ethics. One issue in particular has preoccupied commentators, namely, the capacity of this method to provide a convincing account of the objectivity of our moral beliefs. Call this the Lack-of-Objectivity Charge. One aim of this article is to examine the charge within the context of Rawls’s later philosophy, and I claim that the lack-of-objectivity charge remains unanswered. A second (...) aim of this article is to examine the extent to which, despite Rawls’s express intention to avoid reliance on Kant’s moral philosophy, supplementing Rawls’s political constructivism with some Kantian elements, in particular Kant’s idea of a universal principle of right, not only addresses some of the issues raised by the lack-of-objectivity charge, but also does so without compromising the ability of the Rawlsian account to accommodate the pluralism of conceptions of the good, which he takes to be a fact of modern democracies. I argue for a revised justificatory methodology, which combines Rawls’s MRE and Kant’s Critical Method. (shrink)

I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.