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.
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)
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.
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)
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.
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.
Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a (...) persistent myth, we conclude that having an exact solution is not generally more epistemologically beneficial than lacking one. (shrink)
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)
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: [email protected]. (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)
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.
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)
ABSTRACT 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.
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.
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)
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)
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.
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)
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)
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)
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.
: 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)
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.
When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival (...) of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ – but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation. (shrink)
The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of (...) the Wigner problem. (shrink)
The paper focuses on the lectures on the philosophy of mathematics delivered by Wittgenstein in Cambridge in 1939. Only a relatively small number of lectures are discussed, the emphasis falling on understanding Wittgenstein’s views on the most important element of the logicist legacy of Frege and Russell, the definition of number in terms of classes—and, more specifically, by employing the notion of one-to-one correspondence. Since it is clear that Wittgenstein was not satisfied with this definition, the aim of the essay (...) is to propose a reading of the lectures able to clarify why that was the case. This reading shows that his better known views on language and mind expressed in Philosophical Investigations illuminate his conception of mathematics. (shrink)
Table of Contents: Preface.- 1. Introduction; Mauricio Suárez.- PART I: PROBABILITIES.- 2. Probability and time symmetry in classical Markov processes; Guido Bacciagaluppi.- 3. Probability assignments and the principle of indifference: An examination of two eliminative strategies; SorinBangu.- 4. Why typicality does not explain the approach to equilibrium; Roman Frigg; PART II: CAUSES.- 5. From metaphysics to physics and back: The example of causation; Federico Laudisa.- 6. On explanation in retro-causal interpretations of quantum mechanics; Joseph Berkovitz.- 7. Causal (...) completeness in general probability theories; Balasz Gyenis, Miklós Rédei.- 8. Causal Markov, robustness and the quantum correlations; Mauricio Suárez, Iñaki San Pedro.- PART III: PROPENSITIES.- 9. Do dispositions and propensities have a role in the ontology of quantum mechanics? Some critical remarks; Mauro Dorato.- 10. Is the quantum world composed of propensitons?; Nicholas Maxwell.- 11. Derivative dispositions and multiple derivative levels; Ian Thompson. (shrink)
Formalised knowledge systems, including universities and research institutes, are important for contemporary societies. They are, however, also arguably failing humanity when their impact is measured against the level of progress being made in stimulating the societal changes needed to address challenges like climate change. In this research we used a novel futures-oriented and participatory approach that asked what future envisioned knowledge systems might need to look like and how we might get there. Findings suggest that envisioned future systems will need (...) to be much more collaborative, open, diverse, egalitarian, and able to work with values and systemic issues. They will also need to go beyond producing knowledge about our world to generating wisdom about how to act within it. To get to envisioned systems we will need to rapidly scale methodological innovations, connect innovators, and creatively accelerate learning about working with intractable challenges. We will also need to create new funding schemes, a global knowledge commons, and challenge deeply held assumptions. To genuinely be a creative force in supporting longevity of human and non-human life on our planet, the shift in knowledge systems will probably need to be at the scale of the enlightenment and speed of the scientific and technological revolution accompanying the second World War. This will require bold and strategic action from governments, scientists, civic society and sustained transformational intent. (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)
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)