A theory of causality: Causality=interaction (as defined by a suitable quantum field theory)

In this paper I put forward a suggestion for identifying causality in micro-systems with the specific quantum field theoretic interactions that occur in such systems. I first argue — along the lines of general transference theories — that such a physicalistic account is essential to an understanding of causation; I then proceed to sketch the concept of interaction as it occurs in quantum field theory and I do so from both a formal and an informal point of view. Finally, I present reasons for thinking that only a quantum field theoretic account can do the job — in particular I rely on a theorem by D. Currie and to the effect that interaction cannot be described in (a Hamiltonian formulation of) Classical Mechanics. Throughout the paper I attempt to suggest that the widespread scepticism about the ability of quantum theory to support a theory of causality is mistaken and rests on several misunderstandings.     

Arrow’s theorem and theory choice

In a recent paper (Okasha, Mind 120:83–115, 2011), Samir Okasha uses Arrow’s theorem to raise a challenge for the rationality of theory choice. He argues that, as soon as one accepts the plausibility of the assumptions leading to Arrow’s theorem, one is compelled to conclude that there are no adequate theory choice algorithms. Okasha offers a partial way out of this predicament by diagnosing the source of Arrow’s theorem and using his diagnosis to deploy an approach that circumvents it. In this paper I explain why, although Okasha is right to emphasise that Arrow’s result is the effect of an informational problem, he is not right to locate this problem at the level of the informational input of a theory choice rule. Once the informational problem is correctly located, Arrow’s theorem may be dismissed as a problem.     

Proving quadratic reciprocity: explanation,disagreement, transparency and depth

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late eighteenth century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

     

Some Applications of Lawvere’s Fixpoint Theorem

The famous diagonal argument plays a prominent role in set theory as well as in the proof of undecidability results in computability theory and incompleteness results in metamathematics. Lawvere (1969) brings to light the common schema among them through a pretty neat fixpoint theorem which generalizes the diagonal argument behind Cantor’s theorem and characterizes self-reference explicitly in category theory. Not until Yanofsky (2003) rephrases Lawvere’s fixpoint theorem using sets and functions, Lawvere’s work has been overlooked by logicians. This paper will continue Yanofsky’s work, and show more applications of Lawvere’s fixpoint theorem to demonstrate the ubiquity of the theorem. For example, this paper will use it to construct uncomputable real number, unnameable real number, partial recursive but not potentially recursive function, Berry paradox, and fast growing Busy Beaver function. Many interesting lambda fixpoint combinators can also be fitted into this schema. Both Curry’s Y combinator and Turing’s Θ combinator follow from Lawvere’s theorem, as well as their call-by-value versions. At last, it can be shown that the lambda calculus version of the fixpoint lemma also fits Lawvere’s schema.     

RELIGION AND THE THEORIES OF SCIENCE: A RESPONSE TO BARBOUR

Abstract. This paper offers a detailed response to “Religion and the Theories of Science” in Barbour's Gifford Lectures I. Topics include: complementarity, indeterminacy, parts and wholes, and Bell's theorem in quantum theory; metaphysical issues raised by relativity theory and thermodynamics, principally the problem of temporality and “top-down” versus “bottom—up” causality; design arguments and the origins of the universe in astronomy and creation; and God's action in the context of evolution and continuing creation. Areas of agreement and disagreement between Barbour and myself over philosophical and theological implications are presented, and endnotes indicate further areas of conversation.     

Plato on the role of contradiction in education

In this paper, I will look at two passages from the discussion of education in Book VII of Plato’s Republic: 523b-524d and 537e-539d. These passages, when taken together, present a puzzle for the coherency of the educational programme Socrates describes. Both discuss contradiction. One says that contradiction is educationally edifying, the other, that it is corrupting. This sounds like a contradiction about contradiction. As far as I know, no one has noticed this puzzle before. By the end of this paper, I hope to have not only provided a solution to the apparent contradiction about contradiction that is compelling, but also one that shows that this puzzle, which might at first have seemed restricted to a textual issue about the educational programme in the Republic, is in fact one that has far reaching implications for a range of Plato’s theories across several dialogues. Along with education, corruption, and contradiction, I will discuss Plato’s theory of psychology, and his theory of forms.     

Haag’s Theorem and its Implications for the Foundations of Quantum Field Theory

Although the philosophical literature on the foundations of quantum field theory recognizes the importance of Haag’s theorem, it does not provide a clear discussion of the meaning of this theorem. The goal of this paper is to make up for this deficit. In particular, it aims to set out the implications of Haag’s theorem for scattering theory, the interaction picture, the use of non-Fock representations in describing interacting fields, and the choice among the plethora of the unitarily inequivalent representations of the canonical commutation relations for free and interacting fields.     

Quasi-Truth In Quasi-Set Theory

Throughout the last two decades, Newton da Costa and his collaborators have developed some frameworks to help the interpretation of science. Two of them are particularly noteworthy: partial structures and quasi-truth (that provide a way of accommodating the openness and partiality of scientific activity), and quasi-set theory (that allows one to take seriously the idea, put forward by several physicists, that we can't meaningfully apply the notion of identity to quantum particles). In this paper I explore the interconnection between these two frameworks. After reviewing the extant formulations of quasi-truth and quasi-set theory, I suggest a way of combining them, advancing a formulation of quasi-truth in quasi-set theory. In this way, a good sense can be made of the idea that quantum mechanics, if not true, is at least quasi-true. I then explore an application of this combined framework, arguing that it provides a conceptual setting appropriate to overcome two (philosophical) difficulties in van Fraassen's modal interpretation of quantum mechanics.     

Mereological nihilism: quantum atomism and the impossibility of material constitution

Mereological nihilism is the philosophical position that there are no items that have parts. If there are no items with parts then the only items that exist are partless fundamental particles, such as the true atoms (also called philosophical atoms) theorized to exist by some ancient philosophers, some contemporary physicists, and some contemporary philosophers. With several novel arguments I show that mereological nihilism is the correct theory of reality. I will also discuss strong similarities that mereological nihilism has with empirical results in quantum physics. And I will discuss how mereological nihilism vindicates a few other theories, such as a very specific theory of philosophical atomism, which I will call quantum abstract atomism. I will show that mereological nihilism also is an interpretation of quantum mechanics that avoids the problems of other interpretations, such as the widely known, metaphysically generated, quantum paradoxes of quantum physics, which ironically are typically accepted as facts about reality. I will also show why it is very surprising that mereological nihilism is not a widely held theory, and not the premier theory in philosophy.     

KAUFMAN'S RESPONSE TO LUCRETIUS

Abstract: The symmetry argument is an objection to the ‘deprivation approach’– the account of badness favored by nearly all philosophers who take death to be bad for the one who dies. Frederik Kaufman's recent response to the symmetry argument is a development of Thomas Nagel's suggestion that we could not have come into existence substantially earlier than we in fact did. In this paper, I aim to show that Kaufman's suggestion fails. I also consider several possible modifications of his theory, and argue that they are unsuccessful as well.     

Making Sense of Non-refuting Anomalies

As emphasized by Larry Laudan in developing the notion of non-refuting anomalies (Laudan 1977; Nola and Sankey 2000), traditional analyses of empirical adequacy have not paid enough attention to the fact that the latter does not only depend on a theory’s empirical consequences being true but also on them corresponding to the most salient phenomena in its domain of application. The purpose of this paper is to elucidate the notion of non-refuting anomaly. To this end, I critically examine Laudan’s account and provide a criterion to determine when a non-refuting anomaly can be ascribed to the applicative domain of a theory. Unless this latter issue is clarified, no proper sense can be made of non-refuting anomalies, and no argument could be opposed to those cases where an arbitrary restriction in a theory’s domain of application dramatically reduces the possibilities for its empirical scrutiny. In arguing for the importance of this notion, I show how several semanticist resources can help to reveal its crucial implications, not only for theory evaluation, but also for understanding the nature of a theory’s applicative domain.     

The Intrinsic Quantum Nature of Nash Equilibrium Mixtures

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics (QM), which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics.     

Evidence-Responsiveness and Autonomy

It is plausible to think that part of what it is to be an autonomous agent is to adequately respond to important changes in one’s circumstances. The agent who has set her own course in life, but is unable to recognize and respond appropriately when evidence arises indicating the need to reconsider and perhaps adjust her plan, lacks an important form of personal autonomy. However, this “evidence-responsiveness” aspect of autonomy has not yet been adequately analyzed. Most autonomy theorists ignore it altogether and the few who have addressed it have failed to give a satisfactory account. In this paper, I first examine an evidence-responsiveness condition proposed by Arneson. I argue there that while Arneson’s condition provides a valuable framework in which to examine evidence-responsiveness, there are several crucial issues that it either fails to address at all or else fails to adequately resolve. That condition is therefore in need of further elaboration and refinement. I then examine a recent article in this journal by Blöser, Schöpf, and Willaschek which develops an account of autonomy that I argue can usefully be understood as employing and elaborating upon the general framework offered by Arneson. I argue that while the elaboration Blöser and her co-authors provide Arneson’s condition is instructive, it is inadequate in several important ways which indicate the form a more satisfactory evidence-responsiveness condition will take. I go on to develop such a condition and conclude by highlighting the advantages to be gained by including that condition in a complete theory of autonomy.     

Forgiveness and Moral Development

Forgiveness is clearly an important aspect of our moral lives, yet surprisingly Kant, one of the most important authors in the history of Western ethics, seems to have very little to say about it. Some authors explain this omission by noting that forgiveness sits uncomfortably in Kant’s moral thought: forgiveness seems to have an ineluctably ‘elective’ aspect which makes it to a certain extent arbitrary; thus it stands in tension with Kant’s claim that agents are autonomous beings, capable of determining their own moral status through rational reflection and choice. Other authors recognise that forgiveness plays a role in Kant’s philosophy but fail to appreciate the nature of this duty and misrepresent the Kantian argument in support of it. This paper argues that there is space in Kant’s philosophy for a genuine theory of forgiveness and hopes to lay the grounds for a correct interpretation of this theory. I argue that from a Kantian perspective, forgiveness is not ‘elective’ but, at least in some cases, morally required. I claim that, for Kant, we have an imperfect duty of virtue to forgive repentant wrongdoers that have embarked on a project of self-reflection and self-reform. I develop a novel argument in support of this duty by drawing on Kant’s theory of rational agency, the thesis of radical evil, Kant’s theory of moral development, and the formula of humanity. However, it must be noted that this is a conditional duty and Kant’s position also entails that absence of repentance on the part of the wrongdoer should be taken as evidence of a lack of commitment to a project of self-reflection and self-reform. In such cases, Kant claims, we have a perfect duty to ourselves not to forgive unrepentant wrongdoers. I argue that this duty should be understood as one of the duties of self-esteem, which involves the duty to respect and recognise our own dignity as rational beings.     

Do We Need Two Notions of Constitution?

Traditionally, constitutionalists have offered just one notion of constitution to analyse the relation that an object, such as a statue or a chain, bears to the object/s from which it is made: let us say, a piece of marble in the first case or a piece of metal in the second. Robert Wilson proposes to differentiate two notions of constitution and, in this way, to offer constitutionalists a more varied range of metaphysical tools. To justify the introduction of the difference, he presents several phenomena and problems, the explanation of which would justify the distinction he makes. In this paper I argue that Wilson’s proposal would not increase the explanatory power of a theory of constitution as it has traditionally been understood, only its complexity. Increasing the complexity without increasing the explanatory power of a theory, I defend, goes against one, at least prima facie, basic theoretical virtue: parsimony. In my argumentation I crucially use, for the case of Wilson’s first three arguments, the existence of principles of existence?persistence, which constitutionalists, Wilson among them, usually accept. In arguing against Wilson’s fourth argument I use a slightly modified version of Lynne Rudder Baker’s theory of constitution.     

Time's Arrow and Irreversibility in Time‐Asymmetric Quantum Mechanics

The aim of this paper is to analyze time‐asymmetric quantum mechanics with respect to the problems of irreversibility and of time’s arrow. We begin with arguing that both problems are conceptually different. Then, we show that, contrary to a common opinion, the theory’s ability to describe irreversible quantum processes is not a consequence of the semigroup evolution laws expressing the non‐time‐reversal invariance of the theory. Finally, we argue that time‐asymmetric quantum mechanics, either in Prigogine’s version or in Bohm’s version, does not solve the problem of the arrow of time because it does not supply a substantial and theoretically founded criterion for distinguishing between the two directions of time.     

Desires,beliefs and conditional desirability

Does the desirability of a proposition depend on whether it is true? Not according to the Invariance assumption, held by several notable philosophers. The Invariance assumption plays an important role in David Lewis’ famous arguments against the so-called Desire-as-Belief thesis (DAB), an anti-Humean thesis according to which a rational agent desires a proposition exactly to the degree that she believes the proposition to be desirable. But the assumption is of interest independently of Lewis’ arguments, for instance since both Richard Jeffrey and James Joyce make the assumption (or, strictly speaking, accept a thesis that implies Invariance) in their influential books on decision theory. The main claim to be defended in this paper is that Invariance is incompatible with certain assumptions of decision theory. I show that the assumption fails on the most common interpretations of desirability and/or choice-worthiness found in decision theory. I moreover show that Invariance is inconsistent with Richard Jeffrey’s decision theory, on which Lewis’ arguments against DAB are based. Finally, I show that Invariance contradicts how we in general do and should think about conditional desirability.     

Estimating Conditional Chances and Evaluating Counterfactuals

The paper addresses a puzzle about the probabilistic evaluation of counterfactuals, raised by Ernest Adams as a problem for his own theory. I discuss Brian Skyrms’s response to the puzzle. I compare this puzzle with other puzzles about counterfactuals that have arisen more recently. And I attempt to solve the puzzle in a way that is consistent with Adams’s proposal about counterfactuals.     

The limits of unification for theory appraisal: a case of economics and psychology

In this paper I examine Don Ross’s application of unificationism as a methodological criterion of theory appraisal in economics and cognitive science. Against Ross’s critique that explanations of the preference reversal phenomenon by the ‘heuristics and biases’ programme is ad hoc or ‘Ptolemaic’, I argue that the compatibility hypothesis, one of the explanations offerd by this programme, is theoretically and empirically well-motivated. A careful examination of this hypothesis suggests several strengths of a procedural approach to modelling cognitive processes underlying individual decision making, compared to a multiple-agent approach which Ross promotes. I argue that the debate between economists and psychologists are both theoretical and empirical, but cannot be resolved by appealing to the ideal of unification.