How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence (...) on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical (...) depth’. (shrink)

Hilbert took to using ideal elements in the 1890's, in both algebraic number theory and geometry. His Zahlbericht of 1897 popularized the concept of the ideal introduced by Dedekind in 1871 (which in turn formalized the concept of "ideal number" introduced by Kummer in the 1840's). His geometric work likewise followed a long history of ideal elements, some that originated in geometry and others that originated elsewhere and were applied to geometry. Important examples were:Piero della Francesca's Ideal City.Points at infinity (...) in projective geometry, originating in the 1430s from the "vanishing points" in perspective drawing.Imaginary points in algebraic geometry, originating from the imaginary numbers used by .. (shrink)

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse (...) Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. (shrink)

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a (...) large extent, the story of philosophy of mathematics. (shrink)

Ikujiro Nonaka, whose formative experience is Japanese, is an established scholar who has written about large business organizations. He sees knowledge at the heart of the organization and its products and aims to develop Michael Polanyi’s conception of tacit knowledge in a practical direction to enhance organizational “knowledge creation.” For Nonaka, what matters is the practice, the doing, the embodiment of knowledge. An organization can amplify and crystallize individuals’ tacit knowledge in a process that allows them to experience deeper understanding. (...) Nonaka holds that it is iimportant to explore the potential that knowledge holds. His spiral process describes disciplined practices that make tacit knowledge independent and available to restructure the organizational knowledge context. (shrink)

The following three related contributions jointly serve to lift up elements of the thought of the anthropolo­gist Gregory Bateson that can be fruitfully compared with elements of Michael Polanyi’s thought. In a brief introduction, William Stillwell reviews Bateson’s life and developing interests. Stillwell also provides, in a creative dialog form akin to Bateson’s own dialogs, a short review article on Noel Charlton’s Understanding Gregory Bateson: Mind, Beauty and the Sacred Earth. The third piece is Jere Moorman’s short 1991 (...) essay (now out of print) discussing Polanyi’s ideas about tacit knowing and their connection with Bateson’s ideas about the double bind. (shrink)

I present a new interpretation of Wittgenstein's later philosophy of logic and mathematics. This interpretation, like others, emphasizes Wittgenstein's attempt to reconcile platonistic and constructivistic approaches. But, unlike other interpretations, mine explains that attempt in terms of Wittgenstein's position about the relations between our concepts of necessity and provability. If what I say here is correct, then we can rescue Wittgenstein from the charge of naive relativism. For his relativism extends only to provability, and not to necessity.

Many contemporary theorists charge that naïve realists are incapable of accounting for illusions. Various sophisticated proposals have been ventured to meet this charge. Here, we take a different approach and dispute whether the naïve realist owes any distinctive account of illusion. To this end, we begin with a simple, naïve account of veridical perception. We then examine the case that this account cannot be extended to illusions. By reconstructing an explicit version of this argument, we show that it depends critically (...) on the contention that perceptual experience is diaphanous, or more minimally and precisely, that there can be no difference in phenomenal properties between two experiences without a difference in the scenes presented in those experiences. Finding no good reason to accept this claim, we develop and defend a simple, naïve account of both veridical perception and illusion, here dubbed Simple, Austere Naïve Realism. (shrink)

The larger project of which this volume forms part is an attempt to craft a coherent doctrine of divine eternity and God's relationship to time.

This paper discusses the mistake of understanding the laws and concepts of thermodynamics too literally in the foundations of statistical mechanics. Arguing that this error is still made in subtle ways, the article explores its occurrence in three examples: the Second Law, the concept of equilibrium and the definition of phase transitions.

Throughout this century many philosophers and physicists have gone for thc ‘big ki11’ regarding tenses. They have tried to show via McTaggart’s paradox and special relativity that tcnscs arc logically and physically impossible, rcspcctivcly. Ncithcr attempt succccds, though as I argue, both lcavc their mark. In thc iirst two sections of thc paper I introduce some conceptual difficulties for the tensed theory of time. The next section then discusses the standing 0f tenses in light of special relativity, cspccially rcccnt work (...) by Stcin on thc topic. I argue that, Stcin’s possibility thcorcm notwithstanding, special relativity is inconsistent with any philosophicully interesting conception of tense. Finally, I search for help for tenses in the broader context of quantum theory, Lorcntzian interpretations 0f time dilation/length contraction, and gcncral relativistic spacctimcs. I suggest that these avenues d0 not provide tenses the home for which some have hoped. (shrink)

Given his view that the modern world is ‘radically evil’, Adorno is an unlikely contributor to business ethics. Despite this, we argue that his work has a number of provocative implications for the field that warrant wider attention. Adorno regards our social world as damaged, unfree, and false and we draw on this critique to outline why the achievement of good work is so rare in contemporary society, focusing in particular on the ethical demands of roles and the ideological nature (...) of management’s self-understanding. Nevertheless, we show that Adorno’s comments on activities such as art and philosophy mean that it is possible to draw on his work in a way that contributes constructively to the conversation about good and meaningful work within business ethics. (shrink)

For the generalizations of thermodynamics to obtain, it appears that a very ‘special’ initial condition of the universe is required. Is this initial condition itself in need of explanation? I argue that it is not. In so doing, I offer a framework in which to think about ‘special’ initial conditions in all areas of science, though I concentrate on the case of thermodynamics. I urge the view that it is not always a serious mark against a theory that it must (...) posit an ‘improbable’ initial condition. (shrink)

Naïve Realists think that the ordinary mind-independent objects that we perceive are constitutive of the character of experience. Some understand this in terms of the idea that experience is diaphanous: that the conscious character of a perceptual experience is entirely constituted by its objects. My main goal here is to argue that Naïve Realists should reject this, but I’ll also highlight some suggestions as to how Naïve Realism might be developed in a non-diaphanous direction.

We describe a model of iterated belief revision that extends the AGM theory of revision to account for the effect of a revision on the conditional beliefs of an agent. In particular, this model ensures that an agent makes as few changes as possible to the conditional component of its belief set. Adopting the Ramsey test, minimal conditional revision provides acceptance conditions for arbitrary right-nested conditionals. We show that problem of determining acceptance of any such nested conditional can be reduced (...) to acceptance tests for unnested conditionals. Thus, iterated revision can be accomplished in a "virtual" manner, using uniterated revision. (shrink)

Digital medicine is a medical treatment that combines technology with drug delivery. The promises of this combination are continuous and remote monitoring, better disease management, self-tracking, self-management of diseases, and improved treatment adherence. These devices pose ethical challenges for patients, providers, and the social practice of medicine. For patients, having both informed consent and a user agreement raises questions of understanding for autonomy and informed consent, therapeutic misconception, external influences on decision making, confidentiality and privacy, and device dependability. For providers, (...) digital medicine changes the relationship where trust can be verified, clinicians can be monitored, expectations must be managed, and new liability risks may be assumed. Other ethical questions include direct third-party monitoring of health treatment, affordability, and planning for adverse events in the case of device malfunction. This article seeks to lay out the ethical landscape for the implementation of such devices in patient care. (shrink)

Suppose that God or a demon informs you of the following future fact: despite recent cosmological evidence, the universe is indeed closed and it will have a ‘final’ instant of time; moreover, at that final moment, all 49 of the world’s Imperial Faberge eggs will be in your bedroom bureau’s sock drawer. You’re absolutely certain that this information is true. All of your other dealings with supernatural powers have demonstrated that they are a trustworthy lot.

This chapter contains sections titled: * Arguments for Divine Timelessness * Arguments for Divine Temporality * Eternity and the Nature of Time * Notes.

In “Artificial Intelligence, Social Media and Depression: A New Concept of Health-Related Digital Autonomy,” Laacke and colleagues posit a revised model of autonomy when using digital algori...

Imprint covered by label which reads : Barnes & Noble Books, Totowa, N.J. Includes bibliographical references and index.

This paper attempts to show how Adorno’s thought can illuminate our reflections on the future of work. It does so by situating Adorno’s conception of genuine activity in relation to his negativist critical epistemology and his subtle account of the distinction between true and false needs. What emerges is an understanding of work that can guide our aspirations for the future of work, and one we illustrate via discussions of creative work and care work. These are types of work which (...) cater to persistent human needs, albeit ones that are distorted under present social conditions. Adorno’s thought helps us to understand why this is the case. (shrink)

The Past Hypothesis is the claim that the Boltzmann entropy of the universe was extremely low when the universe began. Can we make sense of this claim when *classical* gravitation is included in the system? I first show that the standard rationale for not worrying about gravity is too quick. If the paper does nothing else, my hope is that it gets the problems induced by gravity the attention they deserve in the foundations of physics. I then try to make (...) plausible a very weak claim: that there is a well-defined Boltzmann entropy that *can* increase in *some* interesting self-gravitating systems. More work is needed before we can say whether this claim answers the threat to the standard explanation of entropy increase. (shrink)

Two females, Nadine and Fatu, are the sole surviving Northern White Rhinos (NWR). The subspecies is functionally extinct. Hope for NWR now lies in emerging reproductive and genetic technologies, which could potentially produce NWR from induced pluripotent stem cells. What is the rationale for this project? This question raises almost every philosophical issue facing conservation science today. I argue that NWR recovery is hard to justify via many traditional paths (e.g., historical fidelity, ecosystem health, biodiversity), but if we shift focus (...) to white rhinos in general or even mammals then clear benefits emerge. (shrink)

Most of us would want to say that it is true that Socrates taught Plato. According to realists about past facts,1 this is made true by the fact that there is, located in the past, i.e., earlier than now, at least one real event that is the teaching of Plato by Socrates. Presentists, however, in denying that past events and facts exist2 cannot appeal to such facts to make their past-tensed statements true. So what is a presentist to do?

This paper draws from the fields of history, sociology, psychology, moral philosophy, and organizational theory to establish a theoretical connection between a social/organizational influence (ethical work climate) and an individual cognitive element of moral behavior (moral awareness). The research was designed to help to fill a gap in the existing literature by providing empirical evidence of the connection between organizational influences and individual moral awareness and subsequent ethical choices, which has heretofore largely been merely assumed. Results of the study provide (...) evidence that ethical work climate (EWC) is a primary predictor of individual moral awareness, and that the influence of social factors often overrides the effects of individual differences in a work group setting. Implications for future research are provided. (shrink)

Are the generalizations of classical equilibrium thermodynamics true of self-gravitating systems? This question has not been addressed from a foundational perspective, but here I tackle it through a study of the “paradoxes” commonly said to afflict such systems. My goals are twofold: (a) to show that the “paradoxes” raise many questions rarely discussed in the philosophical foundations literature, and (b) to counter the idea that these “paradoxes” spell the end for gravitational equilibrium thermodynamics.

This paper considers the possibility that nonrelativistic quantum mechanics tells us that Nature cares about time reversal. In a classical world we have a fundamentally reversible world that appears irreversible at higher levels, e.g., the thermodynamic level. But in a quantum world we see, if I am correct, a fundamentally irreversible world that appears reversible at higher levels, e.g., the level of classical mechanics. I consider two related symmetries, time reversal invariance and what I call ‘Wigner reversal invariance.’ Violation of (...) the first is interesting, for not only would it fly in the face of the usual story about temporal symmetry, but it also appears to imply (as I’ll explain) that time is ‘handed’, or as some have misleadingly said in the literature, ‘anisotropic’. Violation of the second is, as I hope to show, even more interesting. The paper also contains a discussion of two mostly neglected topics: what it means to say time is handed and what warrants such an attribution to time. (shrink)

Was the first book to examine the exciting area of overlap between philosophy and quantum mechanics with chapters by leading experts from around the world.

In October 2021, the Allen Institute for Artificial Intelligence publicly released Delphi, an artificial intelligence system trained to make general moral decisions (Allen Institute for Artifi...

ABSTRACT:We combine prior research on ethical decision-making in organizations with a rational choice theory of corporate crime from criminology to develop a model of corporate offending that is tested with a sample of U.S. managers. Despite demands for increased sanctioning of corporate offenders, we find that the threat of legal action does not directly affect the likelihood of misconduct. Managers’ evaluations of the ethics of the act, measured using a multidimensional ethics scale, have a significant effect, as do outcome expectancies (...) that result from being associated with the misconduct but not facing formal sanctions. The threat of formal sanctions appears to operate indirectly, influencing ethical evaluations and outcome expectancies. Obedience to authority also affects illegal intentions, with managers reporting higher prospective offending when they are ordered to engage in misconduct by a supervisor. (shrink)

Over 50% of the world population will develop a psychiatric disorder in their lifetime.1 In the realm of psychiatric treatment, two primary modalities have been established: pharmacotherapy and psychotherapy. Yet, pharmacological interventions often take precedence as the initial treatment choice despite their comparable outcomes, severe side effects and disputed evidence of their efficacy. This preference for medication foregrounds a vital re-examination of what it means to be invasive in medical treatments, namely in psychiatric care. De Marco et al challenge the (...) standard account of invasiveness, presenting considerations that transcend the traditional criteria of physical intrusion.2 Through case studies centred on treatments for schizophrenia and depression, we aim to (1) reinforce De Marco et al ’s challenge to the standard account of invasiveness and (2) extend the dialogue to the moral imperative of redefining mental invasiveness. Our aims are to broaden the definition of mental invasiveness, aligning with De Marco et al ’s emphasis on context sensitivity, and to include the severity of side effects as a critical factor in this assessment. Redefining what is considered invasive in psychiatric treatment has profound implications for patient care and medical decision-making. Labelling treatments, namely medications, as invasive encourages both patients and healthcare providers to approach these options with a heightened level of scrutiny and care. This is …. (shrink)

Personification has received little philosophical attention, but Daniel Nolan has recently argued that it has important ramifications for the relationship between fictional representation and possibility. Nolan argues that personification involves the representation of metaphysically impossible identities, which is problematic for anyone who denies that fictions can have impossible content. We develop an account of personification which illuminates how personification enhances engagement with fiction, without need of impossible content. Rather than representing an identity, personification is something that is done with representations—a (...) matter of use rather than content—and involves only a comparison of possibilities. We illustrate our account using the personification of death in the film Meet Joe Black, and show that there are no grounds for taking it to be fictionally true that there is a metaphysically impossible identity between Death and death. (shrink)