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.

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 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)

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)

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)

How can pain complaints be elicited and analyzed so as to increase the empathic bond between patient and clinician? I will argue that though Wierzbicka’s approach to this question is useful—an exploration of certain abstract dimensions of pain’s meaning—it fails to examine key aspects that are the most useful and crucial for cultural analysis and for building empathic bonds between the clinician and patient. Not just a grammar of pain is needed; rather a biological philology of pain.

ABSTRACTConcerned with how individuals assess scientific experts on the Internet, our research investigates the virtual r/science subreddit and their popular Ask-Me-Anything series, where sci...

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)

In recent years, a distinction between two concepts of equality has been much discussed: 'distributive’ equality involves people having equal amounts of a good such as welfare or resources, and ‘social’ or ‘relational’ equality involves the absence of social hierarchy and the presence of equal social relations. This contrast is commonly thought to have important implications for our understanding of the relationship between equality and justice. But the nature and significance of the distinction is far from clear. I examine several (...) accounts of this issue and argue none are entirely satisfactory. In turn, I offer an alternative proposal. Relational equality, on my account, involves a concern with each person having an equal 'civic status'. I characterize this concern and show it has distinctive and normatively significant positional and temporal aspects. (shrink)

Several philosophers argue for the ‘convergence thesis’ for positional goods: prioritarians, sufficientarians, and egalitarians may converge on favouring an equal (or not too unequal) distribution of goods that have positional aspects. I discuss some problems for this thesis when applied to two key goods for which it has been proposed: education and wealth. I show, however, that there is a variant of the thesis that avoids these problems. This version of the thesis is significant, I demonstrate, because it applies to (...) a person’s status as a citizen, which I suggest is the central concern of social or ‘relational’ egalitarianism. (shrink)

This paper examines the historiography of literacy between 1780 and 1830 in Britain. The paper first explores the challenges faced by historians in measuring literacy and examines the lenses through which historians have interpreted the available data. Factors thought to contribute to rising literacy rates in this time period include access to education, the Industrial Revolution, religious movements and gender norms.

This paper examines the following distinctive republican claims: (1) goodwill and virtuous self-restraint are insufficient to realize freedom; and (2) suitable law is constitutive of freedom. In the contemporary literature, these claims are commonly defended in connection with the conception of freedom as nondomination. This account, however, is often rejected on the grounds that freedom as nondomination is moralized and impossible to realize. In response, I propose that the point of protecting people from domination is better understood not as realizing (...) freedom, but instead as giving persons a civic status as equals. Giving persons a civic status as equals, I claim, realizes an important form of social or 'relational' equality, and I distinguish this aim from giving persons a 'status as nondominated’, as some republicans require. I argue that this account vindicates alternative claims about the insufficiency of goodwill and constitutive importance of suitable law—understood in terms of equality rather than freedom—while avoiding the moralization and impossibility objections. I conclude by suggesting some further advantages of the proposed account versus the standard republican view. (shrink)

Paul Ekman is an American psychologist who pioneered the study of facial behaviour. Bringing together disciplinary history, life study, and history of science, this paper focuses on Ekman’s early research during the twenty-year period between 1957 and 1978. I explicate the historical development of Ekman’s semiotic model of facial behaviour, tracing the thread of iconicity through his life and works: from the iconic coding of rapid signs; through the eventual turn from classifying modes of iconic signification using gestalt categories to (...) classifying modes of producing iconic sign-functions using minimal units; to the role and importance of iconicity for the study of the facial expression of emotion, both in terms of the similarities between iconic and analogue signs as well as the differences between facial coding and linguistic signification. In this intellectual genealogy, I argue not only that Ekman relied extensively upon conceptualizations and terminologies from semiotic thought for the creation of the Facial Action Coding System (FACS), but also that the question of iconicity is the pivotal problem across the many discoveries and innovations in what I term ‘Ekmanian faciasemiotics’. (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)

Describes God in simple, easy terms for children.

Throughout Supernatural, we watch the Winchesters resist, embrace, and redefine their roles in the family business, “saving people, hunting things.” These tensions echo a topic that philosophers have explored for thousands of years—free will. According to the existentialist philosopher Jean‐Paul Sartre, each person is in a constant state of shaping himself and his place in the world through free will. Dean is admirable in his ability to resist bad faith and act as captain for Team Free Will. In Supernatural, the (...) choices people make are impacted by their soul. Knowing that Sam could simply walk away from hunting to pursue a normal life shows Dean that they no longer share the same sense of moral responsibility. For good or for ill, Sam and Dean are responsible for their actions, which is better than you'd expect from an ex‐blood junkie and a high school dropout with six dollars in his pocket. (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)

A philosophical analysis of the pessimistic and nihilistic conditions of the existential possibilities for blackness and antiblack racism in 21st Century America.

John Rawls’s thesis that a certain package of basic liberties should be given lexical priority is of great interest for legal and political philosophy, but it has received relatively little defense from Rawls or his supporters. In this paper, I examine three arguments for the thesis: the first is based on the two ‘moral powers’; the second, on the social bases of self-respect; and the third, on a Kantian notion of autonomy. I argue none of these accounts successfully establishes 1) (...) the distinct claim of lexical priority, 2) for the complete package of basic liberties, on the basis of reasons that are appropriately public. In turn, I propose an alternative argument, in support of those two claims, based on the social or ‘relational’ conception of equality. (shrink)

The call for a biosemiotic perspective within medical semiotics has been steadily increasing over the late twentieth and early twenty-first centuries. In _Food and Medicine: A Biosemiotic Perspective_, Yogi Hale Hendlin, Johnathan Hope, and the nine contributions in their edited volume boldly seek to bridge the segregation between nature and culture in the medical sciences as well as in the medical humanities. To a large extent, they achieve this aim by explicating the sign relations _in_ food and medicine, the sign (...) relations _of_ medical theory and practice, and the sign relations _between_ the biology in medicine and medicine of society. Taking up a semio-historical approach, I contextualize two select contributions from Hendlin and Hope’s _Food and Medicine_ with the medical semiotics of the Hippocratic tradition. By comparing the biological semiotics from the contributions to the medical semiotics from the Corpus, I critically explicate the ways in which biosemiotics moves this subdiscipline forward and why the perspective is significant not only for the health of humans, but also for the health of other animals, and indeed for the health of the planet that we all inhabit together. On these grounds, I propose a turn from medical semiotics to _health semiotics_. This program for semiotics would encompass not only food and medicine, but also lifestyle and wellbeing, as well as the subjective, qualitative perspectivism that makes biosemiotics frontier research, thereby constituting a biosemioethics and promoting a semiotic fitness. (shrink)

There is limited evidence that socially responsible investment (SRI) strategies can resolve persistent concerns brought up in scholarship on the industry, particularly as it relates to considerations of justice. It is critical that SRI initiatives be interrogated about their broader impacts on environmental inequality and justice in the context of global power relations. Drawing upon environmental justice (EJ) theory, we propose a framework for transformative investment to halt the exploitation of humans and environment in pursuit of profit. We posit that (...) transformative investment initiatives, including those related to strategies of screening and shareholder resolutions, would reflect three general conditions of environmental justice. They would require (1) cumulative responsibility that focuses on holistic gains so that investment initiatives do not replace one harm with another; (2) embedded accountability that is rooted in the decision-making and concerns of directly impacted communities; and (3) counter-hegemonic practices that are integrated with broader strategies and movements that challenge structures of oppression and exploitation. Through these principles, we address problems with SRI that have already been identified in scholarship on the subject. Moreover, we draw upon Antonio Gramsci’s concept of ‘strategic power’ to conceptualize the conditions and actions necessary for transformative investment to take root. (shrink)

This paper summarizes and discusses three studies of patterns in the recall of persons in socially bounded communities. Individual sin three different communities (a graduate academic program, a religious fellowship, and a department in a formal organization) free-listed the names of persons in their respective communities. Results indicate that the individuals in each community share a common cognitive structure of community members that is based on the community’s social structure. These studies, combined with the results of other research, strongly suggest (...) that persons are organized in memory according to social structural principles and that affiliation and dominance are the principal dimensions of social cognition. Suggestions are offered for future research to test the generality of these findings. (shrink)

The distinction between ‘relational' and ‘distributive’ equality has come to play an important role in discussions of equality and justice. But the nature of the distinction is not as clear as we might hope. In this regard, Juliana Bidadanure makes an interesting and important proposal: the two views involve differing kinds of temporal concern. The distributive approach, she suggests, is concerned with equality over people’s complete lives (diachronic equality), whereas the relational approach is concerned with egalitarian social relations at each (...) moment in time (synchronic equality). I argue that this suggestion, while insightful, is not entirely satisfactory. Both relational and distributive equality, I demonstrate, have diachronic and synchronic aspects. Nonetheless, they do so in distinctive ways. So, while the diachronic/synchronic distinction does not provide a clean cut between relational and distributive equality, examining the temporal aspects of each approach helps to illuminate their distinctiveness. (shrink)

(This book, in Portuguese translation, is an accessible introduction to the philosophy of equality and social justice, with a focus on contemporary theories, especially relational egalitarianism.) -/- O que significa para uma sociedade ser igualitária? A igualdade é amplamente reconhecida como valiosa, seja por cidadãos, políticos, juristas ou filósofos. Ademais, a ideia de igualdade desempenha um papel de grande relevância em muitos movimentos sociais do passado e do presente. Há bastante discordância, porém, acerca de como, exatamente, devemos compreender a natureza (...) e a importância da igualdade. O presente ensaio examina estas questões e apresenta um panorama sumário das mais centrais e influentes abordagens filosóficas à igualdade no quadro de diferentes teorias da justiça, concentrando-se, em particular, na ideia de justiça baseada na igualdade de relações sociais. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Practice of SurgeryKaren DevonThere’s no one on the snowy road driving beside me. It’s Christmas Eve, the night the newest attending surgeon on staff gets to be on call. Tonight feels like an anniversary of sorts. The first time I performed an appendectomy “alone” was on Christmas Eve. I can’t recall if it was snowing back then since I hadn’t left the hospital in days. I had assumed (...) that while I was operating on that 26–year–old woman with new abdominal pain that moved towards the left lower quadrant, my attending staff on call was sleeping soundly. I didn’t even know where he lived, or how long it would take if something catastrophic were to happen. I phoned the white–haired man to get his blessing just before midnight. “Have you done this operation before?” he asked. “Yes, several, but not completely alone” I confessed. “Go for it and call me when you are done. Oh, and be safe!” Click. Gulp.As I drive in tonight, again past midnight, I reflect upon that night. Was it practice? A right of passage? The most appropriate treatment for the patient? Perhaps it was even safer this way since my attending really had never done this operation the way it is now done—laparoscopically—by residents. I clearly recall telling the patient and her mother, “We do this all the time! The staff surgeon on call will come in if I need him, but the chief residents always do this operation and I feel comfortable taking care of you.” Sign on the dotted line and make sure you read the part where you are told this is a teaching hospital. Not only was independence an expectation, but I was also supposed to teach the more junior house staff how to emulate what I was so expertly performing.That night I took out the rotting appendix without incident and called the grey haired surgeon in the wee hours. “Merry Christmas Doctor. Sorry to bother you. I just wanted to let you know that the patient is awake and doing well.” “Yes I know. Good work” he replied. “I was in my office. Just in case.”As a resident, I was often torn between the need to gain independence in order to learn and grow as a surgeon, and the idea that I was practicing. As most budding surgeons do, I wanted to be the person doing the surgery. Working on my technical skills was clearly an important part of expertise. Surgery is inherently different than other types of medicine. It seemed slightly more difficult for someone to stop my cutting, than to catch the error in a prescription, before some damage is done. Perhaps a little more like driving a car, than learning to use a calculator. Recently a more modern surgical education is evolving, whereby students practice by simulations on machines. Yet educators know that ultimately many “firsts” need to occur with real patients. That proficiency and expertise are more than a diagnosis, a steady hand, or even the two combined. If the experience of operating as the primary surgeon without immediate supervision, on a patient with acute appendicitis never occurred until we passed licensing exams wouldn’t that be more harmful? I may have been the least legally responsible for outcomes, that night of my own first, when the grey–haired surgeon took the [End Page E1] responsibility of trusting me. But in fact, the obligation I felt was enormous. We cannot simulate the weight of responsibility that comes when responsible for another human being, who you have met and who provides their trust in you. Furthermore, as the teachers of this art and science, we need to balance our obligations to each individual patient with our obligation to society to train competent and caring future surgeons. Surgery would have no history if the “practice” of surgery had nothing to do with practice, defined in its simplest form, yet how do we cover all of this during the late night conversation that is called informed consent. Interestingly since that December evening and for the next 200 laparoscopic appendectomies before my graduation in to... (shrink)

The notion that genetics, through natural selection, determines innate traits has led to much debate and divergence of thought on the impact of innate traits on the human phenotype. The purpose of this synthesis was to examine how innate theory informs genetic research and how understanding innate theory through the lens of Martha Rogers’ theory of unitary human beings can offer a contemporary view of how innate traits can inform epigenetic and genetic research. We also propose a new conceptual model (...) for genetic and epigenetic research. The philosophical, theoretical, and research literatures were examined for this synthesis. We have merged philosophical and conceptual phenomena from innate theory with the theory of unitary beings into the University of Illinois at Chicago model for genetic and epigenetic research. Innate traits are the cornerstone of the framework but may be modified epigenetically by biological, physiological, psychological, and social determinants as they are transcribed. These modifiers serve as important links between the concept of innate traits and epigenetic modifications, and, like the theory of unitary human beings, the process is understood in the context of individual and environmental interaction that has the potential to evolve as the determinants change. (shrink)