Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- and fourthorder number (...) theory. The mathenatics of many structures of central importance in functional analysis, measure theory, and topology can be recovered within essentially these frameworks. (shrink)

Drawing on philosophical thought from the eighteenth century as well as conceptual frameworks developed in the twenty-first century, the essays in Beyond Sense and Sensibility examine moral formation as represented in or implicitly produced by literary works of late eighteenth-century British authors.

This paper investigates how coercive and non-coercive power impacts on the successful diffusion of sustainability within supply networks. The paper reports on four in-depth case studies of the development of sustainability initiatives, each case based on data collection from focal companies and suppliers. The four case studies are based on 38 semi-structured interviews in total and supported by secondary data. The case studies indicate that both coercive and non-coercive power impact suppliers’ engagement in sustainability initiatives and its wider diffusion in (...) supply networks. However, where the use of coercive power facilitates diffusion to immediate suppliers, the use of non-coercive (reward and expert) power leads to sustainability diffusion beyond the dyadic level into wider supply networks. The study provides rich insights into understanding sustainability diffusion in supply networks and the perceptions of multiple supply network actors on the role of different types of power on the diffusion process. We elaborate existing theory and formulate propositions to guide future research into the role and coexistence of different types of power in diffusing sustainability in supply networks. (shrink)

Columella in his Res Rustica always speaks of the bailiff as the slave of the owner of the farm, but in his Preface he states that the owner sometimes sent a mercenarius to be bailiff, and this has by some scholars been taken to mean that a freeborn labourer could be appointed. Since such a possibility is not mentioned by Columella elsewhere or by any other Roman writer, it is probable that the term mercenarius in the Preface has been misunderstood. (...) Hired labourers in ancient Rome included slaves as well as freemen. (shrink)

My purpose is to compare the different explanations that have been offered of the expression propius periclo it timor. This is its context. Evander, king of Pallanteum, has decided to send cavalry under the command of his son Pallas to assist the Trojans and Etruscans in the war against Turnus. When a report spreads that the cavalry are about to set out, the mothers of the soldiers are alarmed. uota metu duplicant matres, propiusque periclo it timor et maior Martis iam (...) apparet imago. ‘The mothers redouble their vows in fear, fear goes closer to the danger, and the image of Mars appears greater.’ James Henry writes: ‘The women, who had previously … made vows, felt fear, and seen the image of war before them , now, when the war has thus come to their very door, … double their vows, feel a sharper fear, and see their picture, image, or idea of war larger than it was before. ‘But propius periclo it has been explained in other ways. Some think periclo dative, other ablative. Conington increases the confusion by citing the explanations of Heyne and Wagner as if they were compatible with his own translation. (shrink)

Juturna drives Turnus’ chariot now here now there, hoping to throw off Aeneas’ pursuit, but he follows the twisted circles of her course. Virgil compares her to a black hirundo flying through a rich man's house out into the colonnades and then round the pools or fishtanks. Hirundo can mean swallow, martin, or even swift. All these birds eat insects and air-borne spiders; they do not eat human food. The common swallow chiefly eats flies, and feeds the nestlings on flies; (...) it also eats wasps and bees. Its average prey size is much greater than the house martin's. Virgil's hirundo gathers pabula parua for the nestlings. W F. J. Knight in the Penguin translation writes ‘tiny scraps of food’; C. Day Lewis translates ‘crumbs of food’. If Virgil meant scraps of meat or crumbs of bread, stolen from the rich man's dinner table, then Virgil did not know what these birds eat. (shrink)

This article examines the complaint that arbitrary algorithmic decisions wrong those whom they affect. It makes three contributions. First, it provides an analysis of what arbitrariness means in this context. Second, it argues that arbitrariness is not of moral concern except when special circumstances apply. However, when the same algorithm or different algorithms based on the same data are used in multiple contexts, a person may be arbitrarily excluded from a broad range of opportunities. The third contribution is to explain (...) why this systemic exclusion is of moral concern and to offer a solution to address it. (shrink)

There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...) when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’. (shrink)

This book explores the research of Professor Hilary Putnam, a Harvard professor as well as a leading philosopher, mathematician and computer scientist. It features the work of distinguished scholars in the field as well as a selection of young academics who have studied topics closely connected to Putnam’s work. It includes 12 papers that analyze, develop, and constructively criticize this notable professor's research in mathematical logic, the philosophy of logic and the philosophy of mathematics. In addition, it features a short (...) essay presenting reminiscences and anecdotes about Putnam from his friends and colleagues, and also includes an extensive bibliography of his work in mathematics and logic. The book offers readers a comprehensive review of outstanding contributions in logic and mathematics as well as an engaging dialogue between prominent scholars and researchers. It provides those interested in mathematical logic, the philosophy of logic, and the philosophy of mathematics unique insights into the work of Hilary Putnam. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

In this article, we investigate the role an ethic of care might play in constructing a normative model of ethical practice for journalism. How would practice be changed if the goal of journalism shifted from the traditional epistemological understanding to an ontological-ethical orientation? What would it mean for journalism to think of itself as an institution committed to aiding in the construction of a community defined by the solidarity of its citizens with one another?

Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal (...) language, and we show that the two approaches are equivalent. (shrink)

Paradoxically, DNA sequence polymorphisms in cancer risk loci rarely correlate with the expression of cancer genes. Therefore, the molecular mechanism underlying an individual's susceptibility to cancer has remained largely unknown. However, recent evaluations of the correlations between DNA methylation and gene expression levels across healthy and cancerous genomes have revealed enrichment of disease‐related DNA methylation variations within disease‐associated risk loci. Moreover, it appears that transcriptional enhancers embedded in cancer risk loci often contain DNA methylation sites that closely define the expression (...) of prominent cancer genes, despite the lack of significant correlations between gene expression levels and the surrounding disease‐associated polymorphic sequences. We suggest that DNA methylation variations may obscure the effect of co‐residing risk sequence alleles. Analysis of enhancer methylation data may help to reveal the regulatory circuits underlying predisposition to cancers and other common diseases.Editor's suggested further reading in BioEssays DNA methylation reprogramming in cancer: Does it act by re‐configuring the binding landscape of Polycomb repressive complexes? Abstract. (shrink)

Through the years there have been subtle changes in the evaluations of the work of Tycho Brahe. As one examines the tracts dealing with novae and comets in which reference is made to the nova of 1572 or the comet of 1577, it becomes quite evident that in different parts of Europe and in the Near East and at different periods of time and among men of different religious convictions different values were placed on his work. The extent of his (...) influence should be distinguished from the measure of his achievements. Moreover, his importance cannot be completely separated from that of Kepler and the horde of other writers who furnished more than a mere background for the display of Tycho's brilliance. Here, as always, there is the danger of assigning to one man innovations that were, so to speak, in the air. (shrink)

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...) relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or space-time.

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality (...) of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to other systems, without regard to the particular nature of (...) the objects themselves. Geometric spaces need not be made up of spatial or temporal points or other intrinsically geometric objects; as Hilbert famously put it, items of furniture suitably interrelated could satisfy all the relevant axiomatic conditions as far as pure mathematics is concerned. A group, for instance, can be any multiplicity of objects with operations fulfilling the basic requirements of the binary group operation; indeed the very abstractness of the group concept allows for its remarkably wide applicability in pure and applied mathematics. Similar remarks can be made regarding other algebraic structures, and the many spaces of analysis, differential geometry, topology, etc. Of course, mathematicians distinguish between “abstract structures” and “concrete ones”, e.g. made up of familiar, basic items such as real or complex numbers or functions of such, or rationals, or integers, etc. (For example, the space L2 of square-integrable functions from R (or Rn) to C, with inner product (f, g) =. (shrink)

Some remarks on determination, physicalism, model theory, and logical truth.//An attempt to defend physicalism against objections that its bases are indeterminate.

We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...) of "indecomposability" from a non-punctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own. (shrink)

The essays in this volume apply philosophical analysis to address three kinds of questions: What are the implications of genetic science for our understanding of nature? What might it influence in our conception of human nature? What challenges does genetic science pose for specific issues of private conduct or public policy?

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, (...) and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives. (shrink)

Standard proofs of generalized Bell theorems, aiming to restrict stochastic, local hidden-variable theories for quantum correlation phenomena, employ as a locality condition the requirement of conditional stochastic independence. The connection between this and the no-superluminary-action requirement of the special theory of relativity has been a topic of controversy. In this paper, we introduce an alternative locality condition for stochastic theories, framed in terms of the models of such a theory (§2). It is a natural generalization of a light-cone determination condition (...) that is essentially equivalent to mathematical conditions that have been used to derive Bell inequalities in the deterministic case. Further, it is roughly equivalent to a condition proposed by Bell that, in one investigation, needed to be supplemented with a much stronger assumption in order to yield an inequality violated by some quantum mechanical predictions. It is shown here that this reflects a very general situation: from the proposed locality condition, even adding the strict anticorrelation condition and the auxiliary hypotheses needed to derive experimentally useful (and theoretically telling) inequalities, no Bell-type inequality is derivable. (These independence claims are the burden of §4.) A certain limitation on the scope of the proposed stochastic locality condition is exposed (§5), but it is found to be rather minor. The conclusion is thus supported that conditional stochastic independence, however reasonable on other grounds, is essentially stronger than what is required by the special theory.Our results stand in apparent contradiction with a class of derivations purporting to obtain generalized Bell inequalities from locality alone. It is shown in Appendix (B) that such proofs do not achieve their goal. This fits with our conclusion that generalized Bell theorems are not straightforward generalizations of theorems restricting deterministic hidden-variable theories, and that, in fact, such generalizations do not exist. This leaves open the possibility that a satisfactory, non-deterministic account of the quantum correlation phenomena can be given within the framework of the special theory. (shrink)

This article argues that the fact that an action will compound a prior injustice counts as a reason against doing the action. I call this reason The Anti-Compounding Injustice principle or aci. Compounding injustice and the aci principle are likely to be relevant when analyzing the moral issues raised by “big data” and its combination with the computational power of machine learning and artificial intelligence. Past injustice can infect the data used in algorithmic decisions in two distinct ways. Sometimes prior (...) injustice undermines the accuracy of the data itself. In these contexts, improving accuracy will also help to avoid compounding injustice. Other times, past injustice produces real-world differences among people with regard to skills, health, wealth, and other traits that employers, lenders, and others seek to measure. When decisions are based on accurate data that itself results from prior injustice, these decisions can also compound injustice. This second dynamic has received less attention than the first but is especially important because improving the accuracy of data will not mitigate this unfairness. (shrink)

Exploring the philosophical foundations of discrimination law as it exists in several jurisdictions, this collection of all new essays bridges the gap between abstract philosophical work on justice and fairness and legal work on specific types of discrimination.