After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( 1930 ) to secure (...) ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair. (shrink)

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)

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis ('SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ('CA') without resort to the method of limits. Formally, however, unlike Robinsonian 'nonstandard analysis', SIA conflicts with CA, deriving, e.g., 'not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this 'change of logic', (...) arguing that standard arguments based on 'smoothness' requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism. (shrink)

This paper presents parts of a theory of radical translation with applications to the problem of construing reference. First, in sections 1 to 4 the general standpoint, inspired by Goodman's approach to induction, is set forth. Codification of sound translational practice replaces the aim of behavioral reduction of semantic notions. The need for a theory of translational projection (manual construction on the basis of a finite empirical correlation of sentences) is established by showing the anomalies otherwise resulting (e.g. from Quine's (...) position). Then in section 5 a partial characterization of what constitutes an "admissible translation manual" is developed in the form of rules of translational projection governing correspondence of sentence-parts of a language-pair. Finally, in section 6, we give applications of the theory to the problem of construing reference. The general rules of section 5 suffice to rule out deviant referential schemes which have been held inscrutable alternatives to standard schemes. (shrink)

Although ‘most contemporary analytic philosophers [endorse] a physicalist picture of the world’ (A. Newen; V. Hoffmann; M. Esfeld, ‘Preface to Mental Causation, Externalism and Self‐Knowledge’, Erkenntnis, 67 (2007), p. 147), it is unclear what exactly the physicalist thesis states. The response that physicalism is the thesis that everything is physical does not solve the problem but is a precise statement of the problem because ‘the claim is hopelessly vague’ (G. Hellman; F. Thompson, ‘Physicalism: Ontology, Determination, and Reduction’, Journal of (...) Philosophy, 72 (1975), p. 552). I argue that physicalism in fact should be the thesis that every existing particular essentially exemplifies properties the exemplification of which does not conceptually entail the existence of conscious beings. Physicalism thus is a purely philosophical thesis with no intrinsic relation to physics.1. (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.

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)

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)

This is a welcome reprint of a book that continues to grow in importance.

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)

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

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)

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)

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.

Together with relativity theory, quantum mechanics stands as the conceptual foundation of modern physics. It forms the basis by which we understand the minute workings of the subatomic world. But at its core lies a paradox--it is unmeasurable. This book presents a powerful and energetic new approach to the measurement dilemma.

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)

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.

This book presents G. A. Cohen's Gifford Lectures, delivered at the University of Edinburgh in 1996. Focusing on Marxism and Rawlsian liberalism, Cohen draws a connection between these thought systems and the choices that shape a person's life. In the case of Marxism, the relevant life is his own: a communist upbringing in the 1940s in Montreal, which induced a belief in a strongly socialist egalitarian doctrine. The narrative of Cohen's reckoning with that inheritance develops through a series of sophisticated (...) engagements with the central questions of social and political philosophy. In the case of Rawlsian doctrine, Cohen looks to people's lives in general. He argues that egalitarian justice is not only, as Rawlsian liberalism teaches, a matter of rules that define the structure of society, but also a matter of personal attitude and choice. Personal attitude and choice are, moreover, the stuff of which social structure itself is made. Those truths have not informed political philosophy as much as they should, and Cohen's focus on them brings political philosophy closer to moral philosophy, and to the Judeo-Christian ethical tradition, than it has recently been. (shrink)

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

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)