The importance of Georg Cantor’s religious convictions is often neglected in discussions of his mathematics and metaphysics. Herein I argue, pace Jan ́e (1995), that due to the importance of Christianity to Cantor, he would have never thought of absolutely infinite collections/inconsistent multiplicities,as being merely potential, or as being purely mathematical entities. I begin by considering and rejecting two arguments due to Ignacio Jan ́e based on letters to Hilbert and the generating principles for ordinals, respectively, showing that my reading (...) of Cantor is consistent with that evidence. I then argue that evidence from Cantor’s later writings shows that he was still very religious later in his career, and thus would not have given up on the reality of the absolute, as that would imply an imperfection on the part of God. (shrink)

This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. A series of chapters all set in the eighteenth century consider topics such as John Marsh’s techniques (...) for the computation of decimal fractions, Euler’s efforts to compute the surface area of scalene cones, a little-known work by John Playfair on the practical aspects of mathematics, and Monge’s use of descriptive geometry. After a brief stop in the nineteenth century to consider the culture of research mathematics in 1860s Prussia, the book moves into the twentieth century with an examination of the historical context within which the Axiom of Choice was developed and a paper discussing Anatoly Vlasov’s adaptation of the Boltzmann equation to ionized gases. The remaining chapters deal with the philosophy of twentieth-century mathematics through topics such as an historically informed discussion of finitism and its limits; a reexamination of Mary Leng’s defenses of mathematical fictionalism through an alternative, anti-realist approach to mathematics; and a look at the reasons that mathematicians select specific problems to pursue. Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics. (shrink)

In Winter 2017, the first author piloted a course in formal logic in which we aimed to (a) improve student engagement and mastery of the content, and (b) reduce maths anxiety and its negative effects on student outcomes, by adopting student oriented teaching including peer instruction and classroom flipping techniques. The course implemented a partially flipped approach, and incorporated group-work and peer learning elements, while retaining some of the traditional lecture format. By doing this, a wide variety of student learning (...) preferences could be provided for. (shrink)

G. Genzten’s 1938 proof of the consistency of pure arithmetic was hailed as a success for finitism and constructivism, but his proof requires induction along ordinal notations in Cantor normal form up to the first epsilon number, ε0. This left the task of giving a finitisically acceptable proof of the well-ordering of those ordinal notations, without which Gentzen’s proof could hardly be seen as a success for finitism. In his seminal book Proof Theory G. Takeuti provides such a proof. After (...) a brief philosophical introduction, we provide a reconstruction of Takeuti’s proof including corrections, comments, re-organization and notational adjustments for the sake of clarity. The result is a much longer, but much more tractable proof of the well-ordering of ordinal notations in Cantor normal form less than ε0, that nevertheless follows Takeuti’s strategy closely. We end with some more general comments about that proof strategy and the notion of accessibility more generally. (shrink)

David Hilbert sought to secure the epistemic foundations of mathematics by providing consistency proofs of axiomatized mathematical theories from within the finite standpoint. This standpoint requires concrete constructions without reference to completed infinities. In 1938, Gerhardt Gentzen proved the consistency of first-order Peano Arithmetic relying on the well-ordering of certain ordinal notations. This was thought by Gentzen and Paul Bernays to be finitistically acceptable. However, a finitistically acceptable proof of the relevant well-ordering was not available until Gaisi Takeuti’s proof in (...) 1978. Beginning with the proposition that Takeuti’s proof conforms to the finite standpoint, this chapter explores the limits of a finitism based on the methods of Gentzen and Takeuti. It is argued that the upper limit is the Feferman-Schütte ordinal, and thus that finitism may be extended to strong subsystems of second-order arithmetic-systems strong enough to provide a theory of the ordinals for which Gentzen employed notations. (shrink)

The question of the analyticity of Hume’s Principle (HP) is central to the neo-logicist project. We take on this question with respect to Frege’s definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege’s definition of number, it isn’t analytic, and if HP is taken to (...) be primitive there is only a very narrow range of circumstances where it might be taken to be analytic. The latter discussion also sheds some light on the connections between the Bad Company and Caesar objections. (shrink)

If one of Gentzen’s consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert’s program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen’s second proof can be finitistically justified. In particular, the focus is on Takeuti’s purportedly finitistically acceptable proof of the well ordering of ordinal notations in Cantor normal form.The paper begins with a historically informed discussion of finitism and its limits, before (...) introducing Gentzen and Takeuti’s respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti’s proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti’s proof, and therefore Gentzen’s proof, conforms to. (shrink)

If one of Gentzen’s consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert’s program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen’s second proof can be finitistically justified. In particular, the focus is on Takeuti’s purportedly finitistically acceptable proof of the well ordering of ordinal notations in Cantor normal form.The paper begins with a historically informed discussion of finitism and its limits, before (...) introducing Gentzen and Takeuti’s respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti’s proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti’s proof, and therefore Gentzen’s proof, conforms to. (shrink)

If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of (...) finitism and its limits, before introducing Gentzen and Takeuti's respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti's proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti's proof, and therefore Gentzen's proof, conforms to. (shrink)

forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, soundness, and functional completeness. Exercises with solutions are available. (...) It is provided in PDF (for screen reading, printing, and a special version for dyslexics), HTML, and in LaTeX source code. (shrink)

forall x: Dortmund is an adaptation and German translation of forall x: Calgary. As such, it is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity, the syntax of truth-functional (propositional) logic and truth-table semantics, the syntax of first-order (predicate) logic with identity and first-order interpretations, formalizing German in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as the (...) syntax and (Kripke-)semantics of modal logic. The book is provided in PDF and in LaTeX source code. A booklet with solutions for the exercises in the book is available. (shrink)

Livro-texto de introdução à lógica, com (mais do que) pitadas de filosofia da lógica, produzido como uma versão revista e ampliada do livro Forallx: Calgary. Trata-se da versão de 13 de outubro de 2022. Comentários, críticas, correções e sugestões são muito bem-vindos.

As sharing and secondary research use of biospecimens increases, IRBs and researchers face the challenge of protecting and respecting donors without comprehensive regulations addressing the human subject protection issues posed by biobanking. Variation in IRB biobanking policies about these issues has not been well documented.

Reformed Scholastic John Owen's appropriation and adaptation of Thomas Aquinas’ development of the classical ‘disposition’ (Latin: habitus) concept offers practical insight into seventeenth century faculty psychology. This article argues that Owen not only borrows deliberately from Aquinas, he also attempts to simplify and even improve upon Aquinas’ more complicated theological, philosophical, and psychological insights in this important area. While he deals with dispositions of the mind, will, and affections in a way that is broadly similar to Aquinas’ ontological understanding, (...) Owen's most significant contribution to seventeenth century faculty psychology and its theological use is a sustained and consistent emphasis on the necessity of virtuous affections in the pursuit of communion with God. Examining this concept also provides greater context for how the Reformed Scholastics were able to interact with their medieval counterparts. In this we see the Reformed Scholastics’ continuity with the Christian tradition and their depth of understanding regarding human nature. (shrink)

Antisocial behaviors including bullying, violence, and aggression have been an area of intense interest among researchers, clinicians, policy makers, and the general public because of their grievous consequences on individuals and society. Our understanding of the origins and development of these behaviors in individuals has recently progressed with the application of new scientific advancements and technologies such as neuroimaging, genomics, and research methods that capture behavioral changes in the first few years of life.The Origins of Antisocial Behavior: A Developmental Perspective (...) provides an overview of the recent research on the development of antisocial behavior and synthesizes this information to inform readers not only of the risks, but also how they interact, to result in antisocial and aggressive behavior. The volume is divided into three sections: advances in neuroscience, advances in behavioral and clinical research, and legal and policy implications. Specific topics include genetic markers and aggressive behavior, the use of fMRI to track adolescent brain development, the role of peer influences on aggression, parenting and temperament, screening tools for diagnosing antisocial behavior in toddlers and adolescents, and how new research will influence interventions, policy, and future study. Experts from genetics, neuroimaging, and developmental science discuss the insights these scientific approaches have provided in understanding how nature and the environment interact in the emergence of antisocial behavior. The Origins of Antisocial Behavior is an important and unique resource that will be of use to developmental scientists, mental health professionals, and policymakers involved in the juvenile justice system. (shrink)

Meta-learning is even more broadly relevant to the study of inductive biases than Binz et al. suggest: Its implications go beyond the extensions to rational analysis that they discuss. One noteworthy example is that meta-learning can act as a bridge between the vector representations of neural networks and the symbolic hypothesis spaces used in many Bayesian models.

Volume 20, Issue 7, July 2020, Page W15-W16.

We might begin a search for the possibility of translation with a search for a common moral foundation for both the Eastasian world and the Western world. Answers come easily with a qualified yes or no; for example, we might make a list or table comparing how East and West think about adultery. This is unsatisfying; we ourselves are often unclear what we think and feel about these issues. So, a few of the more circumspect might ask where our own (...) sense of the moral arises. Perhaps we will seek the source of the OUGHT. Our friends from the Orient might ask us what these words moral or OUGHT mean. I do not think the West can as easily convey to the East what it means by sense of obligation as the West imparts technology. Try to translate our word obligation by the Japanese word giri; this translation is suspect since for the West, obligations are obligations only if they grate against the well ingrained inclinations of our being. The Kantian necessary test of authenticity of the true moral act will be if we act against the grain of our inclinations. (shrink)

THE MOST SACRED FREEDOM includes eight essays that were first presented at the 2014 A.V. Elliott Conference on Great Books and Ideas, the seventh annual conference sponsored by Mercer Universitys Thomas C. and Ramona E. McDonald Center for Americas Founding Principles. Together, these essays explore the great principle of religious liberty by charting its development in the Western tradition and reconsidering its place at Americas founding. The book begins with a comparison between the flood accounts in Genesis and the (...) Mesopotamian Atra-Has's and advances all the way to the 2014 Supreme Court case Burwell v. Hobby Lobby. The intervening chapters examine the contributions of figures such as Emperor Julian, Roger Williams, Cecilius Calvert, John Locke, Jean-Jacques Rousseau, and the American Founders. Contributors to the volume are Steven Grosby, Jeremiah H. Russell, Maura Jane Farrelly, Daniel Cullen, John Witte, Jr., Scott Yenor, David Ramsey, and Michael Novak. (shrink)

This study looked for evidence of biases in the allocation and disengagement of attention in dysphoric individuals. Participants studied images for a recognition memory test while their eye fixations were tracked and recorded. Four image types were presented (depression-related, anxiety-related, positive, neutral) in each of two study conditions. For the simultaneous study condition, four images (one of each type) were presented simultaneously for 10 seconds, and the number of fixations and the total fixation time to each image was measured, similar (...) to the procedure used by Eizenman et al. (2003) and Kellough, Beevers, Ellis, and Wells (2008). For the sequential study condition, four images (one of each type) were presented consecutively, each for 4 seconds; to measure disengagement of attention an endogenous cuing procedure was used (Posner, 1980). Dysphoric individuals spent significantly less time attending to positive images than non-dysphoric individuals, but there were no group differences in attention to depression-related images. There was also no evidence of a dysphoria-related bias in initial shifts of attention. With respect to the disengagement of attention, dysphoric individuals were slower to disengage their attention from depression-related images. The recognition memory data showed that dysphoric individuals had poorer memory for emotional images, but there was no evidence of a conventional mood-congruent memory bias. Differences in the attentional and memory biases observed in depressed and dysphoric individuals are discussed. (shrink)

Revisits, reassesses, and reclaims the legacy of May '68 in light of our present cultural and historical emergency.

Revisits, reassesses, and reclaims the legacy of May '68 in light of our present cultural and historical emergency.

Description: The book is, so to say, a bouquet in two respects. It is, first, a presentation of academic tributes, in the form of a festschrift, to a well-known Indian philosopher Professor Margaret Chatterjee; and, second, a hand-picked collection of original essays of multifaceted reflection for serious students of philosophy. Areas of study covered are various-metaphilosophy, philosophy or religion, metaphysics, aesthetics, existentialism, and Indian and comparative philosophies; and so are the lands of the philosophers who have contributed to the making (...) of this volume: India, England, Greece, America, Canada, and Japan. The work is a signal product of international cooperation and philosophy-a cause which Professor Chatterjee has been actively pursuing for long and with great success. (shrink)

Emotional reactivity has been implicated in the development and maintenance of anxiety. The metacognitive model suggests that maladaptive metacognitive beliefs may inc...

People diagnosed with cancer typically want information from their doctor or nurse. However, many individuals now turn to the Internet to tackle unmet information needs and to complement healthcare professional information. The purpose of this study was to qualitatively explore the content of commonly searched cancer websites from a critical nursing perspective, as this information is accessible, and allows patients to address their information needs in ways that healthcare professionals cannot. This qualitative examination of websites is informed by Carper's fundamental (...) patterns of knowing and complemented with the critical view to technology espoused by the philosophy of technology. We conducted a review of 20 websites using a two‐step interpretive descriptive approach and thematic analysis. We identified the dominant discourse to be focused on empirical information on treatment, prognosis, and cure, and a paucity of sociopolitical, ethical, personal, and esthetic information. In place of holistic, nuanced, and accurate knowledge nurses may provide, patients find predominantly empirical and biomedical information online. Discussion explores and critiques online cancer content, gaps in information, and the importance of information diversity. Implications focus on needed discourse around pervasive technologies and the nursing role in assessing and directing patients to holistic information. (shrink)

This article questions the extent to which interpretation explains literature, arguing that interpretation and explanation are not the same thing. It first engages with recent critical discussions of method and explanation in literary studies, finding that they are not much about method at all. It then offers a methodological framework that goes beyond various "method wars" over "critique" and "postcritique," and toward ways of explaining literature that are not reducible to matters of interpretation.