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 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)

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)

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)

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.

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)

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)

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)

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)

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.

Reform of the federal income tax system has become a perennial item on the domestic policy agenda of the United States, although there is considerable uncertainty over specifics. Indeed the recent report of the President's Advisory Panel on Federal Tax Reform recommended not one but two divergent policy directions. In Fundamental Tax Reform, top experts in tax policy discuss a wide range of issues raised by the prospect of significant tax reform, identifying the most critical questions and considering whether the (...) answers are known, unknown--or unknowable. The debates over tax reform usually concern the advantages and disadvantages of income-based taxation as opposed to any of the several alternative forms of consumption-based taxation. The book opens with chapters that discuss the strengths, weaknesses, and political feasibility of these options. Other chapters consider the effect of tax reform on businesses, especially their investment behavior, and include a discussion of possible problems in any transition to a consumption-based tax; international taxation issues arising in an era of globalization; and individual behavioral response to tax reform, including a view of the topic from the perspective of the relatively new field of behavioral economics. ContributorsRosanne Altshuler, Alan J. Auerbach, John W. Diamond, Harry Grubert, Arnold C. Harberger, Kevin A. Hassett, Thomas J. Kniesner, Laurence J. Kotlikoff, Edward J. McCaffery, Kathryn Newmark, David Rapson, Daniel Shaviro, Joel Slemrod, James P. Ziliak, George R. ZodrowDiscussants James Alm, Henry J. Aaron, Charles L. Ballard, Leonard E. Burman, Robert S. Chirinko, Robert D. Dietz, Malcolm Gillis, Roger H. Gordon, Jane G. Gravelle, Timothy S. Gunning, James M. Poterba, Thomas S. Neubig, Alan Viard, George Yin John W. Diamond is Edward and Hermena Kelly Fellow in Tax Policy and Adjunct Professor of Economics at the Baker Institute for Public Policy, Rice University. George R. Zodrow is Professor of Economics and Rice Scholar at the Baker Institute and International Research Fellow at the Centre for Business Taxation, Oxford University. (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.

Locke is the first English philosopher to be considered in this series, and that fact of itself is worthy of attention. Philosophy, of course, like science, knows no frontiers and no national boundaries. Yet it is true to say that Locke’s contribution to philosophy is typically and peculiarly English. His moderation, his emphasis upon experience, his tolerant spirit of compromise, his dislike of mystical extravagance and of metaphysical speculation, even that elusive quality of his which people call his “common sense”, (...) are English traits. His very defects, illogicality leading to inconsistency and an awkward disorderliness of thought so abhorrent to the. (shrink)

There can be no doubt that we do know one another. We know that others exist and we know a good deal about others. The question is how we know others. To say that others do not exist would be to assert a solipsism—a theory which no serious philosopher has ever maintained. Solipsism is absurd. Not because it is self-contradictory, for there is nothing self-contradictory in the notion that I alone exist having the experiences and thoughts which I do have (...) and that apart from me nothing and no one else exists. It is absurd simply because others do exist and I know this; because, that is to say, it contradicts the known evidence. This is the sole—but the adequate—ground for concluding solipsism to be absurd. Any discussion of this present problem, therefore, must begin with, the recognition of the fact that knowledge of others occurs. (shrink)

Locke is the first English philosopher to be considered in this series, and that fact of itself is worthy of attention. Philosophy, of course, like science, knows no frontiers and no national boundaries. Yet it is true to say that Locke’s contribution to philosophy is typically and peculiarly English. His moderation, his emphasis upon experience, his tolerant spirit of compromise, his dislike of mystical extravagance and of metaphysical speculation, even that elusive quality of his which people call his “common sense”, (...) are English traits. His very defects, illogicality leading to inconsistency and an awkward disorderliness of thought so abhorrent to the. (shrink)