Henry at Work invites readers to rethink how we work today by exploring an aspect of Henry David Thoreau that has often been overlooked: Thoreau the worker. John Kaag and Jonathan van Belle overturn the popular misconception of Thoreau as a navel-gazing recluse who was scornful of work and other mundanities. In fact, Thoreau worked hard—surveying land, running his family’s pencil-making business, writing, lecturing, and building his cabin at Walden Pond—and thought intensely about work in its many dimensions. And (...) his ideas about work have much to teach us in an age of remote work and automation, when many people are reconsidering what kind of working lives they want to have. -/- Through Thoreau, readers will discover a philosophy of work in the office, factory, lumber mill, and grocery store, and reflect on the rhythms of the workday, the joys and risks of resigning oneself to work, the dubious promises of labor-saving technology, and that most vital and eternal of philosophical questions, “How much do I get paid?” In ten chapters, including “Manual Work,” “Machine Work,” and “Meaningless Work,” this personal, urgent, practical, and compassionate book introduces readers to their new favorite coworker: Henry David Thoreau. (shrink)

THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world— explicating such facts presents philosophy with some of its subtlest problems. Let me begin by reminding you of some celebrated past attempts made by philosophers and mathematicians to explicate the nature of mathematics.

Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, an important form of type theory (...) based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

Most countries appear to believe that their health system is in a state of semi-crisis with expenditures rising rapidly, with the benefits of many services unknown and with pressure from the public to ensure access to a comprehensive range of services. But whose values should inform decision-making in the health area, and should the influence of different groups vary with the level of decision-making? These questions were put to 54 members of the public and health professionals in eight focus groups. (...) Adopting a different perspective from other studies, participants were not asked if particular groups should be involved in decisions but rather through deliberation and discussion nominated their own potential decision makers. This delivered a clear message that participants saw a legitimate role for a broad range of stakeholders in priority-setting decisions so as to incorporate a diversity of expertise and opinion. Companion themes were the acknowledgment that decisions involve ethical judgments and are not purely technical, that the power of special interest groups (such as clinicians) should be kept in check, and that the process by which decisions are reached is important. The results suggest that qualitative methods of investigation have the potential to improve the legitimacy of policy decisions by contributing to a better understanding of the values of the public and health professionals, and by expanding the range of options available for further research. (shrink)

Tracking theories of knowledge are widely known to have the consequence that knowledge is not closed. Recent arguments by Vogel and Hawthorne claim both that there are no legitimate examples of knowledge without closure and that the costs of theories that deny closure are too great. This paper considers the tracking theories of Dretske and Nozick and the arguments by Vogel and Hawthorne. We reject the arguments of Vogel and Hawthorne and evaluate the costs of closure denial for tracking theories (...) of knowledge. (shrink)

This chapter traces the historical and conceptual development of the idea of the continuum and the allied concept of real number. Particular attention is paid to the idea of infinitesimal, which played a key role in the development of the calculus during the seventeenth and eighteenth centuries, and which has undergone a revival in the later twentieth century.

THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. We shall discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: (...) Plato, Aristotle, Leibniz, and Kant. We conclude with a more detailed discussion of the three “schools” of mathematical philosophy which have emerged in the twentieth century: Logicism, Formalism, and Intuitionism. (shrink)

The article presents information on the capabilities of Dretske-style analysis of knowing (DAK) and of several competing analyses with respect to accounting for the apparent facts. It informs that the DAK can ground plausible verdicts about knowledge and ignorance in cases involving lotteries. It further informs that the knowledge-efficacy donor imply the implausible thesis.

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...) continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit. (shrink)

This study examines the metajournalistic discourse reflected in the use of corpse images from the DusitD2 terror attack in Nairobi, Kenya, in January 2019. Drawing from concepts such as responsibility and resistance ethics, this study explores the viewpoints of Kenyan journalists and bloggers. Situated within qualitative research methodology, the findings suggest that the New York Times’ use of victims’ corpse images reflects a double standard and visual bias, and its defense of the news report can be considered an example of (...) professional posturing. Conversely, sharing of perpetrators’ postmortem pictures produced mixed findings among audiences. While online circulation was blamed on bloggers’ inexperience, it was also seen as a sign of victory for the value of visual evidence. This study also contributes to the scholarship on use of graphic images by drawing on African ethical systems and just war theory. (shrink)

This monograph is a follow up to the author's classic text Boolean-Valued Models and Independence Proofs in Set Theory, providing an exposition of some of the most important results in set theory obtained in the 20th century--the independence of the continuum hypothesis and the axiom of choice. Aimed at research students and academics in mathematics, mathematical logic, philosophy, and computer science, the text has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory, and (...) includes recent developments in the field. Numerous exercises, along with the enlarged and entirely updated background material, make this an ideal text for students in logic and set theory. (shrink)

Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.

One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem — employed by Cantor himself in showing that the set of real numbers is uncountable — is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. (...) Classically, these two methods are equivalent. But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e., employs intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. I describe the ways in which these two methods.. (shrink)

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.

I show the inadequacy of whitrow's recent argument ("british journal for the philosophy of science", Volume 29, Pages 39-45) against the possibility of an infinite past. I argue that it is impossible to prove "a priori" the non-Existence of an infinite past or future.

The opposed concepts of continuity and discreteness have figured prominently in the development of mathematics, and have also commanded the attention of philosophers. Continuous entities may be characterized by the fact that they can be divided indefinitely without altering their essential nature. So, for instance, the water in a bucket may be indefinitely halved and yet remain water. (For the purposes of illustration I ignore the atomic nature of matter which has been established by modern physics.) Discrete entities, on the (...) other hand, typically cannot be divided without effecting a change in their nature: half a wheel is plainly no longer a wheel. Thus we have two contrasting properties: on the one hand, the property of being indivisible, separate or discrete, and, on the other, the property of being indefinitely divisible and continuous although not actually divided into parts. (shrink)

Two new intuitionistic equivalents to Zarn’s Lemma are stated and proved.

V. Hendricks and H. Leitgeb, eds., Automatic Press, 2007.

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting (...) mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers.. (shrink)

This paper is concerned with Wittgenstein's early doctrine of the independence of elementary propositions. Using the notion of a free generator for a logical calculus–a concept we claim was anticipated by Wittgenstein–we show precisely why certain difficulties associated with his doctrine cannot be overcome. We then show that Russell's version of logical atomism–with independent particulars instead of elementary propositions–avoids the same difficulties.

is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...) of transformation of structure (homomorphism) was central from the beginning, and it soon became apparent to algebraists that its most important concepts and constructions were in fact formulable in terms of that idea alone. Thus emerged the view that the essence of a mathematical structure is to be sought not in its internal constitution, but rather in the nature of its relationships with other structures of the same kind, as manifested through the network of transformations. This idea has achieved its fullest expression in category theory, an axiomatic framework within which the notions of transformation (as morphism or arrow) and composition (and also structure, as object) are fundamental, that is, are not defined in terms of anything else. (shrink)

Bell, John I've been invited to share with you my experiences of a lifetime's acquaintance with Shakespeare, and how that acquaintance has led to what might be loosely termed a humanist philosophy.

The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on the foundations of (...) mathematics. (shrink)

This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong (1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation crucially does not. (...) The argument is conducted via a thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices (3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic of bivalence realism (i.e., of truth and falsity), although it is consistent with some form of ontological realism (4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks. (shrink)

A formal, logical, theory of events is developed and used as the basis for a definition of causation and to provide a pragmatics for causal counterfactuals. The theory begins with with a logical formalization of events as represented in the planner strips. The resulting inertial theories include a common sense law of inertia and their pragmatics is based on the principle of chronological minimization. The theory of events is then developed by removing some of the simplifying assumptions of the strips (...) representation. To begin with the assumption that events always succeed, given that their preconditions are true when they occur, is dropped. Instead a common sense law of change is introduced with the effect that events normally succeed but may fail if some qualification (some unusual or unexpected condition) arises. The pragmatics of the resulting causal theories is provided by refining the pragmatics of inertial theories. In particular, where there is a conflict between change and inertia, preference is given to change. The resulting defeasibility of events makes it possible to remove the strips assumption that only one event occurs at any given time; as conflicting events now result in failure rather than inconsistency. However it is desirable to provide a means for adjudicating between conflicting events. Consequently the theory is extended so that qualification preferences can be specified and the pragmatics are refined accordingly. A further strips assumption is that the effects of an event are invariant. In order to remove this assumption a distinction is drawn between primary events, events of the kind described above, and secondary events. Secondary events are invoked by primary events in appropriate contexts, they are causally dependent on primary events and when successful produce the additional, variant, effects of the primary events which invoke them. As a result, both the context-dependent effects of events (their ramifications) and non-deterministic events (events whose outcome is uncertain) can be represented. The theory of primary and secondary events is then used as the basis for a definition of causation which can be seen as providing a formal account of Mackie's definition of a cause as an inus condition. A major theme of this paper is the conflict between the opposite but complementary principles of change and inertia. For much of the paper the emphasis is on restricting the effects of inertia and favouring change. The resulting imbalance becomes unnatural when considering the combined effects of several events. However the definition of causation makes it possible to state a law of change and inertia, which is used to regulate change and inertia and thus ensure a proper balance between them: while caused change is preferred to inertia, inertia is preferred to uncaused change. The theory is then used to provide a pragmatics for causal counterfactuals, thereby extending the representation of actual causal reasoning to counterfactual causal reasoning. This is illustrated by the use of causal counterfactuals in order to obtain explanations. (shrink)

Contexte : L’ensemble des troubles causés par l’alcoolisation fœtale (ETCAF), un diagnostic complexe qui comprend une vaste gamme de troubles neurodéveloppementaux, résulte de l’exposition à l’alcool dans l’utérus. L’ETCAF demeure mal compris par les Canadiens, ce qui pourrait contribuer à la stigmatisation dont souffrent les personnes atteintes d’ETCAF et les femmes qui consomment de l’alcool pendant leur grossesse. Méthodes : Pour mieux comprendre comment l’information sur l’ETCAF est présentée dans la sphère publique, nous avons analysé le contenu de 286 articles (...) tirés de dix grands journaux canadiens de langue anglaise (2002-2015). Nous avons utilisé le codage inductif pour établir une grille de codage à partir des données, puis nous avons appliqué de façon itérative des codes identifiés sur l’échantillon, en vérifiant la fiabilité intercodeurs. Résultats : Nous avons identifié six grands thèmes liés au contenu cliniques et scientifiques des médias : 1) prévalence de l‘ETCAF et de la consommation d’alcool chez les femmes ; 2) recherche en lien avec l’ETCAF ; 3) diagnostic d’ETCAF ; 4) traitement de l’ETCAF et de l’abus de substances par les mères ; 5) incapacités primaires associés à l’ETCAF ; et 6) effets de l’alcool pendant la grossesse. Discussion : Dans le cadre de ces six thèmes, nous examinons trois types d’exagération et de fausse représentation qui ont des conséquences sur le plan éthique : 1) l’exagération des taux d’ETCAF dans les communautés autochtones ; 2) la contradiction entre les articles sur les effets de l’exposition prénatale à l’alcool ; et 3) l’information scientifiquement exacte qui néglige le contexte social de la consommation et de l’abus d’alcool par les femmes. Respectivement, ces représentations pourraient mener à des croyances stéréotypées préjudiciables au sujet des peuples autochtones, pourraient créer de la confusion quant aux choix sains pendant la grossesse et risqueraient d’enflammer inutilement les débats sur des questions délicates concernant les choix des femmes. (shrink)

In 1949 the great logician Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least, possible. Within the framework of Einstein’s general theory of relativity Gödel produced cosmological solutions to Einstein’s field equations which contain closed time-like curves, that is, curves in spacetime which, despite being closed, still represent possible paths of bodies. An object moving along such a path would travel back into its own past, to the very (...) moment at which it “began” the journey. More generally, Gödel showed that, in his “universe”, for any two points P and Q on a body’s track through spacetime, such that P temporally precedes Q, there is a timelike curve linking P and Q on which Q temporally precedes P. This means that, in principle at least, one could board a “time machine” and travel to any point of the past. Gödel inferred, in consonance with the views of Parmenides, Kant and the modern idealists, that under these circumstances there could be no such thing as an objective lapse of time, that time or, more generally, change, is an illusion arising from our special mode of perception. For consider an observer initially at point P. At point Q he boards a time machine and travels back to point P, taking time t′′ to do so. In that case, according to his own clock, t′ – t + t′′ > 0 seconds have elapsed, and yet an identical clock left at P would show that 0 seconds have elapsed. In short, there has been no “objective” lapse of time at all. Gödel remarks that in his universe this situation is typical: for every possible definition of an “objective” time one could travel into regions which are past according to that definition. He continues. (shrink)

ABSTRACT Drawing inspiration from Fred Dretske, L. S. Carrier, John A. Barker, and Robert Nozick, we develop a tracking analysis of knowing according to which a true belief constitutes knowledge if and only if it is based on reasons that are sensitive to the fact that makes it true, that is, reasons that wouldn’t obtain if the belief weren’t true. We show that our sensitivity analysis handles numerous Gettier-type cases and lottery problems, blocks pathways leading to skepticism, and (...) validates the epistemic closure thesis that correct inferences from known premises yield knowledge of the conclusions. We discuss the plausible views of Ted Warfield and Branden Fitelson regarding cases of knowledge acquired via inference from false premises, and we show how our sensitivity analysis can account for such cases. We present arguments designed to discredit putative counterexamples to sensitivity analyses recently proffered by Tristan Haze, John Williams and Neil Sinhababu, which involve true statements made by untrustworthy informants and strange clocks that sometimes display the correct time while running backwards. Finally, we show that in virtue of employing the paradox-free subjunctive conditionals codified by Relevance Logic theorists instead of the paradox-laden subjunctive conditionals codified by Robert Stalnaker and David Lewis. (shrink)

subsets X of A for which ∃x (x ∈ A). The set of functions from A to B is denoted by BA.

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

5.5. Every topos is linguistic: the equivalence theorem.

Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In Oppositions and Paradoxes, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, (...) Bell not only motivates abstract conceptual thinking about the paradoxes at issue, he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life and he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or imagine an English professor who time-travels back to 1599 to offer a printing of Hamlet to Shakespeare, so as to help the Bard overcome writer's block and author the play which will centuries later inspire an English professor to travel back in time. These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts. (shrink)

Logical Options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature: many-sorted logic, second-order logic, modal logics, intuitionistic logic, three-valued logic, fuzzy logic, and free logic. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and proof methods are employed to facilitate comparison of the various systems. The book is designed to be useful for philosophy students and professional philosophers who have learned some (...) classical first-order logic and would like to learn about other logics important to their philosophical work. (shrink)

In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s account of discrete spacetime as sets “evolving” over a causal set. We observe that Markopoulou’s proposal may be effectively realized by working within an appropriate frame-valued model of set theory. We go on to show that, within (...) this framework, cover schemes may be used to force certain conditions to prevail in the associated models: for example, rendering the universe timeless, obliterating a given event or forcing it to become the universe’s “beginning”. (shrink)