How do people understand science? How do they feel about science, how do they relate to it, what do they hope from it and what do they fear about it? Science of the People: Understanding and using science in everyday contexts helps answer these questions as the result of painstaking interviewing by Professor Joan Solomon of all and sundry in a fairly atypical small town. The result is a unique overview of how a very wide range of adults, (...) united only by local geography, relate to science. Many of the findings run contrary to what is widely believed about how science is learnt and about how people view it. Chapters include:An Approach to AwarenessPublics for Science?Ethics and ActionInterpretation and ChangeJoan Solomon, who sadly died before this book could be published, enjoyed an international reputation in science education. After a long career teaching science in secondary schools she moved into the university sector and ending up holding chairs of science education at the Open University, King's College London and the University of Plymouth. She was a world leader in her subject and inspired classroom teachers and wrote a number of very influential papers with some of them. She produced many important books, booklets and other resources to help science teachers and science educators get to grips with the history and philosophy of science and the teaching of energy, amongst other topics. This book is essential reading for those involved in Science education and educational policy. (shrink)

John Ziman, like most other scientists, learnt about the social nature of science by becoming a scientist. He travelled through the various stages of passing examinations, having articles that he had written reviewed by peers, and applying for academic posts. Better still he was made a Fellow of the Royal Society because of the problems he had solved-- at least for the time being-- in several landmark papers. There was little written about the social nature of science at this time (...) so when he was asked to give a talk on the radio in 1960 he chose the title 'Science is Social' and began as follows: I am going to try to justify the following proposition 'A scientific truth is statement that has been publicly accepted by the experts'. (shrink)

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...) objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)

This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.

Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.

What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...) so I shall give them special attention. NB. Ahistorically, modern logical and set-theoretical notation will be used throughout, as long as it does not conflict with original intentions. (shrink)

In 1991, Jones developed an issue-contingent model of ethical decision making in which moral intensity is posited to affect the four stages of Rest's 1986 model. Jones claimed that moral intensity, which is "the extent of issue-related moral imperative in a situation", consists of six characteristics: magnitude of consequences, social consensus, probability of effect, temporal immediacy, proximity, and concentration of effect. This article reports the findings of two studies that analyzed the factor structure of moral intensity, operationalized by a 12-item (...) Perceived Moral Intensity Scale adapted from the work of Sing-hapakdi et al. [1996, Journal of Business Research, 36, 245-255] and Frey [2000, Journal of Business Ethics, 26, 181-195]. The two items that were purported to measure CE were dropped due to their inability to effectively tap into the characteristic proposed by Jones. Factor analyses of the remaining 10 items supported a 3-factor structure, with the MC, PE, and TI items loading on the first factor, the PX items loading on the second factor, and the SC items loading on the third factor. These factors were labeled: Probable Magnitude of Consequences, Proximity, and Social Consensus. The authors conclude that moral intensity consists of three characteristics, rather than the six posited by Jones. (shrink)

Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize (...) almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments. (shrink)

This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results depending (...) on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).

Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.

elaboration of the last part of my Tarski Lecture, “Truth unbound”, UC Berkeley, 3 April 2006, and of the lecture, “A nicer formal theory of non-hierarchical truth”, Workshop on Mathematical Methods in Philosophy, Banff , 18-23 Feb. 2007.

Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set induction (...) is proof-theoretically equivalent to Peano arithmetic PA; BON plus formula induction is proof-theoretically equivalent to the system <0 of second-order arithmetic. (shrink)

This chapter is a detailed study of predicativity in mathematics. It presents a number of historical versions predicativity requirements, looking for unifying ideas. The further development of the notions and requirements up to the present is traced, articulating connections among the different ideas. One underlying theme of the chapter is the motivations for the various requirements for rejecting impredicativity and the various ways of stating the requirement.

From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be (...) a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. (shrink)

_A new, definitive translation of Heidegger's most important work._.

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman , Gödel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic . In particular, we examine two restricted unfoldings and , as well as a full unfolding, . The principal results then state: is equivalent to ; is equivalent to ; is equivalent to . Thus is proof-theoretically equivalent (...) to predicative analysis. (shrink)

The factor structure of the Multidimensional Ethics Scale (MES; Reidenbach and Robin: 1988, Journal of Business Ethics 7, 871–879; 1990, Journal of Business Ethics 9, 639–653) was examined for the 8-item short form (N = 328) and the original 30-item pool (N = 260). The objectives of the study were: to verify the dimensionality of the MES; to increase the amount of true cross-scenario variance through the use of 18 scenarios varying in moral intensity (Jones: 1991, Academy of Management Review (...) 16, 366–395); and, to examine the items for measurement precision using item-response theory (IRT) methods. Results of confirmatory and exploratory factor analysis failed to conclusively support the hypothesized 3- (short form) or 5-factor (long form) structure; both instruments were instead dominated by a general factor. Item response theory analyses using Samejima’s (1969, Psychometrika Monograph Supplement 34, (4, Pt. 2)) graded response model revealed that many items in the 30-item pool performed very well, and suggested that a different collection of items be used to form a short-form version of the MES. Our proposed 10-item instrument includes more discriminating items than the 8-item version, and has the added advantage of including two items from each of the five ethical philosophies represented in the original 30-item pool. (shrink)

The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.

Ernest Nagel Lecture, Columbia University, Sept. 27, 2007.

The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, (...) recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper,various reduction relations betweensystems are explained and compared, and arguments against proof-theoretic reduction as a ``good'' reducibilityrelation are taken up and rebutted. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Questions of definedness are ubiquitous in mathematics. Informally, these involve reasoning about expressions which may or may not have a value. This paper surveys work on logics in which such reasoning can be carried out directly, especially in computational contexts. It begins with a general logic of partial terms, continues with partial combinatory and lambda calculi, and concludes with an expressively rich theory of partial functions and polymorphic types, where termination of functional programs can be established in a natural way.

✤ It is the characterization of those forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter.

A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems)met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i)conceptual, (ii)pedagogical and (iii)practical. The system FS0 serves under (i)and (ii)as a theoretical framework for the formalization of metamathematics. The general approach (...) may be used under (iii)for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way. (shrink)

This paper makes the case for a pluralist, contextualist view of nursing ethics. In defending this view, I briefly outline two current perspectives of nursing ethics – the Traditional View and the Theory View. I argue that the Traditional View, which casts nursing ethics as a subcategory of healthcare ethics, is problematic because it (1) fails to sufficiently acknowledge the unique nature of nursing practice; and (2) applies standard ethical frameworks such as principlism to moral problems which tend to alienate (...) or undermine nursing ethical concerns. Alternatively, the Theory View, which aims to build an independent and comprehensive theory of nursing ethics, is also found wanting because it (1) fails to sufficiently acknowledge the heterogeneous nature of nursing practices; (2) overemphasizes the differences and undervalues the similarities between nurses and other health professionals; and (3) assumes that one ethical framework can be meaningfully applied across diverse moral problems and contexts.My alternative, is to argue that nursing ethics inquiry should take a pluralist and critical stance towards available ethical frameworks and the negotiation of the ethical realm. On this view, the search for moral consensus or a unique ethical framework for nursing is replaced by the task of working strategically with multiple frameworks in order to expand the moral agency of nurses and empower them to positively engage with moral uncertainty as an inevitable feature of living a moral life. I conclude by indicating some of the implications that this has for the teaching of nursing ethics. (shrink)

This is a sequel to our article “Predicative foundations of arithmetic” (1995), referred to in the following as [PFA]; here we review and clarify what was accomplished in [PFA], present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by [PFA] was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George and Daniel (...) Velleman (1996) have examined [PFA] closely in the context of a general discussion of different philosophical approaches to the foundations of arithmetic. (shrink)

Ambiguity is a property of syntactic expressions which is ubiquitous in all informal languages–natural, scientific and mathematical; the efficient use of language depends to an exceptional extent on this feature. Disambiguation is the process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical if the process of disambiguation can be carried out in some systematic way. Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency (“having (...) the cake”) with the freedom of the informal untyped theory of classes and relations (“eating it too”). The paper begins with a brief tour of Russell’s uses of typical ambiguity, including his treatment of the statement Cls ∈ Cls. This is generalized to a treatment in simple type theory of statements of the form A ∈ B where A and B are class expressions for which A is prima facie of the same or higher type than B. In order to treat mathematically more interesting statements of self membership we then formulate a version of typical ambiguity for such statements in an extension of Zermelo-Fraenkel set theory. Specific attention is given to how the“naive” theory of categories can thereby be accounted for. (shrink)

Does the “free market” foster more freedom for individuals generally and less coercion? Libertarians and other market advocates argue that the unfettered market maximizes freedom and hence has less coercion than any feasible alternative. Welfare liberals, Socialist, and Marxists, in different ways, argue against the claim that the unrestricted market maximizes freedom generally. Both supporters and critics agree that coercion undermines freedom and that that is what is ultimately prima facie wrong with it. Further, they agree that the extent to (...) which an economic system includes coercion provides a serious strike against it. The problem is that there is little consensus on the necessary conditions for coercion. In this paper I will be concerned with the nature of coercive relationships in the market. Market interactions have a particular character and occur within a specific institutional framework. Other accounts of coercion have failed to capture the unique character of coercion in the market because they have failed to take into account certain nontrivial facts about the market which are essential to the analysis of coercive market relations. (shrink)