The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested (...) and illustrated by considering a case study discussed in the literature. On the view put forward here one can obtain a manner of informal, rigorous mathematics founded on any of a variety of proof systems. The latter part of the thesis is concerned with the question of how we should decide which of these competing proof systems we should base our mathematics on: i.e., the question of which proof system we should take as a foundation for our mathematics. A novel answer to this question is proposed, in which the key property we should require of a proof system is that for as many different kinds of structures as possible, when the proof system allows a generalization about that kind of structure to be proved, the generalization actually holds of all real examples of that kind of structure which exist. This is the requirement of soundness of the proof system. It is argued that the best way to establish the soundness of a proof system may be by giving an interpretation of its axioms on which they are established as true. As preparation for this discussion, the thesis first investigates the logical and conceptual basis of various arithmetic concepts, with the results obtained used in the final discussion of soundness. (shrink)

Neo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume’s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume’s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic (...) concepts, including finiteness, equinumerosity and addition and multiplication of cardinality—revealing these to be logical concepts, and obtaining much of ordinary arithmetic knowledge as logical knowledge. Supplementing this with an abstraction principle and a simple axiom of infinity (known either empirically or modally) we obtain a full interpretation of arithmetic. (shrink)

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

This paper sketches an account of the standard of acceptable proof in mathematics—rigour—arguing that the key requirement of rigour in mathematics is that nontrivial inferences be provable in greater detail. This account is contrasted with a recent perspective put forward by De Toffoli and Giardino, who base their claims on a case study of an argument from knot theory. I argue that De Toffoli and Giardino’s conclusions are not supported by the case study they present, which instead is a very (...) good illustration of the kind of view of proof defended here. (shrink)

Although Peano arithmetic is necessarily incomplete, Isaacson argued that it is in a sense conceptually complete: proving a statement of the language of PA that is independent of PA will require conceptual resources beyond those needed to understand PA. This paper gives a test of Isaacon’s thesis. Understanding PA requires understanding the functions of addition and multiplication. It is argued that grasping these primitive recursive functions involves grasping the double ancestral, a generalized version of the ancestral operator. Thus, we can (...) test Isaacon’s thesis by seeing whether when we phrase arithmetic in a context with the double ancestral operator, the result is conservative over PA. This is a stronger version of the test given by Smith, who argued that understanding the predicate “natural number” requires understanding the ancestral operator, but did not investigate what is required to understand the arithmetic functions. (shrink)

This paper gives a semantics for schematic logic, proving soundness and completeness. The argument for soundness is carried out in ontologically innocent fashion, relying only on the existence of formulae which are actually written down in the course of a derivation in the logic. This makes the logic available to a nominalist, even a nominalist who does not wish to rely on modal notions, and who accepts the possibility that the universe may in fact be finite.

This chapter describes the potential far‐reaching consequences of contemplative higher education for the fields of science and medicine.

The adaptationist framework is necessary and sufficient for unifying the social and natural sciences. Gintis's “beliefs, preferences, and constraints” (BPC) model compares unfavorably to this framework because it lacks criteria for determining special design, incorrectly assumes that standard evolutionary theory predicts individual rationality maximisation, does not adequately recognize the impact of psychological mechanisms on culture, and is mute on the behavioural implications of intragenomic conflict. (Published Online April 27 2007).

1 Evidence-based policy requires researchers to provide the answers to ecological questions that are of interest to policy makers. To find out what those questions are in the UK, representatives from 28 organizations involved in policy, together with scientists from 10 academic institutions, were asked to generate a list of questions from their organizations. 2 During a 2-day workshop the initial list of 1003 questions generated from consulting at least 654 policy makers and academics was used as a basis for (...) generating a short list of 100 questions of significant policy relevance. Short-listing was decided on the basis of the preferences of the representatives from the policy-led organizations. 3 The areas covered included most major issues of environmental concern in the UK, including agriculture, marine fisheries, climate change, ecosystem function and land management. 4 The most striking outcome was the preference for general questions rather than narrow ones. The reason is that policy is driven by broad issues rather than specific ones. In contrast, scientists are frequently best equipped to answer specific questions. This means that it may be necessary to extract the underpinning specific question before researchers can proceed. 5 Synthesis and applications. Greater communication between policy makers and scientists is required in order to ensure that applied ecologists are dealing with issues in a way that can feed into policy. It is particularly important that applied ecologists emphasize the generic value of their work wherever possible. (shrink)

On one hand, Chinese consumers are well known for conspicuous consumption and the adoption of luxury products and named brands. On the other hand, they also have a bad reputation for buying counterfeit products. Their simultaneous preferences for two contrasting types of product present a paradox that has not been addressed in the literature. This study attempts to present an explanation of this paradox by examining the effects of traditional Chinese cultural values and consumer values on consumers’ deontological judgment of (...) pirated CDs and the amount of social benefits they perceive they gain from them. We interviewed 300 Hong Kong Chinese consumers, and found that face consciousness increased materialism and risk aversion, thereby producing a favorable deontological judgment of pirated CDs. Face consciousness also has a direct effect on the amount of social benefits perceived in pirated CDs. Both favorable deontological judgment and perceived social benefits contributed to a strong intention to buy pirated CDs. The results are discussed in a cultural perspective. (shrink)

Background While prenatal surgery historically was performed exclusively for lethal conditions, today intrauterine surgery is also performed to decrease postnatal disabilities for non-lethal conditions. We sought to describe physicians' attitudes about prenatal surgery for lethal and non-lethal conditions and to elucidate characteristics associated with these attitudes. Methods Survey of 1200 paediatric surgeons, neonatologists and maternal–fetal medicine specialists. Results Of 1176 eligible physicians, 670 responded. In the setting of a lethal condition for which prenatal surgery would likely result in the child (...) surviving with a severe disability, most respondents either disagreed or strongly disagreed that they would recommend the surgery. Male physicians were twice as likely to recommend surgery for the lethal condition, as were physicians who believe that abortion is morally wrong. Older physicians were less likely to recommend surgery. For non-lethal conditions, most respondents agreed that they would recommend prenatal surgery, even if the surgery increases the risk of prematurity or fetal death. Compared with MFMs, surgeons were less likely to recommend such surgery, as were physicians not affiliated with a fetal centre, and physicians who were religious. Conclusion Physician’s attitudes about prenatal surgery relate to physicians’ beliefs about disability as well as demographic, cultural and religious characteristics. Given the variety of views, parents are likely to receive different recommendations from their doctors about the preferable treatment choice. (shrink)

Bovine fetuin, the first fetal protein to be described, has recently been shown to be a species homologue of a well known human plasma protein ‐ α2HS glycoprotein (α2HS). The fetuins are now known to be members of the cystatin superfamily. The structural properties of the six fetuins that have been fully sequenced are compared. Despite the structural homology of these proteins, their described properties in the literature make them appear to be quite different. The diverse in vitro properties claimed (...) for fetuin/α2HS are reviewed. In vivo, fetuins are involved in the acute phase response. In development, in all species studied so far, fetuins are present in a specific cell population that forms the developing neocortex. The possible functional significance of this distribution is discussed. (shrink)

Guilt and shame are self-conscious emotions with implications for mental health, social and occupational functioning, and the effectiveness of sports practice. To date, the assessment and role of athlete-specific guilt and shame has been under-researched. Reporting data from 174 junior elite cricketers, the present study utilized exploratory factor analysis in validating the Athletic Perceptions of Performance Scale, assessing three distinct and statistically reliable factors: athletic shame-proneness, guilt-proneness, and no-concern. Conditional process analysis indicated that APPS shame-proneness mediated the relationship between general (...) and athlete-specific distress. While APPS domains of guilt-proneness and no-concern were not significant mediators, they exhibited correlations in the expected direction with indices of psychological distress and well-being. The APPS may assist coaches and support staff identify players who may benefit from targeted interventions to reduce the likelihood of experiencing shame-prone states. (shrink)

The Monographs produced by the International Agency for Research on Cancer (IARC) apply rigorous procedures for the scientific review and evaluation of carcinogenic hazards by independent experts. The Preamble to the IARC Monographs, which outlines these procedures, was updated in 2019, following recommendations of a 2018 expert Advisory Group. This article presents the key features of the updated Preamble, a major milestone that will enable IARC to take advantage of recent scientific and procedural advances made during the 12 years since (...) the last Preamble amendments. The updated Preamble formalizes important developments already being pioneered in the Monographs Programme. These developments were taken forward in a clarified and strengthened process for identifying, reviewing, evaluating and integrating evidence to identify causes of human cancer. The advancements adopted include strengthening of systematic review methodologies; greater emphasis on mechanistic evidence, based on key characteristics of carcinogens; greater consideration of quality and informativeness in the critical evaluation of epidemiological studies, including their exposure assessment methods; improved harmonization of evaluation criteria for the different evidence streams; and a single-step process of integrating evidence on cancer in humans, cancer in experimental animals and mechanisms for reaching overall evaluations. In all, the updated Preamble underpins a stronger and more transparent method for the identification of carcinogenic hazards, the essential first step in cancer prevention. (shrink)

This paper reports the introduction of electronic handsets, like those used on the television show 'Who Wants To Be A Millionaire?' into the teaching of philosophical logic. Logic lectures can provide quite a formidable challenge for many students, occasionally to the point of making them ill. Our rationale for introducing handsets was threefold: to get the students thinking and talking about the subject in a public environment; to make them feel secure enough to answer questions in the lectures because the (...) system enabled them to do this anonymously; and to build their confidence about their learning by their being able to see how they were progressing in relation to the rest of the students in the class. We have achieved all of these and more. Our experience has revealed that the use of handsets encourages a more dynamic form of student interaction in an environment - the lecture - that can, in the wrong hands, be utterly enervating, but they also provide an opportunity for the lecturer to respond to students' difficulties at the time when they really matter. In this paper, we discuss our case of rapid adoption, our grounds for judging it a success, and what that success seems to have depended on. (shrink)

For almost 30 years the ion channel that initiates the ON visual pathway in vertebrate vision has remained elusive. Recent findings now indicate that the pathway, which begins with unbinding of glutamate from the metabotropic glutamate receptor 6 (mGluR6), ends with the opening of the transient receptor potential (TRP)M1 cation channel. As a component of the mGluR6 signal transduction pathway, mutations in TRPM1 would be expected to cause congenital stationary night blindness (CSNB), and several such mutations have already been identified (...) in CSNB families. Furthermore, expression of TRPM1 in both the retina and skin raises the possibility that a genetic link exists between certain types of visual and skin disorders. (shrink)