This article is devoted to the historical roots of the quest for rigor associated with analytic philosophy. Starting out from distinctions between different senses of “rigourµ, it considers the rather diverse conceptions and pursuits of rigour in Frege, Russell, Moore, Wittgenstein and Carnap. On that basis it diagnoses a potential conflict that some of them were aware of, but that has been ignored by recent commentators, namely between certain kinds of rigour on the hand, clarity and surveyability on (...) the other. That conflict is related to the contrast between formal and non-formal ways of pursuing rigour. Frege, Wittgenstein and Carnap occupy distinct positions concerning both issues. Nevertheless, all three are united with n the hope to achieve philosophical progress by means of clear questions and statements put in the service of rigorous argument. (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 thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on (...) the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour. (shrink)

This article discusses the important and influential views of John Burgess on the nature of mathematical rigour and John Norton on the nature of thought experiments. Their accounts turn out to be surprisingly similar in spite of different subject matters. Among other things both require a reconstruction of the initial proof or thought experiment in order to officially evaluate them, even though we almost never do this in practice. The views of each are plausible and seem to solve interesting (...) problems. However, both have problems and would seem not able to do justice to some interesting examples. They fail in similar ways. More pluralistic accounts of proof and of thought experiment could embrace aspects of each, while rejecting their claims to universality. An ideal account (not provided here) would contribute to _explanation_ and _understanding_ as well as evidence. These are important topics for future work. (shrink)

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)

This paper defends the need for evidential diversity and the mix of methods that that can in train require. The focus is on causal claims, especially ‘singular’ claims about the effects of causes in a specific setting—either what will happen or what has happened. I do so by offering a template that categorises kinds of evidence that can support these claims. The catalogue is generated by considering what needs to happen for a causal process to carry through from putative cause (...) at the start to the targeted effect at the end. The usual call for mixed methods focusses on a single overall claim and argues that we increase certainty by the use of different methods with compensating strengths and weaknesses. My proposals instead focus on the evidence that supports the great many subsidiary claims that must hold if the overall one is to be true. As is typical for singular causal claims, the mix of methods that will generally be required to collect the kinds of evidence I urge will usually have little claim to the kind of rigour that is now widely demanded in evidencing causal claims, especially those for policy/treatment effectiveness. So I begin with an exploration of what seems to be intended by ‘rigour’ in such discussions, since it is seldom made clear just what makes the favoured methods especially rigorous. I then argue that the emphasis on rigour can be counterproductive. Rigour is often the enemy of evidential diversity, and evidential diversity—lots of it—can make for big improvements in the reliability of singular causal predictions and post hoc evaluations. I illustrate with the paragon of rigour for causal claims, randomised controlled trials (RCTs), rehearsing at some length what they can and cannot do to make it easier to assess the importance of rigour in warranting singular causal claims. (shrink)

The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

Built in the centre of Copenhagen, and noted for its equestrian stairway, the Rundetaarn, was intended as an astronomical observatory. Part of a complex of buildings that once included a university library, it affords expansive views of the city in every direction, towering above what surrounds it. The metaphor of the towering figure, who sees what others might not, whose vantage point allows him to visualize how things fit together, and who has an earned-stature of respect and authority, fits another (...) Danish stalwart, Hans Vilhelm Hansen, whose contributions to the fields of informal logic and argument theory have earned the gratitude of his colleagues, and inspired this collection of essays, written to express the appreciation of its authors and of the many, many colleagues they represent. (shrink)

During the early years of the COVID-19 pandemic, preclinical and clinical research were sped up and scaled up in both the public and private sectors and in partnerships between them. This resulted in some extraordinary advances, but it also raised a range of issues regarding the ethics, rigour, and integrity of scientific research, academic publication, and public communication. Many of the failures of scientific rigour and integrity that occurred during the pandemic were exacerbated by the rush to generate, (...) disseminate, and implement research findings, which not only created opportunities for unscrupulous actors but also compromised the methodological, peer review, and advisory processes that would usually identify sub-standard research and prevent compromised clinical or policy-level decisions. While it would be tempting to attribute these failures of science and its translation solely to the “unprecedented” circumstances of the COVID-19 pandemic, the reality is that they preceded the pandemic and will continue to arise once it is over. Existing strategies for promoting scientific rigour and integrity need to be made more rigorous, better integrated into research training and institutional cultures, and made more sophisticated. They might also need to be modified or supplemented with other strategies that are fit for purpose not only in public health emergencies but in any research that is sped-up and scaled up to address urgent unmet medical needs. (shrink)

This paper begins with the ‘ancient quarrel’ between philosophy and literature, which, with the subsequent splitting of logos into word and reason, comes to mark philosophy's self-conception and much other thinking besides—compartmentalising, in the process, what is understood by ‘literature’. Philosophy, thus separated becomes atemporal and abstract, preoccupied with propositions rather than statements or sentences, and, in some of its incarnations, aligning itself with science. Language, thus separated, becomes ‘literary’—that is, it comes to be epitomised by self-consciousness about literary form (...) and style; and a casualty of this is the ‘poetic’, a term whose origins in poiesis (production of meaning) are forgotten. But the relationship has never been as settled as it may have seemed, and a series of examples from classical and contemporary philosophy and literature helps to demonstrate this. At stake in these examples are the ways in which reason requires that one means what one says. A classic expression of commitment to this view is provided in Hamlet, by the much-quoted words of Polonius to his son, who is about to travel to Germany to study philosophy: ‘This above all, to thine own self be true’. The implications of this in relation to the claims of reason are developed with reference to moral education and the development of judgement for learners and teachers. (shrink)

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)

Built in the centre of Copenhagen, and noted for its equestrian stairway, the Rundetaarn (Round Tower), was intended as an astronomical observatory. Part of a complex of buildings that once included a university library, it affords expansive views of the city in every direction, towering above what surrounds it. The metaphor of the towering figure, who sees what others might not, whose vantage point allows him to visualize how things fit together, and who has an earned-stature of respect and authority, (...) fits another Danish stalwart, Hans Vilhelm Hansen, whose contributions to the fields of informal logic and argument theory have earned the gratitude of his colleagues, and inspired this collection of essays, written to express the appreciation of its authors and of the many, many colleagues they represent. (shrink)

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By (...) bringing considerations of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures. (shrink)

Science is remarkably reliable. It puts people on the moon, performs laser eye surgery, tells us about ancient civilisations and species, and predicts the future of our climate. What underwrites this reliability? This book argues that the standard answers—the scientific method, rigour, and objectivity—are insufficient for the job. Here we propose a new model of science that places its products front and centre. This is the ‘Tangle of Science’. In this book we show how any reliable piece of science (...) is underpinned by a vast, diverse, and thick network of other scientific products. In doing so we bring back into focus areas of science that have been long neglected, emphasising how every product, from the screws that hold the space shuttle together to ways of measuring the consumer price index to Einstein’s theory of general relativity, work together to support results we can trust. (shrink)

This essay claims that Stiegler’s sense of metaphor gives his work an overlooked rigour. Part one argues that La Faute d’Epiméthée’s key claim opens an e...

Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible candidates for the epistemic value arising from atomization and (...) finds that none of them fits the bill. In particular, atomized arguments do not ground the correctness of their unatomized counterparts; they are typically not credence-preserving; and they need not reveal the source of inferential disagreement. The moral this suggests is that complete rigour is not even a defeasible epistemic ideal. (shrink)

Carnap's position on mathematical truth in The Logical Syntax of Language has been attacked from two sides: Kreisel argues that it is formalistic but should not be, and Friedman argues that it is not formalistic but needs to be. In this paper I argue that the Carnap of Syntax does not eliminate our ordinary notion of mathematical truth in favour of a formal analogue; so Carnap's notion of mathematical truth is not formalistic. I further argue that there is no conflict (...) between Carnap's use of informal notions and his principle of tolerance; so Carnap's definition of mathematical truth need not be formalistic. CiteULike Connotea Del.icio.us What's this? (shrink)

In this paper we discuss a particular example of the passage from the informal, but rigorous description of a concept to the axiomatic formulation of principles holding for the concept; in particular, we look at the principles of continuity and lawlike choice in the theory of lawless sequences. Our discussion also leads to a better understanding of the rôle of the so-called density axiom for lawless sequences.

(1997). How to Give Analytical Rigour to ‘Soupy’ Metaphysics∗. Inquiry: Vol. 40, No. 1, pp. 95-113.

This paper develops a framework for establishing rigour for a discourse analysis of professional transition in midwifery, theorised as a ‘female professional project’. Discourse analysis has gained recognition as a useful approach in nursing and midwifery research. It provides an alternative to those qualitative approaches that propose to reveal a ‘reality’ from the perspective of the individual experience, and that this lived experience can be directly represented in language. There are multiple discourse analytic approaches, and often researchers are not (...) explicit about what type they are employing. Furthermore, to date there are few clear guidelines for what constitutes rigour in a discourse analytic study. Rigour in qualitative research broadly falls into two categories: the replication perspective that argues for the maintenance of criteria of validity and reliability, usually with an underpinning assumption (often unquestioned) that a ‘truth’ can be revealed. The second is the parallel perspective that questions the ontological and epistemological assumptions of the replication perspective and argues for the development of criteria that reflect the assumed questionable nature of truth and reality. Transferring these criteria to a discourse analytic study revealed some problems. A common challenge identified in achieving rigour in discourse analysis is the maintenance of congruity between the epistemological and ontological basis of a piece of research and the actual analysis conducted or reported. A framework is proposed to address these incongruities and includes six elements. As nurses and midwives increasingly employ discourse analysis in their research, attempts to clarify how rigour will be determined are important if the outcomes of such research are to be seriously considered for their implications for theory, policy and practice. (shrink)

I argue that algebraic quantum field theory (AQFT) permits an undisturbed view of the right ontology for fundamental physics, whereas standard (or Lagrangian) QFT offers different mutually incompatible ontologies.My claim does not depend on the mathematical inconsistency of standard QFT but on the fact that AQFT has the same concerns as ontology, namely categorical parsimony and a clearly structured hierarchy of entities.

This paper describes the process by which female typists became participants of sorts while transcribing audiotaped interviews. The primary data included sensitive and sexual subject matter about women's Pap smear (PS) experiences. Besides overcoming minor technical difficulties, die major discovery was that the typists had a tendency to interpolate and even normalize the data. A feminist post‐structuralist perspective was extended to examine these interpolations as secondary data. This revealed that the typists were becoming unexpected participants as commentators, validators, analysts and (...) normalizers of the data. From the subjective positions of both medical typists and recipients of PS, the typists provided data validation that included contradictions and transgressions from the cancer screening scheme propaganda. Furthermore, both typists were exposed to other women's PS stories and experienced vicarious consciousness raising. For one typist this included embodiment of other women's distress, raising questions about the occupational hazards associated with transcribing distressing data. Thus the social and relational aspects of those involved in the transcription process proved influential to both the data (and analysis), and to the practical aspects of data management. I argue that details of such influences need to be submitted for scrutiny as additional aspects of rigour. (shrink)

Between 1906 and 1911, as a response to Betrand’s Russell’s review of La Science et l’Hypothèse, Henri Poincaré launched an attack on the movement to formalise the foundations of mathematics reducing it to logic. The main point is the following: the universality of logic is based on the idea that their truth is independent of any context including epistemic and cultural contexts. From the free context notion of truth and proof it follows that, given an axiomatic system, nothing new can (...) follow. One of the main strategies of Poincaré’s solution to this dilemma is based on the notions of understanding and of grasping the architecture of the propositions of mathematics. According to this view mathematic rigour does not reduce to “derive blindly” without gaps from axioms,mathematical rigour is, according to Poincaré, closely linked to the ability to grasp the architecture of mathematics and contribute to an extension of the meaning embedded in structures that constitute the architecture of mathematical propositions. The focus of my paper relates precisely to the notion of architecture and to the notion of understanding. According to my reconstruction, Poincaré’s suggestions could be seen as pointing out that understanding is linked to reason not only within a structure but reasoning about the structure. (shrink)

The recent interest in systematic review methods in bioethics has highlighted the need for greater transparency in all literature review processes undertaken in bioethics projects. In this article, I articulate features of a good bioethics literature review that does not aim to be systematic, but rather to capture and analyse the key ideas relevant to a research question. I call this a critical interpretive literature review. I begin by sketching and comparing three different types of literature review conducted in bioethics (...) scholarship. Then, drawing on Dixon-Wood's concept of critical interpretive synthesis, I put forward six features of a good critical interpretive literature review in bioethics: answering a research question, capturing the key ideas relevant to the research question, analysing the literature as a whole, generating theory, not excluding papers based on rigid quality assessment criteria, and reporting the search strategy. (shrink)

We are told that there are standards of rigour in proof, and we are told that the standards have increased over the centuries. This is fairly clear. But rigour has also changed its nature. In this paper we as-sess where these changes leave us today.1 To motivate making the new assessment, we give two illustra-tions of changes in our conception of rigour. One, concerns the shift from geometry to arithmetic as setting the standard for rig-our. The other, (...) concerns the notion of effective proof or compu-tations. To make the assessment, we look at one motivation for increasing the rigour of a mathematical argument: explicitness and honesty. We then present a standard of rigour by means of a characterisation developed with reference to what we call ‘an account-proof’. We evaluate the standard with reference to the motivation. With the analysis we discover that, regardless of the motivations for rigour, the standard is almost never met, and that the motivations are not all satisfied. It follows that, in some sense, the motivations have misfired. The misfiring suggests to us that we re-assess our notion of rigour. We think of rigour as a relative term. Moreover, the standard for rigour depends on philosophical underpinnings. The strength of the argument of this paper rests on the plausibility of our selection of motivations and on the plausibility of our standard. (shrink)

The Faculty of Medicine and Health Sciences of the University of Sherbrooke has observed year after year, that certain students have not started and or completed their immunizations for common infectious diseases, which in effect makes them inadmissible for their clinical internships in healthcare establishments. The program administrators have posed a series of questions on the best way to proceed with these students as, a certain number remain reluctant to vaccination. They are often confronted with ethical dilemmas, are not necessarily (...) comfortable with their rights and responsibilities and wish to be guided throughout the process. This article aims to put forth a procedure that would be put in place to support the administrators in a way for them to obtain a more favorable vaccination level for all of their students in the most reasonable amount of time possible. The interest of the procedure is to move forward in a few simple and well defined steps that are supported by a rigorous legal framework, while respecting the rights and freedoms of the students. It is also designed so that it could be replicated in other faculties of medicine. (shrink)