Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

We present a taxonomy of errors in the scientific literature and an account of how the errors are distributed over the categories. We have developed the taxonomy by studying substantial errors in the scientific literature as described in retraction notices published in the journal Science over the past 35 years. We then examine how the sorts of errors that lead to retracted papers can be prevented and detected, considering the perspective of collaborating scientists, journal editors and referees, and readers of (...) the published articles. (shrink)

The authorship policies of scientific journals often assume that in order to be able to properly place credit and responsibility for the content of a collaborative paper we should be able to distinguish the contributions of the various individuals involved. Hence, many journals have introduced a requirement for author contribution statements aimed at making it easier to place credit and responsibility on individual scientists. We argue that from a purely descriptive point of view the practices of collaborating scientists are at (...) odds with the requirement for author contribution statements. We also argue that from a normative point of view the authorship policies may be unnecessary. Our arguments draw on an examination of 35 years of retraction notices in the journal Science. (shrink)

We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was very knowledgeable about the subject of the paper and had good presentational skills but, as a new PhD student, did not yet (...) have experience writing research papers for mathematicians. Thus, one main purpose of revising the paper was to make it take into account the intended audience. For this reason, the changes made to the initial draft and the authors’ purpose in making them provide a window for viewing how mathematicians write for mathematicians. We examined how their paper attracts the interest of the reader and prepares their proofs for validation by the reader. Among other findings, we found that their paper prepares the proofs for two types of validation that the reader can easily switch between. (shrink)

Retractions are rare in science, but there is growing concern about the impact retracted papers have. We present data on the retractions in the journal Science, between 1983 and 2017. Each year, approximately 2.6 papers are retracted; that is about 0.34% of the papers published in the journal. 30% of the retracted papers are retracted within 1 year of publication. Some papers are retracted almost 12 years after publication. 51% of the retracted papers are retracted due to honest mistakes. Smaller (...) research teams of 2–4 scientists are responsible for a disproportionately larger share of the retracted papers especially when it comes to retractions due to honest mistakes. In 60% of the cases all authors sign the retraction notice. (shrink)

Mathematicians appear to have quite high standards for when they will rely on testimony. Many mathematicians require that a number of experts testify that they have checked the proof of a result p before they will rely on p in their own proofs without checking the proof of p. We examine why this is. We argue that for each expert who testifies that she has checked the proof of p and found no errors, the likelihood that the proof contains no (...) substantial errors increases because different experts will validate the proof in different ways depending on their background knowledge and individual preferences. If this is correct, there is much to be gained for a mathematician from requiring that a number of experts have checked the proof of p before she will rely on p in her own proofs without checking the proof of p. In this way a mathematician can protect her own work and the work of others from errors. Our argument thus provides an explanation for mathematicians’ attitude towards relying on testimony. (shrink)

It has been a common belief among scientists, including mathematicians, that young scientists are especially good at bringing about scientific change. A number of studies suggest, however, that older scientists are not more resistant to change than young scientists are. It is nonetheless worth examining why a scientist’s or mathematician’s outsider status – due to age, educational background, or something else – can sometimes be effective in enabling scientific change. This paper focuses on the case of the solving of the (...) Four Color Problem by Wolfgang Haken and Kenneth Appel. Building on Donald MacKenzie’s paper ‘Slaying the kraken: The sociohistory of a mathematical proof’, I argue that Haken’s outsider status is central to understanding his success with the problem. On this basis, I offer an argument against Margaret Gilbert’s account of why a scientist’s outsider status can be effective in enabling scientific change. (shrink)

In his account of scientific revolutions, Thomas Kuhn suggests that after a revolutionary change of theory, it is as if scientists are working in a different world. In this paper, we aim to show that the notion of world change is insightful. We contrast the reporting of the discovery of neon in 1898 with the discovery of hafnium in 1923. The one discovery was made when elements were identified by their atomic weight; the other discovery was made after scientists came (...) to classify elements by their atomic number. By considering two instances of the reporting of the discovery of a new chemical element 25 years apart, we argue that it becomes clear how chemists can be said to have been responding to different worlds as a result of the change in the concept of a chemical element. They saw, did, and reported different things as they conducted their research on the new chemical elements. (shrink)

The phenomenon of shared intention has received much attention in the philosophy of mind and action. Margaret Gilbert (1989, 2000c, 2014b) argues that a shared intention to do A consists in a joint commitment to intend to do A. But we need to know more about the nature of joint commitments to know what exactly this implies. While the persistence of joint commitments has received much attention in the literature, their impersistence has received very little attention. In this paper, we (...) shed light on the impersistence of joint commitments by showing how joint commitments can be dissolved by unexpected events. (shrink)

The authorship policies of scientific journals often assume that in order to be able to properly place credit and responsibility for the content of a collaborative paper we should be able to distinguish the contributions of the various individuals involved. Hence, many journals have introduced a requirement for author contribution statements aimed at making it easier to place credit and responsibility on individual scientists. We argue that from a purely descriptive point of view the practices of collaborating scientists are at (...) odds with the requirement for author contribution statements. We also argue that from a normative point of view the authorship policies may be unnecessary. Our arguments draw on an examination of 35 years of retraction notices in the journal Science. (shrink)

Human communication is by defaultpolysemiotic: it involves the spontaneous combination of two or moresemiotic systems, the most important ones beinglanguage,gesture, anddepiction. We formulate an original cognitive-semiotic framework for the analysis of polysemiosis, contrasting this with more familiar systems based on the ambiguous term “multimodality.” To be fully explicit, we developed a coding system for the analysis of polysemiotic utterances containing speech, gesture, and drawing, and implemented this in the ELAN video annotation software. We used this to analyze 23 video-recordings of (...) sand drawing performances on Paama, Vanuatu and 20 sand stories of the Pitjantjatjara culture in Central Australia. Methodologically we used theconceptual-empirical loopof cognitive semiotics: our theoretical framework guided general considerations, such as distinguishing between the “tiers” of gesture and depiction, and the three kinds of semiotic grounds (iconic, indexical, symbolic), but the precise decisions on how to operationalize these were made only after extensive work with the material. We describe the coding system in detail and provide illustrative examples from the Paamese and Pitjantjatjara data, remarking on both similarities and differences in the polysemiosis of the two cultural practices. We conclude by summarizing the contributions of the study and point to some directions for future research. (shrink)