Mikkel Johansen

From a purely formalist viewpoint on the philosophy of mathematics, experiments cannot (and should not) play a role in warranting mathematical statements but must be confined to heuristics. Yet, due to the incorporation of new mathematical methods such as computer-assisted experimentation in mathematical practice, experiments are now conducted and used in a much broader range of epistemic practices such as concept formation, validation, and communication. In this article, we combine corpus studies and qualitative analyses to assess and categorize the epistemic roles experiments are seen—by mathematicians—to have in actual mathematical practice. We do so by text-mining a corpus of reviews from the _Mathematical Reviews_, which include the indicator word “experiment”. Our qualitative, grounded classification of samples from this corpus allows us to explore the various roles played by experiments. We thus identify instances where experiments function as references to established knowledge, as tools for heuristics or exploration, as epistemic warrants, as communication or pedagogy, and instances simply proposing experiments. Focusing on the role of experiments as epistemic warrants, we show through additional sampling that in some fields of mathematics, experiments can warrant theorems as well as methods. We also show that the expressed lack of experiments by reviewers suggests concordant views that experiments could have provided epistemic warrants. Thus, our combination of corpus studies and qualitative analyses has added a typology of roles of experiments in mathematical practice and shown that experiments can and do play roles as epistemic warrants depending on the mathematical field.

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.

In this paper, we wish to explore the role that textual representations play in the creation of new mathematical objects. We do so by analyzing texts by Joseph-Louis Lagrange (1736–1813) and Évariste Galois (1811–1832), which are seen as central to the historical development of the mathematical concept of groups. In our analysis, we consider how the material features of representations relate to the changes in conceptualization that we see in the texts.Against this backdrop, we discuss the idea that new mathematical concepts, in general, are increasingly abstract in the sense of being detached from material configurations. Our analysis supports the opposite view. We suggest that changes in the material aspects of textual representations (i.e., the actual graphic inscriptions) play an active and crucial role in conceptual change.We employ an analytical framework adapted from Bruno Latour’s 1999 account of intertwined material and representational practices in the empirical sciences. This approach facilitates a foregrounding of the interconnection between the conceptual development of mathematics, and the construction, (re-)configuration, and manipulation of the materiality of representations. Our analysis suggests that, in mathematical practice, distinctions between the material and structural features of representations are not permanent and absolute. This problematizes the appropriateness of the distinction between concrete inscriptions and abstract relations in understanding the development of mathematical concepts.

Background Minimal research has been done to determine how well European nursing students understand the core principles of academic integrity and how often they deviate from good academic practice. Aim The aim of this study was to find out what educational needs nursing students have in terms of academic integrity. Research design A quantitative cross-sectional study in the form of a survey of nursing students was conducted via questionnaire in the fall of 2020. Participants The sample was composed of 79 students in the BScN and MScN programs at Zürich University of Applied Sciences. Ethical considerations An application for a non-competence clearance was approved by the Ethics Committee in Zurich (BASEC No. Req-2020-00868). The survey was anonymous, and informed consent was obtained prior to participation. Results The participants had a high level of confidence in their own knowledge but were in many cases unable to correctly identify clear-cut examples of misconduct and to differentiate them from questionable practices. About 13% of the participants admitted that during their university education they had copied shorter passages from other sources into their own text without marking them as quotes. Conclusions The study documents extensive knowledge gaps among nursing students regarding both academic misconduct and questionable practices and indicates a need for improved academic integrity training.

This article aims to explain the behaviour of public intellectuals, celebrities, and media audiences in the construction of anti-emotional narratives in the online culture wars. In the investigation of how these narratives are constructed on YouTube, the article focuses on the rhetorical juxtaposition of rationality and emotionality surrounding the viral argument between public intellectual Sam Harris and Hollywood star Ben Affleck on HBO’s Real Time with Bill Maher, uploaded to YouTube. The video is an apt example of the positioning dynamics on YouTube, where an increased integration of media content with its context of reception unsettles the positioning process by which the authority of both intellectuals and celebrities is negotiated and culture war narratives constructed. In conclusion, the article reflects on the relationship between the increasingly affective nature of online communication and the simultaneous emergence of a strict dichotomy between rationality and emotionality in anti-woke culture war narratives.

Plagiarism and other transgressions of the norms of academic integrity appear to be a persistent problem among upper secondary students. Numerous surveys have revealed high levels of infringement of what appear to be clearly stated rules. Less attention has been given to students’ understanding of academic integrity, and to the potential misconceptions and false beliefs that may make it difficult for them to comply with existing rules and handle complex real-life situations.In this paper we report findings from a survey of European upper secondary students’ views on issues relating to academic integrity. We relate these findings to the students’ training about academic integrity, self-reported level of questionable behavior and country of study.A total of 1654 students at 51 institutions located in 6 European countries participated in the study. The participants generally believed they had a good understanding of the rules applying to them and knew how to behave in compliance with norms of academic integrity. The results indicate, however, that often, in practice, this belief was mistaken. Many students had an inadequate understanding of core elements of academic integrity. They were uncertain about how to act, and they struggled in the handling of complex situations that require context-sensitive judgement. While some differences between countries were identified, they were modest and exhibited no clear pattern. Our results also suggest that reducing students’ level of uncertainty and, to a lesser degree, improving their level of knowledge could lead them to engage less in certain types of questionable behaviours. Surprisingly, the effect of academic training is modest and ambiguous. The study also confirms that perception of peer behaviour has the strongest association with student engagement in questionable behaviours. Thus, academic integrity at the upper secondary level cannot be explained simply in terms of individual ethics or knowledge.

Good academic practice is more than the avoidance of clear-cut cheating. It also involves navigation of the gray zones between cheating and good practice. The existing literature has left students’ understanding of gray zone practices largely unexplored. To begin filling in this gap, we present results from a questionnaire study involving N = 1639 undergraduate students from seven European countries representing all major disciplines. We show that large numbers of these students are unable to identify gray area issues and lack sensitivity to the context dependence of these. We also show that a considerable proportion of students have a poor understanding of concepts like plagiarism and falsification, not only in gray zone scenarios, but also in cases of relatively clear-cut cheating. Our results are similar across the faculties and countries of study, and even for students who have attended academic integrity training. We discuss the implications of this for academic integrity training.

Over the last century, there have been considerable variations in the frequency of use and types of diagrams used in mathematical publications. In order to track these changes, we developed a method enabling large-scale quantitative analysis of mathematical publications to investigate the number and types of diagrams published in three leading mathematical journals in the period from 1885 to 2015. The results show that diagrams were relatively common at the beginning of the period under investigation. However, beginning in 1910, they were almost completely unused for about four decades before reappearing in the 1950s. The diagrams from the 1950s, however, were of a different type than those used earlier in the century. We see this change in publication practice as a clear indication that the formalist ideology has influenced mathematicians’ choice of representations. Although this could be seen as a minor stylistic aspect of mathematics, we argue that mathematicians’ representational practice is deeply connected to their cognitive practice and to the contentual development of the discipline. These changes in publication style therefore indicate more fundamental changes in the period under consideration.

The “practice turn” in philosophy of science has strengthened the connections between philosophy and scientific practice. Apart from reinvigorating philosophy of science, this also increases the relevance of philosophical research for science, society, and science education. In this paper, we reflect on our extensive experience with teaching mandatory philosophy of science courses to science students from a range of programs at University of Copenhagen. We highlight some of the lessons we have learned in making philosophy of science “fit for teaching” outside of philosophy circles by taking selected cases from the students’ own field as the starting point. We argue for adapting philosophy of science teaching to particular audiences of science students, and discuss the benefits of drawing on research within science education to inform curriculum and course design. This involves reconsidering teaching resources, assumptions about students, intended learning outcomes, and teaching formats. We also argue that to make philosophy of science relevant and engaging to science students, it is important to consider their potential career trajectories. By anticipating future contexts and situations in which methodological, conceptual, and ethical questions could be relevant, philosophy of science can demonstrate its value in the education of science students.

This paper presents and discusses empirical results from a survey about the research practice of Danish chemistry students, with a main focus on the question of anomalous data. It seeks to investigate how such data is handled by students, with special attention to so-called ‘questionable research practices’ where anomalous data are simply deleted or discarded. This question of QRPs is of particular importance as the educational practices students experience may influence how they act in their future professional careers, for instance in research. The ethical evaluation of QRPs however is not univocal. In parts of the literature QRPs are seen as unquestionably bad, while in other parts of the literature certain QRPs are seen as a necessary aspect of scientific practice. Results from the survey of Danish chemistry students shows that many students engage in certain types of questionable practices, and that a large minority of the students have been actively encouraged by their teachers to engage in such practices. The paper discusses to what extent and under what circumstances such instructional practices can be defended and suggests how the instructional practice connected to the handling of anomalous data can be improved.

Mathematicians’ use of external representations, such as symbols and diagrams, constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition and cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians’ use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process.

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions.

This paper investigates the notion of semiotic scaffolding in relation to mathematics by considering its influence on mathematical activities, and on the evolution of mathematics as a research field. We will do this by analyzing the role different representational forms play in mathematical cognition, and more broadly on mathematical activities. In the main part of the paper, we will present and analyze three different cases. For the first case, we investigate the semiotic scaffolding involved in pencil and paper multiplication. For the second case, we investigate how the development of new representational forms influenced the development of the theory of exponentiation. For the third case, we analyze the connection between the development of commutative diagrams and the development of both algebraic topology and category theory. Our main conclusions are that semiotic scaffolding indeed plays a role in both mathematical cognition and in the development of mathematics itself, but mathematical cognition cannot itself be reduced to the use of semiotic scaffolding