Since 2004, it has been mandated by law that all Danish undergraduate university programmes have to include a compulsory course on the philosophy of science for that particular program. At the Faculty of Science and Technology, Aarhus University, the responsibility for designing and running such courses were given to the Centre for Science Studies, where a series of courses were developed aiming at the various bachelor educations of the Faculty. Since 2005, the Centre has been running a dozen different courses (...) ranging from mathematics, computer science, physics, chemistry over medical chemistry, biology, molecular biology to sports science, geology, molecular medicine, nano science, and engineering. -/- We have adopted a teaching philosophy of using historical and contemporary case studies to anchor broader philosophical discussions in the particular subject discipline under consideration. Thus, the courses are tailored to the interests of the students of the particular programme whilst aiming for broader and important philosophical themes as well as addressing the specific mandated requirements to integrate philosophy, some introductory ethics, and some institutional history. These are multiple and diverse purposes which cannot be met except by compromise. -/- In this short presentation, we discuss our ambitions for using case studies to discuss philosophical issues and the relation between the specific philosophical discussions in the disciplines and the broader themes of philosophy of science. We give examples of the cases chosen to discuss various issues of scientific knowledge, the role of experiments, the relations between mathematics and science, and the issues of responsibility and trust in scientific results. Finally, we address the issue of how and why science students can be interested in and benefit from mandatory courses in the philosophy of their subject. (shrink)

During the first half of the nineteenth century, mathematical analysis underwent a transition from a predominantly formula-centred practice to a more concept-centred one. Central to this development was the reorientation of analysis originating in Augustin-Louis Cauchy's (1789–1857) treatment of infinite series in his Cours d’analyse. In this work, Cauchy set out to rigorize analysis, thereby critically examining and reproving central analytical results. One of Cauchy's first and most ardent followers was the Norwegian Niels Henrik Abel (1802–1829) who vowed to (...) shed some light on the vast darkness in analysis.This paper investigates some important aspects of Abel's contribution to the reorientation in analysis. In particular, it stresses the role for critical revision in the process of rigorization. By critically examining past practice, the new practice sought to explain the relative success of the previous—now outdated—approach. This is illustrated by discussing a number of issues related to Abel's new proof of the binomial theorem (1826) including his reactions to the exception that he encountered to one of the central theorems of Cauchy's theory.Following this discussion, the formation of new concepts as the result of critical revisions is illustrated by analysing the early history of the concept of absolute convergence. Thereby, it is shown how a new concept was distilled, investigated, put to use and eventually superseded. (shrink)

In the present paper, I go beyond these examples by bringing into play an example that I nd more experimental in nature, namely that of the use of the so-called PSLQ algorithm in researching integer relations between numerical constants. It is the purpose of this paper to combine a historical presentation with a preliminary exploration of some philosophical aspects of the notion of experiment in experimental mathematics. This dual goal will be sought by analysing these aspects as they are presented (...) by some of the protagonists of the eld and discussing them using notions from contemporary philosophy of science. (shrink)

200 years ago, on August 5, 1802, Niels Henrik Abel was born on Finnøy near Stavanger on the Norwegian west coast. During a short life span, Abel contributed to a deep transition in mathematics in which concepts replaced formulae as the basic objects of mathematics. The transformation of mathematics in the 1820s and its manifestation in Abel’s works are the themes of the author’s PhD thesis. After sketching the formative instances in Abel’s well-known biography, this article illustrates two aspects (...) of the transformation which concern the introduction of concept based mathematics and the related shift in standards of mathematical rigor. Furthermore, the article outlines some of the many bicentennial celebrations in Norway and gives a short, thematic introduction to the literature on Abel and his work. (shrink)

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. (shrink)

In this paper, I describe some aspects of the phenomenon of "experimental mathematics" in order to discuss whether it constitutes a subdiscipline or a particular style of mathematics. My conclusion is that neither of these notions accurately capture the complex culture of experimental mathematics.

This chapter is devoted to discussions about the nature of experiments in mathematics during the formative period of experimental mathematics and to a quantitative and qualitative analysis of the journal Experimental Mathematics founded in 1992. Particular attention is paid to interpret the roles of experiment and observation in the mathematical practices surrounding Experimental Mathematics.

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. (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)

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)

Although the use of mathematical models is ubiquitous in modern science, the involvement of mathematical modeling in the sciences is rarely seen as cases of interdisciplinary research. Often, mathematics is “applied” in the sciences, but mathematics also features in open-ended, truly interdisciplinary collaborations. The present paper addresses the role of mathematical models in the open-ended process of conceptualizing new phenomena. It does so by suggesting a notion of cultures of mathematization, stressing the potential role of the mathematical model as a (...) boundary object around which negotiations of different desiderata can take place. This framework is then illustrated by a case study of the early efforts to produce a mathematical model for quasi-crystals in the first two decades after Dan Shechtman’s discovery of this new phenomenon in 1984. (shrink)

La théorie du pronostic est un des éléments les plus importants (et les plus discutés) de la théorie de la science du droit d'Alf Ross. Cet article veut montrer que cette théorie du pronostic est incompatible avec les propres fondements théoriques de Ross, et notamment avec sa théorie des sources du droit et de l'interprétation. Elle doit être comprise comme une tentative réductionniste de la validité normative, qui a échoué par son incapacité à fournir à la recherche juridique les concepts (...) qui lui sont nécessaires. (shrink)

Artikeln är en översikt av den tysk-judiska invandringen till Sverige från 1770-talet och framåt. Till en början skedde invandringen till stor del i form av kedjemigration från Mecklenburg. Många i pionjären Aaron Isaacs släktkrets invandrade, likaså i släktkretsen kring hans kompanjon Abraham Aaron. Under 1800-talet fortsatte frivilliginvandringen men nu i mindre form av kedjemigration och med mer differentierad geografisk bakgrund än tidigare. Så småningom kom invandringen från Tyskland numerärt att överflygas av den så kallade östjudiska invandringen; av de judar som (...) bodde i Sverige 1880 var bara åtta procent födda i Tyskland och 1920 mindre än fem procent. Många tysk-judiska invandrare vid denna tid, såsom bankmannen Louis Frænckel, gjorde betydelsefulla insatser i det svenska industriundret. Trots den restriktiva invandringspolitiken på 1930-talet kom ett antal tysk-judiska flyktingar till Sverige, såväl individuellt som genom särskilda kvoter. Många blev betydande aktörer i olika sektorer av samhället, och några blev internationellt kända namn som författaren och Nobelpristagaren Nelly Sachs och kärnfysikern Lise Meitner. Många flyktingar fick dock inte arbete som motsvarade deras utbildningar. Bland ”1945 års räddade” fanns tämligen få från Tyskland. (shrink)

Patients in specialist mental healthcare services who are at risk of suicide may experience their struggles as existential in nature. Yet, research on meaning in life has been relatively scarce in suicidology. This qualitative study aimed to explore how patients at risk of suicide perceived their encounters with specialist healthcare professionals after a suicide attempt (SA), with special reference to meaning in life experiences. The study was conducted in specialised mental healthcare services in Norway. Data were collected via individual interviews (...) with eight patients aged 20–75 years. Using a four-step procedure, the interviews were analysed by systematic text condensation. The participants understood their feelings of shame, self-contempt and challenging life experiences as contributing factors to their SA. They perceived that existential themes in relation to financial difficulties, shame and trauma were resolved, while issues associated with the SA, such as death, loss and beliefs, were given less attention. The participants were either ambivalent about continuing to live or wished to rebuild a meaningful life. Overall, their experiences of meaningfulness were hampered. Assisting patients with meaning in life experiences may help them alter their life interpretations and increase their ability to rebuild their lives as meaningful. The present study should be seen as a contribution to meaning-informed approaches in specialist mental healthcare services. More research is needed to equip healthcare personnel in their overall aims of preventing suicide and supporting patients at risk in their efforts to live a meaningful life. (shrink)

This wide-ranging collection of essays contains eighteen original articles by authors representing some of the most important recent work on Wittgenstein. It deals with questions pertaining to both the interpretation and application of Wittgenstein s thought and the editing of his works. Regarding the latter, it also addresses issues concerning scholarly electronic publishing. The collection is accompanied by a comprehensive introduction which lays out the content and arguments of each contribution. Contributors: Knut Erik Tranoy, Lars Hertzberg, Georg Henrik von (...) Wright, Marie McGinn, Cora Diamond, James Conant, David G. Stern, Eike von Savigny, P.M.S. Hacker, Hans-Johann Glock, Allan Janik, Kristof Nyiri, Antonia Soulez, Brian McGuinness, Anthony Kenny, Joachim Schulte, Herbert Hrachovec, Cameron McEwen.". (shrink)

IN 1959 and 1960 I gave the Gifford Lectures in the University of St. Andrews. The lectures were called 'Norms and Values, an Inquiry into the Conceptual Foundations of Morals and Legislation'. The present work is substantially the same as the content of the second series of lectures, then advertised under the not very adequate title 'Values'. It is my plan to publish a revised version of the content of the first series of lectures, called 'Norms', as a separate book. (...) The two works will be independent of one another. (shrink)

In diesem Artikel geht es um die Beziehungen zwischen der von Jürgen Habermas entwickelten Diskursethik und den Studierendenseminaren, die der Ulmer Arbeitskreis „Ethik in der Medizin“ veranstaltet. Zunächst erfolgt eine Darstellung der Kernaussagen der philosophischen Diskursethik. Sie liefert eine formale Argumentationsprozedur, mit der es möglich ist, die Legitimität strittig gewordener Normen im Diskurs zu prüfen. Anschließend wird das in Ulm von Baitsch und Sponholz entwickelte Modell der Seminare „Ethische Entscheidungskonflikte im ärztlichen Alltag“ vorgestellt. Da sich die Ulmer Seminare unter anderem (...) auf die Diskursethik als theoretischen Hintergrund berufen, werden die Gemeinsamkeiten und Unterschiede beider Konzepte untersucht. Der Hauptunterschied ist der zwischen einem Begründungs- und einem Anwendungsdiskurs. Gemeinsam sind den Ansätzen die grundsätzlich kognitivistische Einstellung und die „Spielregeln“ des Diskurses. (shrink)

This paper argues that ‘the Anthropocene’ is a deeply depoliticizing notion. This de-politicization unfolds through the creation of a set of narratives, what we refer to as ‘AnthropoScenes’, which broadly share the effect of off-staging certain voices and forms of acting. Our notion of the Anthropo-obScene is our tactic to both attest to and undermine the depoliticizing stories of ‘the Anthropocene’. We first examine how various AnthropoScenes, while internally fractured and heterogeneous, ranging from geo-engineering and earth system science to more-than-human (...) and object-oriented ontologies, place things and beings, human and non-human, within a particular relational straitjacket that does not allow for a remainder or constitutive outside. This risks deepening an immunological biopolitical fantasy that promises adaptive and resilient terraforming, an earth system management of sorts that permits life as we know it to continue for some, while turning into a necropolitics for others. Second, we develop a post-foundational political perspective in relation to our dramatically changing socio-ecological situation. This perspective understands the political in terms of performance and, in an Arendtian manner, re-opens the political as forms of public-acting in common that subtracts from or exceeds what is gestured to hold socio-ecological constellations together. We conclude that what is off-staged and rendered obscene in ‘the AnthropoScenes’ carries precisely the possibility of a return of the political. (shrink)

Many scientists regard mass and energy as the primary currency of nature. In recent years, however, the concept of information has gained importance. In this book, eminent scientists, philosophers, and theologians chart various aspects of information, from quantum information to biological and digital information, in order to understand how nature works. Beginning with a historical treatment of the topic, the book also examines physical and biological approaches to information, and the philosophical, theological, and ethical implications.

The idea that technologies can change moral beliefs and practices is an old one. But how, exactly, does this happen? This paper builds on an emerging field of inquiry by developing a synoptic taxonomy of the mechanisms of techno-moral change. It argues that technology affects moral beliefs and practices in three main domains: decisional (how we make morally loaded decisions), relational (how we relate to others) and perceptual (how we perceive situations). It argues that across these three domains there are (...) six primary mechanisms of techno-moral change: (i) adding options; (ii) changing decision-making costs; (iii) enabling new relationships; (iv) changing the burdens and expectations within relationships; (v) changing the balance of power in relationships; and (vi) changing perception (information, mental models and metaphors). The paper also discusses the layered, interactive and second-order effects of these mechanisms. (shrink)

Technologies can have profound effects on social moral systems. Is there any way to systematically investigate and anticipate these potential effects? This paper aims to contribute to this emerging field on inquiry through a case study method. It focuses on two core human values—truth and trust—describes their structural properties and conceptualisations, and then considers various mechanisms through which technology is changing and can change our perspective on those values. In brief, the paper argues that technology is transforming these values by (...) changing the costs/benefits of accessing them; allowing us to substitute those values for other, closely-related ones; increasing their perceived scarcity/abundance; and disrupting traditional value-gatekeepers. This has implications for how we study other, technologically-mediated, value changes. (shrink)

A wide range of non-pharmaceutical interventions (NPIs) have been introduced to stop or slow down the COVID-19 pandemic. Examples include school closures, environmental cleaning and disinfection, mask mandates, restrictions on freedom of assembly and lockdowns. These NPIs depend on coercion for their effectiveness, either directly or indirectly. A widely held view is that coercive policies need to be publicly justified—justified to each citizen—to be legitimate. Standardly, this is thought to entail that there is a scientific consensus on the factual propositions (...) that are used to support the policies. In this paper, we argue that such a consensus has been lacking on the factual propositions justifying most NPIs. Consequently, they would on the standard view be illegitimate. This is regrettable since there are good reasons for granting the state the legitimate authority to enact NPIs under conditions of uncertainty. The upshot of our argument is that it is impossible to have both the standard interpretation of the permissibility of empirical claims in public justification and an effective pandemic response. We provide an alternative view that allows the state sufficient room for action while precluding the possibility of it acting without empirical support. (shrink)