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)

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)

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.

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)

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)

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)

This study investigates the prevalence of ‘Seeking God's Help’, its relation to time since diagnosis, and its association with Life Satisfaction for all cancer types. This study also investigates Disease-Specific Quality of Life for patients with breast, prostate, and colorectal cancers. Data were obtained from the third wave of the Nord-Trøndelag Health Study of Norway, with 2,086 cancer patients identified by the Cancer Registry of Norway and 6,258 cancer-free controls. Our results indicate a higher prevalence of ‘Seeking God's Help’ after (...) a shorter time since diagnosis among men. No association was observed in multivariate analyses between ‘Seeking God's Help’ and ‘Life Satisfaction’ or ‘Disease-Specific QoL’ in long-term cancer survivors. Longitudinal investigations are needed to elucidate the relationship between the ‘Seeking God's Help’ variable and Life Satisfaction and Disease-Specific QoL among cancer patients in a Norwegian context. (shrink)

In Capitalism, Alienation and Critique Asger Sørensen offers a wide-ranging argument for the classical Critical Theory of the Frankfurt School, thus endorsing the dialectical approach of the original founders (Horkheimer, Adorno, Marcuse) and criticizing suggested revisions of later generations (Habermas, Honneth). Being situated within the horizon of the late 20th century Cultural Marxism, the main issue is the critique of capitalism, emphasizing experiences of injustice, ideology and alienation, and in particular exploring two fundamental subject matters within this horizon, namely economy (...) and dialectics. Apart from in-depth discussions of classical political economy and Hegelian dialectics, the explorative and inclusive argument also takes issues with Émile Durkheim’s theory of value, the general economy of Georges Bataille and the dialectics of Mao Zedong. -/- - See below link to the book's homepage at the publisher Brill. - See also link to a Youtubevideo from a seminar on the book in Belgrade, November 2019. (shrink)

The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to (...) term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning. (shrink)

To study whether the distinction between introspective and non-introspective states of mind is an empirical reality or merely a conceptual distinction, we measured event-related potentials elicited in introspective and non-introspective instruction conditions while the observers were trying to detect the presence of a masked stimulus. The ERPs indicated measurable differences related to introspection in both preconscious and conscious processes. Our data support the hypothesis that introspective states empirically differ from non-introspective states.

This study investigates the prevalence of ‘Seeking God’s Help’, its relation to time since diagnosis, and its association with Life Satisfaction for all cancer types. This study also investigates Disease-Specific Quality of Life for patients with breast, prostate, and colorectal cancers. Data were obtained from the third wave of the Nord-Trøndelag Health Study of Norway , with 2,086 cancer patients identified by the Cancer Registry of Norway and 6,258 cancer-free controls. Our results indicate a higher prevalence of ‘Seeking God’s Help’ (...) after a shorter time since diagnosis among men. No association was observed in multivariate analyses between ‘Seeking God’s Help’ and ‘Life Satisfaction’ or ‘Disease-Specific QoL’ in long-term cancer survivors. Longitudinal investigations are needed to elucidate the relationship between the ‘Seeking God’s Help’ variable and Life Satisfaction and Disease-Specific QoL among cancer patients in a Norwegian context. (shrink)

This article analyses the general economy of Georges Bataille (1897–1962) in relation to political economy. In the first section I present a critical perspective on economy that is necessary in order to appreciate Bataille’s conception of general economy, which is presented in the second section. The general economy is first considered in a macro-perspective, which comprises the whole of the universe, second in a micro-perspective, where the subjective aspect of economy is maintained as non-objectified desire and inner experience. In the (...) third section I turn to the general economy as it was explicitly intended, namely as a political economy. First I argue that the suggestions that Bataille himself presents are apolitical in an ordinary sense of politics, and that this can be shown to be due to some conceptual slides between nature and society and between history and ontology. I then sketch some postmodern attempts to legitimize respectively capitalism and communism, which refer to the general economy, but argue finally that Bataille can escape both, since he maintains the important distinction between need and desire. Although Bataille’s conception of economy thus reminds us of aspects often overlooked by economy in an ordinary sense, it also contains some serious aporias, which means that it cannot constitute the theoretical basis of a new general political economy, as Bataille had hoped. (shrink)

There is a simple technique, due to Dragalin, for proving strong cut-elimination for intuitionistic sequent calculus, but the technique is constrained to certain choices of reduction rules, preventing equally natural alternatives. We consider such a natural, alternative set of reduction rules and show that the classical technique is inapplicable. Instead we develop another approach combining two of our favorite tools—Klop’s ι-translation and perpetual reductions. These tools are of independent interest and have proved useful in a variety of settings; it is (...) therefore natural to investigate, as we do here, what they have to offer the field of sequent calculus. (shrink)

There is a simple technique, due to Dragalin, for proving strong cut-elimination for intuitionistic sequent calculus, but the technique is constrained to certain choices of reduction rules, preventing equally natural alternatives. We consider such a natural, alternative set of reduction rules and show that the classical technique is inapplicable. Instead we develop another approach combining two of our favorite tools—Klop's l-translation and perpetual reductions. These tools are of independent interest and have proved useful in a variety of settings; it is (...) therefore natural to investigate, as we do here, what they have to offer the field of sequent calculus. (shrink)

How do historians understand the minds, motivations and intentions of historical agents? What might evolutionary and cognitive theorizing contribute to this work? What is the relation between natural and cultural history? Historians have been intrigued by such questions ever since publication in 1859 of Darwin's The Origin of Species, itself the historicization of biology. This interest reemerged in the latter part of the twentieth century among a number of biologists, philosophers and historians, reinforced by the new interdisciplinary finding of cognitive (...) scientists about the universal capacities of and constraints upon human minds. The studies in this volume, primarily by historians of religion, continue this discussion by focusing on historical examples of ancient religions as well as on the theoretical promises and problems relevant to that study. (shrink)

This article reintroduces a phenomenological experiment designed in the early 1960’s, The Alien-Hand Experiment (TAHE), and it illustrates how phenomena denoted by theoretical concepts like body image, body schema and agency can be studied via the experiment. An analysis of the verbal reports from 26 subjects who participated in TAHE is presented in this article. Subjects were divided into three groups: A group of non-bulimic men, a group of non-bulimic women and a group of female bulimics. The group of (female) (...) bulimics was studied due to the widely spread notion that subjects suffering from eating disorders have a distorted body image. TAHE can be thought of as both a qualitative and quantitative way to study the phenomena arising when the normal relationship between motor behaviour and body experience is disrupted. The present investigation is not an operational definition of body schema and body image, but the two concepts offer a useful interpretive framework. (shrink)

Today the idea of cosmopolitanism has become widely accepted as an appropriate answer to what we now call globalization. A key reference is Kant who argues for a Recht of the world citizen, and this is normally understood as a cosmopolitan law. Apparently Kant lets the law of the world citizen be limited to a right to visit, but somehow his peace project must imply something more than just this very modest claim. Following a hint from Kant himself I take (...) a closer look at the material aspect of cosmopolitanism, i.e. the economy of travelling, and it appears that cosmopolitanism can function as an ideology for letting possible investors look for business cases, that is, for promising places for capitalist exploitation. As an answer to such strong material processes cosmopolitanism is insufficient, both as a moral imperative and as a stipulated right. What we need is a more comprehensive political and legal perspective and the solutions this indicates. To get inspiration for this, however, we can turn to Kant again, since it turns out that he does not argue for cosmopolitanism at all, neither as a programme nor as an ideology. For Kant the overall goal is perpetual peace, and the law of the world citizen represents only one subordinate element. As an answer to globalization we should thus drop the limited ideal of cosmopolitanism and follow Kant in his ambition of a threefold political constitution comprising state law, law of the people and law of the world citizen. (shrink)

The dialectical aspect in the work of Georges Bataille is often neglected. At the suggestion of Foucault and Derrida, Bataille is most often even taken to be a non-dialectical thinker. But Bataille worked intensely with Hegel's ideas, his thought was expressed in Hegelian terms, and both his epistemology and his ontology can be considered a determinate negation of Hegel's position in the Phenomenology. This is shown, first, by analysing Bataille's notions of `inner experience', and, second, by showing how Bataille extends (...) dialectics to the natural, non-human realm, and even conceives the link between the human and non-human as itself dialectical. However, once we see the dialectical nature of his theoretical stance, we are struck by a great vagueness in his practical conception of where society ought to be going. (shrink)

Beck uses the term `subpolitics' to refer to forms of politics outside and beyond the representative institutions of the political system of nation-states. From the perspective of the theory of reflexive modernization, the proliferation of subpolitics indicates a weakening of the `iron cage' of bureaucratic, state-oriented politics. We argue that subpolitics does indeed challenge conventional notions of politics. It mobilizes sources of societal influence that transcend the formal political system. In particular, subpolitics correlates with the command over positive or negative (...) sanctions and the capacity for uncertainty absorption. This can be illustrated in the areas of scientific expertise, corporate decision-making and political consumption. Those cases show that the political system cannot monopolize the means of societal influence. For a political system that regards itself as the centre of society, this must be irritating. For sociology, however, it provides the opportunity to part with such an inappropriate account of the role of politics in modern society. (shrink)

Value is a basic concept in economics, ethics and sociology. Locke made labour the source of value, whereas Smith referred to an ideal exchange and Kant specified that commodities only have a market price, no intrinsic value. One can distinguish two modern concepts of value, an economic one trying to explain value in terms of utility, interest or preferences, and an ideal one considering values as ends in themselves. On this basis, Durkheim constructed his theory of value, which was developed (...) by his followers Mauss and Bouglé and further by Bataille. Their line of thought makes it possible to develop a conceptual framework, which can be used to criticise neo-liberalism, big business and the effects of globalisation, while at the same time defending the moral value of business and giving an interpretation of the anti-globalisation protests. (shrink)

: C. S. Peirce had no theory of metaphor and provided only few remarks concerning the trope. Yet, some of these remarks seem to suggest that Peirce saw metaphor as fundamental to consciousness and thought. In this article we sketch a possible connection between metaphor and cognition; we understand Peircean metaphor as rooted in abduction; it is part of an intricate relation between experience, body, sign and guessing instinct as a semeiotic mechanism which can convey new insights.

The first part of the article demonstrates how the concept of mental resilience has developed from the 1970s to the 2010s, spreading from the field of developmental psychopathology to a wide range of psychological and psychotherapeutic disciplines. Today, there are many varied definitions of mental resilience, ranging from a relation to the concept of mental vulnerability to a related concept of mental strength. The second part of the article demonstrates how the current popularisation of the idea of individual resilience is (...) associated with a social and cultural problematisation of the vulnerable life as well as with a psychopolitical and governmental focus on enhancing the mental capacity of citizens. (shrink)

The philosophical situation at Copenhagen University in the 1960’s was dominated by two positivists. Th elogical positivist Jørgen Jørgensen – who had written the history of the “movement” – and the legal positivistAlf Ross. There were also two “outsiders”: Peter Zinkernagel, who did more analytical philosophy of language in the British style, and K. Grue Sørensen who was working in the traditions of neo-Kantianism. In 1955 Grue-Sørensen was hired as the first professor in education – after a long controversy about (...) the scientific status ofeducation as a discipline – but with a focus on the history of education. He had received a doctoral degree in philosophy in 1950 with a dissertation on refl exivity as a philosophical concept and a thesis about the reflexivity of consciousness. He was also an objectivist in ethics, and had been critical of the prevalent moral relativism and subjectivism found in recent philosophy. Jørgensen and Ross had done important work on moral argumentation with more technical work on the logic of imperatives and norms. Moral objectivism was not only wrong but in a way also “immoral” because it undermined their belief in democracy. Especially Jørgensen also thought that the idea of reflexivity was wrong when applied to consciousness. Neither statements nor consciousness could be reflexive – that is refer to themselves/itself. The reflexivity of consciousness is – according to Jørgensen – simply not an empirical psychological fact. Grue-Sørensen tried to establish the foundation of a theory of education based both on conceptions of consciousness and of the relation between scientific knowledge – facts – and moral values – in a neo-Kantian fashion. For him the interplay between ethics and knowledge was a central part of a theory of education – a belief due to which he never became a professor of philosophy – having tried many times. These debates in philosophy and in education were superseded in the 1970’s by the rise in influence of the German inspiration from Critical Theory and the demise of logical positivism. (shrink)

Grue-Sørensen’s concept of ’educational teaching’ is traced back to an original infl uence from Herbart and Kant. On this background the article attempts to interpret, how one can understand a concept of educationalteaching today. With that, the concept is shown to have its root in a tradition of general education and Grue-Sørensen is shown to be a Danish representative of this. However, in research programs as well as educational programs this tradition has generally been under increasing pressure the last approximately (...) 30 years. Grue-Sørensen and his possible relevance today is discussed in connection with a potential revitalization of a general educational thinking in our current postmodern epoche of higher education. (shrink)