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)
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)
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)
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)
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)
I sin udmærkede kommentar til vores artikel «En etisk diskussion af screening for kræftsygdomme» beskriver Geir Hoff den udtalte mangel på evidens vedrørende nytteværdien af screeningsprogrammer for kræftsygdomme baseret på randomiserede studier. Ydermere fremhæver Geir Hoff misforholdet mellem den manglende evidens ved screening og de strenge krav, der er til evidensen i den farmaceutiske industri. Dette er en velkommen kritik, pga. en udtalt ukritisk og uvidenskabelig tilgang til anvendelse af screening for denne eller hin sygdom eller risikofaktor.
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)
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)
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)
In Capitalism, Alienation and Critique, part of the development of Asger S?rensen?s overall argument is a disagreement with Georges Bataille. The crux of the argument is that Bataille?s thinking - especially his conception of subjectivity - is?apolitical?. The aim of this paper is to investigate the force of this claim. What does it mean for a position - albeit a philosophical one - to be?apolitical??
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 External Links to the book's homepage at the publisher Brill and to the Introduction. - See also External Links to a Youtubevideo from a seminar on the book in Belgrade, November 2019 and two Author Meet Critics sections from 2020 and 2021. (shrink)
As a primary school teacher in Copenhagen, and, simultaneously, as a student of philosophy at The Universityof Copenhagen, Grue-Sörensen became so well acquainted with contemporary psychology that he, in theyears 1941-55, worked as a school psychologist in Copenhagen. Furthermore, from 1934 until 1955, he published 22 articles or chapters about psychological issues. The present contribution presents and characterizes the 22 publications categorized in works about developmental psychology, and works about central psychological issues – motivation, learning, and cognition. In the last (...) Section, Grue-Sörensen’s understanding of the relationships between the fields of education, philosophy and psychology is discussed, especially the question of the importance of psychology to education. Finally, it is concluded that, according to Grue-Sörensen, the value of philosophy to psychology is not to off er insight into theories of science but, rather, inspiration to use words and concepts in a clear and careful way. (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)
This is a comment to Graham Harman’s 2019 response to an article by Þóra Pétursdóttir and Bjørnar Olsen (2018) in which they propose that a materially grounded, archaeological perspective might complement Harman’s historical approach in Immaterialism (2016). Harman responds that his book is indeed already more archaeological than historical, stipulating that history is the study of media with a high density of information, whereas archaeology studies media with a low density of information. History, Harman holds, ends up in too much (...) detail, while archaeology has the advantage of lending itself to the imagination. Hence, his reading of history had the aim of tempering the historical information overload, in effect making the book a work of archaeology. In this comment, I want to do three things: (1) critique the idea that archaeological and historical media are inherently different with regard to their densities of information, (2) discuss how archaeology and history approach their media, and (3) reflect on conceptualisations of “archaeology” outside the discipline itself. (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)
When K. Grue-Sørensen became a professor of pedagogy at the University of Copenhagen in 1955, he was inline with the dominant historical-hermeneutical approach to humanities. From the late 1960s until retirementin 1974, his approach was challenged by both technical and critical alternatives. Both these alternative havesince grown steadily, while the historical-hermeneutical view has been in the defensive. But Grue-Sørensenand the tradition he represented have three signifi cant points for today’s pedagogy, whether it is technicalor critical: pedagogy can and should not (...) deliver effi ciency technology, pedagogy should as far as possible useeveryday language, and fi nally that the educational history can make us wiser. (shrink)