When did the concept of model begin to be used in mathematics? This question appears at first somewhat surprising since “model” is such a standard term now in the discourse on mathematics and “modelling” such a standard activity that it seems to be well established since long. The paper shows that the term— in the intended epistemological meaning—emerged rather recently and tries to reveal in which mathematical contexts it became established. The paper discusses various layers of argumentations and reflections in (...) order to unravel and reach the pertinent kernel of the issue. The specific points of this paper are the difference in the epistemological concept to the usually discussed notions of model and the difference between conceptions implied in mathematical practices and their becoming conscious in proper reflections of mathematicians. (shrink)

This comment is analysing the last section of a paper by Piotr Blaszczyk, Mikhail G. Katz, and David Sherry on alleged misconceptions committed by historians of mathematics regarding the history of analysis, published in this journal in the first issue of 2013. Since this section abounds of wrong attributions and denouncing statements regarding my research and a key publication, the comment serves to rectify them and to recall some minimal methodological requirements for historical research.

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of these conferred an only (...) marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

Enno Heeren Dirksen (1788–1850) was one of several mathematicians originating from Eastern Frisia at the end of the eighteenth century to acquire prominence in neohumanist reformed Prussia. While striving—like so many of his contemporaries—for more rigour in the foundations of calculus, his work is significant due to his unusual emphasis on the semiotic aspects of the basic concepts. As a clear-sighted propagator of Cauchy's conceptual innovations in Germany, his achievements in improving sign expressions of such concepts enabled him to clarify (...) hitherto not differentiated multiple limit processes—although less successfully in constructive concept formation itself. The contribution is based on the analysis of Dirksen's published texts and on the numerous texts which remained drafts for his projected voluminousOrganon der gesamten Analysis. (shrink)

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of these conferred an only (...) marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

The life and work of Hermann Günther Graßmann (1809–1877) attract not only ever again the attention of mathematicians, mathematical historians and those interested in the history of mathematics, they constitute also a challenge for the methodology of historiographical research. This challenge persists since Friedrich Engel’s biography of 1911; there, two sources were presented and interpreted in a not legitimate manner which even mislead since then various scholars. This paper faces the intricate task to unravel not only the methodological shortcomings of (...) Engel’s biography, but also to re-assess the misinterpretations – particularly of alleged influences on Graßmann’s approaches – induced by Engel’s misleading claims. Based upon broader historically contextualised analyses of Graßmann’s innovative elements in his theory of extension and upon new primary sources for his student times at the Gymnasium and Berlin University, the paper presents a novel assessment of Graßmann’s approaches. (shrink)

The mathematical and epistemological origins of Hermann Graßmann’s innovative work have always attracted the interest of mathematicians and historians. Since Friedrich Engel’s biography, a favourite source for these interpretations has been two curriculum vitae, which were, however, only known from several excerpts. The complete texts are edited here for the first time. They are presented and commented on in their respective contexts, namely the examinations required for a career as Gymnasium teacher and as Protestant pastor. Graßmann’s relations to Schleiermacher’s theology (...) and to his philosophy, as well as Graßmann’s philosophical and mathematical motivations, are discussed in the light of these two documents. (shrink)

ArgumentThis paper studies the evolution of mathematics teaching in France and Germany from 1900 to about 1980. These two countries were leading in the processes of international modernization. We investigate the similarities and differences during the various periods, which showed to constitute significant time units and this in a remarkably parallel manner for the two countries. We argue that the processes of reform concerning the teaching of this major school subject are not understandable from within mathematics education or even within (...) the school system. Rather, the evolution of the processes of reform prove to be intimately tied to changing conceptions of modernity according to respective social and cultural values and to changing epistemological conceptions of mathematics. It is particularly novel that we show the key impact of the changing social status of primary schooling for these modernization processes. (shrink)

An updated and expanded successor to Culver and Gert's Philosophy in Medicine, this book integrates moral philosophy with clinical medicine to present a comprehensive summary of the theory, concepts, and lines of reasoning underlying the ...

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...) in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)

Prologue: on the weakness of education -- Creativity -- Communication -- Teaching -- Learning -- Emancipation -- Democracy -- Virtuosity -- Epilogue: for a pedagogy of the event -- Appendix: coming into the world, uniqueness, and the beautiful risk of education: an interview with Gert Biesta by Philip Winter.

This book offers the fullest and most sophisticated account of Gert's influential moral theory, a model first articulated in the classic work The Moral Rules: A New Rational Foundation for Morality, published in 1970. In this final revision, Gert makes clear that the moral rules are only one part of an informal system that does not provide unique answers to every moral question but does always provide a range of morally acceptable options. A new chapter on reasons includes (...) an account of what makes one reason better than another and a second new chapter is devoted to the question of justifying violations of the rules. Moral impartiality, the moral ideals, and virtue and vice, are all treated in greater detail. Throughout, Gert attempts to answer all of the challenges that his work has provoked. (shrink)

The second volume in Joel Feinberg's series The Moral Limits of the Criminal Law, Offense to Others focuses on the "offense principle," which maintains that preventing shock, disgust, or revulsion is always a morally relevant reason for legal prohibitions. Feinberg clarifies the concept of an "offended mental state" and further contrasts the concept of offense with harm. He also considers the law of nuisance as a model for statutes creating "morals offenses," showing its inadequacy as a model for understanding "profound (...) offenses," and discusses such issues as obscene words and social policy, pornography and the Constitution, and the differences between minor and profound offenses. (shrink)

Joshua Gert presents an original account of color properties, and of our perception of them. He employs a general philosophical strategy - neo-pragmatism - which challenges an assumption made by virtually all other theories of color: he argues that colors are primitive properties of objects, irreducible to physical or dispositional properties.

Moral problems do not always come in the form of great social controversies. More often, the moral decisions we make are made quietly, constantly, and within the context of everyday activities and quotidian dilemmas. Indeed, these smaller decisions are based on a moral foundation that few of us ever stop to think about but which guides our every action. Here distinguished philosopher Bernard Gert presents a clear and concise introduction to what he calls "common morality" -- the moral system (...) that most thoughtful people implicitly use when making everyday, common sense moral decisions and judgments. Common Morality is useful in that -- while not resolving every disagreement on controversial issues -- it is able to distinguish between acceptable and unacceptable answers to moral problems. In the first part of the book Gert lays out the fundamental features of common morality: moral rules, moral ideals, and a two-step procedure for determining when a violation of a moral rule is justified. Written in a non-technical style, the ten general moral rules include rules on which everyone can agree, such as "do not kill," "do not deceive," and "keep your promises." The moral ideals include similarly uncontroversial precepts such as "Relieve pain" and "Aid the needy." In the second part of the book Gert examines the underlying concepts that justify common morality, such as the notions of rationality and impartiality. The distillation of over 40 years of scholarship, this book is the most accessible version of Gert's influential theory of morality as well as an eye-opening look at the moral foundations of our everyday actions. Throughout the discussion is clear enough for a reader with little or no philosophy background. (shrink)

This volume is a revised, enlarged, and broadened version of Gert's classic 1970 book, The Moral Rules. Advocating an approach he terms "morality as impartial rationality," Gert here presents a full discussion of his moral theory, adding a wealth of new illuminating detail to his analysis of the concepts--rationality/irrationality, good/evil, and impartiality--by which he defines morality. He constructs a "moral system" that includes rules prohibiting the kinds of actions that cause evil, procedures for determining when violation of the (...) rules is permitted, and ideals which encourage actions that prevent or relieve suffering. To be valid, Gert argues, any such system must be "a public system that applies to all rational persons." The book concludes with a discussion of medical ethics, demonstrating the link between moral theory and its application to real moral problems. (shrink)

Whether or not one endorses realism about colour, it is very tempting to regard realism about determinable colours such as green and yellow as standing or falling together with realism about determinate colours such as unique green or green31. Indeed some of the most prominent representatives of both sides of the colour realism debate explicitly endorse the idea that these two kinds of realism are so linked. Against such theorists, the present paper argues that one can be a realist about (...) the determinable colours of objects, and thus hold that most of the colour ascriptions made by competent speakers are literally true, while denying that there are any positive facts of the matter as to the determinate colours of objects. The result is a realistic colour realism that can certify most of our everyday colour ascriptions as literally correct, while acknowledging the data regarding individual variation. (shrink)

In this paper I explore the relationship between teaching and learning. Whereas particularly in the English language the relationship between teaching and learning has become so intimate that it often looks as if ‘teaching and learning’ has become one word, I not only argue for the importance of keeping teaching and learning apart from each other, but also provide a number of arguments for suggesting that learning may not be the one and only option for teaching to aim for. I (...) explore this idea through a discussion of the relationship between teaching and learning, both at a conceptual and at an existential level. I discuss the limitations of the language of learning as an educational language, point at the political work that is being done through the language of learning, and raise epistemological and existential questions about the identity of the learner, particularly with regard to the question what it means to be in and with the world in terms of learning as comprehension and sense making. Through this I seek to suggest that learning is only one possible aim for teaching and that the learner identity and the learning way of engaging with the world puts the learner in a very specific position vis-à-vis the world, one where the learner remains in the centre and the world appears as object for the learner’s acts of learning. That it is possible to teach without requesting from students that they learn, comprehend and make sense, is demonstrated through a brief account of a course in which students were explicitly asked to refrain from learning and were instead asked to adopt a concept. I show how this request opened up very different existential possibilities for the students and argue that if we value such existential possibilities, there may be good reasons for freeing teaching from learning. (shrink)

Joshua Gert offers an original account of normative facts and properties, those which have implications for how we ought to behave. He argues that our ability to think and talk about normative notions such as reasons and benefits is dependent on how we respond to the world around us, including how we respond to the actions of other people.

Modern Christianity has failed to update its myths and has even eliminated them, thus, excluding the metaphysical experience indispensable to religion. Myths should be interpreted, not eliminated. Answering the question about how to interpret myths without eliminating them or their intended effect is the object of this paper. The study investigates the possibility of interpreting myths as metaphors, thus, in a non-literal way. Various definitions of metaphor and myth, and theories for their interpretation are discussed, with focus on their relationship (...) to symbolic universes. Finally, a non-mythical symbolic universe structured by root-metaphors is suggested as a framework for the existential interpretation of mythical concepts in the New Testament. (shrink)

This book is the result of over 30 years of collaboration among its authors. It uses the systematic account of our common morality developed by one of its authors to provide a useful foundation for dealing with the moral problems and disputes that occur in the practice of medicine. The analyses of impartiality, rationality, and of morality as a public system not only explain why some bioethical questions, such as the moral acceptability of abortion, cannot be resolved, but also provide (...) a method for determining the correct answer for those occasions when a bioethical question has a unique correct answer. This new edition includes an entire chapter that has been added to address the controversial issue of abortion within the authors' distinct framework. This book presents the latest revisions of the authors' orignial analyses of the concepts of death and disease, analyses that have had a significant impact on the field of bioethics. It also includes an added chapter on mental disorders, where the authors' definition influenced what psychiatry classifies as a mental disorder, and so has had an impact that reveals beyond the field of bioethics. In this edition, the authors also offer a new, more developed perspective on the concept of valid or informed consent by considering what information physicians should be required to know before proposing screening, diagnostic testing, prescribing medications, or performing surgery. The book also integrates some of the important insights of the field of clinical epidemiology into its discussion of valid consent. Its account of paternalism and its justification, perhaps the most ubiquitous moral problem in medical ethics, has had considerable influence. Its discussion of euthanasia and physician assisted suicide challenges the standard views that have been put forward by both proponents and opponents of physician assisted suicide and voluntary active euthanasia. (shrink)

This book presents an account of normative practical reasons and the way in which they contribute to the rationality of action. Rather than simply 'counting in favour of' actions, normative reasons play two logically distinct roles: requiring action and justifying action. The distinction between these two roles explains why some reasons do not seem relevant to the rational status of an action unless the agent cares about them, while other reasons retain all their force regardless of the agent's attitude. It (...) also explains why the class of rationally permissible action is wide enough to contain not only all morally required action, but also much selfish and immoral action. The book will appeal to a range of readers interested in practical reason in particular, and moral theory more generally. (shrink)

In §50 of Philosophical Investigations Wittgenstein wrote the sentence, “There is one thing of which one can say neither that it is one metre long, nor that it is not one metre long, and that is the standard metre in Paris.” Although some interpreters have claimed that Wittgenstein’s statement is mistaken, while others have proposed various explanations showing that this must be correct, none have questioned the fact that he intended to assert that it is impossible to describe the standard (...) meter as being a meter long. Given that Wittgenstein introduces this sentence as analogous to the claim that “existence cannot be attributed to an element,” and that the preceding passages discuss a language-game the simples of which can be described by their own names, there is good reason to think that Wittgenstein did not intend to assert this infamous sentence. (shrink)

_Obstinate Education: Reconnecting School and Society_ argues that education is not just there to give individuals, groups and societies what they want from it, but that education has a duty to resist.

In §50 of Philosophical Investigations Wittgenstein wrote the sentence, “There is one thing of which one can say neither that it is one metre long, nor that it is not one metre long, and that is the standard metre in Paris.” Although some interpreters have claimed that Wittgenstein's statement is mistaken, while others have proposed various explanations showing that this must be correct, none have questioned the fact that he intended to assert that it is impossible to describe the standard (...) meter as being a meter long. Given that Wittgenstein introduces this sentence as analogous to the claim that “existence cannot be attributed to an element,” and that the preceding passages discuss a language‐game the simples of which can be described by their own names, there is good reason to think that Wittgenstein did not intend to assert this infamous sentence. (shrink)

This paper offers one formal reason why it may often be inappropriate to hold, of two conflicting desires, that the first must be weaker than, stronger than, or of the same strength as the second. The explanation of this fact does not rely on vagueness or epistemological problems in determining the strengths of desires. Nor does it make use of the problematic notion of incommensurability. Rather, the suggestion is that the motivational capacities of many desires might best be characterized by (...) two values, neither of which should be interpreted as strength. (shrink)

I see someone who looks vaguely familiar. I wonder if he could be the same person I had a fight with on my birthday ten years ago. I hear that some scientist has received the Nobel prize in chemistry for some work very similar to that which interested the brightest boy in my chemistry class twenty years ago. I wonder if that scientist and the brightest boy might be the same person. What facts, if they could be discovered, would be (...) decisive in answering these questions? Are there any facts that, if it turns out that they do not obtain, would show conclusively that the two people are not the same person? That is, are there any necessary conditions for personal identity? Are there any facts that, if it turns out that they do obtain, would show conclusively that the two people are the same person? That is, are there any sufficient conditions for personal identity? (shrink)

Despite the urgency of this issue, AI still struggles to represent social life. This article presents a comprehensive agent-based model that investigates status-power dynamics in groups. Kemper’s sociological status–power theory of social relationships, and a literature review on school children in middle youth, is its basis. The model allows us to investigate causation of the near-ubiquitous phenomenon that females have lower social status on average than males. Possible causes included in the model are children’s dispositional traits (kindness, beauty, and physical (...) power), schoolyard culture (social acceptability of fighting), behavioural strategy (amount of rough-and-tumble play) and the balance between public and dyadic sources of status. An agent-based model of a virtual schoolyard was created in which the children assemble in changing groups and mutually confer status. The status conferred upon a child modifies the status it holds. Rough-and-tumble is modelled as ambiguous: it is intended as a status conferral, but may be perceived as a power move. Running many trials of the model we found that in time, depending on the parameter settings, a gender-based status gap emerged. Rough-and-tumble play had more impact on emergent status differences than did physical power differences. Social acceptability of fighting also strongly moderated the resulting status gap. Placing more weight on dyadic relationship could alleviate status loss. All these model behaviours are in line with empirical findings of child behaviour studies at schools. They have face validity for social status issues in the adult world. We conclude from this that this kind of agent-based model merits use in studying the status–power dynamics of other issues in child behaviour, or indeed in social behaviour in general. (shrink)