In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.

PurposeIn this article, we aim to present and defend a contextual approach to mathematical explanation.MethodTo do this, we introduce an epistemic reading of mathematical explanation.ResultsThe epistemic reading not only clarifies the link between mathematical explanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematical explanation can be read in this approach.ConclusionThe contextual approach defended here clears up the notion of explanation and pushes us towards a pluralist vision (...) on mathematical explanation. (shrink)

Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements about explanatoriness in mathematics by identifying differences in the background knowledge, skill corpus, or epistemic aims of (...) mathematicians or mathematical communities. We focus on the relation between explanation, epistemic aims and diverging explanatory assessments by looking at cases from mathematical practice. (shrink)

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our (...) story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like. (shrink)

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.

It has been observed many times before that, as yet, there are no encompassing, integrated theories of mathematical practice available.To witness, as we currently do, a variety of schools in this field elaborating their philosophical frameworks, and trying to sort out their differences in the course of doing so, is also to be constantly reminded of the fact that a lot of epistemic aspects, extremely relevant to this task, remain dramatically underexamined. This volume wants to contribute to the stock of (...) studies filling this perceived lacuna. It contains papers by established, upcoming, as well as beginning scholars, covering general, metaphilosophical themes such as naturalism, semiotics, pragmaticism, or empiricism, next to more specific topics including the unity of mathematical theories, thruth-flow in mathematics, diagrammatic reasoning, erroneous argumentation, or numerical analysis. (shrink)

As far as ‘modern’ logical theories of vagueness are concerned, a main distinction can be drawn between ‘semantical’ ones and ‘pragmatical’ ones. The latter are defended here, because they tend to retake into account important contextual dimensions of the problem abandoned by the former. Their inchoate condition seems not alarming, since they are of surprisingly recent date. This, however, could very well be an accidental explanation. That is, the true reason for it might sooner or later turn out to be (...) bearing exactly on the fundamental human limitations, when it comes to theorizing, that these approaches are urging us to appreciate. (shrink)

Context: As one of the major approaches within the philosophy of mathematics, constructivism is to be contrasted with realist approaches such as Platonism in that it takes human mental activity as the basis of mathematical content. Problem: Mathematical constructivism is mostly identified as one of the so-called foundationalist accounts internal to mathematics. Other perspectives are possible, however. Results: The notion of “meaning finitism” is exploited to tie together internal and external directions within mathematical constructivism. The various contributions to this issue (...) support our case in different ways. Constructivist content: Further contributions from a multitude of constructivist directions are needed for the puzzle of an integrative, overarching theory of mathematical practice to be solved. (shrink)

Understanding is a valued trait in any epistemic practice, scientific or not. Yet, when it comes to characterizing its nature, the notion has not received the philosophical attention it deserves. We have set ourselves three tasks in this paper. First, we defend the importance of this endeavor. Second, we consider and criticize a number of proposals to this effect. Third, we defend an alternative account, focusing on abilities as the proper mark of understanding.

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.

This title offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

This paper defends a pragmatical approach to vagueness. The vagueness-adaptive logic VAL is a good reconstruction of and an excellent, instrument for human reasoning processes in which vague predicates are involved. Apart from its proof-theory and semantics, a Sorites-treating model based on it is presented, disarming the paradox. The paper opens perspectives with respect to the construction of theories by means of vague predicates.

This volume focuses on the importance of historical enquiry for the appreciation of philosophical problems concerning mathematics.

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

The paper gives an impression of the multi-dimensionality of mathematics as a human activity. This 'phenomenological' exercise is performed within an analytic framework that is both an expansion and a refinement of the one proposed by Kitcher. Such a particular tool enables one to retain an integrated picture while nevertheless welcoming an ample diversity of perspectives on mathematical practices, that is, from different disciplines, with different scopes, and at different levels. Its functioning is clarified by fitting in illustrations based on (...) actual research. (shrink)

Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of (...) explicit and implicit, formal and informal background knowledge. (shrink)

Philosophy of mathematics today has transformed into a very complex network of diverse ideas, viewpoints, and theories. Sometimes the emphasis is on the "classical" foundational work (often connected with the use of formal logical methods), sometimes on the sociological dimension of the mathematical research community and the "products" it produces, then again on the education of future mathematicians and the problem of how knowledge is or should be transmitted from one generation to the next. The editors of this book felt (...) the urge, first of all, to bring together the widest variety of authors from these different domains and, secondly, to show that this diversity does not exclude a sufficient number of common elements to be present. In the eyes of the editors, this book will be considered a success if it can convince its readers of the following: that it is warranted to dream of a realistic and full-fledged theory of mathematical practices, in the plural. If such a theory is possible, it would mean that a number of presently existing fierce oppositions between philosophers, sociologists, educators, and other parties involved, are in fact illusory. (shrink)

We can understand and act upon the beliefs of other people, even when these conflict with our own beliefs. Children’s development of this ability, known as Theory of Mind, typically happens around age 4. Research using a looking-time paradigm, however, established that toddlers at the age of 15 months old pass a non-verbal false-belief task (Onishi and Baillargeon in Science 308:255–258, 2005). This is well before the age at which children pass any of the verbal false-belief tasks. In this study (...) we present a more complex case of false-belief reasoning with older children. We tested second-order reasoning, probing children’s ability to handle the belief of one person about the belief of another person. We find just the opposite: 7-year-olds pass a verbal false-belief reasoning task, but fail on an equally complex low-verbal task. This finding suggests that language supports explicit reasoning about beliefs, perhaps by facilitating the cognitive system to keep track of beliefs attributed by people to other people. (shrink)

The epistemologists Biro and Siegel have raised two objections against the pragma-dialectical approach to argumentation. According to the first objection the pragma-dialectical theory is not genuinely normative. According to the second objection the rejection of justificationism by pragma-dialecticians is unwarranted: they reject justificationism prematurely and they are not consistent in accepting some arguments (‘justifications’) as sound. The first objection is based on what we regard as the misconception that the goal of resolving differences of opinion cannot provide a normative approach. (...) In response to the second objection we argue that in pragma-dialectics, the notion of argument, and related notions, are defined in a non-justificatory manner. (shrink)

The domain of a quantifier is determined by a variety of factors, which broadly speaking fall into two types. On the one hand, the context of utterance plays a role: if the focus of attention is on a particular collection of kangaroos, for example, then “Q kangaroos” is likely to range over the individuals in that set. On the other hand, the utterance itself will help to establish the quantificational domain, inter alia through presuppositions triggered within the sentence. In this (...) paper, we concentrate our attention on constructions like the following, in which “the square to which... ” is the critical presupposition trigger: Q circles... Q of these circles... Q of these five circles...... have the same colour as the square to which they are connected. Many theories predict that all instances of these schemata will give rise to the presupposition that every circle is connected to a square. We present an analysis which predicts that these sentences should generally be accepted in a context in which not all the circles are connected to a square, with one exception only: if a quantified sentence is of type and Q is non-intersective, then the sentence should be more likely to be rejected. Furthermore, we predict that manipulating the context so as to make the connected circles more or less salient should have an effect on statements with non-intersective quantifiers only. These predictions were tested in a series of experiments. (shrink)

Responding to Open Philosophy’s call ‘Does public art have to be bad art?’, in this paper we argue that this discussion should pay attention to the consequences of structural transformations that guide the production and presentation of public art in today’s increasingly private city. While entrepreneurial governance and corporate branding strategies generate new opportunities, they might also result in increased risk averseness and control over the content of public art, thus putting its critical potential at risk. That observation ushers in (...) urgent questions about control, complicity and criticality. We aim to reflect on those questions through two public art projects in Hong Kong: Antony Gormley’s Event Horizon (2015) and Our 60-second friendship begins now (2016) by Sampson Wong Yu-hin and Jason Lam Chi-fai. After drawing conclusions on the justification of public funding for co-productions, the legitimacy for artists to sometimes not ‘follow the rules’, and the problematic nature of a narrow definition of professionalism as a means to discredit artists, our analysis underlines the urgent need to develop a framework that can guide discussions on the consequences of control and complicity for the critical potential of public art. (shrink)

This study aims to increase insight into the uses of experts’ references to physically absent technology users in government-funded plant science. A discursive psychological analysis of expert board meetings shows that experts invoke various forms of reported dialogue/thoughts and dispositional statements when problems with technology and with program funding are discussed. Forms of reported dialogue serve to demonstrate that experts engage in dialogue with users, understand and are reasonable about users’ concerns, and that the content of user concerns does not (...) agree with expert views. Dispositional statements allow users’ feelings rather than users’ knowledge to be acknowledged as relevant. By establishing that user concerns contrast with expert concerns in type and content and by not discussing how users’ feelings may be incorporated into technology, experts shelve user concerns. This practice may hinder the development of user-friendly technologies. (shrink)

This volume comprises a selection of contributions to the theorizing about argumentation that have been presented at the 9th conference of the International Society for the Study of Argumentation, held in Amsterdam in July 2018. The chapters included provide a general theoretical perspective on central topics in argumentation theory, such as argument schemes and the fallacies. Some contributions concentrate on the treatment of the concept of conductive argument. Other contributions are dedicated to specific issues such as the justification of questions, (...) the occurrence of mining relations, the role of exclamatives, argumentative abduction, eudaimonistic argumentation and a typology of logical ways to counter an argument. In a number of cases the theoretical problems addressed are related to a specific type of context, such as the burden of proof in philosophical argumentation, the charge of committing a genetic fallacy in strategic manoeuvring in philosophy, the necessity of community argument, and connection adequacy for arguments with institutional warrants. The volume offers a great deal of diversity in its breadth of coverage of argumentation theory and wide geographic representation from North and South America to Europe and China. (shrink)

This edited volume covers new ground by bringing together perspectives from symbolic legislation theory on the one hand, and from biolaw and bioethics on the other hand. Symbolic legislation has a bad name. It usually refers to instances of legislation which are ineffective and that serve other political and social goals than the goals officially stated. Recently, a more positive notion of symbolic legislation has emerged in legislative theory. From this perspective, symbolic legislation is regarded as a positive alternative to (...) the more traditional, top-down legislative approach. The legislature no longer merely issues commands backed up with severe sanctions, as in instrumental legislation. Instead, lawmakers provide open and aspirational norms that are meant to change behavior not by means of threat, but indirectly, through debate and social interaction. Since the 1990s, biomedical developments have revived discussions on symbolic legislation. One of the reasons is that biomedical legislation touches on deep-rooted, symbolic-cultural representations of the biological aspects of human life. Moreover, as it is often impossible to reach consensus on these controversial questions, legislators have sought alternative ways to develop legal frameworks. Consequently, communicative and interactive approaches to legislation are prominent within the governance of medical biotechnology. The symbolic dimensions of biolaw are often overlooked. Yet, it is clear that the symbolic is at the heart of many legal-political debates on bioethical questions. Since the rise of biomedical technologies, human body materials have acquired a scientific, medical and even commercial value. These new approaches, which radically question existing legal symbolizations of the human body, raise the question whether and how the law should continue to reflect symbolic values and meanings. Moreover, how can we decide what these symbolic values are, given the fact that we live in a pluralistic society? (shrink)

How do Dutch people let each other know that they disagree? What do they say when they want to resolve their difference of opinion by way of an argumentative discussion? In what way do they convey that they are convinced by each other’s argumentation? How do they criticize each other’s argumentative moves? Which words and expressions do they use in these endeavors? By answering these questions this short essay provides a brief inventory of the language of argumentation in Dutch.

The study of argumentation is prospering. After its brilliant start in Antiquity, highlighted in the classical works of Aristotle, after an alternation of ups and downs during the following millennia, in the post-Renaissance period its gradual decline set in. Revitalization took place only after Toulmin and Perelman published in the same year their landmark works The Uses of Argument and La nouvelle rhétorique. The model of argumentation presented by Toulmin and Perelman’s inventory of argumentation techniques inspired a great many scholars (...) in various ways to take up the study of argumentation in a serious manner. Nowadays there are well-established logical approaches to argumentation, but also social and socio-psychological, linguistic, juridical and other approaches. In most of these approaches traces can be found of the influence of the classical and neo-classical argumentation theories just mentioned. (shrink)

New Perspective scholars challenge Protestant interpretations of Paul. It used to be the case, they state, that Protestants assumed that Paul was to Judaism as Luther was to Medieval Catholicism. Both men supposedly reacted against legalistic religions and championed grace-based faiths. However, in 1977, E.P. Sanders wrote Paul and Palestinian Judaism, arguing that Judaism is not a legalistic but a grace-based faith. Assuming that Sanders is correct, New Perspectivists claim that Paul’s and Luther’s theologies and experiences were thus not parallel. (...) Hence, Luther misunderstood Paul. Additionally, New Perspectivists challenge Protestant understandings of ‘justification’. In New Perspective thought, Paul uses the term ‘justification’ primarily to describe how people, particularly Gentiles, join the church Christians without following Jewish ritual laws. ‘Justification’, then, does not describe how people ‘stay in’ the covenant and receive salvation, as Protestants think. However, this article maintains that while New Perspectivists have some knowledge of Paul and Judaism, they are much less knowledgeable regarding Luther, Medieval Catholicism and Luther’s reaction to it. Greater scrutiny of these latter areas reveals large difficulties with New Perspective arguments. In addition, a review of relevant passages from Paul’s letters demonstrates that Protestants have not misunderstood Paul’s use of the term ‘justification’. Many Pauline passages show that when Paul discusses justification he is also thinking about ‘staying in’, not just ‘getting in’ the covenant. (shrink)

The intention of this paper is to relate the various standpoints regarding anxiety and uncertainty. Within the humanities and social sciences, research is pursued in many different disciplines without much interaction between them. Everyone's thinking is based on concepts which are domain-specific, and the distinctions, methods and arguments used are the ones that are generally accepted within the discipline. The divergent conclusions constitute pieces of a puzzle that are seldom if ever put together. There are even doubts about whether such (...) a puzzle can be put together. It can hardly be denied, however, that inspiration is to be found in a consideration of the views and findings of colleagues from other disciplines who are working on the same problems. Now that we have as our guests two prominent academics, each with specific views about our subject, we should take this opportunity to discuss together some of the sociological, anthropological, economic, philosophical and theological approaches to anxiety and uncertainty.If a number of well-known authors are to be believed, lack of trust, uncertainty and insecurity affect the most affluent and modern societies. In French one speaks of précarité , in German of Unsicherheit and Risikogesellschaft , and in English of “fear”, “risk” and “insecurity” . All modern Western societies seem to be afflicted with the same dismal and frustrating human experience.Belgian society is no exception to this. In recent years we have regularly been subjected to the emotional reactions provoked by sensational events in the areas of politics , justice and public health . Less spectacular but no less real were the results of opinion polls which showed that confidence in public institutions in general, and politics in particular, is at an all-time low. (shrink)

In the early months of 2020, the deadly Covid-19 disease spread rapidly around the world. In response, national and regional governments implemented a range of emergency lockdown measures, curtailing citizens’ movements and greatly limiting economic activity. More recently, as restrictions begin to be loosened or lifted entirely, the use of so-called contact tracing apps has figured prominently in many jurisdictions’ plans to reopen society. Critics have questioned the utility of such technologies on a number of fronts, both practical and ethical. (...) However, little has been said about the ways in which the normative design choices of app developers, and the products that result therefrom, might contribute to ethical reflection and wider political debate. Drawing from scholarship in critical design and human–computer interaction, this paper examines the development of a QR code-based tracking app called Zwaai (‘Wave’ in Dutch), where its designers explicitly positioned the app as an alternative to the predominant Bluetooth and GPS-based approaches. Through analyzing these designers’ choices, this paper argues that QR code infrastructures can work to surface a set ofethical–political seams, two of which are discussed here—responsibilizationandnetworked (im)permanence—that more ‘seamless’ protocols like Bluetooth actively aim to bypass, and which may go otherwise unnoticed by existing ethical frameworks. (shrink)

In this article we examine the effectiveness of consent in data protection legislation. We argue that the current legal framework for consent, which has its basis in the idea of autonomous authorisation, does not work in practice. In practice the legal requirements for consent lead to ‘consent desensitisation’, undermining privacy protection and trust in data processing. In particular we argue that stricter legal requirements for giving and obtaining consent as proposed in the European Data protection regulation will further weaken the (...) effectiveness of the consent mechanism. Building on Miller and Wertheimer’s ‘Fair Transaction’ model of consent we will examine alternatives to explicit consent. (shrink)

The recent chiming of Simone de Beauvoir's legacy by French feminists for a policy of assimilation of Muslim women to Western models of self and society reduces the complexity and richness of Beauvoir's views in inacceptable ways. This article explores to what extent a politics of difference that challenges the ideals and political strategies of abstract liberalism can be extracted from and legitimized by the philosophies of Simone de Beauvoir and Jean-Paul Sartre. Without assuming their thought is identical, we can (...) read them as elucidating each other and as implicitly exposing weak and strong points in their respective philosophies on ethnocultural rehtions and social identities. (shrink)