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In this paper we show how simple text-driven variations of given statements in mathematics can lead to interesting new problems and push forward a whole theory around simple initial questions. We exemplify this in two cases. Case 1 deals with problem-posing activities suitable for pupils and case 2 is a rational reconstruction of the organisation of mathematical knowledge within problems of graph colorings. Mathematicians learn to systematically look for subsequent problems around a given problem. We argue that this toy-model captures a nontrivial part of professional mathematical research within the pure fields and conjecture that it even grasps high level developments in mathematics. By doing this, we implicitly encourage a very simplistic view on criteria, so to speak a “cowpath” approach to progress in mathematics. The term “cowpath” is borrowed from architecture and software design, where it is commonly used. While we can contemplate which pathways are ideal, we may also just plant grass and see where people choose to walk. Those pathways are also self-enforcing, since we are less hesitant to walk on those rather than criss-cross the landscape.

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.

Our proposal of this chapter is to analyze aspects of indigenous school education as organized by Brazilian public policy and implemented as a result of struggles, organizations and mobilizations of social movements and indigenous communities to having their civil rights recognized and implemented. We will use the term indigenous peoples, which contains a plural notion of collective identity and ethnic diversity. The use of this term is not to homogenize nor relativize all subjects and groups to the same dimension, as it is impossible to include all indigenous peoples in one single category, due to the pluricultural existence that different peoples have (Maracci, 2012). Even recognizing that some authors (Minaya & Roque, 2015) from other parts of the world also use the term originary peoples, in this chapter we will use the terms indigenous peoples and indians, considering the construction of meaning for these words throughout the trajectories of these peoples' struggles in Brazil. In this chapter we first have a brief contextualization about the original inhabitants of Brazil and their trajectory to conquer the right of formal education which respects their history and cultures. Secondly, we present aspects of the trajectory of indigenous school education in Brazil based on a literature review, including an overview of contents and ideas related to official policy regulations concerning indigenous school education, and the principles of intercultural education. We also discuss some implications from the recent regulations of Brazilian curriculum (BNCC, 2018). Finally, we will provide an example of an empirical investigation on indigenous school education developed in the period 2014-2016. The study aimed to analyze the teaching of statistical topics in indigenous schools of the Xukuru of Ororubá people. The investigation was set up in collaboration with the local community, especially as to how the method was implemented in the school practice (Oliveira, 2016, Oliveira et al., 2018).

In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections.

The chapter aims to develop four lines of discussion on people's relationships with statistics. A first line problematizes historical aspects related to the origins of statistics and its relationship with people's lives. A second line discusses social aspects which enable people's empowerment through the use of statistics. In the third line we address elements of the cognitive, socio-cultural, and affective dimensions of critical issues associated with people interpreting statistical data. In the fourth line, we present three examples of educational projects that involve teachers, students, and communities with the aim of employing Civic Statistics to initiate social actions. These projects are related to environmental themes and use Civic Statistics as a tool to empower people to achieve cultural changes in their social contexts.

ArgumentLambert ten Kate (1674-1731), the scholar of language, religious writer, art theoretician and collector, and natural philosophy enthusiast, was part of an informal network of Amsterdam-based mathematics and natural philosophy enthusiasts who played a pivotal role in the early diffusion of Newton’s natural philosophical ideas in the Dutch Republic. Because Ten Kate contributed to several areas of research, it is worth asking whether connections can be found between his different scholarly activities and, more specifically, whether his oeuvre as a whole was shaped by his religious views, as has been suggested. In this essay, I shall argue that his oeuvre was indeed shaped by his religious beliefs, which reflect elements typical of Dutchdoperdom, but also reflect a more general Christian orientation that transcends confessional divides. More particularly, I aim to show that, if we want to understand why Ten Kate was drawn to the natural philosophy of his day, and especially to Isaac Newton’s (1642-1727), and why he sought to promote it, we also need to pay attention to this broader Christian orientation in his thought. Along the way, I shall add nuance to earlier characterizations of how ten Kate mobilized Isaac Newton’s natural philosophy according to his own agenda.

According to the naturalist Charles Bonnet (1720–1793), an “art of observing” was sorely needed to stimulate further progress in natural history. Although he never published on the subject, Bonnet proposed a prize competition on the “art of observing” to Hollandsche Maatschappij der Wetenschappen (the Dutch Society of Sciences) in Haarlem of which he was a member. Jean Senebier, a pastor and librarian who later became a skilled scientific observer in his own right, took part in this competition (1768) with an essay on the art of observing that influenced, embodied and codified the advanced scientific observational practices of the Genevan naturalists in the eighteenth century.

This paper investigates the impact of publication pressure on the ethics and the scientific integrity in the domain of mathematics and of the arts. Both research fields are specific in their methodology, being that they do not start from a classical hypothesis and researchers in these areas are not knowing what the outcome will be. The research design is open, and creativity is a main part of the research investigation. Both research fields do not rely on empirical cases nor on data collection or data handling. This could be a reason why mathematics and arts seem to be less subject to scientific misconduct. After presenting the (inter)national regulations on ethics and integrity we will investigate a philosophical analysis in which we consider possible influences from publication pressure that became widespread over all disciplines. We will clarify if and how mathematics and the arts are sensitive to scientific research misconduct or questionable research practices.

In the field of research integrity, there has been increasing interest in the influence of culture on research misconduct. Yet, there is a lack of empirical study on how differently researchers from distinct cultural backgrounds perceive research misconduct. Hence, we used the qualitative approach to investigate and compare perspectives concerning research misconduct between Chinese and Flemish researchers. By means of 45 interviews we explored their perspectives on types, determinants of, and solutions to research misconduct. Our findings show elements of scientific culture shared by both groups, but also clear dissimilarities. The Chinese participants more often mentioned plagiarism and inappropriate authorship as examples of research misconduct, while Flemish respondents brought up fabrication and falsification. Research misconduct was associated with diverse determinants, though the Chinese favored the “bad apple” theory (more intrinsic), while the Flemish pointed to the “publish or perish” pressure and attributed misconduct to situational aspects (rather extrinsic). Accordingly, they proposed diverse strategies to handle this issue, yet education and training were recommended by both groups. This study confirms that research misconduct is the result of many factors intertwining and interacting, and suggests that cultural traits contribute to our perceptions of research misconduct and subsequent bias in that context.

Research institutions are responsible for promoting research integrity and handling allegations of research misconduct. Due to various cultural and social contexts, institutional policies from different cultural backgrounds exhibit many differences, such as their primary concern and mechanisms for dealing with allegations of research misconduct. This comparative study analyses research misconduct policies from 21 Chinese and 22 European universities. The results show that definitions of research misconduct from all retrieved policies go beyond fabrication, falsification, and plagiarism but include different types of questionable research practices. Their procedures for handling research misconduct differ in, for example, confidentiality and disclosure of conflict of interest. Differences can also be found in their governance approaches (“bottom-up” versus “top-down”).

One of the central aims of the philosophical analysis of mathematical explanation is to determine how one can distinguish explanatory proofs from non-explanatory proofs. In this paper, I take a closer look at the current status of the debate, and what the challenges for the philosophical analysis of explanatory proofs are. In order to provide an answer to these challenges, I suggest we start from analysing the concept understanding. More precisely, I will defend four claims: (1) understanding is a condition for explanation, (2) unificatory understanding is a type of explanatory understanding, (3) unificatory understanding is valuable in mathematics, and (4) mathematical proofs can contribute to unificatory understanding. As a result, in a context where the epistemic aim is to unify mathematical results, I argue it is fruitful to make a distinction between proofs based on their explanatory value.

The Amsterdam-based self-educated polymath Lambert ten Kate (1674-1731) was well versed in different subjects. His contributions to the study of language and the new methods he employed to pursue it are the focus of this article. Ten Kate was well in tune with the natural philosophical developments of his day. Unsurprisingly, it has been argued that the methods he developed in the study of language were inspired by Isaac Newton’s (1643-1727) natural philosophical approach. I argue that a more nuanced understanding of the relation between Ten Kate’s methods in the study of language and Newton’s natural philosophical approach is called for. De Amsterdamse autodidact Lambert ten Kate (1674-1731) was bedreven in verschillende disciplines. In dit artikel staan zijn bijdragen aan de studie van de taal en de nieuwe methodes die hij daartoe ontwikkelde centraal. Ten Kate volgde de ontwikkelingen binnen de natuurfilosofie op de voet. In de literatuur wordt betoogd dat de methodes die hij aanwendde bij het bestuderen van de taal geïnspireerd waren door Isaac Newtons (1643-1727) natuurfilosofische aanpak. Ik argumenteer echter dat de relatie tussen de door Ten Kate gebruikte methodes bij de studie van de taal en de natuurfilosofische benadering van Newton genuanceerder ligt.

In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can know a mathematical proposition based solely on a probabilistic proof, one is then forced to admit that such an epistemic subject can know several lottery propositions based solely on probabilistic evidence. Insofar as knowledge of lottery propositions on the basis of probabilistic evidence alone is denied by the vast majority of epistemologists, it is concluded that this constitutes an epistemic reason for rejecting probabilistic proofs as a means of acquiring mathematical knowledge.

In what follows, the essentials of Willem Jacob ’s Gravesande’s (1688-1742) treatment of space and time, body, and force will be addressed, respectively.

In the age of the Anthropocene - an era in which technological intervention plays an ever more central role in the way we build, develop and attempt to maintain our increasingly fragile and risk-prone human and natural world, what are the implications of the hegemonic epistemic status of mathematics in those processes? Does mathematics define the conditions of possibility of all knowledge, whether expressed in a theory or silently invested in a practice? Does or can mathematics and its presumed value-neutrality serve to limit, constrain, suppress, and even preclude other, perhaps more valuable forms of knowledge? Alternatively, can philosophical dialogue about mathematics serve to clarify, unmask, reframe and recreate our understanding of mathematics and its symbolic power in the human and material world, and act as an emancipatory form of knowledge in culture and society? What would such dialogues look like in the mathematics classroom?

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.

The distinction between distributive and non-distributive profiles figures prominently in current evaluations of the ethical and epistemological risks that are associated with automated profiling practices. The diagnosis that non-distributive profiles may coincidentally situate an individual in the wrong category is often perceived as the central shortcoming of such profiles. According to this diagnosis, most risks can be retraced to the use of non-universal generalisations and various other statistical associations. This article develops a top-down analysis of non-distributive profiles in which this fallibility of non-distributive profiles is no longer central. Instead, it focuses on how profiling creates various asymmetries between an individual data-subject and a profiler. The emergence of informational, interest, and perspectival asymmetries between data-subject and profiler explains how non-distributive profiles weaken the epistemic position of a profiled individual. This alternative analysis provides a more balanced assessment of the epistemic risks associated with non-distributive profiles.

In this paper I propose to reflect on contemporary philosophy of mathematics. i argue that the impartiality to methodology that is present in the current practice of philosophy of mathematics is an intellectual vice of our research community because it leaves our philosophical methods to engage with mathematics implicit. This hides difficulties our research community faces, difficulties our students can reasonably expect to learn about and which we need to engage with to overcome them. to be virtuously open-minded about methodology is not to be impartial about methodology but rather to critically assess the proposals on offer in a public debate.

In this paper we argue that the choice of research methods reflects the theoretical framework even before these methods have been put to use in case studies. We understand the term ‘case study’ broadly in this paper and argue that neither thinking of them as cherry-picked cases to support preconceived ideas about mathematical practices nor thinking of them as inductive leaps from (too) few cases to general features is suitable. By realising the deep entanglement of our case studies with our theoretical framework we propose to view case studies as an invitation for critical reflection upon one’s own assumptions. We discuss an example taken from the philosophy of mathematical practices. The upshot is threefold: (1) we provide an argument that case study based research strategies can be successful; (2) we delineate how an awareness of the methodological difficulties of case study based research strategies can positively influence the way case studies are conducted; (3) we suggest that case studies are not dispassionate examinations that deliver cold facts.

The Polymath Project is an online collaborative enterprise that was initiated in 2009, when Timothy Gowers asked whether and how groups could work together to solve mathematical problems that “do not naturally split up into a vast number of subtasks.” Gowers proposed to answer this question himself by actually trying to set up such a collaboration, based on interactions taking place in the comment-threads of a series of posts on a WordPress blog. Hence, the first project officially started in early 2009, to be proclaimed successful only 6 weeks later (Gowers and Nielsen 2009). From its inception until April 2018, 15 more Polymath problems (and a handful of smaller or related ones) have been launched. These projects have attracted attention from different scholarly communities, including the philosophy of mathematical practices, from the perspective of which the Polymath Project can be seen as a vast repository of mathematics in action. This chapter continues previous work by its authors on the topic in question and topics related. More specifically, for the purposes of this volume, it is our aim to both summarize and expand upon these earlier contributions. The starting point is the above observation that in the past decade, the issue of “massively collaborative mathematics” (to be qualified below) has drawn quite some attention. However, does it also warrant intensive philosophical attention in particular? For, as exciting as these developments might be from mathematical and other points of view, the enhanced possibilities that have come with it do not per se give rise to any philosophical import. In Van Bendegem (2011), one of us has indeed tentatively argued for the philosophical relevance of these new dynamics of proof construction flowing from the wide availability and efficiency of Internet technology. We shall below briefly rehearse and update the argument given there.

This chapter surveys logical methods and their philosophical relevance

This document contains the appendix to a paper of ours that was published in 2021 in the European Journal of Science and Theology, vol. 17(3). It presents the manuscript material we have analysed in this paper and it also provides evidence for the relative datings we have offered there.

In this essay, we document the development of five important methodological passages in Isaac Newton’s (1642-1727) Queries to the Opticks and explore some of the reasons why he continuously revised and reformulated them. The Queries have a complex and fascinating editorial history, appearing in both English and Latin forms. Here we analyse the extant manuscript material in relation to the published versions of Opticks (1704, 1717, 1721, and 1730) and Optice (1706 and 1719) and, additionally, to other relevant sources such as Newton’s ‘an account of the book called Commercium epistolicum’ and Cambridge University Library, add. ms. 3970, f. 479rv and f. 480v in which he addressed a number of principles of philosophy. More specifically, we analyse five passages in which Newton clarified his methodological views to his readers: in the first passage, he demarcated “the main Business of natural Philosophy” as arguing from phenomena without feigning hypotheses; in the second, he justified his frequent use of the word “attraction”; in the third, he briefly discussed under which conditions a property, especially hardness, can be considered universal; in the fourth, he defended his own method by refuting the view that gravity and other active principles are “occult Qualities”; and, finally, in the fifth, he described his own method in terms of analysis and composition (or synthesis). Based on our analyses of these passages, we are able to trace Newton’s changing methodological ideas from the early 1700s to 1717. Furthermore, we uncover how the erstwhile natural philosophical debates informed his writing of the Queries and what passages acted as precursors to his famous regulae philosophandi.

How is science performative and how do the arts make science perform? Performance scholar Bleeker and mathematician and philosopher of science Van Bendegem explore various examples of scientific processes that expose performative mechanisms resulting from the entanglement of technological tools and human agency. A dynamic that characterizes scientific practices such as the hunt for the Higgs particle, but also the presentation of proof and artistic practices such as that of Iannis Xenakis and Annie Dorsen. Van Bendegem and Bleeker enter into dialogue starting from a series of questions, concerned with the performative dimension of the generation, transmission and notation of knowledge in scientific practices. Thus going back and forth between areas of research in philosophy, sciences and the humanities, a surprising reciprocity with regard to these issues emerges.

This article challenges the scholarly consensus on Willem Jacob 's Gravesande’s philosophy, namely the view that it was predominantly influenced by Newton, Locke, and Descartes. More generally, it opposes the tendency to read the work of eighteenth-century natural philosophers as being part of well-defined strands of Newtonianism. I argue that we will understand the intellectual history of the early eighteenth century better if we let go of essentialist taxonomic views and instead pay attention to the historical trajectories of various actors and ideas. In the case of 's Gravesande (1688–1742), it has often been assumed that he used a particular form of Newtonianism to combat rationalist metaphysics, particularly Spinozism. Here I show that this view is not tenable when we take account of 's Gravesande’s metaphysical writings. On topics such as liberty, necessity, and theodicy, these writings deviate from what Newtonianism is usually taken to stand for. Instead, 's Gravesande’s views on such matters correspond more closely to certain interpretations of Leibniz’s Théodicée.

John Theophilus Desaguliers (1683–1744) was a French-born English Huguenot who made his name as a public lecturer in London and a demonstrator at the Royal Society, writing a very popular introduction to Isaac Newton's natural philosophy, the two-volume A course of experimental philosophy (1734–1744). This paper looks at the influence of three French natural philosophers, Edme Mariotte (1620–1684), Antoine Parent (1666–1716) and Bernard Forest de Bélidor (1698–1761), on the account of waterwheel functioning in the second volume of that work. The aim of the paper is to show that, although Desaguliers demonstrated a commitment to Newton's work, his own natural philosophical objectives also led him to borrow ideas from natural philosophers outside Newton's direct sphere of influence. To do this I shall give an account of what Desaguliers appropriated from Newton's Principia , how it fitted in with his own project and how he also made use of other natural philosophers' theories in his discussion of fluid mechanics. This will hopefully result in a more nuanced conception of Desaguliers' ‘Newtonianism’ that takes into account the diverse sources and influences in his work.

The Amsterdam-based merchant and mathematics enthusiast Adriaen Verwer (1654/5–1717) was one of the few in the Dutch Republic to respond to the first edition of Newton's Principia (1687). Based on a close study of his published work, his correspondence with the Scottish mathematician and astronomer David Gregory (1659–1708), and his annotations in his own copy of the first edition of the Principia , I shall scrutinize the impact of Newton's ideas on Verwer's thinking. The proposed analysis, which will add nuance to earlier findings, also has broader implications for our understanding of the introduction of Newton's ideas in the Dutch Republic, as will be shown.

In the literature, there have been several methods and definitions for working out if two theories are "equivalent" (essentially the same) or not. In this article, we do something subtler. We provide means to measure distances (and explore connections) between formal theories. We define two main notions for such distances. A natural definition is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable definition is that of conceptual distance which measures the minimum number of concepts that separate two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. We also develop further notions of distance, and we include a number of suggestions for applying and extending our project. We end with a philosophical discussion of the significance of these approaches.

The camera obscura was a simple astronomical tool that evolved into an important optical instrument in the seventeenth century. During its development, the camera obscura generated various philosophical discussions about perception and image formation.

The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, we tested whether the processing of metric vs topological relations yielded the same hemispheric specialization as the processing of coordinate vs categorical relations. In the second part, we investigated the specific performance patterns for the processing of five pairs of exact/co-exact relations, where stimuli for the co-exact relations were divided into three categories depending on their distance from the exact case. Regarding the processing of metric vs topological relations, hemispheric differences were found for only a few of the stimuli used, which may indicate that other processing mechanisms might be at play. Regarding the processing of exact vs co-exact relations, results show that the level of agreement among participants in judging co-exact relations decreases with the distance from the exact case, and this for the five pairs of exact/co-exact relations tested. The philo- sophical implications of these empirical findings for the epistemological analysis of Euclid's diagram-based geometric practice are spelled out and discussed.

This paper discusses the concept of probability literacy in teacher education, based on an empirical study with in-service teachers that relies on a theoretical framework which analysed the concept of probability related to education. The study is also based on a theorical probability literacy model comprised of two types of elements: cognitive and dispositional components, which can enable people to understand statistical concepts. The mathematics teachers were able to realise the importance of probability literacy in decision making and for a critical citizenship. The results suggested that an important strategy on these teacher education meetings was to raise different arguments and questioning, problematizing the teachers’ answers. The analysis of data suggested that the questioning approach with teachers may provide their reflections about concepts of probability and how to teach critically these concepts

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to in- tertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we intro- duce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson generalization in the case of disjoint languages. We show that the Andréka–Németi generaliza- tion is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize themfor theories in non-disjoint languages.

China has achieved a dramatic development in scientific research over the last few decades. However, just like many other countries, it has also seen a surge of scientific misconducts. With its expansion of international publications, retractions due to suspected research misconduct are also on the rise. A transcultural case study was conducted by investigating the perception of research misbehaviors by Chinese researchers compared to their Belgian Flemish colleagues. The study was designed to find out variation in research practices in different countries and to see how research was shaped and influenced by cultural contexts. An online questionnaire was sent to 3,236 researchers by e-mails and it received a response rate of 13.09%. They were asked to score 32 research misbehaviors on a 5-point scale. The findings suggested that compared to Flemish respondents, the Chinese had a significantly higher acceptance toward most research misbehaviors. To be more specific, the Chinese respondents felt less unacceptable toward behaviors violating such values as honesty, fairness, and verifiability than the Flemish, while their perceptions of behaviors that violated such values as responsibility, objectivity, and truth were not different compared to the Flemish. This case study implies that the Chinese research community is in an urgent need of training in responsible conduct of research and strong as well as sound guidelines for responsible research practices in place.

The deviation ofmathematical proof—proof in mathematical practice—from the ideal offormal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inferenceandlogical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Suchdifferentiating claimsare, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of themeaningof the differentiating claims—through the properties that occur in them—as well as thereasonsthat support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties offormality,generality, andmechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain.

This study investigates perceptions of European and Chinese stakeholders on doctoral education (DE) in China and Europe, particularly the cooperation between the two sides. Data were collected through online and paper survey from both European and Chinese stakeholders (N = 946). The results provide insights for policy-makers, university administrators, doctoral students and their supervisors when planning and engaging in DE in collaboration between European and Chinese higher education institutions (HEIs). Various cooperation models are currently implemented between Chinese and European HEIs. European respondents reported a higher awareness level of cooperation issues compared to the Chinese respondents. Significant differences were found between the Chinese and European respondents regarding their perceptions, perceived advantages and challenges, as well as expectations on cooperation in DE.

Could groups ever be an understanding subject (an epistemic agent ascribed with understanding) or should we keep our focus exclusively on the individuals that make up the group? The way this paper will shape an answer to this question is by starting from a case we are most willing to accept as group understanding, then mark out the crucial differences with an unconvincing case, and, ultimately, explain why these differences matter. In order to concoct the cases, however, we need to elucidate what it means to be attributed with understanding and what makes up an epistemic agent. I shall argue that it is abilities, above all, that guide our attributions of understanding. While it is true that understanding must go beyond single acts, this is not going beyond as in going behind them (to private occurrences which are impossible to discern and don’t themselves contribute anything of value), but beyond as in considering what people could and would do: their discernable and valuable abilities. To conceptualize whom the abilities belong to, I will specify what it means to be an epistemic agent. I shall argue that it is being a successful target of the epistemic stance. The epistemic stance (heavily inspired by Dennett’s intentional stance) forms an instrumental abstraction by attributing epistemic properties (i.e., beliefs, epistemic aims, problem-solving tactics) to an entity to explain or predict its behavior. Macro-systematicity (a higher-level pattern which a theory can exploit) is what makes the epistemic stance’s abstraction an explanatory powerful one (regardless of how that macro-systematicity is realized) and emergence (the lack of a straightforward mapping-relation between the micro and the macro) is what makes its power unique to a particular level of explanation. To end, I suggest two kinds of mapping-relations and walk through the shift from reducibility to emergence.

The Western history of aesthetics is characterised by tension between theory and practice. Musicians listen, play, and then listen more profoundly in order to play differently, adapt the body, and sense the environment. They become deeply involved in the sensorial qualities of music practice. Artistic practice refers to the original meaning of aesthetics―the senses. Whereas Baumgarten and Goethe explored the relationship between sensibility and reason, sensation and thinking, later philosophers of aesthetics deemed the sensorial to be confused and unreliable and instead prioritised a cognitive or objective approach. Written by authors from the fields of philosophy, composition, performance, and artistic practice, Sensorial Aesthetics in Music Practices repositions aesthetics as a domain of the sensible and explores the interaction between artists, life, and environment. Aesthetics becomes a field of sensorial and embodied experience involving temporal and spatial influences, implicit knowledge, and human characteristics.

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

The mathematician, physicist and philosopher W. J. 's Gravesande is particularly known for his adherence to ‘Newtonian philosophy’. Currently, it is widely held that 's Gravesande got his main inspiration for his scholarly calling from Newton himself, whom he met in 1715 during a first career as a lawyer; and that it was mainly Newton's own intervention that ensured the appointment of the unqualified 's Gravesande at Leiden University. I challenge these views by bringing together all currently known information about 's Gravesande, including a number of as yet unused documents. I show that 's Gravesande's appointment resulted from a very carefully built up reputation in scholarly circles rather than from accidental meetings and patronage. 's Gravesande had written several innovative papers and was in contact with both leading mathematicians and local political and patrician figures already before 1715. This article therefore explains the rationale behind his appointment in Leiden.

This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails.

The aim of this paper is to present an in-depth inquiry into one of the most disregarded dimensions of Fichte's philosophy, i.e., the systematic place of embodiment in his transcendental epistemology. Highlighting the necessarily embodied nature of the constitution of the notion of thinghood or being in Fichte's philosophy could not only help pave the way for a more elegant understanding of the relation between idealism's and phenomenology's subject views, it likewise enables a more comprehensive insight into Fichte's much debated theory of subjectivity. Furthermore, Fichte's transcendental account of the body provides one with a new vantage point from which to consider some classical interpretive issues, most notably those pertaining to Fichte's peculiar methodology, his Ideal-Realism, and his take on the problem of explanatory circularity in trying to tackle the problem of the genesis of (bodily) self-consciousness.