Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical practice. Hosted by (...) the Rheinische Friedrich- Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008. (shrink)

The focus on discourse and communication in the recent sociology of scientific knowledge offers new perspectives for an integration of qualitative and quantitative approaches in science studies. The common point of interest is the question of how reflexive communication systems communicate. The elaboration of the mathematical theory of communication into a theory of potentially self-organizing entropical systems enables us to distinguish the various layers of communication, and to specify the dynamic changes in these configurations over time. For example, (...) a paradigmatic discourse can be considered as a virtual communication system at the supra-individual level. Communication systems, however, cannot be directly observed. One observes only their instantaneous operations. The reflexive analyst is able to attribute the observed uncertainty to hypothesized systems that interact in the events. The implications of this perspective for various programmes in the sociology of scientific knowledge are discussed. (shrink)

Antony Flew's ?A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365?78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But (...) this analysis only succeeds if the rejection of an explanatory theory entails that there are reasonable grounds for the rejection of the set of propositions which that theory was intended to explain. I argue against such an entailment, and thus against Bloor's attempt to relativize mathematical knowledge. (shrink)

This paper considers whether philosophy of mathematics could benefit by the introduction of some sociology. It begins by considering Lakatos's arguments that philosophy of science should be kept free of any sociology. An attempt is made to criticize these arguments, and then a positive argument is given for introducing a sociological dimension into the philosophy of mathematics. This argument is illustrated by considering Brouwer's account of numbers as mental constructions. The paper concludes with a critical discussion of Azzouni's (...) view that mathematics differs fundamentally from other social practices. (shrink)

Affect Control Theory was developed to address some issues in role theory. However, a mathematical formulation allowed the theory to expand rapidly to a variety of substantive issues, such as labeling, attributions, emotions, and the impact of settings on social interaction. Formalization raised theoretical issues that might have been neglected otherwise, and helped in defining the boundaries of the theory. A reasonable lesson to draw from development of affect control theory is that even modest mathematical analyses can expand (...) theoretical productivity. (shrink)

Sociology originated in the mid-nineteenth century from a new confidence in the power of science to explain the world on a mathematical foundation. Both mathematics and sociology transformed over the ensuing century, inverting the hierarchical relationship from sociology as a mathematics-based science of complex human configurations to mathematics as a complex science based on social institutions. That is, where sociology began as the hard case for mathematics, it became possible to see mathematics as the hard (...) case for sociology. In this light, we examine a number of motivations and provocations for the sociology of mathematics, from language and social cognition to Marxist and materialist skepticism of ideology, and sample the evidence and arguments brought to bear on the sociology of mathematics at a range of scales. (shrink)

The idea that sociology has the status of a strict science—that is, that sociology, like mathematics, has at its disposal a well-founded, deductive system of propositions—is nowadays rejected even more by its pragmatic advocates than by its skeptical practitioners; it is refuted both by the arbitrary manipulation of sociology’s internally constitutive, theoretical interconnections at the hands of practical interests and technocratic utility, and by the resultant increasing relativization of its findings. However, as we shall see, the arbitrariness (...) of the treatment of sociology does not correspond with any arbitrariness in its object. On the contrary, with all empirical relativity a structural constancy persists in the agency constructing, comparing, and evaluating the experience of the everyday world, as well as scientific experience. This was precisely Kant’s discovery, that the conditions making possible our primary experience of objects must be, at the same time, the constitutive structure of the objects themselves, and also the framework for the scientifically systematic treatment of them, because otherwise there could be no knowledge in the sense of the structural identification of object, experience, and reflection. This Kantian principle guarantees, then, that sociology is indeed possible as a strict science; that is, it is completely possible as an integral, deductive aspect of theory, as long as we reconstruct, by means of axioms and postulates, the constitution of the object of theory out of the structure of our direct interpersonal experience. In this vein, Rene König, a pioneer in modern German social science, states. (shrink)

As sociologists deepen their examinations of human rights in their teaching, research, and thinking, it is essential that such work is conducted in a manner that is both mindful and critical of the knowledge we are building upon in sociology and human rights. As the authors of this volume reveal, creating sociological knowledge that examines human rights for the expansion of human rights is something that sociologists are well equipped to undertake, whether through the use of mathematics, comparative-historical analysis, (...) the study of emotions, conversations, or social psychology. In these chapters you will find the roots of the study of human rights deep within sociological research and thinking as well as emerging techniques that will push the discipline as it seeks to expand understanding of human rights together with so many other aspects of the social condition. (shrink)

Records of online collaborative mathematical activity provide us with a novel, rich, searchable, accessible and sizeable source of data for empirical investigations into mathematical practice. In this paper we discuss how the resources of crowdsourced mathematics can be used to help formulate and answer questions about mathematical practice, and what their limitations might be. We describe quantitative approaches to studying crowdsourced mathematics, reviewing work from cognitive history (comparing individual and collaborative proofs); social psychology (on the prospects for (...) a measure of collective intelligence); human–computer interaction (on the factors that led to the success of one such project); network analysis (on the differences between collaborations on open research problems and known-but-hard problems); and argumentation theory (on modelling the argument structures of online collaborations). We also give an overview of qualitative approaches, reviewing work from empirical philosophy (on explanation in crowdsourced mathematics); sociology of scientific knowledge (on conventions and conversations in online mathematics); and ethnography (on contrasting conceptions of collaboration). We suggest how these diverse methods can be applied to crowdsourced mathematics and when each might be appropriate. (shrink)

The fact that sociology was born during the period of the Industrial Revolution does not authorize us to consider its discourse as lacking in philosophical elements that are rooted in a previous age. Neither can we consider as fully accomplished its role for modernity, nonetheless today, in an after-modern climate (in the sense of Donati 2009), sociology is trying to escape the prejudice of modern ethics to go beyond the clichés of postmodernity (Ardigò 1989). Filled with self-reflexivity and (...) reductionist dichotomies, the twenty-first-century sociologist feels the need to ?own factual reality again? and to rediscover ?a new metaphysics of the social world? (Donati 1993). If self-consciousness is in the world, sociology, perhaps, has to go beyond science and turn into ?globology? (Arnason 1990), or into a sociology on a global scale, which looks at how world unification has occurred. In order to accomplish this, it has to be careful about what it was able to do best in the past: ?to foresee and to enhance sustainable change,? to be aware of the ?relational connections,? which no mathematics will ever be able to show, to build new ?memes,? and to decide to accelerate or to go against the phenomena it encounters in its observation. Society in the twenty-first century will go beyond postmodern stagnation and turn into something new (After-modernity? Hyper-modernity? Trans-modernity?) if it is to be helped by the interpretations of sociology. Notwithstanding the endeavors to change, most Westernized countries are trapped in the lib-lab model, while China argues for a complete reconfiguration of the concepts of public and private, states and market, freedom and controls, copyright and copyleft. What is going to happen in the future? Are we going to fall into a technocratic and authoritarian form of neo-modernization? Are we going to rediscover the system of exchanging gifts? Are we going to create a fully ?relational? society, going beyond the Hegelian categories of right and left? It will be the role of a ?strong and relational? sociology to identify all the ?viable? scenarios and to prepare its advent in symbolic terms. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also (...) concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth. (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)

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but (...) have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

Contemporary sociology of scientific knowledge (SSK) is defined by its relativist trend. Its programme often calls for the support of philosophers, such as Duhem, Quine, and Wittgenstein. A critical re-reading of key texts shows that the main principles of relativism are only derivable with difficulty. The thesis of the underdetermination of theory doesn't forbid that Duhem, in many places, validates a correspondence-consistency theory of truth. He never said that social beliefs and interests fill the lack of underdetermination. Quine's idea (...) of a selective revision of hypotheses, as well as the neat incompatibility between holism and conventionalism, openly challenges the principles of relativism. Wittgenstein's work, which is not presented in book-form but rather as a tree, forces us to avoid aphoristic choices that credit any text-excising. This remark allows us to tackle the passages that sociological relativism is based on.Mathematical conventions are not anthropological objects. When Wittgenstein examines the "language-games," he only speaks of the functioning of natural language, not to be confused with scientific formal languages. We then should render the formula "language-game" by "well-defined, explicit and compulsory rules of communication", which is a much less attractive formula for relativism. Consequently, there does not exist a real continuity between the epistemologies of Duhem, Quine and Wittgenstein, and the recent works of the SSK. (shrink)

Luhmann’s theory has been commonly considered as a radical overcoming of the traditional philosophy. The interpreters often refer to the non-ontological background of the theory as the criticism of the conscience's centrality, the emphasis in conflict and distinction and the influence of sciences as cybernetic, biology and mathematics. In the present paper we try to demonstrate that there is also an important philosophical heritage in the Luhmann’s sociological work: the Hegelian heritage. We refer to four main points: the congruence of (...) being and thought; the being as the unity of unity/distinction; the rationality as auto-reference and the congruence between theory and society. The focus of our analysis is on the complications of the system theory when it tries to eliminate the idea of spirit. La teoría de Luhmann ha sido considerada comúnmente como una radical superación de la filosofía tradicional. Los interpretes se refieren frecuentemente a los antecedentes no ontológicos de esta, como la crítica a la centralidad de la conciencia, el énfasis en el conflicto y la distinción y la influencia de ciencias como la cibernética, la biología y las matemáticas. En el presente artículo se intenta demostrar que existe también otra importante herencia en la obra sociológica de Luhmann: la herencia hegeliana. Los principales puntos analizados son: la congruencia entre ser y pensamiento; el ser como unidad de unidad/distinción; la racionalidad como auto-referencia y la congruencia entre teoría y sociedad. El enfoque de nuestro análisis radica en las dificultades de la teoría de sistemas al intentar eliminar la idea de espíritu. (shrink)

In the present paper, I go beyond these examples by bringing into play an example that I nd more experimental in nature, namely that of the use of the so-called PSLQ algorithm in researching integer relations between numerical constants. It is the purpose of this paper to combine a historical presentation with a preliminary exploration of some philosophical aspects of the notion of experiment in experimental mathematics. This dual goal will be sought by analysing these aspects as they are presented (...) by some of the protagonists of the eld and discussing them using notions from contemporary philosophy of science. (shrink)

Responding to widespread interest within cultural studies and social inquiry, this book addresses the question 'what is a mathematical concept?' using a variety of vanguard theories in the humanities and posthumanities. Tapping historical, philosophical, sociological and psychological perspectives, each chapter explores the question of how mathematics comes to matter. Of interest to scholars across the usual disciplinary divides, this book tracks mathematics as a cultural activity, drawing connections with empirical practice. Unlike other books in this area, it is highly (...) interdisciplinary, devoted to exploring the ontology of mathematics as it plays out in different contexts. This book will appeal to scholars who are interested in particular mathematical habits - creative diagramming, structural mappings, material agency, interdisciplinary coverings - that shed light on both mathematics and other disciplines. Chapters are also relevant to social sciences and humanities scholars, as each offers philosophical insight into mathematics and how we might live mathematically. (shrink)

During the second half of the 20th century, attempts were made to operationally redefine various social activities, including those related to science, the military, administration and industry. These attempts were aided by scientific and technical innovations developed in the Second World War, and subsequently by the increase in use of automation in various domains. This Ph.D. thesis addresses these attempts from a sociohistorical perspective, focusing on the specific case of archaeology. During this period, the domain of archaeology underwent a process (...) of disciplinarisation and professionalisation. The same occurred in applied mathematics and then computer science: this thesis focuses on the relationships between these three domains. In France, during the 1950's and 1960's, there were significant methodological and conceptual innovations. Their subsequent scientific recognition, was, however, relatively minor. In archaeology, innovations related to applied mathematics and automatics did not lead to the emergence of an archaeological speciality based on computation. This situation was in striking contrast to what happened in other scientific domains and in archaeology in other countries, where new theoretical and methodological Anglophone definitions in ‘New Archaeology’ were spreading worldwide. This thesis explores three collective attempts to redefine the conceptual and methodological basis of archaeology, led by Georges Laplace, Jean-Claude Gardin and Jean Lesage, across France, Spain and Italy. These cases are completed by other people who had significant careers in both engineering and archaeology. In general, this thesis studies a scientific activity by investigating the cognitive and social aspects of peoples’ methodological contributions. Three models of the relationships between experts in a scientific domain and experts in an applied science (here mathematics and computing) are empirically identified and described. The effects of introducing mathematical and automation procedures on the division of labour and the distribution of recognition are analysed. The success or failure of the methodological propositions are discussed with reference to several factors and models of scientific innovation. This thesis generates new information on the development of rescue and preventive archaeology and on the use of digital technologies in human sciences. The analysis draws on 82 interviews, 23 archives and several bibliometric datasets (extracted from pre-existing databases or constructed for the purpose of this research). Mirroring the archaeological propositions under study, this research also intends to illustrate the possible use of computing and formalised procedures in social sciences. The documentation and demonstrative principles underlying this work, implemented by using Wiki, the methods of literate programming and reproducible research, are themselves analysed. (shrink)

Contemporary analytic philosophy recognizes few principled constraints on its subject matter. When other disciplines also lay claim to a particular topic, however, important questions arise concerning the relation between these other disciplines and philosophy. A case in point is mathematics: traditional philosophy of mathematics defines a set of problems and certain general answers to those problems. However, mathematics is a subject matter that can be studied in many other ways: historically, sociologically, or even aesthetically, for example. Given this, we may (...) ask what implications discoveries in these disciplines have for the philosophy of mathematics. (shrink)

In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of ω? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. Many (...) mathematicians believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity. (shrink)

Co-learning of multiple agents has been studied in co-learning settings, and how do they help, or many different disciplines under various guises. For hamper, learning and cooperation? example, the issue has been tackled by distributed • How do we characterize the process and the artificial intelligence, parallel and distributed com- dynamics of co-learning, conceptually, mathe- puting, cognitive psychology, social psychology, matically, or computationally? game theory (and other areas of mathematical econ- • how do social structures and relations interact omics), (...) sociology, anthropology, and many other with co-learning of multiple agents? related disciplines. (shrink)

The 1990s could be called The Decade of Sociology in mathematics education. It was during those years that the sociology of mathematics became a core ingredient of discourse in mathematics education and the philosophy of mathematics and mathematics education. Unresolved questions and uncertainties have emerged out of this discourse that hinge on the key concept of social construction. More generally, what is at issue is the very idea of “the social”. Within the framework of the general problem of (...) “the social”, we want to open a discussion of boundaries and margins in mathematics and mathematics education. By theorizing the divisions of purity and danger, we will be able to better understand the intersection of logic, mathematics, and thinking with gender, race, and class, and morals, ethics, and values in the classroom. The process of transforming the sociology of mathematics and the sociology of mind into pedagogical tools for mathematics educators and philosophers of education has already begun. One of the tasks before us is the development of a more profound and at the same time more practical grasp of “the social”. Our objective in this paper is to move ourselves and our readers in the direction of just such a grasp of the social. (shrink)

An international group of distinguished scholars brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field traditionally dominated (...) by platonic perspectives. In a section on mathematics, politics, and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics, and with mathematics and social change. Contributors include Philip J. Davis, Helga Jungwirth, Nel Noddings, Yehuda Rav, Michael D. Resnik, Ole Skovsmose, and Thomas Tymoczko. (shrink)

Evidence Theory is a branch of mathematics that concerns combination of empirical evidence in an individual’s mind in order to construct a coherent picture of reality. Designed to deal with unexpected empirical evidence suggesting new possibilities, evidence theory is compatible with Shackle’s idea of decision-making as a creative act. This essay investigates this connection in detail, pointing to the usefulness of evidence theory to formalise and extend Shackle’s decision theory. In order to ease a proper framing of the issues involved, (...) evidence theory is compared with sub-additive probability theory and Ewens’s infinite alleles model. Furthermore, the original version of evidence theory is presented along with its most recent developments. (shrink)

In this paper, I argue that recent sociological theory has become increasingly bifurcated into two mutually incompatible styles of theorizing that I label formalist and behavioral-realist. Formalism favors mathematization and proposes an instrumentalist ontology of abstract processes while behavioral-realist theory takes at its basis the "real" physical individual endowed with concrete biological, cognitive and neurophysiological capacities and constraints and attempts to derive the proper conceptualization of social behavior from that basis. Formalism tends to lead toward a conceptually independent sociology (...) that in principle requires only minimal reference to the empirical and ontological storehouse of other disciplines, while behavioral-realist theory leads to an interdisciplinary sociology that can be located within a hierarchy of behavioral sciences, leading to questions regarding the relationship between sociology and other disciplines as well as issues of transdisciplinary unification and possible interdisciplinary reduction. I explore the consequences of this split for the project of explanatory sociological theory within the context of how it has manifested itself in sociological network theory and social psychology. I close with a critique and assessment of formalist tendencies in sociological theorizing. (shrink)

Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as a case study of (...) ... (shrink)

This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disci¬plines, which since the fourth century BC were called mathēmata or mathē¬ma¬tikai epistē¬mai. In the group of mathēmata that in Antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics and optics, we have also included geography. Using a dataset based on The Encyclopaedia of Ancient Natural Scientists, our essay considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are defined for (...) the purposes of the present paper) from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities, ancient scientists and philosophers, allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major centers – Athens, Alexandria, Rhodes, Rome and Byzantium/Constantinople – that appear and replace one another in succession as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 38.1 (2004) 12-22 [Access article in PDF] Art of a Child with Autism:Drawing Systems and Proto Mathematics Julia Kellman Sung, a five year old girl with autism, was enrolled by her mother in the university Saturday art program with the hope that Sung's favorite church school teacher, a graduate student in art education, would be able to tutor her daughter during the weekly classes. (...) Though hesitant because of her inexperience with autistic children and because of Sung's frequently angry tantrums on Sundays and the little girl's inability to speak, the young teacher, nonetheless, agreed to the plan. Her concerns proved unfounded, for Sung was able to work in the midst of the churning mass of noisy pre-K classmates without any signs of the fury that drove her to screaming, biting fits in church school. Additionally, and to everyone's delight, Sung's ability to develop as an image-maker was itself confirmed within the first few weeks of art class, dispelling any remaining doubts of her art aptitude or the value of art classes for the little girl.Additional questions and considerations arose about Sung and her art during the classes. How might one understand Sung's changing imagery? What might such explanations and images suggest about Sung herself and her activities? What might her art indicate about her similarity to other children and their art making? The possible answers to these and related queriespresented themselves during the semester, suggesting new avenues for inquiry surrounding both the wordless little girl and her Saturday morning creations.What follows is the story of Sung and her graphic development as it unfolded during the art class. Further, it is an attempt to uncover what Sung's drawings suggest about methods that are useful in the examination of the art of an autistic child. Documentation was carried out with a 35 mm. camera and color copier throughout the semester. Ethnographic methods including participant observation and interviews were employed to gather contextual data. Alfred Schutz's phenomenological sociology allowed for [End Page 12] consideration of nonverbal and verbal social interactions and behaviors. 1 A search of the literature of early mathematical development and drawing systems provided the underpinnings for early insights and later hypotheses. Here is Sung's story. Sung Sung, thick black hair swung forward to hide her face, lies prone in the middle of the busy classroom oblivious to the surging, chattering children around her, so intent is she on the sketchbook in which she draws. Next to her left elbow teeters a stack of a half dozen boxes of thick craypas from which she snatches colors, never looking at the hue she grabs. Her drawing behavior has a pattern; it looks like this. Shuffle through the identical boxes. Choose a box. Open it. Snatch out a bright waxy craypa. Hold it close to the eyes. Sniff it. Lick it. If the craypa passes muster, scribble with it in rich heavy lines — banana-like arcs, large solid dots the size of cherries, rectangular patches, and a variety of forms that come and go from one drawing to the next, like heat lightening on a summer evening. Her less frequent forms include: an enclosed yellow irregular shape with a dark dot in the middle suggesting a fried egg; a two-sided zig-zag; X forms; and solid and empty trapezoids made sail-like with superimposed crosses serving as mast and boom.Unlike the chosen craypas, however, those that do not pass Sung's inspection suffer a terrible fate; they are broken in two and their pieces tossed without ceremony to opposite sides of the room to be trodden into flat, bright smears by the other children in the class, pathetic traces of Sung's art. The craypas boxes themselves also play a role. For when they are closed they are used as blocks or toys, objects to be neatly stacked and restacked or placed side by side on the floor. When they are open, however, they are mere receptacles, passively holding... (shrink)

ABSTRACTThis paper offers a re-interpretation of the development of practical mathematics in Elizabethan England, placing artisanal know-how and the materials of the discipline at the heart of analysis, and bringing attention to Tudor economic policy by way of historical context. A major new source for the early instrument trade is presented: a manuscript volume of Chancery Court documents c.1565–c.1603, containing details of a patent granting a monopoly on making and selling mathematical instruments, circa 1575, to an unnamed individual, identified (...) here as the instrument maker Humphrey Cole. Drawing on economic and legal history, the paper argues that practical mathematics needs to be understood as one ‘project’ among many, at a time when monopoly patents were used to advance industry, lower unemployment, secure the realm and reward invention. Drawing on the history and sociology of technology, it argues that the management and control of materials – mathematical instruments themselves, and the local socio-legal context within which they could be made – needs to be understood as prior to and separate from the rhetoric of mathematical authors, which is of interest in its own right but which may not have a direct relationship to mathematical practice. (shrink)

This article deals with the public debates on the genealogy of Polish social sciences after the Second World War. The author shows how the changes in political conditions in the period between the end of the war (1945) and the ‘Stalinisation’ of Polish science at the First Congress of Polish Science (1951) influenced the ‘limits of the permissible’ in public discussions about the scientific identity of sociology. The article describes the stages in the development of public discourse on the (...) genealogy of the social sciences and thematises the increasing influence of the political agenda on the public practices of academics. When, immediately after the war, continuity with inter-war tradition, and the limited character of the Marxist approach, were particularly emphasised in the public discourse on the genealogy of social sciences, the beginning of the ‘Stalinisation’ of public space in 1948 forced the scholars to put emphasis on the political role of sociology, which, according to them, belonged to the ‘progressive camp’ together with Marxism. The closing of the sociological departments, and a journal, between 1950 and 1951 started a new stage of the debates within the preparatory meetings for the scientific congress. At this congress, the task of the social scientists was to construct, together with philosophers, a ‘progressive scientific tradition’ in Poland. For the Polish social scientists, the congress was a task to write a ‘mathematical equation’ with a known result and several mandatory elements such as ‘the progressive national tradition,’ ‘Marxism-Leninism,’ and ‘six-year plan.’ However, even under strong political pressure, the participants in the discussions did not come to a compromise concerning the progressive tradition of the Polish social sciences, and did not achieve the necessary ‘unity.’. (shrink)

To discuss the developments of mathematics that have to do with the introduction of new objects, we distinguish between ‘Aristotelian’ and ‘non-Aristotelian’ accounts of abstraction and mathematical ‘top-down’ and ‘bottom-up’ approaches. The development of mathematics from the 19th to the 20th century is then characterized as a move from a ‘bottom-up’ to a ‘top-down’ approach. Since the latter also leads to more abstract objects for which the Aristotelian account of abstraction is not well-suited, this development has also lead to (...) a decrease of visualizations in mathematical practice. (shrink)

This paper provides a brief overview of findings in mathematical cognition and how a human-like AI in mathematics may look like. Then, it provides six reasons in favor of a human-like AI for mathematics: (1) human cognition, with all its limits, creates mathematics; (2) human mathematics is insightful, not merely deductive steps; (3) human cognition detects structure in the real world; (4) human cognition can tackle and detect complex problems; (5) human cognition is creative; (6) human cognition considers ethical (...) issues. The paper provides a tentative frame to the question whether human mathematical cognition is relevant for designing an artificial intelligence that works on and creates mathematics. (shrink)

The volume before us is the fourth in the series of proceedings of what used to be the Israel Colloquium for the History, Philosophy and Sociology of Science. This Colloquium has in the meantime been renamed. It now bears the name of Yehoshua Bar-Hillel (1915-1975). Bar-Hillel was an eminent philosopher of science, language, and cognition, as well as a fearless fighter for enlightenment and a passionate teacher who had a durable influence on Israeli philosophical life. The essays collected in (...) this volume have of course this much in common, that they are all in, of, and pertaining to science. They also share the property of having all been delivered before live, and often lively, audiences in Jerusalem and in Tel Aviv, in the years 1984-1986. As is customary in the volumes of this series, the essays and commentaries presented here are intended to strike a rather special balance between the disciplines to which the Colloquium is dedicated. The historical and sociological vantage point is addressed in Kramnick's and Mali's treatment of Priestley, in Vickers' and Feldhay's studies of the Renaissance occult, and in Warnke's and Barasch's work on the imagination. From a philosophical angle several concepts, all material to the methodology of science, are taken up: rule following, by Smart and Margalit; analysis, by Ackerman; explanation, by Taylor; and the role of mathematics in physics, by Levy-Leblond and Pitowsky. (shrink)

A response to complex problems spanning disciplinary boundaries, Worlds of ScienceCraft offers bold new ways of conceptualizing ideas of science, sociology, and philosophy. Beginning with the historical foundations of civilization and progress, assumptions about the categories we use to talk about minds, identities, and bodies are challenged through case studies from mathematics, social cognition, and medical ethics.