An inclusive vision of mathematics-- its beauty, its humanity, and its power to build virtues that help us all flourish. For mathematician Francis Su, a society without mathematical affection is like a city without museums. To miss out on mathematics is to live without experiencing some of humanity's most beautiful ideas. In this profound book, written for a diverse audience but especially for those disenchanted by their past experiences, an award-winning mathematician and educator weaves personal reflections, puzzles, and (...) stories to show how mathematics meets basic human desires and cultivates virtues essential for human flourishing. Readers will explore mathematical concepts-- and see how mathematical thinking can even fulfill such longings as for love, play, freedom, justice, and community. Some lessons come from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother's, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher Jackson's letters to the author appear throughout the book and show how this intellectual pursuit can-- and must-- be open to all. (shrink)

Stereotypes of social construction suggest that the existence of social constructs is accidental and that such constructs have arbitrary and subjective features. In this paper, I explore a conception of social construction according to which it consists in the collective imposition of function onto reality and show that, according to this conception, these stereotypes are incorrect. In particular, I argue that the collective imposition of function onto reality is typically non-accidental and that the products of such imposition (...) frequently have non-arbitrary and objective features. These conclusions are interesting in and of themselves since they debunk important aspects of our socially constructed conception of social construction. Yet, additionally, they have important implications for the viability of mathematical social constructivism since resistance to such constructivism is frequently grounded in the observation that mathematics is non-accidental, non-arbitrary, and objective. As a secondary focus, I explore these implications in this paper. (shrink)

The term “rationality” is applied to many different things, from beliefs and preferences to decisions and choices, actions and behaviors, people, collectivities, andinstitutions. Therefore this paper will limit its considerations only to social preferences and choices in order to clarify the role of rationality in social explanation. The paper will focus on degrees of rationality, calling upon the concept of transitivity for help.

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)

This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way (...) as the set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we present two case studies from 17th century France. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it, a case of (im)purity in mathematics. Then, we compare Descartes' and Newton's mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas (Ayres-Bennett 1987). We argue that these early modern normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political program: the development of a new bourgeois intellectual class. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods, such as 17th century France, open up a new area of inquiry at the borders of linguistics and the philosophy of mathematics. (shrink)

The semantic inferentialist account of the social institution of semantic meaning can be naturally extended to account for social ontology. I argue here that semantic inferentialism provides a framework within which mathematical ontology can be understood as social ontology, and mathematical facts as socially instituted facts. I argue further that the semantic inferentialist framework provides resources to underpin at least some aspects of the objectivity of mathematics, even when the truth of mathematical claims is understood as (...) socially instituted. (shrink)

Philosopher Rene Descartes visualized a world unified by mathematics, in which all intellectual issues could be resolved rationally by local computation. This series of provocative essays takes a modern look at the seventeenth-century thinker’s dream, examining the physical and intellectual influences of mathematics on society, particularly in light of technological advances. They survey the conditions that elicit the application of mathematic principles; the effectiveness of these applications; and how applied mathematics constrain lives and transform perceptions of reality. (...) Highly suitable for browsing, the essays require different levels of mathematical knowledge that range from popular to professional. 1987 ed. (shrink)

In this paper I study the activity of mathematical problem-solving in scientific practice, focussing on enquiries in mathematical social science. I identify three salient phases of mathematical problem-solving and adopt them as a reference frame to investigate aspects of applications that have not yet received extensive attention in the philosophical literature.

A traditional problem of ethics in mathematics is the denial of social responsibility. Pure mathematics is viewed as neutral and value free, and therefore free of ethical responsibility. Applications of mathematics are seen as employing a neutral set of tools which, of themselves, are free from social responsibility. However, mathematicians are convinced they know what constitutes good mathematics. Furthermore many pure mathematicians are committed to purism, the ideology that values purity above applications in (...) class='Hi'>mathematics, and some historical reasons for this are discussed. MacIntyre’s virtue ethics accommodates both the good mathematician and the ethics of the social practice of mathematics. It demonstrates that purism is compatible with acknowledging the social responsibility of mathematics. Four aspects of this responsibility are mentioned, two concerning the impact of mathematics via education, and two concerning explicit and implicit applications of mathematics. The last of these opens up the performativity of mathematical and measurement applications in society, which change the very processes they are supposed to measure. Although these applications are not explored in detail, they illustrate the importance of considering the ethics and social responsibility of mathematics in society. MacIntyre’s virtue theory opens a broad approach to the controversial topic of the ethics of mathematics encompassing purism, and absolutist and social constructivist philosophies of mathematics, but still enabling ethical critiques of the impact of mathematics on society. (shrink)

This book is a mathematical potpourri. Its material originated in classroom presentations, formal lectures, sections of earlier books, book reviews, or just things written by the author for his own pleasure. Written in a nontechnical fashion, this book expresses the unique vision and attitude of the author towards the role of mathematics in society. It contains observations or incidental remarks on mathematics, its nature, its impacts on education and science and technology, its personalities and philosophies. The book is (...) directed towards the math buffs of the world and, more generally, towards the literate and interested public. Philip Davis is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics. Currently a Professor Emeritus from the Division of Applied Mathematics at Brown University, Davis is known for his books both in the areas of computational mathematics and approximation theory and for books exploring certain questions in the philosophy of mathematics and the role of mathematics in society. (shrink)

This book explains how limitations in the movement and perception of information constrain human behavior, cognition, interaction, and perspective. How fast can we learn? How much? Why are habits and biases unavoidable? Aspects considered include: how much information can one human absorb in a lifetime? How far does a process of perturbation propagate? How do specialization or generalization, critical thinking or belief, influence what people accomplish? It is aimed at general readers and scientists with an interest in how limitations of (...) the physical sciences affect society and human behavior. (shrink)

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...) of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

The present yearbook (which is the fourth in the series) is subtitled Trends & Cycles. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics. An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of mathematical (...) methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the yearbook consists of three main sections: (I) Long-Term Trends in Nature and Society; (II) Cyclical Processes in Pre-industrial Societies; (III) Contemporary History and Processes. We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (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)

This book uses Ludwig Wittgenstein's philosophical methodology to solve a problem that has perplexed thinkers for thousands of years: 'how come (abstract) mathematics applies so wonderfully well to the (concrete, physical) world?' The book is distinctive in several ways. First, it gives the reader a route into understanding important features of Wittgenstein's writings and lectures by using his methodology to tackle this long-standing and seemingly intractable philosophical problem. More than this, though, it offers an outline of important (sometimes little-known) (...) aspects of the development of mathematical thought through the ages, and an engagement of Wittgenstein's philosophy with this and with contemporary philosophy of mathematics on its own terms. A clear overview of all this in the context of Wittgenstein's philosophy of mathematics is interesting in its own right; it is also just what is needed to solve the problem of mathematics and world. (shrink)

Some courses achieve existence, some have to create Professional Issues and Ethics in existence thrust upon them. It is normally Mathematics; but if you don’t do it, we will a struggle to create a course on the ethical be.” I accepted. or social aspects of science or mathematics. The gift of a greenfield site and a bull- This is the story of one that was forced to dozer is a happy occasion, undoubtedly. But exist by an unusual (...) confluence of outside cirwhat to do next? It seemed to me I should cumstances. ensure the course satisfied these require- In the mid 1990s, the University of New ments: South Wales instituted a policy that all its • It should look good to students, to staff. (shrink)

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from (...) both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics. (shrink)

In the 1980s, philosophical, historical and social studies of science underwent a change which later evolved into a turn to practice. Analysts of science were asked to pay attention to scientific practices in meticulous detail and along multiple dimensions, including the material, social and psychological. Following this turn, the interest in scientific practices continued to increase and had an indelible influence in the various fields of science studies. No doubt, the practice turn changed our conceptions and approaches of (...) science, but what did it really teach us? What does it mean to study scientific practices? What are the general lessons, implications, and new challenges? This volume explores questions about the practice turn using both case studies and theoretical analysis. The case studies examine empirical and mathematical sciences, including the engineering sciences. The volume promotes interactions between acknowledged experts from different, often thought of as conflicting, orientations. It presents contributions in conjunction with critical commentaries that put the theses and assumptions of the former in perspective. Overall, the book offers a unique and diverse range of perspectives on the meanings, methods, lessons, and challenges associated with the practice turn. (shrink)

We propose simple nonlinear mathematical models for the legal concept of balancing of interests. Our aim is to bridge the gap between an abstract formalisation of a balancing decision while assuring consistency and ultimately legal certainty across cases. We focus on the conflict between the rights to privacy and to the protection of personal data in Art. 7 and Art. 8 of the EU Charter of Fundamental Rights (EUCh) against the right of access to information derived from Art. 11 EUCh. (...) These competing rights are denoted by ( \(i_1\) ) _right to privacy _ and ( \(i_2\) ) _access to information_; mathematically, their indices are respectively assigned by \(u_1\in [0,1]\) and \(u_2\in [0,1]\) subject to the constraint \(u_1+u_2=1\). This constraint allows us to use one single index _u_ to resolve the conflict through balancing. The outcome will be concluded by comparing the index _u_ with a prior given threshold \(u_0\). For simplicity, we assume that the balancing depends on only selected legal criteria such as the social status of affected person, and the sphere from which the information originated, which are represented as inputs of the models, called legal parameters. Additionally, we take “time” into consideration as a legal criterion, building on the European Court of Justice’s ruling on the right to be forgotten: by considering time as a legal parameter, we model how the outcome of the balancing changes over the passage of time. To catch the dependence of the outcome _u_ by these criteria as legal parameters, data were created by a fully-qualified lawyer. By comparison to other approaches based on machine learning, especially neural networks, this approach requires significantly less data. This might come at the price of higher abstraction and simplification, but also provides for higher transparency and explainability. Two mathematical models for _u_, a time-independent model and a time-dependent model, are proposed, that are fitted by using the data. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, (...) scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Professor Michael Edgeworth McIntyre is an eminent scientist who has also had a part-time career as a musician. From a lifetime's thinking, he offers this extraordinary synthesis exposing the deepest connections between science, music, and mathematics, while avoiding equations and technical jargon. He begins with perception psychology and the dichotomization instinct and then takes us through biological evolution, human language, and acausality illusions all the way to the climate crisis and the weaponization of the social media, and beyond (...) that into the deepest parts of theoretical physics - demonstrating our unconscious mathematical abilities. He also has an important message of hope for the future. Contrary to popular belief, biological evolution has given us not only the nastiest, but also the most compassionate and cooperative parts of human nature. This insight comes from recognizing that biological evolution is more than a simple competition between selfish genes. Rather, he suggests, in some ways it is more like turbulent fluid flow, a complex process spanning a vast range of timescales. Professor McIntyre is a Fellow of the Royal Society of London (FRS) and has worked on problems as diverse as the Sun's magnetic interior, the Antarctic ozone hole, jet streams in the atmosphere, and the psychophysics of violin sound. He has long been interested in how different branches of science can better communicate with each other and with the public, harnessing aspects of neuroscience and psychology that point toward the deep 'lucidity principles' that underlie skilful communication. (shrink)

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different (...) communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

Philosophy of Science, understood as a special philosophical discipline, was born only at the beginning of the twentieth century as part of the effort for overcoming the “foundational crisis” that had affected especially mathematics and physics. Therefore, it was conceived as an investigation about the features and reliability of scientific knowledge and for a few decades was deeply marked by the philosophical approach of logical empiricism. This cognitive point of view persisted also when, after Kuhn’s work, the attention focused (...) on the scientific activity in order to understand scientific change and a sociological model replaced the view that empirical adequacy and logical consistency are the factors that determine the change of scientific theories. Ethical, social and political considerations regarding science ware considered inappropriate and potentially dangerous since they violate the alleged “neutrality of science” with respect to values. Nevertheless, the strict intertwining of science and technology in contemporary “technoscience” has produced a wide debate regarding the practical aspect of technoscientific activity that has the intrinsic features of a philosophical debate. Therefore, it is natural and advisable that the entire wealth of the philosophical disciplines be called to contribute to the specific complex discourse of the Philosophy of Science. (shrink)

Comparisons made between art and mathematics so often centre on the beauty of mathematics and how its forms might be seen as aesthetically pleasing. Yet the prominence of beauty as an attribute is less prevalent in contemporary art. Rather, art has a much broader scope of concern, perhaps with a greater emphasis on providing apparatus through which we might better understand who we are. This paper considers some performative aspects of contemporary art and draws parallels with some examples (...) of mathematical activity within educative contexts. It argues that the performative dimension of mathematics is underemphasised in school activity and that the social and linguistic conditioning of mathematics within performance is a crucial aspect of the discipline being addressed in school and vocational courses. In particular, proficiency with concretisations of mathematics and the social dynamics that attend these is integral to the broader proficiency of moving between concrete and abstract domains. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Handbook of Cognitive Mathematics ed. by Marcel DanesiNathan HaydonMarcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including indexFor one acquainted with C.S. Peirce, it is hard to see Springer's recent Handbook of Cognitive Mathematics (editor: Marcel Danesi) through none other than a Peircean lens. Short for the cognitive science of mathematics, such a modern, scientific pursuit into (...) the nature and study of mathematical practice would no doubt be found agreeable to Peirce. The fact that references to Peirce appear often throughout the Handbook is a welcome find, with Peirce's ideas being a key subject of half a dozen chapters, and where a reader of any other chapter may well find further connections to Peirce's ideas. After spending time with the Handbook, it is clear that cognitive mathematics has not only embraced some of Pierce's ideas but may be at an important forefront of Peirce studies. In the end, the field may well be an area a Peircean should pay attention to.The book itself is pitched as a reference volume to the field (p. v) with the necessary background to familiarize oneself with the aims and results. The connection to cognitive science may call to mind detailed cognitive models [Ch. 10–14], theories on the origins of numeracy and other theories behind the biological and evolutionary requirements of mathematical thought [Ch.15–18], and the like. These are all present. But a key theme of the Handbook is to situate mathematics not just within more traditional 'cognitive' faculties and the more formal, i.e. algebraic, presentations of mathematics, but also to place mathematics within other human faculties and practices, from the arts to language [Ch. 19–22], within education and learning more broadly [Ch. 23–26], and in relation to the significant, though at first perhaps less quantitative parts of mathematics, like the use of metaphor, gesture, analogy, [End Page 243] abstraction, as well as further cultural and ethnographic considerations [Ch. 5–8]. The Handbook has explicitly taken a broader, more interdisciplinary approach (p. vi–vii) towards the scientific aspects of mathematical practice—choosing to regulate the study not by antecedently drawn opinions about what mathematics is (or has traditionally been taken to be) but by what future quantifiable and diverse study may come to bear on the practice and have to say about those engaging in it.This broad interdisciplinarity has a pragmatist ring, where theory cannot so easily be separated from the normative, social, and, more generally, the more thoroughly human aspects that we encounter and employ when we engage in it. Two further commitments of cognitive mathematics steer us even closer towards Peirce's views. The first—going back, for example, to Lakoff and Núñez's Where Mathematics Comes From (2000), which is taken to be a key early work in shaping the field—is that mathematics is taken to be a language like any other and as such must be learned and situated within our other cognitive faculties (p. vi, Ch. 4). The second—and largely, one imagines, following from a concern for seeing how mathematics is actually practiced—is that mathematics is essentially diagrammatic, involving as it does experimenting in diagrams and employing other signs (one may think here even of the mathematician sketching equations or geometric figures on a sheet of paper). What is perhaps initially most noteworthy about Peirce's philosophy of mathematics is his early concern for both these aspects of the practice.The book (more like a tome of over 1300 pages) is too large to be covered here in all its aspects, hence the review will focus on those aspects a Peircean or Pragmatist may find of interest. In particular, this review focuses on several of the themes above—situating mathematics more broadly within our everyday (cognitive) lives, the necessarily diagrammatic, fallible, and creative aspects of the practice, and, finally, what these aspects may have to say about Peirce's philosophy more generally, particularly with respect to (scientific) inquiry.A Peircean reader may well begin with Pietarinen's "Pragmaticism as Philosophy of Cognitive Mathematics" [Ch. 39] and "Peirce on Mathematical... (shrink)

Le but de cette encyclopédie en deux volumes, rédigée à l’instigation du président Hoover, était d’exposer tous les aspects de la vie sociale américaine, dans le dessein bien arrêté de saisir les rapports existant entre les différentes sphères sociales et de déterminer les transformations qui se produisent au sein de la société. Ces transformations devaient non seulement être constatées, mais on voulait encore essayer de trouver leur tendance générale, leur „trend“, c’est-à-dire la loi qui les détermine. Dans une critique des (...) résultats et des méthodes de cette „expression collective de la recherche sociale aux États-Unis“, M. G. s’efforce de montrer qu’en dépit de l’ampleur considérable de l’ouvrage, qui est également, en quelque sorte, une encyclopédie de la société américaine, le but scientifique visé n’a pas été atteint, et qu’il ne pouvait pas l’être. En effet, la théorie d’Ogburn, qui est à la base de l’ouvrage, et d’après laquelle les crises sont dues à des disproportions dans la vitesse de développement des différentes régions sociales, est insuffisante. Elle présuppose un ordre social relativement stable, dans lequel les dérangements peuvent être considérés comme de simples oscillations autour d’un noyau sain en lui-même. Comme le phénomène fondamental, la crise économique, n’est pas traité, on renonce à reconnaître méthodiquement que tous les processus sociaux sont incontestablement influencés par les faits économiques. Une autre objection importante, illustrée par M. G. au moyen d’une série d’exemples, concerne la notion du „trend“, empruntée à l’étude de la conjoncture. Le „trend“ ne peut pas avoir une loi sociale adéquate, parce qu’il tend à transformer tous les rapports sociaux qualitatifs en rapports mathématico-statistiques de valeurs. Le système actuel de la société, avec ses multiples ramifications, ne pourra toutefois jamais être saisi à i’aide de méthodes uniquement quantitatives ; il exige au contraire, dans son ensemble, une interprétation dynamique de la société. Comme le „Report“ y a renoncé, son intention théorique devait nécessairement échouer. Plus encore : les propositions pratiques qu’il fait pour l’élimination des défauts de la société américaine n’arrivent pas à la racine du mal.The joint report written at the request of president Hoover undertakes to present every aspect of social life in America. It has more over the special purpose of comprehensing the inter-relationships between these separate social compartments, and determining what social changes are taking place. Not only are these changes to be ascertained but the attempt made to grasp their trend, or in other words, to grasp the law determining these changes. G. seeks to show in a criticism of the results and methods of this „collective expression of sociological research in the United States“, that notwithstanding the extraordinary experience these 40 leading social scientists could draw upon, and the admirable comprehensiveness of this work-indeed an encyclopedia of American social life-, the scientific aim was not achieved, and could not be achieved. This is due to the insufficiency of Ogburn’s theory which underlies the work, that phenomena of crisis are to be traced to the disproportionate rates of change of the various provinces of social life. It presupposes a relatively stable social order in which upheavals are felt as mere oscillations around an intrinsically firm nucleus. Because the fundamental phenomenon of the economic crisis is not considered, the Report dispenses with the approach to a methodological standpoint, according to which all social processes are decisively determined by economic processes. Another important objection, which G. illustrates by a series of examples relates to concept of the trend borrowed from the business cycle. This can in no way furnish a suitable social law, because it tends to resolve all qualitative social relationships into mathematical- statistical formulae. Present day society, with its manifold divisions, can never be understood by quantitative methods only, but presumes a dialectical point of view for social life as a whole, which is ever in a state of flux. Since such a theoretical aim vitiated, but even its practical proposals, which it puts forward to rectify the shortcomings of American social life, fail to touch the root of the evil. (shrink)

In this paper, the authors try to clarify the relations between Meinong’s and Russell's thoughts on the ontological ideas of existence. The Meinongian theory on non-existent objects does not in itself violate the principle of non-contradiction, since the problem that this hypothesis offers to the theory of definite descriptions is not so much a logical problem as an ontological problem. To demonstrate this we will establish what we believe are the two main theses basic to the theory of descriptions: the (...) epistemological thesis and logical thesis. (shrink)

This article reclaims mathematics from the measures of profit and control by first presenting an anarchist analysis of mathematics? status quo societal uses and pedagogic activities. From this analysis, a vision for an anarchist math education is developed, as well as suggestions for how government school practitioners sympathetic to anarchism can insert this vision into their current work. Aspects to this vision include teacher autonomy, freedom from hierarchical curriculum structure and math class as a non-coercive, happy place. Finally, (...) mathematics is argued to be essential knowledge for anarchistic society for three potentialities: in solving social and technological problems through application, as an analytic technology and for increasing individual happiness via the aesthetic dimension. (shrink)

In this paper, we suggest the broader concept of proof-event, introduced by Joseph Goguen, as a fundamental methodological tool for studying proofs in history of mathematics. In this framework, proof is understood not as a purely syntactic object, but as a social process that involves at least two agents; this highlights the communicational aspect of proving. We claim that historians of mathematics essentially study proof-events in their research, since the mathematical proofs they face in the extant sources (...) involve many informal components, often not completely formalizable, and convey some kind of semantic content calling for understanding and verification. We illustrate the application of this methodological approach in some outstanding historical cases, paying particular attention to the process of proof interpretation that makes a proof-event alive. Finally, we suggest a classification of proof-events, according to the conditions imposed upon problem-solving. This enables us to speak about broad classes of proof-events in history of mathematics that share a common characteristic. (shrink)

Comparisons made between art and mathematics so often centre on the beauty of mathematics and how its forms might be seen as aesthetically pleasing. Yet the prominence of beauty as an attribute is less prevalent in contemporary art. Rather, art has a much broader scope of concern, perhaps with a greater emphasis on providing apparatus through which we might better understand who we are. This paper considers some performative aspects of contemporary art and draws parallels with some examples (...) of mathematical activity within educative contexts. It argues that the performative dimension of mathematics is underemphasised in school activity and that the social and linguistic conditioning of mathematics within performance is a crucial aspect of the discipline being addressed in school and vocational courses. In particular, proficiency with concretisations of mathematics and the social dynamics that attend these is integral to the broader proficiency of moving between concrete and abstract domains. (shrink)

Science and Technology Studies has attained a strong international profile in recent decades. Science Studies incorporates work in the History and Philosophy of Science, but emphasizes the social, cultural, and political implications of developments in the natural sciences, mathematics, engineering, and medicine. The Sociology of Science remains a vital part of Science Studies, but many other key contributors in the field identify more strongly with core disciplines such as Anthropology, Political Science, and Communication Studies. Edited by a leading (...) scholar in the field, this new four-volume collection from Routledge brings together classic work and the very best contemporary and cutting-edge scholarship. It provides researchersâe"and advanced studentsâe"with easy access to the key items of scholarly literature, material that is otherwise inaccessible or scattered throughout a variety of specialist journals and books. In particular, the collection enables users to gain a deep understanding of the current controversies surrounding science. The materials gathered exemplify important arguments, empirical studies, perspectives, and controversies (such as âe~the science warsâe(tm) of the 1990s). The collection also includes a full index, together with a comprehensive introduction, newly written by the editor, which places the collected material in its historical and intellectual context. Science and Technology Studies is an essential work of reference and will be welcomed as a vital one-stop research and pedagogic resource. (shrink)

To better conceive the socializing and pragmatic aspects of mathematics, it can be useful to use a process ontology, which allows, starting from an analysis of the processes of conversations, to compare their recourse, from degree to degree, to supposedly common “virtualities”, in particular in argumentative conversations, with the construction of more complex mathematical entities that allow new symmetries, but also with controversies between mathematicians on the use of these entities.

The complexity of social data has been a barrier which sociology has seemingly been unable to surmount. Consequently sociology, like other social sciences, has tended to divide itself into groups advocating different emphases in approach, concerning themselves respectively with the “quantitative” or “qualitative” aspects of social data. These two camps may be seen to diverge along distinct lines, the former approaching material from what is conceived to be a “scientific” frame of reference, posing problems which—it is hoped—will (...) be “explained” by the mathematical arrangement of materials; whereas the latter constructs questions the answer to which will make for “understanding”—ordinarily of behavior—through verbal symbolizations. Explanation usually is accomplished through the achievement of a formula—if couched in mathematical terms it is viewed as the highest order—and this group views the achievement of “understanding” a useless procedure, for—since “understanding” is not mathematically exposited—it is necessarily incapable of being validated. On the other hand, he of the group desirous to understand behavior, finds the problems posed by the “positivist” irrelevant to the larger problems in which his interest lies, and unenlightening for all of the devious steps taken to arrive at a solution; more in the manner than in the spirit of science. (shrink)

This interdisciplinary volume brings together expert researchers coming from primatology, anthropology, ethology, philosophy of cognitive sciences, neurophysiology, mathematics and psychology to discuss both the foundations of non-human primate and human social cognition as well as the means there currently exist to study the various facets of social cognition. The first part focusses on various aspects of social cognition across primates, from the relationship between food and social behaviour to the connection with empathy and communication, offering (...) a multitude of innovative approaches that range from field-studies to philosophy. The second part details the various epistemic and methodological means there exist to study social cognition, in particular how to ascertain the proximal and ultimate mechanisms of social cognition through experimental, modelling and field studies. In the final part, the mechanisms of cultural transmission in primate and human societies are investigated, and special attention is given to how the evolution of cognitive capacities underlie primates’ abilities to use and manufacture tools, and how this in turn influences their social ecology. A must-read for both, young scholars as well as established researchers! (shrink)

Newton's use of mathematics in mechanics was justified by him from his neo-platonician conception of the physical world that was going along with his «absolute, true and mathematical concepts» such as space, time, motion, force, etc. But physics, afterwards, although it was based on newtonian dynamics, meant differently the legitimacy of being mathematized, and this difference can be seen already in the works of eighteenth century «Geometers» such as Euler, Clairaut and d'Alembert (and later on Lagrange, Laplace and others). (...) Despite their inheritance of Newton's achievements, they understood differently the meaning and use of mathematical quantities for physics, in a way that was more neutral to metaphysics. The continental reception and assimilation of Newton's Principia had indeed occured as its budding onto Leibniz' calculus and a cartesian conception of rationality (spread in particular by the malebranchist disciples of Leibniz). This new thought of the legitimacy of mathematization is clearly at variance with Descartes' identification of physics with geometry, but it nevertheless can be traced back to Descartes' conception of magnitudes, as it was developed and analyzed from the notion of dimension in his Regulæ ad directionem ingenii (in particular, rule 14). This idea can be followed afterwards with further philosophical or mathematical specifications through authors such as Kant, Riemann and others. This inquiry into the original thought of magnitudes, and of physical magnitudes conceived through mathematization, leads us to suggest an extension of meaning for the concept of physical magnitude that puts emphasis on its relational and structural aspects rather than restraining it to a simple «numerically valued» acception. Such a broadening would have immediate implications on our comprehension of «non classical» aspects of contemporary physics in the quantum area and in dynamical systems. (shrink)

This book discusses both the philosophy of mathematics and of mathematics education. The first part is a critique of existing approaches and a new philosophy of mathematics. Chapters include: (1) "A Critique of Absolutist Philosophies of Mathematics," (2) "The Philosophy of Mathematics Reconceptualized," (3) "Social Constructivism as a Philosophy of Mathematics," (4) "Social Constructivism and Subjective Knowledge," and (5) "The Parallels of Social Constructivism." The second part of the book explores the (...) philosophy of mathematics education and shows that many aspects of mathematics education rest on underlying, usually implicit, philosophical assumptions. Making these assumptions explicit puts a critical tool into the hands of teachers and researchers. Chapters in this section are: (6) "Aims and Ideologies of Mathematics Education," (7) "Groups with Utilitarian Ideologies," (8) "Groups with Purist Ideologies," (9) "The Social Change Ideology of the Public Educators," (10) "Critical Review of Cockcroft and the National Curriculum," (11) "Hierarchy in Mathematics, Learning, Ability and Society," (12) "Mathematics, Values and Equal Opportunities," and (13) "Investigation, Problem Solving and Pedagogy." Each chapter contains notes. (Contains 654 references.) (MKR). (shrink)

Isaac Newton's (1642-1727) contribution to the quantitative aspects of mathematics are well known compared to his views on it's qualitative aspect. In this paper, the author attempts to examine Newton.s position with regard to the orientation of mathematical problems based on some of his own writings on the subject.

The purpose of this paper is to discuss the role of financial practice in the development of mathematics as applied in human judgement. The basis of the paper is in historical research from the 1990s that argues that the monetisation of western commerce, which abstracted value into quantified price, was synthesised with scholastic analysis resulting in a “mathematical mechanistic world picture” that led to the widespread use of mathematics in science from the seventeenth century. An aspect of this (...) process was the quantification of chance that led to the development of mathematical probability, the branch of mathematics most relevant to judgement and the focus of this paper. Ideas from this historical research are related to the fundamental theorem of asset pricing, the foundational theory of contemporary financial mathematics. The paper observes that vestiges of medieval scholastic attitudes to financial ethics can be discerned in the FTAP, offering a novel interpretation of the mathematical theorem. The paper then considers the Dutch book argument, the most popular justification for subjective probability. The paper’s main contribution is in describing the significance of financial practice in validating the DBA and the paper explains how the FTAP addresses some criticisms of how the DBA represents beliefs. The conclusions emphasise the distinction between pure and practical reasoning and that this should be mirrored in a distinction between the mathematics of physica and practica. This point is important as mathematics is becoming more widespread in modelling, and directing, social systems. (shrink)

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, discovery, (...) and creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

This book treats so‐called Greek mathematics, developed in the Greek‐speaking world between about 600 b.c. and 600 a.d. It consists of four parts: early Greek mathematics, Hellenistic mathematics, Graeco‐Roman mathematics, and late ancient mathematics. Each part is divided into two chapters, “The Evidence” and “The Questions.”This separation of evidence and questions is significant. Serafina Cuomo has refused to follow the familiar method of weaving an apparently seamless history of Greek mathematics out of fragmentary and (...) heterogeneous documents and conjectures about them. The chapters of questions, where she points to issues that remain open, are very suggestive. For example, most important documents about the early development of Greek mathematics derive from a single lost work, the History of Geometry by Eudemus. Cuomo dares to cast doubt on its authenticity. Though her reservations seem extreme, we should remember that no document is neutral: Eudemus's history is a compilation, which involved choices, and the fragments we have now are the result of selection, partly intentional, partly by chance.Another merit of this book is that it considers a wide range of activity as mathematics. Practical mathematics—such as land surveying and accounting, with their sociopolitical importance—is emphasized.Cuomo's chief claim is that the standard historiography that associates the development of Greek mathematics with Plato's philosophy is only the version promulgated by Proclus; other descriptions are also possible. In fact, Pappus, Iamblichus, and others had their own versions. This claim is reasonable and contributes to a better understanding of Greek mathematics and the authors of late antiquity.In her citations, the author tries to let the text speak for itself, allowing as much as a full page to a passage or a proposition, and she refrains from using modern symbolism to explain mathematical content. Though this attitude is admirable, its cost is not negligible. Readers expecting to acquire a basic knowledge of Greek mathematics may find themselves at a loss when faced with highly technical propositions presented without elucidation.What this compact book does not include should also be mentioned. In contrast to her enthusiasm for the social and political dimensions of ancient mathematics, the author seems somewhat indifferent to its technical and theoretical aspects. Archimedes and Apollonius command only 16 pages—less than 7 percent of the text—whereas T. L. Heath dedicated 150 pages of his 1,000‐page history to them . Though documents showing the importance of land surveying are frequently quoted, little is said about the practice and the technical development of this art. The technical details of Ptolemy's works are practically dismissed—but should he not have an especially important role in alternative versions of the history of ancient mathematics because of his ingenious reconciliation of rigorous theory and the limitations imposed by reality in fields like astronomy and geography?The scantiness of the technical ingredients makes Ancient Mathematics more a history of discourses about mathematics than a history of mathematics—though this is to some extent inevitable given the character of late ancient mathematics, as Cuomo correctly emphasizes. The plan of the series to which this book belongs may be too modest to accommodate the author's ambition. Another problem attributable to the publisher is that the notes appear at the end of each chapter, so checking the references is annoying. Short references could be put in parentheses, and footnotes would be more convenient for longer notes. (shrink)

Since the methods employed during teacher-learner interchange are constrained by the internal structure of a discipline, a study of the interaction amongst verbal language, technical language and structure of disciplines is at the heart of the classic problem of transfer in teaching-learning situations. This paper utilizes the analytic method of philosophy to explore aspects of the role of language in mathematics education, and attempts to harmonize mathematical meanings exposed by verbal language and the precise meanings expressed by the (...) class='Hi'>mathematics register (MR) formulated in verbal language. While focusing on the integration of language use and meaning construction in mathematics education, the paper explores the relationship between the conceptual understanding revealed by the mathematics register and the procedural knowledge that refers to the mathematical content through ordinary discourse. (shrink)

While Garfinkel’s early work, captured in Studies in Ethnomethodology, has received a lot of attention and discussion, this has not been the case for his later work since the 1970s. In this paper, we critically examine the aims of Garfinkel’s later ethnomethodological studies of work programme and evaluate key ideas such as the ‘missing what’ in the sociology of work, ‘the unique adequacy requirements of methods’, and the notion of ‘hybrid studies’. We do so through a detailed engagement with a (...) study that has frequently been singled out as exemplary by Garfinkel for his studies of work programme, namely Livingston’s The Ethnomethodological Foundations of Mathematics. We show how Livingston uses the proof of Gödel’s Incompleteness Theorem as a way to exhibit the work involved in understanding mathematical proofs. We then discuss how Livingston uses this example to introduce a distinction between the written ‘proof account’ and the associated ‘lived work’ of working through that proof, which we argue allows Livingston to provide a powerful critique of a formalist understanding of the objectivity of mathematics. We then discuss three aspects that we find problematic in Livingston’s and Garfinkel’s claims about Livingston’s study. Firstly, we question whether written proofs are best conceived of as descriptions or accounts. Secondly, we interrogate whether an ethnomethodological study could teach mathematicians how to make discoveries. Thirdly, we throw doubt on Garfinkel’s claim that Livingston’s results are results in mathematics. We conclude this paper with a discussion of how Livingston’s study Gödel’s proof highlights key tensions in Garfinkel’s later work. Firstly, we argue that there exists an ambiguity in Garfinkel’s treatment of texts as ‘incompetent’. Secondly, we show that Garfinkel’s attempt to extend his idea of classical studies from sociology to other professions and disciplines is problematic. Finally, we question Garfinkel’s proposals to reorient ethnomethodological studies away from sociological audiences and ask whether ethnomethodological studies promise the delivery of a ‘large prize’ or, rather, provide something like ‘helpful therapy’. (shrink)

ArgumentThis paper challenges the use of the notion of “culture” to describe a particular organization of mathematical knowledge, shared by a few mathematicians over a short period of time in the second half of the nineteenth century. This knowledge relates to “geometrical equations,” objects that proved crucial for the mechanisms of encounters between equation theory, substitution theory, and geometry at that time, although they were not well-defined mathematical objects. The description of the mathematical collective activities linked to “geometrical equations,” and (...) especially the technical aspects of these activities, is made on the basis of a sociological definition of “culture.” More precisely, after an examination of the social organization of the group of mathematicians, I argue that these activities form an intricate system of patterns, symbols, and values, for which I suggest a characterization as a “cultural system.”. (shrink)

Le fruit d'une enquête sociologique, menée dans les années 1990 aux Etats-Unis, sur la forme d'activité scientifique réputée la plus pure et la plus rebelle à l'investigation des sciences sociales : celle des logiciens et des chercheurs en intelligence artificielle.

The logician Kurt Gödel published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and (...) philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues. (shrink)

Les mathématiques sont bien plus qu’une science, tant elles ont acquis un statut central et tout à fait particulier. Elles sont à la fois un langage permettant d’articuler la réalité et un outil qui façonne le réel, et jouent à ce titre un rôle clé dans le développement des sciences de la nature, des sciences humaines et sociales, et de l’industrie. Leur position centrale dans les mécanismes de sélection et de reproduction qui caractérisent l’enseignement en France en fait par ailleurs (...) un enjeu social majeur. L’essence des mathématiques, c’est la liberté, disait Georg Cantor. Quelles sont les nouvelles frontières, les nouveaux défis de cette science qui ne s’impose aucune limite? Comment cette science de l’abstraction s’inscrit-elle au cœur de la réalité, au point d’être incontournable? Quel rôle, quelle place, quelle responsabilité s’assignent les mathématiciens? Deux mathématiciens, Pierre Cartier qui fut l’un des piliers du fameux groupe Bourbaki et Cédric Villani, un des représentants les plus brillants de sa génération (médaille Fields en 2010), le mathématicien et historien des sciences Jean Dhombres, et le philosophe des sciences Gerhard Heinzmann nous invitent à débattre avec eux de l’activité mathématique et de ses conséquences sociales, tant dans l’histoire de l’humanité que dans le monde contemporain. (shrink)