A characteristic feature of the philosophy of mathematical practice is to attend to what people do when they do mathematics. But what does it mean to do mathematics? This question raises several issues regarding the nature of action, activity, and agency in mathematical practice. The present chapter reviews contributions in the field that have attempted to theorize about these notions. It begins with some motivations for taking agents seriously in the philosophical study of mathematical practice. The core (...) of the chapter discusses, in turn, what it means to carry out mathematical activities, do things with mathematical artifacts, engage with mathematical proofs, and perform mathematical actions prescribed by mathematical texts. Taken together, the various lines of work reported here provide an initial, but already sophisticated, picture of what it means to do mathematics. The chapter ends with some suggestions for future research on agency in mathematical practice. (shrink)

Mathematical proofs are not sequences of arbitrary deductive steps—each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities—that is, sequences of deductive inferences—and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by specific norms of rational planning agency, (...) and that a deductive step in a mathematical proof qualifies as rational whenever the corresponding deductive inference in the associated proof activity figures in a plan that has been constructed rationally. It is then shown that mathematical proofs whose associated proof activities violate these norms are likely to be judged as defective by mathematical agents, thereby providing evidence that these norms are indeed present in mathematical practice. We conclude that, if mathematical proofs are not mere sequences of deductive steps, if they possess a rational structure, it is because they are the product of rational planning agents. (shrink)

Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational reconstruction, building in part on (...) Bratman's theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The (...) starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

In a recent paper, Negri and Pavlović (Studia Logica 1–35, 2020) have formulated a decidable sequent calculus for the logic of agency, specifically for a deliberative see-to-it-that modality, or dstit. In that paper the adequacy of the system is demonstrated by showing the derivability of the axiomatization of dstit from Belnap et al. (Facing the future: agents and choices in our indeterminist world. Oxford University Press, Oxford, 2001). And while the influence of the latter book on the study of (...) logics of agency cannot be overstated, we note that this is not the only axiomatization of that modality available. In fact, an earlier (and arguably purer) one was offered in Xu (J Philosophical Logic 27(5):505–552, 1998). In this article we fill this lacuna by proving that this alternative axiomatization is likewise readily derivable in the system of Negri and Pavlović (Studia Logica 1–35, 2020). (shrink)

The themes of connectedness and continuity, which are also mathematical properties, have run like a red thread through the last fifty years of History and Theory, notably in the theory of the narration of action in history. In this essay I review various answers to the question of the driving force that motivates action and that propels a sequence, continuous or discontinuous. These answers underpin narrative strategies intended to solve the problem of human agency and thereby to provide the (...) basis for historical narratives. I argue that both continuous and discontinuous conceptions of history have to do with the restrictive concept of trajectory. Most of our familiar concepts such as trajectory, equilibrium, optimum, probability, and sensitivity to initial conditions are taken from mathematics and physics, but how well adapted are they to dealing with the time of actors, whose actions are intermingled with uncertainty? I shall present alternative concepts of dynamics, ones that no longer lead toward just one particular future or that reflect a single past. On this basis I suggest reorganizing our view of historical time along the principles of maintenance, acquisition, and victory. Historical sources trace back not to one story or one process that is plausible or that appeals to common sense, but to a whole family of processes. From the past, we can obtain sets of constraints that circumscribe sets of stories rather than a single scenario. I shall then propose a topology of the time of action based on my alternative conception of dynamics, a topology made of what I call viability kernels, capture basins, and victory domains. (shrink)

The primary aim of this article is to bring the work of Deleuze and Guattari to bear on the question ofcommunication in the classroom. I focus on the mathematics classroom, where agency and subjectivity are highly regulated by the rituals of the discipline, and where neoliberal psychological frameworks continue to dominate theories of teaching and learning. Moreover, the nature ofcommunication in mathematics classrooms remains highlyelusive and problematic, due in part to the distinct relationship the discipline has with (...) verbal language and thought. I first discuss current attempts to better address the embodied nature ofcommunication in mathematics classrooms, and argue that these remain overly logo-centric and language-centric in their conception of thinking. I then show how the work of Deleuze and Guattari on thought as a radical disruptive event can be used effectively to critique current pedagogical practices that privilege a narrow conception of communication in the classroom. I examine a set of exemplary classroom videos used in mathematics teacher education to argue that the current approach fails to honor the highly creative and disruptive nature ofthinking. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

The ArgumentWhen economists report on research using mathematical models, they use a literary form similar to the experimental report in the laboratory sciences. This form consists of a narrative of a series of events, with a clear temporal segregation of the agency of the author and the agency of the objects of study. Existing explanations of this literary form treat it as a rhetorical device that either conceals the agency of the author in constructing and interpreting the (...) findings, or simply appropriates the appearance of accepted scientific method. This article — based on analysis of a research program in economics, a single article that issued from that program, and in-depth interviews with the authors — proposes an alternate interpretation. Drawing on the praxeological “laboratory studies” tradition in science studies, we treat work with mathematical models as involving the interaction of economists with objects that act independently of the analyst's will. The clear separation of the economist's and the models agency, as depicted in the published report, is not the result of a rhetorical rewriting of actual events, but is a practical accomplishment. Every step in the analytical work that preceded the paper is devoted to developing a procedure in which the economists' agency will be completely accountable in terms of accepted practices, and the performance of the model will be distinct and compelling. (shrink)

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

In this book, Binder shows that at the heart of the most prominent arguments in favour of value-neutral approaches to overall freedom lies the value freedom has for human agency and development. Far from leading to the adoption of a value-neutral approach, however, ascribing importance to freedom’s agency value requires one to adopt a refined value-based approach. Binder employs an axiomatic framework in order to develop such an approach. She shows that a focus on freedom’s agency value (...) has far reaching consequences for existing results in the freedom ranking literature: it requires one to move beyond a person’s given all-things-considered preferences to the values underlying a person’s preference formation. Furthermore, it requires, as Binder argues, one to account for those differences between choice options which really matter to people. Binder illustrates the implications of her analysis for the evaluation of public policy and human development with the capability approach: only if sufficient importance is ascribed to freedom’s agency value can the capability approach keep its promises. ​. (shrink)

This text looks at the construction of the Semantic Web, which will enable computers to automatically and independently consume Web-based information. With numerous programming examples, it is ideal for undergraduates and graduates in mathematics, computer science and logic and researchers interested in Multi-Agent Systems and the Semantic Web.

This ambitious book by one of the most original and provocative thinkers in science studies offers a sophisticated new understanding of the nature of scientific, mathematical, and engineering practice and the production of scientific knowledge. Andrew Pickering offers a new approach to the unpredictable nature of change in science, taking into account the extraordinary number of factors—social, technological, conceptual, and natural—that interact to affect the creation of scientific knowledge. In his view, machines, instruments, facts, theories, conceptual and mathematical structures, disciplined (...) practices, and human beings are in constantly shifting relationships with one another—"mangled" together in unforeseeable ways that are shaped by the contingencies of culture, time, and place. Situating material as well as human agency in their larger cultural context, Pickering uses case studies to show how this picture of the open, changeable nature of science advances a richer understanding of scientific work both past and present. Pickering examines in detail the building of the bubble chamber in particle physics, the search for the quark, the construction of the quarternion system in mathematics, and the introduction of computer-controlled machine tools in industry. He uses these examples to address the most basic elements of scientific practice—the development of experimental apparatus, the production of facts, the development of theory, and the interrelation of machines and social organization. (shrink)

Doctoral programs at U.S. universities play a critical role in the development of human resources both in the United States and abroad. This volume reports the results of an extensive study of U.S. research-doctorate programs in five broad fields: physical sciences and mathematics, engineering, social and behavioral sciences, biological sciences, and the humanities. Research-Doctorate Programs in the United States documents changes that have taken place in the size, structure, and quality of doctoral education since the widely used 1982 editions. (...) This update provides selected information on nearly 4,000 doctoral programs in 41 subdisciplines at 274 doctorate-granting institutions. This volume also reports the results of the National Survey of Graduate Faculty, which polled a sample of faculty for their views on the scholarly quality of program faculty and the effectiveness of doctoral programs in preparing research scholars/scientists. This much-anticipated update of such an essential reference will be useful to education administrators, university faculty, and students seeking authoritative information on doctoral programs. (shrink)

In his comments, Daniel Nicholls succeeds in saying more than a few things that I had scarcely realized about the ways in which I write and, therefore, of what I tend to take for granted. He sees in what I write a capacity ‘to utilize the “obvious” whilst at the same time saying something about it.’ Not every philosopher would take that as a compliment. Many philosophers and philosophies have quite other pretensions – to transcend the illusions of common thought (...) and perception towards a higher plane of metaphysical philosophy – or of physics and mathematics. I find that Nicholls’s remark both compliments and complements my work, however. The use of everyday language and example to disturb that everyday world to ‘bring it into high relief’ as Nicholls puts it – is a significant part of philosophical method. To juxtapose the esoteric language of metaphysics (being-in-itself, being-for-itself, being-for-others) with the everyday (what there is, living for oneself, living for others) is to haunt the everyday with a metaphysical imagination even as it reveals what metaphysics had erased from the coinage of common words. (I am thinking, of course, of Derrida’s remarks about Anatole France’s satirical view of philosophical language as coinage from which all marking has been erased). (shrink)

In this article, Kurt Stemhagen reconstructs mathematics education in light of Dewey's democratic theory and his ideas about mathematics and mathematics education. The resulting democratic philosophy and pedagogy of mathematics education emphasizes agency and the connections between mathematics and students' social experiences. Stemhagen considers questions about the disconnect between constructivist reformers and critical mathematics educators, and he positions Dewey's ideas as a way to draw on the best of both to create an active (...) and more democratic school math experience. (shrink)

Book synopsis: The London School of Economics and Political Science has embraced the full range of the social sciences and its related disciplines. Contributors to this book were invited to write on the subject of freedom. The volume is an exemplary reflection of the variety, the individuality, the different interests, and the range of assumptions found in the scholars of the LSE. The authors come from varied backgrounds - linguistics, mathematics, computer science, sociology, geography, economics, industrial relations, anthropology, political (...) science. They provide a stimulating array of viewpoints on the universally discussed issue of freedom. (shrink)

In this paper we consider logical inference as an activity that results in proofs and hence produces knowledge. We suggest to merge the semantical analysis of deliberatively seeing-to-it-that from stit theory and the semantics of the epistemic logic with justification from. The general idea is to understand proving that A as seeing to it that a proof of A is available. We introduce a semantics of various notions of proving as an activity and present a number of valid principles that (...) relate the various notions of proving to each other and to notions of justified knowledge, implicit knowledge, and possibility. We also point out and comment upon certain principles our semantics fails to validate. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both (...) disciplines. (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)

This section aims to summarize and conclude Part I in the form of a taxonomy of legitimate and illegitimate corporate moral responsibility attributions. I believe we can categorise four types of corporate moral responsibility attributions two of which are legitimate and two which are illegitimate with regard to our concept of moral agency and our moral intuition of fairness.

For many, the two key thinkers about science in the twentieth century are Thomas Kuhn and Karl Popper, and one of the key questions in contemplating science is how to make sense of theory change. In Creatively Undecided, philosopher Menachem Fisch defends a new way to make sense of the rationality of scientific revolutions. He argues, loosely following Kuhn, for a strong notion of the framework dependency of all scientific practice, while at the same time he shows how such frameworks (...) can be deemed the possible outcomes of keen rational deliberation along Popperian lines. Fisch's innovation is to call attention to the importance of ambiguity and indecision in scientific change and advancement. Specifically, he backs the problem up, looking not at how we might communicate rationally across an already existing divide but at the rational incentive to create an alternative framework in the first place. Creatively Undecided will be essential reading for philosophers of science, and its vivid case study in Victorian mathematics will draw in historians. (shrink)

Having defined a notion of homology for paired graphs, Métayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document} there exists a Jordan-Hölder decomposition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathsf H}_0(G)$\end{document}. This decomposition is determined by a certain enumeration of the pairs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document}. We correct his (...) proof of this fact and show that there exists a 1-1 correspondence between these Jordan-Hölder decompositions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathsf H}_0(G)$\end{document} and the possible ‘construction-orders’ of the par-net underlying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document}. (shrink)

A sense of agency (SoA) is the experience of subjective awareness regarding the control of one’s actions. Humans have a natural tendency to generate prediction models of the environment and adapt their models according to changes in the environment. The SoA is associated with the degree of the adaptation of the prediction models, e.g., insufficient adaptation causes low predictability and lowers the SoA over the environment. Thus, identifying the mechanisms behind the adaptation process of a prediction model related to (...) the SoA is essential for understanding the generative process of the SoA. In the first half of the current study, we constructed a mathematical model in which the SoA represents a likelihood value for a given observation (sensory feedback) in a prediction model of the environment and in which the prediction model is updated according to the likelihood value. From our mathematical model, we theoretically derived a testable hypothesis that the prediction model is updated according to a Bayesian rule or a stochastic gradient. In the second half of our study, we focused on the experimental examination of this hypothesis. In our experiment, human subjects were repeatedly asked to observe a moving square on a computer screen and press a button after a beep sound. The button press resulted in an abrupt jump of the moving square on the screen. Experiencing the various stochastic time intervals between the action execution button-press) and the consequent event square jumping caused gradual changes in the subjects' degree of their SoA. By comparing the above theoretical hypothesis with the experimental results, we concluded that the update (adaptation) rule of the prediction model based on the SoA is better described by a Bayesian update than by a stochastic gradient descent. (shrink)

In this paper we show that the Hilbert system of agency and ability presented by Dag Elgesem is incomplete with respect to the intended semantics. We argue that completeness result may be easily regained. Finally, we shortly discuss some issues related to the philosophical intuition behind his approach. This is done by examining Elgesem's modal logic of agency and ability using semantics with different flavours.

If you have ever had to move house, you will know this: the worst part is the sofa. You cannot do it alone. Nor will it be enough for me to just lift one end waiting for you to lift the other. We will have to work together to get the job done. If spaces are tight, we will even have to find a practical solution to a tantalizing mathematical puzzle: the moving sofa problem.Joint actions like that are part and (...) parcel of everyday life. But what exactly is special about acting together? After all, the actions of two strangers also depend on one another when one exits and the other enters through a revolving door, when they happen to walk side-by-side along a forest path, or when they exchange blows in a pub brawl.The problem is that two patterns of social behavior might look identical, even though one is a case of joint action and the other is not. There need not be an observable difference between the movements of old friends taking their morning constitutional together, on the one hand, a .. (shrink)

Predictive policing has become a new panacea for crime prevention. However, we still know too little about the performance of computational methods in the context of predictive policing. The paper provides a detailed analysis of existing approaches to algorithmic crime forecasting. First, it is explained how predictive policing makes use of predictive models to generate crime forecasts. Afterwards, three epistemologies of predictive policing are distinguished: mathematical social science, social physics and machine learning. Finally, it is shown that these epistemologies have (...) significant implications for the constitution of predictive knowledge in terms of its genesis, scope, intelligibility and accessibility. It is the different ways future crimes are rendered knowledgeable in order to act upon them that reaffirm or reconfigure the status of criminological knowledge within the criminal justice system, direct the attention of law enforcement agencies to particular types of crimes and criminals and blank out others, satisfy the claim for the meaningfulness of predictions or break with it and allow professionals to understand the algorithmic systems they shall rely on or turn them into a black box. By distinguishing epistemologies and analysing their implications, this analysis provides insight into the techno-scientific foundations of predictive policing and enables us to critically engage with the socio-technical practices of algorithmic crime forecasting. (shrink)

Provability, Computability and Reflection.

Rational agents are important objects of study in several research communities, including economics, philosophy, cognitive science, and most recently computer science and artificial intelligence. Crudely, a rational agent is an entity that is capable of acting on its environment, and which chooses to act in such a way as to further its own best interests. There has recently been much interest in the use of mathematical logic for developing formal theories of such agents. Such theories view agents as practical reasoning (...) systems, deciding moment by moment which action to perform next, given the beliefs they have about the world and their desires with respect to how they would like the world to be. In this article, we survey the state of the art in developing logical theories of rational agency. Following a discussion on the dimensions along which such theories can vary, we briefly survey the logical tools available in order to construct such theories. We then review and critically assess three of the best known theories of rational agency: Cohen and Levesque's intention logic, Rao and Georgeff's BDI logics, and the KARO framework of Meyer et al. We then discuss the various roles that such logics can play in helping us to engineer rational agents, and conclude with a discussion of open problems. (shrink)

The paper presents an infinite hierarchy of sound and complete axiomatic systems for Two-Dimensional Modal Tense Logic with Historical Necessity, Agents and Acts. A main novelty of these logics is their capacity to represent formally (i) basic action-sentences asserting that such and such an act is performed/omitted by an agent, as well as (ii) causative action-sentences asserting that by performing/omitting a certain act, an agent causes that such and such a state-of-affairs is realized (e.g. comes about/ceases/remains/remains absent). We illustrate how (...) the formal machinery of our systems can be used to reconstruct a number of interesting ideas in the Logic of Agency and Action that have been proposed by authors like von Wright, von Kutschera, Belnap and Segerberg. (shrink)

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version (...) of mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

The massive industrialization of World War I resulted in a previously unimaginable number of casualties. The military and civil agencies in Germany that managed the welfare systems for veterans collaborated with companies, engineers and physicians to produce prosthetic arms, hands and legs that would allow disabled former soldiers to re-enter the factory as productive workers. This article focuses on the one of the best known arm prosthesis, the Siemens-Schuckert-Arm. While other historians have argued that the military command “recycled” disabled soldiers (...) effectively, this article argues instead that this “recycling” was incomplete due to three factors: not all offices and individuals worked together effectively; the price of the arm was too high; and employers thought the disabled would be less productive then female or elderly workers. In other words, while military and civil welfare agencies attempted to re-purpose disabled soldiers using advanced prosthetic technology, their efforts fell far short of fully reintegrating these soldiers into the war effort. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, (...) but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...) be applied to mathematics as well as science. Michael Grove declared that revolutions never occur in mathematics, while Joseph Dauben argued that there have been mathematical revolutions and gave some examples. This book is the first comprehensive examination of the question. It reprints the original papers of Grove, Dauben, and Mehrtens, together with additional chapters giving their current views. To this are added new contributions from nine further experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives. This thought-provoking volume will interest mathematicians, philosophers, and historians alike. (shrink)

A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with (...) the modern structural approach in mathematics. (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)

In a wide range of fields, the word “modular” is used to describe complex systems that can be decomposed into smaller systems with limited interactions between them. This essay argues that mathematical knowledge can fruitfully be understood as having a modular structure and explores the ways in which modularity in mathematics is epistemically advantageous.

A good therapeutic relationship in mental health services is a predictor of positive clinical outcomes for people who seek help for distressing experiences, such as voice hearing and paranoia. One factor that may affect the quality of the therapeutic relationship and raises further ethical issues is the impact of the clinical encounter on users’ sense of self, and in particular on their sense of agency. In the paper, we discuss some of the reasons why the sense of epistemic (...) class='Hi'>agency may be especially fragile in young people with unusual experiences and beliefs. We argue that it is important to identify and avoid behaviours that can undermine young people’s contributions as epistemic agents in the clinical encounter. (shrink)

Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on (...) the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. (shrink)

In this chapter, the possibility of experiments in mathematics is examined. A general scheme is proposed as a tool to handle the different forms of experiments that are being used in mathematical practices: computations, “experimental mathematics” as a new research domain in mathematics and computer science, real-world experiments, and thought experiments. In a final section, extensions of the scheme are proposed that further support the conclusion that mathematical experiments are indeed facts and the future.

Philosophical theories of counterfactuals have had relatively little to say about counterfactual reasoning in mathematics. Partly this is because most mathematical counterfactuals seem also to be counterpossibles, in that their antecedents deny some necessary truth. In this chapter, I delineate several different categories of mathematical counterfactual (or “countermathematical”) and then examine in detail a case study from mathematical practice that features counterfactual reasoning about “spoof perfect” numbers. I argue that reasoning about spoof perfect numbers presents both a challenge to (...) philosophical analyses of counterfactuals and a resource for thinking more productively about the role of counterfactual reasoning in mathematics. (shrink)