In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

Corporate social responsibility is a recognised and common part of business activity. Some of the regularly cited motives behind CSR are employee morale, recruitment and retention, with employees acknowledged as a key organisational stakeholder. Despite the significance of employees in relation to CSR, relatively few studies have examined their engagement with CSR and the impediments relevant to this engagement. This exploratory case study-based research addresses this paucity of attention, drawing on one to one interviews and observation in a large UK (...) energy company. A diversity of engagement was found, ranging from employees who exhibited detachment from the CSR activities within the company, to those who were fully engaged with the CSR activities, and to others who were content with their own personal, but not organisational, engagement with CSR. A number of organisational context impediments, including poor communication, a perceived weak and low visibility of CSR culture, and lack of strategic alignment of CSR to business and personal objectives, served to explain this diversity of employee engagement. Social exchange theory is applied to help explore the volition that individual employees have towards their engagement with CSR activities, and to consider the implications of an implicit social, rather than explicit economic, contract between an organisation and its employees in their engagement with CSR. (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)

Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosophical investigation into motivated proofs. I suggest that a proof is motivated if and (...) only if mathematicians can identify (i) the tasks each step is intended to perform; and (ii) where each step could have reasonably come from. I argue that motivated proofs promote understanding, convey new mathematical resources and stimulate new discoveries. They thus have significant epistemic benefits and directly contribute to the efficient dissemination and advancement of mathematical knowledge. Given their benefits, I also discuss the more practical matter of how we can produce motivated proofs. Finally I consider the relationship between motivated proofs and proofs which are explanatory, beautiful and fitting. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation. In this essay, we (...) describe an approach to the philosophy of mathematics in which it is an important task to understand the roles of our ontological posits and assess the extent to which they enable us to achieve our mathematical goals. We use the history of Dirichlet’s theorem to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment. We also use these considerations to illuminate the formal treatment of functions and objects in Frege’s logical foundation, and we argue that his philosophical and logical decisions were influenced by many of the same factors. (shrink)

Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...) the abstract concept of a simply infinite system, proves the existence of a “model”, insists on the stepwise derivation of theorems, and defines structure-preserving mappings between different systems that fall under the abstract concept. Crucial parts of these considerations were added, however, only to the penultimate manuscript, for example, the very concept of a simply infinite system. The methodological consequences of this radical shift are elucidated by an analysis of Dedekind’s metamathematics. Our analysis provides a deeper understanding of the essay and, in addition, illuminates its impact on the evolution of the axiomatic method and of “semantics” before Tarski. This understanding allows us to make connections to contemporary issues in the philosophy of mathematics and science. (shrink)

Current fears of a “reproducibility crisis” have led researchers, sources of scientific funding, and the public to question both the efficacy and trustworthiness of science. Suggested policy changes have been focused on statistical problems, such as p-hacking, and issues of experimental design and execution. However, “reproducibility” is a broad concept that includes a number of issues. Furthermore, reproducibility failures occur even in fields such as mathematics or computer science that do not have statistical problems or issues with experimental design. Most (...) importantly, these proposed policy changes ignore a core feature of the process of scientific inquiry that occurs after reproducibility failures: the integration of conflicting observations and ideas into a coherent theory. (shrink)

This paper reviews the literature concerning the neural correlates of the self, the relationship between self and memory and the profile of memory impairments in Alzheimer’s disease and explores the relationship between the preservation of the self and anosognosia in this condition. It concludes that a potential explanation for anosognosia in AD is a lack of updating of personal information due to the memory impairments characteristic of this disease. We put forward the hypothesis that anosognosia is due in part to (...) the “petrified self.”. (shrink)

Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In general, (...) I argue that the answer to this question is “no” and demonstrate this in detail for the theory of mathematical explanation developed by Marc Lange. (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)

Gayatri Chakravorty Spivak's original essay "Can the Subaltern Speak?" transformed the analysis of colonialism through an eloquent and uncompromising argument that affirmed the contemporary relevance of Marxism while using deconstructionist methods to explore the international division of labor and capitalism's "worlding" of the world. Spivak's essay hones in on the historical and ideological factors that obstruct the possibility of being heard for those who inhabit the periphery. It is a probing interrogation of what it means to have political subjectivity, to (...) be able to access the state, and to suffer the burden of difference in a capitalist system that promises equality yet withholds it at every turn. Since its publication, "Can the Subaltern Speak?" has been cited, invoked, imitated, and critiqued. In these phenomenal essays, eight scholars take stock of the effects and response to Spivak's work. They begin by contextualizing the piece within the development of subaltern and postcolonial studies and the quest for human rights. Then, through the lens of Spivak's essay, they rethink historical problems of subalternity, voicing, and death. A final section situates "Can the Subaltern Speak?" within contemporary issues, particularly new international divisions of labor and the politics of silence among indigenous women of Guatemala and Mexico. In an afterword, Spivak herself considers her essay's past interpretations and future incarnations and the questions and histories that remain secreted in the original and revised versions of "Can the Subaltern Speak?"—both of which are reprinted in this book. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.

Prominent mathematician William Thurston was praised by other mathematicians for his intellectual generosity. But what does it mean to say Thurston was intellectually generous? And is being intellectually generous beneficial? To answer these questions I turn to virtue epistemology and, in particular, Roberts and Wood's (2007) analysis of intellectual generosity. By appealing to Thurston's own writings and interviewing mathematicians who knew and worked with him, I argue that Roberts and Wood's analysis nicely captures the sense in which he was intellectually (...) generous. I then argue that intellectual generosity is beneficial because it counteracts negative effects of the reward structure of mathematics that can stymie mathematical progress. (shrink)

In evolutionary medicine, researchers characterize some outcomes as evolutionary mismatch. Mismatch problems arise as the result of organisms living in environments to which they are poorly adapted, typically as the result of some rapid environmental change. Depression, anxiety, obesity, myopia, insomnia, breast cancer, dental problems, and numerous other negative health outcomes have all been characterized as mismatch problems. The exact nature of evolutionary mismatch itself is unclear, however. This leads to a lack of clarity about the sorts of problems that (...) evolutionary mismatch can actually explain. Resolving this challenge is important not only for the evolutionary health literature, but also because the notion of evolutionary mismatch involves central concepts in evolutionary biology: fitness, evolution in changing environments, and so forth. In this paper, I examine two characterizations of mismatch currently in the literature. I propose that we conceptualize mismatch as a relation between an optimal environment and an actual environment. Given an organism and its particular physiology, the optimal environment is the environment in which the organism’s fitness is maximized: in other words, the optimal environment is that in which the organism’s fitness is as high as it can possibly be. The actual environment is the environment in which the organism actually finds itself. To the extent that there is a discordance between the organism’s actual and optimal environments, there is an evolutionary mismatch. In the paper, I show that this account of mismatch gives us the right result when other accounts fail, and provides useful targets for investigation. (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)

Professional historians of science generally recognize the importance of Lavoisier's theory of heat. However, it commonly receives scant attention in the historical treatment of his chemical theories except perhaps as an example illustrating his conservatism and giving the impression that the caloric theory, although perhaps important in the development of ideas on the nature of heat, is independent of and bears little relationship to his general chemistry or is incidental to an understanding of that chemistry. An examination of Lavoisier's writings (...) suggests that the caloric theory is not merely a milestone in the development of physics; and rather than an omittable appendage, his concept of heat forms an integral part of his chemical system and plays a central, necessary role in his oxidation theory in particular. The purpose of this paper is to give a general description of Lavoisier's ideas on the nature and action of heat, the origin of these ideas, their development, and their relation to his general chemistry, pointing out his conservatism as well as his innovations. (shrink)

Many writers have attempted to develop criteria to demarcate between competent science and pseudo?science. Such attempts can be aimed at sizeable, organized endeavours, such as mesmerism and astrology, or at the level of individual practice. The latter is seen by some, such as Lugg, as more likely to be feasible and useful. This paper argues that parapsychology, due to its complexity and diversity, illustrates some of the problems of attempting to develop demarcation criteria for extensive endeavours. It is also suggested (...) that parapsychology may offer a productive ground for testing whether demarcation criteria can be successfully applied to practices in respect of predicting which will succeed and which will not. The conclusion is that, for demarcation efforts to be useful, they should pass some of their own criteria, such as falsifiability. (shrink)

Provides an ideology-critique of Whitehead's and Hartshorne's metaphysics and demonstrates how their metaphysical principles reflect their personal commitments to the values and norms of the modern liberal political ideology.

Scaffolds are fundamental to many cellular signaling pathways. In this essay, a novel class of scaffolds are proposed, whose action bears striking resemblance to kinetic proofreading. Commonly, scaffold proteins are thought to work as tethers, bringing different components of a pathway together to improve the likelihood of their interaction. However, recent studies show that the cytoskeletal scaffold, anillin, supports contractile signaling by a novel, non‐tethering mechanism that controls the membrane dissociation kinetics of RhoA. More generally, such proof‐reading‐like scaffolds are distinguished (...) from tethers by a rare type of cooperativity, manifest as a super‐linear relationship between scaffold concentration and signaling efficiency. The evidence for this hypothesis is reviewed, its conceptual ramifications are considered, and research questions for the future are discussed. (shrink)

Mathematics is becoming increasingly specialized, divided into a vast and growing number of subfields. While this division of cognitive labor has important benefits, it also has a significant drawb...

This paper argues that independent auditors have lost sight of their obligation to be truly impartial, and have increasingly adopted an attitude of client advocacy. We argue that auditors have a professional obligation to go beyond merely passing judgment on whether client accounting methods are acceptable under GAAP, and to judge whether the principles adopted are the most appropriate under the circumstances. We then review recent evidence which suggests that auditors have abandoned this objective in favor of advocating client-preferred principles. (...) The results of a survey of public accountants employed by small CPA firms indicates that, in a scenario in which an auditor did not feel that the client's accounting treatment was the most appropriate under the circumstances but it was recognized as an acceptable alternative under GAAP, advocacy of the client-preferred approach is considered appropriate and is very likely to occur in practice. In less subtle cases (e.g., when the auditor did not feel that the client's treatment provided the best reflection of the economic substance of the underlying transactions), survey participants as a group were ambivalent regarding the appropriateness of client advocacy, and felt that subordination of judgment was likely to occur in practice. The results also indicate that engagement risk, or the risk that the firm will be harmed by aggressive behavior, affects judgments of both the appropriateness and likelihood of subordination of judgment. These findings suggest that auditors have adopted an ideology of client advocacy, within the constraints of engagement risk. (shrink)

Models for truth in fiction must be able to account for differing versions and interpretations of a given fiction in such a way that prevents contradictions from arising. I propose an analysis of truth in fiction designed to accommodate this. I examine both the interpretation of claims about truth in fiction (the ‘Interpretation Problem’) and the metaphysical nature of fictional worlds and entities (the ‘Metaphysical Problem’). My reply to the Interpretation Problem is a semantic contextualism influenced by Cameron (2012), while (...) my reply to the Metaphysical Problem involves an extension and generalisation of the counterpart-theoretic analysis put forth by Lewis (1978). The proposed analysis considers interpretive context as a counterpart relation corresponding to a set of worlds, W, and states that a sentence φ is true in interpretive context W iff φ is true at every world (w∈W). I consider the implications of this analysis for singular terms in fiction, concluding that their extensions are the members of sets of counterparts. In the case of pre-existing singular terms in fiction, familiar properties of the corresponding actual-world entities are salient in restricting the counterpart relation. I also explore interpretations of sentences concerning multiple fictions and those concerning both fictional and actual entities. This account tolerates a plurality of interpretive approaches, avoiding contradictions. (shrink)

Consequentialism has been challenged on the grounds that it is too demanding. I will respond to the problem of demandingness differently from previous accounts. In the first part of the paper, I argue that consequentialism requires us to distinguish the justification of an act \ from the justification of an act \, where \ is an act of praise or blame. In the second part of the paper, I confront the problem of demandingness. I do not attempt to rule out (...) the objection; instead, I argue that if certain plausible empirical claims about moral motivation are true, we morally ought not to blame people for failing to meet certain very demanding obligations. With this theory, we create a space in consequentialism for intuitions questioning the plausibility of demanding obligations. I conclude the paper by showing that separate justifications for \ and \ may also give us a theoretical niche for intuitions about supererogation. (shrink)

Proofs are central, and unique, to mathematics. They establish the truth of theorems and provide us with the most secure knowledge we can possess. It is thus perhaps unsurprising that philosophers once thought that the only value proofs have lies in establishing the truth of theorems. However, such a view is inconsistent with mathematical practice. If a proof’s only value is to show a theorem is true, then mathematicians would have no reason to reprove the same theorem in different ways, (...) yet this is a practice they frequently engage in. This suggests that proofs can have a wide variety of values. My purpose in this chapter is to provide a survey of some of them. I will discuss how proofs can have practical value, for example, by helping mathematicians to make new discoveries or learn new mathematical tools, and how they can have abstract value, for example, by being beautiful or explanatory. I will also explore the relationships between different proof values. (shrink)

In the thirty years since its discovery by Terje Lomo and Tim Bliss, Long Term Potentiation has become one of the most extensively studied topics in contemporary neuroscience. In LTP the strength of synapses between neurons is potentiated following brief but intense activation. LTP is thought to play a central role in learning and memory, though the exact nature of its role is less clear. In spite of years of research, there are many questions about LTP regarding its functional relevance (...) that remain unanswered - for example, is it a model of memory formation, or is the actual neural mechanism used by the brain to store information? This volume presents a state of the art account of LTP. It begins with lively accounts, by the scientists most closely involved, of the discovery of LTP and of the experiments that established its basic properties and induction mechanisms. Later contributions contain reviews and new research that cover the range of molecular, cellular, physiological and behavioural approaches to the study of LTP. Provocative, accessible, and authoritative, this book makes it clear why LTP continues in equal measure to puzzle and beguile neuroscientists today. Advance praise for Long Term Potentiation: 'This book provides a definitive overview of the development of ideas about synaptic plasticity and about the wide range of current research in this fascinating field.' Colin Blakemore, University of OxfordReadership: Neuroscientists from postgraduate level upwards. (shrink)

The Computational History of Philosophy of Science Dataset aims to be a comprehensive set of article and book chapter metadata for philosophy of science. The dataset covers the full run of over 40 journals and 3 major book series in the field. An automated author disambiguation script is used to construct canonical names for each author, and a combination of gender attribution methods is used to attribute the gender of each author. The full code used to generate the dataset is (...) available at [URL not allowed]. See the file data_dictionary.txt for data dictionary and additional information. (shrink)

Despite much research on the relationship between awareness and dementia little can be concluded concerning their relationship and the role of other factors. It is likely that studies capture different phenomena of awareness. This study aimed at identifying and delineating such variation by analysing data from three questionnaires obtained during the longitudinal study of awareness in 101 people with early-stage dementia. The data concerned awareness in relation to memory, activities of daily living and socio-emotional function. Significant differences in patterns of (...) discrepancies were obtained. This suggests that the awareness phenomena involved were structurally different; and that, in turn, this may reflect variation in the intrinsic linking between awareness and its ‘object’ . The identification of such differences is necessary so that appropriate methodologies can be applied to the study of awareness in different contexts. (shrink)