A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of propositions and their proofs are considered. In addition I use part of Peirce’s semiotics to characterise different types of signs used in mathematical reasoning, distinguishing between symbolic expressions and 2-dimensional diagrams. As a starting point I examine a proposal by Macbeth. Macbeth explains how it can be that objects “pop up”, e.g., as (...) a consequence of the constructions made in the diagrams of Euclid, that is, why they are fruitful. It turns out, however, that diagrams are not exclusively fruitful in this sense. By analysing the proofs given in the paper I introduce the notion of a ‘faithful representation’. A faithful representation represents as either an image or as a metaphor. Secondly it represents certain relevant relations. Thirdly manipulations on the representations respect manipulations on the objects they represent, so that new relations may be found. The examples given in the paper illustrate how such representations can be fruitful. These examples include proofs based on both symbolic expressions as well as diagrams and so it seems diagrams are not special when it comes to fruitfulness. Having said this, I do present two features of diagrams that seem to be unique. One consists of the possibility of exhibiting the type of relation in a diagram—or simply showing that a relation exists—as a contrast to stating in words that it exists. The second is the spatial configurations possible when using diagrams, e.g., allowing to show multiple relations in a single diagram. (shrink)

This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept (...) formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something. (shrink)

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the (...) second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. (shrink)

. This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiros identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiros claim that it is not possible to identify objects in a (...) structure except through the relations and functions that are defined on the structure in which the object has a place. I argue that, in the case of the definition of the so called direct image of a function, it is possible to individuate objects in structures. (shrink)

This paper takes as a starting point certain notions from Peirce's writings and uses them to propose a picture of the part of mathematical practice that consists of hypothesis formation. In particular, three processes of hypothesis formation are considered: abstraction, generalisation, and an abductive-like inference. In addition Peirce's pragmatic conception of truth and existence in terms of higher-order concepts are used in order to obtain a kind of pragmatic realist picture of mathematics.

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

Cet article discute le rôle des représentations dans les preuves mathématiques. Il est suggéré ici que les représentations nous permettent de diviser une preuve en plusieurs parties plus faciles à traiter. Nous illustrerons cela avec un exemple de la pratique mathématique actuelle qui consiste à trouver la valeur d'une expression en la divisant graduellement en parties plus simples. Par ailleurs, j'explique le rôle que jouent les icônes et les indices dans cette procédure. Les icônes assurent la similarité entre l'expression et (...) les résultats obtenus. En revanche, les indices agissent comme des indicateurs qui permettent de réinsérer les résultats aux emplacements appropriés. (shrink)

This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary (...) ‘structural properties’. Although I grant that mathematical statements could be considered as necessarily true in this sense, I question whether this justifies the claim that mathematics as a whole is necessary. (shrink)

This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on Riemann’s work on complex functions, I (...) propose that mathematicians deal with systems of representations and that truth—or what we can prove—depends on available representations in some context where the problem can be solved. (shrink)

Cet article discute le rôle des représentations dans les preuves mathématiques. Il est suggéré ici que les représentations nous permettent de diviser une preuve en plusieurs parties plus faciles à traiter. Nous illustrerons cela avec un exemple de la pratique mathématique actuelle qui consiste à trouver la valeur d'une expression en la divisant graduellement en parties plus simples. Par ailleurs, j'explique le rôle que jouent les icônes et les indices dans cette procédure. Les icônes assurent la similarité entre l'expression et (...) les résultats obtenus. En revanche, les indices agissent comme des indicateurs qui permettent de réinsérer les résultats aux emplacements appropriés. (shrink)

From the perspective of mathematical practice, I examine positions claiming that mathematical objects are introduced by human agents. I consider in particular mathematical fictionalism and a recent position on social ontology formulated by Cole (2013, 2015). These positions are able to solve some of the challenges that non-realist positions face. I argue, however, that mathematical entities have features other than fictional characters and social institutions. I emphasise that the way mathematical objects are introduced is different and point to the multifaceted (...) role that relations and interconnections play in this context. Finally, I argue that mathematical entities can be considered to be pragmatically real. (shrink)

Representations, in particular diagrammatic representations, allegedly contribute to new insights in mathematics. Here I explore the phenomenon of a “free ride” and to what extent it occurs in mathematics. A free ride, according to Shimojima, is the property of some representations that whenever certain pieces of information have been represented then a new piece of consequential information can be read off for free. I will take Shimojima’s framework as a tool to analyse the occurrence and properties of them. I consider (...) a number of different examples from mathematical practice that illustrate a variety of uses of free rides in mathematics. Analysing these examples I find that mathematical free rides are sometimes based on syntactic and semantic properties of diagrams. (shrink)

The aim of this paper is to identify some of the motivations that can be found for taking a realist position concerning mathematical entities and to examine these motivations in the light of a case study in contemporary mathematics. The motivations that are found are as follows: (some) mathematicians are realists, mathematical statements are true, and finally, mathematical statements have a special certainty. These claims are compared with a result in algebraic topology stating that a certain sequence, the so-called Mayer-Vietoris (...) sequence, has different properties when placed in different categories. The conclusion is that the before mentioned motivations should be modified and it is suggested that they could also be explained by a position claiming that mathematical entities are introduced by mathematicians. (shrink)

This article focuses on particular ways in which visual representations contribute to the development of mathematical knowledge. I give examples of diagrammatic representations that enable one to observe new properties and cases where representations contribute to classification. I propose that fruitful representations in mathematics are iconic representations that involve conventional or symbolic elements, that is, iconic metaphors. In the last part of the article, I explain what these are and how they apply in the considered examples.

Terence Tao New York, Oxford University Press, 2006xii + 103 pp., ISBN 9780199205615, £37.50 (hardback), ISBN 9780199205608, £12.99 (paperback)This is a book of mathematical problems and their solu...

While fallibilism has been the dominant view in epistemology in recent times, the field has witnessed the rise of a new form of infallibilism. In a recent book, Jessica Brown has taken on the task of mounting a systematic defence of fallibilism against this new infallibilism. She argues that new infallibilism incurs several problematic commitments that fallibilism can avoid. In addition, the key data points that infallibilists have adduced in support of their view can be accommodated by fallibilism, giving (...) fallibilism the upper hand. This paper develops a rejoinder on behalf of new infallibilism. It explores ways in which new infallibilists can avoid the problematic commitments Brown identifies and provides reason to think that some of the data points supporting new infallibilism cannot be as readily handled by fallibilism as Brown would have us think. (shrink)

Pt. I. The apparent good. Evaluative cognition -- Perceiving the good -- Phantasia and the apparent good -- pt. II. The apparent good and non-rational motivation. Passions and the apparent good -- Akrasia and the apparent good -- pt. III. The apparent good and rational motivation. Phantasia and deliberation -- Happiness, virtue, and the apparent good -- Practical induction -- Conclusion : Aristotle's practical empiricism.

Suppositional theories of conditionals take apparent similarities between supposition and conditionals as a starting point, appealing to features of the former to provide an account of the latter. This paper develops a novel form of suppositional theory, one which characterizes the relationship at the level of semantics rather than at the level of speech acts. In the course of doing so, it considers a range of novel data which shed additional light on how conditionals and supposition interact.

Is there a need for a distinctively diachronic conception of metaphysical emergence? Here I argue to the contrary. In the main, my strategy consists in considering a representative sample of accounts of purportedly diachronic metaphysical emergence, and arguing that in each case, the purportedly diachronic emergence at issue either can (and should) be subsumed under a broadly synchronic account of metaphysical emergence, or else is better seen as simply a case of causation. In addition, I consider and argue against the (...) suggestion that 'in-principle unpredictability' accounts of diachronic emergence support there being distinctively diachronic metaphysical emergence. (shrink)

Plato's Epistemology presents an original interpretation of one of the central topics in Plato's work: epistemology. Moss argues, against the grain of much modern scholarship, that Plato's epistemology is radically different from our own.

Fallibilists claim that one can know a proposition on the basis of evidence that supports it even if the evidence doesn't guarantee its truth. Jessica Brown offers a compelling defence of this view against infallibilists, who claim that it is contradictory to claim to know and yet to admit the possibility of error.

A core principle of modern science holds that a scientific explanation must not attribute will or agency to natural phenomena.The Restless Clock examines the origins and history of this, in particular as it applies to the science of living things. This is also the story of a tradition of radicals—dissenters who embraced the opposite view, that agency is an essential and ineradicable part of nature. Beginning with the church and courtly automata of early modern Europe, Jessica Riskin guides us (...) through our thinking about the extent to which animals might be understood as mere machines. We encounter fantastic robots and cyborgs as well as a cast of scientific and philosophical luminaries, including Descartes and Leibnitz, Lamarck and Darwin, whose ideas gain new relevance in Riskin's hands. The book ends with a riveting discussion of how the dialectic continues in genetics, epigenetics, and evolutionary biology, where work continues to naturalize different forms of agency.The Restless Clock reveals the deeply buried roots of current debates in artificial intelligence, cognitive science, and evolutionary biology. (shrink)

Abstract: Might knowledge normatively govern conversations and not just their discrete constituent thoughts and (assertoric) actions? I answer yes, at least for a restricted class of conversations I call aimed conversations. On the view defended here, aimed conversations are governed by participatory know-how - viz., knowledge how to do what each interlocutor to the conversation shares a participatory intention to do by means of that conversation. In the specific case of conversations that are in the service of joint inquiry, the (...) view defended is that interlocutors (A, B, ... n) must intentionally inquire together into whether p, by means of an aimed conversation X, only if A,B, ... n know how to use X to find out together whether p. The view is supported by considerations about instrumental rationality, shared intentionality, the epistemology of intentional action, as well as linguistic data. (shrink)

For readers looking for insights into key issues linking current Eastern and Western views on the arts, aesthetics, and philosophy, Unsettled Boundaries offers fresh and insightful perspectives on current issues as seen by leading Chinese and Western scholars. Represented in the volume are previously unpublished essays of Nöel Carroll, Garry Hagberg, Richard Shusterman, and Jason Wirth alongside writings of Chinese peers Gao Jianping, Peng Feng, Liu Yuedi, Wang Chunchen and Cheng Xiangzhan. The essays in this volume draw attention to evolving (...) cultural and philosophical connections linking the cultures of East and West, while taking note of important differences in the respective cultures. These connections draw upon both traditional Confucian ideas and Chinese Marxist-Leninist aesthetics as well as western pragmatism, somaesthetics, and Wittgenstein-based analytic philosophy. Alongside these philosophical currents, are reflections on issues linking globalization to contemporary Balinese, Chinese, and Japanese arts, by Curtis Carter, Stephen Davies, Garry Hagberg, and Mary Wiseman. (shrink)

did plato and aristotle have anything to say about belief? The answer to this question might seem blindingly obvious: of course they did. Plato distinguishes belief from knowledge in the Meno, Republic, and Theaetetus, and Aristotle does so in the Posterior Analytics. Plato distinguishes belief from perception in the Theaetetus, and Aristotle does so in the De anima. They talk about the distinction between true and false beliefs, and the ways in which belief can mislead and the ways in which (...) it can steer us aright. Indeed, they make belief a central component of their epistemologies.The view underlying these claims—one so widespread these days as to remain largely unquestioned—is that when Plato and Aristotle talk... (shrink)

This article critically reflects on the research portfolio by the ecological economist Clive Spash who has helped pinpoint specific and systemic blindspots in a political-economic system that prioritises myopic development trajectories divorced from ecological reality. Drawing on his published work and collaborations it seeks to make sense of the slow, or absent, progress in averting global warming and ecological destruction. Three strands of key concern and influence are identified and discussed with reference to their orientation and explicit expression regarding Ontology, (...) Epistemology and Axiology. Some complementary points about indeterminacy and holism are presented to further strengthen the arguments for a transition towards a social-ecological economic system that puts values and principles back into focus. While Clive Spash's work has made a strong case within the economic community and appealing to ecology professionals, the value-myopia or value-vacuum has to be tackled across all disciplines, politics and society for a meaningful and urgently required transformation in decision making. Hence, the article finishes with some suggestions for the (higher) education system, and highlights the importance of simplicity and sufficiency, as well as strong sustainability-driven citizen and community action as necessary catalysts of change in this social-ecological transformation. (shrink)

According to a number of contemporary theorists, aesthetic reasons can invite or entice us but never compel us. In this paper, I develop a Kantian account of the normative force of aesthetic reasons. While Kant would likely agree that aesthetic reasons do not give rise to obligations, his account nevertheless gives us the resources for explaining how aesthetic reasons can still have more force than merely enticing reasons. This account appeals to the distinct normativity of aesthetic judgments on Kant's theory (...) and the way in which this kind of normativity structures aesthetic exchanges with other judging subjects. I argue that, in addition to the moral reasons we have for appreciating and preserving beauty, Kant’s account can accommodate distinctly aesthetic reasons for doing so that stem from our membership in aesthetic communities. (shrink)

This book presents a contextual study of the life and work of the Cambridge Platonist Ralph Cudworth (1617-1688). Focusing on the theological basis of Cudworth's ethical philosophy, this book unlocks the hitherto ignored political aspect to Cudworth's ethical philosophy. Through a detailed examination of Cudworth's published works - particularly his voluminous "True intellectual system of the Universe" -, his posthumously published writings, and his 'freewill' manuscripts Benjamin Carter argues that the ethical and political arguments in Cudworth's philosophy develop out (...) of Cudworth's Trinitarian theology. Carter traces the link between Cudworth's trinitarianism and his ethical and political ideas by placing Cudworth's work in the turbulent religious and intellectual context of seventeenth-century England, and the University of Cambridge in particular. (shrink)

Inquiry aims at knowledge. Your inquiry into a question succeeds just in case you come to know the answer. However, combined with a common picture on which misleading evidence can lead knowledge to be lost, this view threatens to recommend a novel form of dogmatism. At least in some cases, individuals who know the answer to a question appear required to avoid evidence bearing on it. In this paper, we’ll aim to do two things. First, we’ll present an argument for (...) this novel form of dogmatism and show that it presents a substantive challenge. Second, we’ll consider a way those who take knowledge to be the aim of inquiry can mount a response. In the course of doing so, we’ll try to get clearer on the normative connections between inquiry, knowledge and evidence gathering. (shrink)

Our understanding of what exactly needs protected against in order to safeguard a plausible construal of our ‘freedom of thought’ is changing. And this is because the recent influx of cognitive offloading and outsourcing—and the fast-evolving technologies that enable this—generate radical new possibilities for freedom-of-thought violating thought manipulation. This paper does three main things. First, I briefly overview how recent thinking in the philosophy of mind and cognitive science recognises—contrary to traditional Cartesian ‘internalist’ assumptions—ways in which our cognitive faculties, and (...) even our beliefs, can be materially realised by as well as stored non-biologically and extracranially. Second, and taking brain-computer interface technologies (BCIs) and the associated possibility of ‘extended’ beliefs as a reference point, I propose and defend a sufficient condition on freedom-of-thought violating (extended) thought manipulation. On the view proposed, the right not to have one’s thoughts or opinions manipulated is violated if one is (i) caused to acquire non-autonomous propositional attitudes (acquisition manipulation) or (ii) caused to have otherwise autonomous propositional attitudes non-autonomously eradicated (eradication manipulation). The implications of this view are then illustrated through four thought experiments, which map on to four distinct ways—what I call Type 1-Type 4 manipulation—in which, and with reference to the view defended, one’s freedom of thought is plausibly violated. (shrink)

I argue that the best explanation of the multiplicity of available frameworks for treating any given philosophical topic is that philosophy currently (though not insuperably) lacks fixed standards; I then go on to identify three barriers to philosophical progress associated with our present epistemic situation. First is that the lack of fixed standards encourages what I call “intra‐disciplinary siloing,” and associated dialectical and argumentative failings; second is that the lack of fixed standards makes room for sociological factors (including elite influence (...) and disciplinary inertia) to be determinative of which philosophical frameworks are embraced; third is that the lack of fixed standards encourages (implicit and/or explicit) bias. I close by offering some suggestions about how to move beyond these barriers, even in the absence of fixed standards. (shrink)

This book is a collection of essays written by Bob Hale (three co-authored), with a critical introduction from Kit Fine. They comprise Hale’s final years of work, adding to and extending beyond his landmark monograph Necessary Beings: An Essay on Ontology, Modality, and the Relations Between Them (OUP, 2013, 2nd edition 2015). The essays develop and consolidate several key themes in Hale’s work, most notably the notion of definition, especially as it extends beyond definition of a word to definition of (...) a thing more generally, and its relations to essence and existence; how the recently influential notion of truthmaking relates to and illuminates some of these issues; and ontological questions connected with Hale’s metaphysical commitments on the one hand, and his commitments in the foundations of mathematics on the other. Several of the essays engage with and respond to work by Kit Fine, and contributions to Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (OUP 2018). As such, these essays also demonstrate a rich and fruitful philosophical dialogue between Hale and his critics and friends. (shrink)

What are rules? -- Why are rules important? -- Rules for the classroom -- Rules at school -- Rules in the community -- Follow the law! -- Rules and laws must be fair -- All together now -- It's good to have rules.

A common objection to Sosa’s epistemology is that it countenances, in an objectionable way, unsafe knowledge. This objection, under closer inspection, turns out to be in far worse shape than Sosa’s critics have realised. Sosa and his defenders have offered two central response types to the idea that allowing unsafe knowledge is problematic: one response type adverts to the animal/reflective knowledge distinction that is characteristic of bi-level virtue epistemology. The other less-discussed response type appeals to the threat of dream scepticism, (...) and in particular, to the idea that many of our everyday perceptual beliefs are unsafe through the nearness of the dream possibility. The latter dreaming response to the safety objection to Sosa’s virtue epistemology has largely flown under the radar in contemporary discussions of safety and knowledge. We think that, suitably articulated in view of research in the philosophy and science of dreaming, it has much more going for it than has been appreciated. This paper further develops, beyond what Sosa does himself, the dreaming argument in response to those who think safety (as traditionally understood) is a condition on knowledge and who object to Sosa’s account on the grounds that it fails this condition. The payoffs of further developing this argument will be not only a better understanding of the importance of insights about dreaming against safety as a condition on knowledge, but also some reason to think a weaker safety condition, one that is relativised to SSS (i.e., skill/shape/situation) conditions for competence exercise, gets better results all things considered as an anti-luck codicil on knowledge. (shrink)

The English Neoplatonic philosopher Ralph Cudworth introduced the term ‘consciousness’ into the English philosophical lexicon. Cudworth uses the term to define the form and structure of cognitive acts, including acts of freewill. In this article I highlight the important role of theological disputes over the place and extent of human freewill within an overarching system of providence. Cudworth’s intellectual development can be understood in the main as an increasingly detailed and nuanced reaction to the strict voluntarist Calvinism that is typified (...) in the thought of his near contemporary William Perkins. At the heart of Cudworth’s rejection of Calvinism is the dilemma over whether God is understood primarily in terms of will or justice. In this fleshing-out of the power of consciousness Cudworth moves from an instrumental account of the working of the human mind towards an account of human consciousness that is intrinsic to his definition of human agency. (shrink)