This chapter presents an overview of the contributions to the section on Experimental Mathematics by focusing in particular on how they characterize the phenomenon of “experimental mathematics” and its origins. The second part presents two case studies illustrating how experimental mathematics is understood in contemporary analysis. The third section offers a systematic presentation of the contributions to the section.

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 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)

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.

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)

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 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.

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)

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.

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)

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)

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)

Knowledge, like other things of value, can be faked. According to Hawley (2011), know-how is harder to fake than knowledge-that, given that merely apparent propositional knowledge is in general more resilient to our attempts at successful detection than are corresponding attempts to fake know-how. While Hawley’s reasoning for a kind of detection resilience asymmetry between know-how and know-that looks initially plausible, it should ultimately be resisted. In showing why, we outline different ways in which know-how can be faked even when (...) a given performance is successful; and in doing so, we distinguish how know-how can be faked (no less than know-that) via upstream and downstream indicators of its presence, and within each of these categories, we’ll distinguish (in connection with detection resilience) both faking symptoms and (various kinds of) criteria. The unappreciated resilience of faked knowledge-how to successful detection highlights a largely overlooked dimension of social-epistemic risk – risk we face not just in our capacity as recipients of testimony, but in our capacity as both (would-be) apprentices and clients of knowledge-how. (shrink)

RésuméL'ouvrage de Sophia Moreau, Faces of Inequality, adopte une méthodologie philosophique provocatrice. Moreau puise dans les expériences des victimes de discrimination et à même les contours fondamentaux du droit à la non-discrimination afin d’élaborer sa théorie de la discrimination répréhensible. Cependant, si nous prenons au sérieux les enseignements des études féministes et critiques de la « race », une tension émerge au sein de la dyade méthodologique de Moreau : si les lois contre la discrimination sont souvent elles-mêmes source d'hostilité (...) et d'invalidation pour les victimes de discrimination, comment ces expériences et ces lois peuvent-elles toutes deux faire autorité pour déterminer la nature de la discrimination répréhensible? Cet article explore cette tension à l’œuvre dans Faces of Inequality à la lumière de la théorie de la « conscience multiple » de Mari Matsuda. (shrink)

Epistemology and Relativism Epistemology is, roughly, the philosophical theory of knowledge, its nature and scope. What is the status of epistemological claims? Relativists regard the status of epistemological claims as, in some way, relative— that is to say, that the truths which epistemological claims aspire to are … Continue reading Epistemology and Relativism →.

The World Health Organisation declared COVID-19 a global pandemic on 11th March 2020, recognising that the underlying SARS-CoV-2 has caused the greatest global crisis since World War II. In this chapter, we present a framework to evaluate whether and to what extent the use of digital systems that track and/or trace potentially infected individuals is not only legal but also ethical.

Recently, a debate has developed between those who claim that essence can be explained in terms of de re modality (modalists), and those who claim that de re modality can be explained in terms of essence (essentialists). The aim of this paper is to suggest that we should reassess. It is assumed that either necessity is to be accounted for in terms of essence, or that essence is to be accounted for in terms of necessity. I will argue that we (...) should assume neither. I discuss what role these key notions – essence and necessity – can reasonably be thought to contribute to our understanding of the world, and argue that, given these roles, there is no good reason to think that we should give an account of one in terms of the other. I conclude: if we can adequately explain de re modality and essence at all, we should aim to do so separately. (shrink)

The philosophical significance of attitudinal autonomy—viz., the autonomy of attitudes such as beliefs—is widely discussed in the literature on moral responsibility and free will. Within this literature, a key debate centres around the following question: is the kind of attitudinal autonomy that’s relevant to moral responsibility at a given time determined entirely by a subject’s present mental structure at that time? Internalists say ‘yes’, externalists say ’no’. In this essay, I motivate a kind of distinctly epistemic attitudinal autonomy, attitudinal autonomy (...) that is relevant to knowledge. I argue that regardless of whether we are externalists or internalists about the kind of attitudinal autonomy that is relevant for moral responsibility, we should be externalists about the kind of autonomy that a belief must have to qualify as knowledge. (shrink)

Drawing on recent work in formal pragmatic theory, this paper shows that the manipulation of discourse structure—in particular, by way of shifting the Question Under Discussion mid-discourse—can constitute an act of epistemic injustice. I argue that the “All Lives Matter” response to the “Black Lives Matter” slogan is one such case; this response shifts the Question Under Discussion governing the overarching discourse from Do Black lives matter? to Which lives matter? This manipulation of the discourse structure systematically obscures the intended (...) meaning of “Black lives matter” and disincentivizes future utterances of it. (shrink)

O povo Kaingang é o mais populoso do Sul do Brasil e está entre os mais numerosos povos indígenas do país. Neste trabalho faz-se uma relação entre as informações encontradas na literatura e a atualidade na Terra Indígena Kaingang em relação ao conhecimento que possuem sobre o uso de plantas para a cura de doenças e como interpretam a ação da erva no organismo. Para isto, realizou-se uma pesquisa de campo em uma Terra Indígena Kaingang no Paraná, cujo entrevistas forneceram (...) confirmações históricas em relação aos etnoconhecimentos da etnia, da evolução dos aspectos químicos e das atitudes perante a química e os remédios de origem farmacêutica e natural. (shrink)

Assertion is a fundamental feature of language. This volume will be the place to look for anyone interested in current work on the topic.

Introduction : reading Nietzsche skeptically -- Nietzsche and the Pyrrhonian tradition -- Skepticism in Nietzsche's early work : the case of "on truth and lie" -- The question of Nietzsche's "naturalism" -- Perspectivism and Ephexis in interpretation -- Skepticism and health -- Skepticism as immoralism.

Edited by leading contributors to the literature, Freedom: An Anthology is the most complete anthology on social, political and economic freedom ever compiled. Offers a broad guide to the vast literature on social, political and economic freedom. Contains selections from the best scholarship of recent decades as well as classic writings from Hobbes, Locke, Rousseau and Kant among others. General and sectional introductions help to orient the reader. Compiled and edited by three important contributors to the field.

Should we regard Jennifer Lackey’s (2007) ‘Creationist Teacher’ as understanding evolution, even though she does not, given her religious convictions, believe its central claims? We think this question raises a range of important and unexplored questions about the relationship between understanding, factivity and belief. Our aim will be to diagnose this case in a principled way, and in doing so, to make some progress toward appreciating what objectual understanding—i.e., understanding a subject matter or body of information—demands of us. Here is (...) the plan. After some ground clearing in §1, §2 outlines and motivates a plausible working model—moderate factivity—for characterising the sense in which objectual understanding should be regarded as factive. §3 shows how the datum that we can understand false theories can, despite initial suggestions to the contrary, be assimilated straightforwardly within the moderate factivity model. §4 highlights how the inverse kind of case to that explored in §3—viz., a variant of Lackey’s creationist teacher case—poses special problems for moderate factivity. With reference to recent work on moral understanding by Hills (2009), §5 proposes a solution to the problem, and §6 attempts to diagnose why it is that we might originally have been led to draw the wrong conclusion. (shrink)

Epistemic relativists often appeal to an epistemic incommensurability thesis. One notable example is the position advanced by Wittgenstein in On certainty (1969). However, Ian Hacking’s radical denial of the possibility of objective epistemic reasons for belief poses, we suggest, an even more forceful challenge to mainstream meta-epistemology. Our central objective will be to develop a novel strategy for defusing Hacking’s line of argument. Specifically, we show that the epistemic incommensurability thesis can be resisted even if we grant the very insights (...) that lead Hacking to claim that epistemic reasons are always relative to a style of reasoning. Surprisingly, the key to defusing the argument is to be found in recent mainstream work on the epistemic state of objectual understanding. (shrink)

Recently, a debate has developed between those who claim that essence can be explained in terms of de re modality (modalists), and those who claim that de re modality can be explained in terms of essence (essentialists). The aim of this paper is to suggest that we should reassess. It is assumed that either necessity is to be accounted for in terms of essence, or that essence is to be accounted for in terms of necessity. I will argue that we (...) should assume neither. I discuss what role these key notions – essence and necessity – can reasonably be thought to contribute to our understanding of the world, and argue that, given these roles, there is no good reason to think that we should give an account of one in terms of the other. I conclude: if we can adequately explain de re modality and essence at all, we should aim to do so separately. (shrink)

A persuasive monograph that answers the keyepistemological arguments against anti-individualism in thephilosophy of mind.

Our aim is to provide a topography of the relevant philosophical terrain with regard to the possible ways in which knowledge can be conceived of as extended. We begin by charting the different types of internalist and externalist proposals within epistemology, and we critically examine the different formulations of the epistemic internalism/externalism debate they lead to. Next, we turn to the internalism/externalism distinction within philosophy of mind and cognitive science. In light of the above dividing lines, we then examine first (...) the extent to which content externalism is compatible with epistemic externalism; second, whether active externalism entails epistemic externalism; and third whether there are varieties of epistemic externalism that are better suited to accommodate active externalism. Finally, we examine whether the combination of epistemic and cognitive externalism is necessary for epistemology and we comment on the potential ramifications of this move for social epistemology and philosophy of science. (shrink)

How should physical entities be characterized? Physicalists, who have most to do with the notion, usually characterize the physical by reference to two components: 1. The physical entities are the entities treated by fundamental physics with the proviso that 2. Physical entities are not fundamentally mental (that is, do not individually possess or bestow mentality) Here I explore the extent to which the appeals to fundamental physics and to the NFM (“no fundamental mentality”) constraint are appropriate for characterizing the physical, (...) especially for purposes of formulating physicalism. Ultimately, I motivate and defend a version of an account incorporating both components: The physics-based NFM account: An entity existing at a world w is physical iff (i) it is treated, approximately accurately, by current or future (in the limit of inquiry, ideal) versions of fundamental physics at w, and (ii) it is not fundamentally mental (that is, does not individually either possess or bestow mentality). (shrink)

Research groups around the world are currently busy trying to invent new life in the laboratory, looking for extraterrestrial life, or making machines increasingly more life-like. In the case of astrobiology, any newly discovered life would likely be very old, but when discovered it would be new to us. In the case of synthetic organic life or life-like machines, humans will have invented life that did not exist before. Together, these endeavors amount to what we call the emerging plurality of (...) lives because we see a future where we are surrounded by, and interact with, life of many different origins, as well as entities that we have not traditionally seen as being alive. In the future these entities will likely possess properties that we so far have only associated with living beings, thereby transcending and perhaps blurring the borders between life and non-life. (shrink)

We investigate a novel use of the English temporal modifier ‘now’, in which it combines with a subordinate clause. We argue for a univocal treatment of the expression, on which the subordinating use is taken as basic and the non-subordinating uses are derived. We start by surveying central features of the latter uses which have been discussed in previous work, before introducing key observations regarding the subordinating use of ‘now’ and its relation to deictic and anaphoric uses. All of these (...) data, it is argued, can be accounted for on our proposed analysis. We conclude by comparing ‘now’ to a range of other expressions which exhibit similar behavior. (shrink)